Research ArticleImproved Finite Beam Element Method to Analyze the NaturalVibration of Steel-Concrete Composite Truss Beam

Zhipeng Lai12 Lizhong Jiang12 Wangbao Zhou12 and Xilin Chai1

1School of Civil Engineering Central South University Changsha 410075 China2National Engineering Laboratory for High Speed Railway Construction Changsha 410075 China

Correspondence should be addressed to Wangbao Zhou zhouwangbao163com

Received 30 April 2017 Accepted 18 July 2017 Published 22 August 2017

Academic Editor Nerio Tullini

Copyright copy 2017 Zhipeng Lai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on Hamiltonrsquos principle this study has developed a continuous treatment for the steel-concrete composite truss beam(SCCTB) It has also deduced the SCCTB element stiffness matrix and mass matrix which include the effects of interface slipshear deformation moment of inertia and many other influencing factors A finite beam element method (FBEM) program forSCCTBrsquos natural vibration frequency has been developed and used to calculate the natural vibration frequencies of several SCCTBswith different spans and different degrees of shear connections The FBEMrsquos calculation results of several SCCTBs agree well withthe results obtained from ANSYS Based on the results of this study the following conclusions can be drawn For the SCCTB withhigh-order natural vibration frequency and with short span the effect of the shear deformation is greater Hence the effect of theshear deformation on the SCCTBrsquos high-order natural vibration frequency cannot be ignored On the other hand the effect of theinterface slip on the SCCTBrsquos high-order natural vibration frequency is insignificant However the effect of the interface slip on theSCCTBrsquos low-order natural vibration frequency cannot be ignored

1 Introduction

The SCCTB is a new-type structural member It is developedon the basis of ordinary composite beams and uses shearconnectors to connect the concrete slab to the steel truss intoone entire joint work Comparing with ordinary truss beamsSCCTB is more effective because it uses both the compressivestrength of concrete and the tensile strength of steel Hence itis characterized by high utilization rate of indoor space higheconomic efficiency high bending stiffness and high bearingcapacity [1 2]

Even though a beam is designed as a full composite sec-tion due to the deformation of shear studs it cannot respondwith perfect composite action without slip Further unlikeEulerndashBernoulli beams in which there is infinite shear stiff-ness it may not be appropriate to design other beams byneglecting the shear deformation of the concrete slab andsteel truss beam for example composite beams with a smallspan-to-depth ratio [3ndash6] Therefore the natural vibrationcharacteristics of SCCTBs are affected by the coupled effects

of the shear deformation and the interfacial slip between theconcrete slab and steel truss beam

Hitherto there are many studies on the shear behavior ofhybrid steel-trussed-concrete beams which are constructedby embedding the prefabricated steel trusses into a concretecore cast in situ On the other hand there are few studieson the mechanical properties of SCCTB Using the softwareABAQUS to carry out the finite element (FE) numericalsimulations Monaco [7] investigated the shear behavior ofhybrid steel-trussed-concrete beams Monti and Petrone [8]developed the shear capacity equations from a purposelydeveloped mechanics-based shear model for hybrid steel-trussed-concrete beams Campione et al [9] investigated acalculation method for the prediction of the shear resistanceof precast composite beam both experimental and numericalresults were used to validate the developed analytical expres-sions

Giltner andKassimali [10] developed amethodwhich canreplace the trusses and the beam elements with an equivalentbeam thereby reducing the size of the computer model for

HindawiShock and VibrationVolume 2017 Article ID 5323246 12 pageshttpsdoiorg10115520175323246

2 Shock and Vibration

analysis Machacek and Cudejko [11 12] investigated thedistribution of longitudinal shear along the interface betweenthe steel and the concrete of various composite truss bridgesfrom the elastic phase to the plastic collapse He foundthat the nonlinear distribution of the longitudinal shearsignificantly depends on the rigidity of the shear connectionand the densification of the shear connectors above thetruss nodes By carrying out an experimental study and atheoretical analysis Chan and Fong [13] concluded that theuse of the effective length method in the linear analysis anddesign method was less convenient and less accurate thanthe second-order analysis On the basis of the literature [13]Fong et al [14] further showed that the second-order analysismethod was not only an accurate design method but it canalso avoid the uncertain approximate value of the effectivelength By comparing the results from the finite elementnumerical calculation method with the experimental resultsBujnak and Bouchair [15] found that the local effects of theconcentrated longitudinal shear forces should be examined inSCCTB Siekierski [16] analyzed the effects of the shrinkageof concrete slab in SCCTB and developed a set of linearequations to compute the axial forces in the flange membersof the truss girder and the transverse shear forces in SCCTB

The natural vibration characteristic analysis is the basisto investigate the dynamic characteristics of SCCTB So it isnecessary to investigate its natural vibration characteristicsAs shown by the earlier studies the mechanical properties ofSCCTB are affected by the interface slip shear deformationmoment of inertia and many other factors However thereare only a few studies on the SCCTBrsquos natural vibrationcharacteristics which have taken these factors into accountIn this study a continuous treatment for SCCTB has beendeveloped It has also deduced the SCCTB element stiffnessmatrix and mass matrix with the cubic Hermite polynomialshape function which includes the effects of the interface slipshear deformation moment of inertia and many other influ-encing factors Hence even with fewer degrees of freedomthe precision in the calculation is satisfactory Based on thedeveloped SCCTB element this study has developed a FBEMprogram which can calculate the natural vibration frequencyof SCCTBs commonly used in the engineering practiceFinally it calculates the natural vibration frequencies ofseveral SCCTBs with different spans and different degrees ofshear connections Based on the results of the analyses someconclusions which are related to the engineering design aredrawn

2 Cross-Sectional Analysis of Strain andStress of SCCTB

21 Displacement Model of SCCTB In Figure 1 1198673 1198793 ℎ3and 1199053 are the height and width of the external and internalwalls of the upper chord respectively 1198674 1198794 ℎ4 and 1199054are the height and width of the external and internal wallsof the lower chord respectively 1198675 1198795 ℎ5 and 1199055 are theheight and width of external and internal walls of the obliqueweb member respectively 1198676 1198796 ℎ6 and 1199056 are the heightand width of the external and internal walls of the vertical

0

b2 b1 b1 b2

t2 t1 t2

ℎ3 H3

b6

ℎ4 H4

b7

z w

y

Figure 1 Sectional dimension and the coordinate system of theSCCTB

web member respectively 1198677 1198797 ℎ7 and 1199057 are the heightand width of the external and internal walls of the lowerhorizontal connection member respectively 1198678 1198798 ℎ8 and1199058 are the height and width of the external and internal wallsof the oblique bracing member respectively 1199051 and 1199052 arethe thicknesses of the concrete roof and the cantilever slabsrespectively 21198871 and 1198872 are thewidths of the concrete roof andthe cantilever slabs respectively and 1198873 1198874 1198875 1198876 1198877 and 1198878are the lengths of the upper chord lower chord oblique webmember vertical web member lower horizontal connectionmember and oblique bracing member respectively

The strain of oblique bracingmember can be expressed as

1205761198878 = Δ 11988781198878 = Δ 1198874 cos1205731198874 cos120573 = Δ 1198874cos21205731198874 = 1205761199094cos2120573 (1)

where 120573 is the angle between lower horizontal connectionmember and oblique bracing member Δ 1198878 is the axialdisplacement of the oblique bracing member and Δ 1198874 is theaxial displacement of the lower chord

Further the axial force of the oblique bracing member isgiven by

1198651198878 = 11986411990412057611988781198608 = 11986411990412057611990941198608cos2120573 (2)

The longitudinal component of the axial force of theoblique bracing member is given by

119865119901119909 = 1198651198878 cos120573 = 11986411990412057611990941198608cos3120573 (3)

Therefore the longitudinal equivalent area of the obliquebracing member can be expressed as follows

1198608119867 = 1198608cos3120573 = 1198608cos3120573 (4)

The longitudinal displacement of SCCTB can beexpressed as follows [17]

119906119894 (119909 119910 119911) = 119896119888120585 (119909) minus (119911 minus 119911119888) 120579 (119909) 119894 = 1 2119896119904120585 (119909) minus (119911 minus 119911119904) 120579 (119909) 119894 = 3 4 (5)

119896119888 = minus119860 1199041198600 119896119904 = 119860119888(1198991198600)

(6)

Shock and Vibration 3

where 119911119904 is the 119911-coordinate of the centroid of the steel trussbeam 119911119888 is the 119911-coordinate of the centroid of the concreteslab 120579(119909) = 1199081015840(119909) minus 120574119908(119909) is the cross-sectional rotationangle 119908(119909) is the vertical deflection of the SCCTB 120585(119909)is the difference between the longitudinal displacement ofthe concrete slab centroid and that of the steel truss beamcentroid 119864119904 is the elastic modulus of the steel truss and 119864119888is the elastic modulus of the concrete slab Further 119899 =119864119904119864119888 119860119888 = 1198601 + 1198602 119860 119904 = 1198603 + 1198604 + 1198608119867 1198600 = 119860119888119899 +119860 119904 1198601 = 211988711199051 1198602 = 211988721199052 1198603 = 2(11986731198793 minus ℎ31199053) 1198604 =2(11986741198794 minus ℎ41199054) 1198605 = 2(11986751198795 minus ℎ51199055) and 1198608 = 11986781198798 minus ℎ81199058

The axial displacement of the oblique web member isgiven by

Δ 5 = 1205751 cos1205721205751 = 1198874 (1199081015840 minus 120579) (7)

where 120572 is the angle between the oblique and vertical webmembers

22 Strain Model of SCCTB According to the displacementmodel of SCCTB in Section 21 the longitudinal strain of thecross-section can be expressed as

120576119909119894 = 120597119906119894120597119909 = 1198961198881205851015840 minus (119911 minus 119911119888) 1205791015840 119894 = 1 21198961199041205851015840 minus (119911 minus 119911119904) 1205791015840 119894 = 3 4 (8)

120574119908 = 1199081015840 minus 120579 (9)

120576119891119909 = Δ 51198875 = 1198874 (1199081015840 minus 120579) cos1205721198875 = 1198874119887611988752 (1199081015840 minus 120579) (10)

where 120576119909119894 (119894 = 1 2 3 4) are the longitudinal strains ofthe top slab cantilever slab upper chord and lower chordrespectively 120576119891119909 is the normal strain of the oblique webmember

Derived from (5) the longitudinal relative slip 120577(119909 119905)between the concrete slab and the truss is

120577 (119909 119905) = 120585 + ℎ119888120579 + ℎ119904120579 = 120585 + ℎ120579 (11)

where ℎ = ℎ119888 + ℎ119904 ℎ119888 and ℎ119904 are the distances from thecentroids of the concrete slab and steel truss to the interfacerespectively

23 Stress Model of SCCTB Based on the strain models ofSCCTB in Section 22 the stresses of SCCTB are

120590119909119894 = 119864119888 [1198961198881205851015840 minus (119911 minus 119911119888) 1205791015840] 119894 = 1 2119864119904 [1198961199041205851015840 minus (119911 minus 119911119904) 1205791015840] 119894 = 3 4

120591119908 = 119866119904 (1199081015840 minus 120579) 120590119891119909 = 119864119904 1198874119887611988752 (1199081015840 minus 120579)

(12)

where 120590119909119894 (119894 = 1 2 3 4) are the longitudinal stresses ofthe top slab cantilever slab upper chord and lower chord

respectively 120591119908 is the shear stress of the upper chord and thelower chord respectively and 120590119891119909 is the normal stress of theoblique web member

The shearing force per unit length of the interface is givenby

120589 (119909 119905) = 119896119904119897120577 (119909 119905) = 119896119904119897 (120585 + ℎ120579) (13)

119896119904119897 = 11987011198971199041198701 = 066119899119904119881119906119881119906 = 119860 119904119891119904119903119897119904(119871119899119904)

(14)

where 119896119904119897 is the longitudinal shear stiffness in per unit lengthbetween the concrete slab and the steel truss beam andthe unit is Nmm2 1198701 is the longitudinal shear stiffnessof the shear connections and the unit is Nmm 119899119904 is thenumber of shear connections in a crosswise row 119897119904 is thelongitudinal spacing between two studs119871 is the effective spanof SCCTB 119903 is the degree of shear connection 119891119904 is the yieldstrength of steel material and 119881119906 is the shear strength of thestud

3 Finite Beam Element Method of SCCTB

31 Improved SCCTB Element Stiffness Matrix and MassMatrix Including the Effects of Interface Slip and Shear Defor-mation In order to satisfy the consistency requirementsfor the different displacement functions to contribute to thesame strain function the degree of polynomial has to bethe same after the finite element approximation Otherwisethere are unreal geometric constraint conditions which cansignificantly reduce the accuracy of the finite elementmethodand result in shear locking [5 18] The highest-order deriva-tives of each displacement function in (8)ndash(11) show that thehighest derivatives of 120585(119909) and 120579(119909) are first-order derivativesHence the corresponding shape function can only satisfythe continuity condition for 1198620 Further (9) shows that theshape function of the deflection function 119908(119909) must satisfythe continuity condition of 1198621 According to the principle ofnumerical calculation in order to satisfy the continuity con-dition of 1198621 the Hermite polynomial has to be at least cubicHowever (9) shows that the contribution of the deflectionfunction 119908(119909) to the cross-section rotation function 120579(119909) isa quadratic polynomial Hence in order to satisfy the consis-tency requirements the displacement functions 120585(119909) and 120579(119909)have to be approximated using the quadratic polynomialsFurther the elements satisfying the consistency requirementshave to have at least 10 degrees of freedom as shownin Figure 2 Assume that the nodal displacement vectoris

120585 (119909) = 120593120585q120585120579 (119909) = 120593120579q120579119908 (119909) = 120593119908q119908

(15)

4 Shock and Vibration

kc1 kc2 kc3

ks1 ks2 ks3

1 2 3

w1 w2

w1

w2

Figure 2 Nodal degrees of freedom of the SCCTB element

q120585 = [1205851 1205852 1205853]119879 q120579 = [1205791 1205792 1205793]119879 q119908 = [1199081 11990810158401 1199082 11990810158402]119879

(16)

The Hermite polynomial shape function to satisfy thecontinuity condition of 1198621 at the element boundary is

120593120585 = 120593120579 = [1 minus 3119896 + 21198962 4119896 minus 41198962 minus119896 + 21198962]120593119908 = [1 minus 31198962 + 21198963 (119896 minus 21198962 + 1198963) 119897 31198962

minus 21198963 119897 (minus1198962 + 1198963)] (17)

where 119896 = 119909119897 and 119897 is the element lengthSubstituting (15) into (8) gives

120576119909119894 = 1198961198881205931015840120585q120585 minus (119911 minus 119911119888) 1205931015840120579q120579 119894 = 1 21198961199041205931015840120585q120585 minus (119911 minus 119911119904) 1205931015840120579q120579 119894 = 3 4

120574119908 = 1205931015840119908q119908 minus 120593120579q120579120577 (119909) = 120593120585q120585 + ℎ120593120579q120579120576119891119909 = 1198874119887611988752 (1205931015840119908q119908 minus 120593120579q120579)

(18)

Substituting (15) into (12)-(13) gives

120590119909119894 = 119864119888 [1198961198881205931015840120585q120585 minus (119911 minus 119911119888) 1205931015840120579q120579] 119894 = 1 2119864119904 [1198961199041205931015840120585q120585 minus (119911 minus 119911119904) 1205931015840120579q120579] 119894 = 3 4 (19)

120591119908 = 119866119904 (1205931015840119908q119908 minus 120593120579q120579) 120589 (119909) = 119896119904119897 (120593120585q120585 + ℎ120593120579q120579) (20)

120590119891119909 = 119864119904 1198874119887611988752 (1205931015840119908q119908 minus 120593120579q120579) (21)

The kinetic energy of the SCCTB can be expressed as

119879 = 12 int1198971198982119889119909 + 12

4sum119894=1

int119897int119860119894

120588119894 2119894 119889119860119889119909

+ 12 int119897int1198605

120588119904 2119894 11988751198874 119889119860119889119909 + 12 int119897int1198608

1205881199042cos120573119889119860119889119909

(22)

where 119898 = 120588119888119860119888 + 120588119904119860 119904 120588119888 and 120588119904 are the densities of thesteel material and the concrete material respectively and1205881 = 1205882 = 120588119888 1205883 = 1205884 = 120588119904

The strain energy of SCCTB can be expressed as

119881 = 12 int119897( 4sum119894=1

int119860119894

120590119909119894120576119909119894 119889119860 + int119860119909119892

120591119908120574119908 119889119860

+ int119860119890119902

120590119891119909120576119891119909 119889119860 + int1198608119867

12059011990941205761199094 119889119860 + 120589120577)119889119909(23)

where 119860119890119902 = 1198605 sin120572 and 119860119909119892 = 1198603 + 1198604According to (22) and (23) the variation forms of the

strain energy and the kinetic energy can be expressed as

int11990521199051

120575119879119889119905 = minusint11990521199051

[119898int119897120575119908119889119909

+ int119897(int119860119888

120588119888119888120575119906119888 119889119860 + int119860119909119892

120588119904119904120575119906119904 119889119860)119889119909+ int119897int119860119890119902

120588119904119904120575119906119904 119889119860119889119909+ int119897int1198608

1cos120573120588119904119904120575119906119904 119889119860 119889119909]119889119905

(24)

int11990521199051

120575119881119889119905 = int11990521199051

[int119897int119860119888

120575120576119879119909120590119909 119889119860119889119909+ int119897int119860119909119892

(120575120576119879119909120590119909 + 120575120574119879119908120591119908) 119889119860119889119909+ int119897int1198605

1sin 120572120575120576119879119891119909120590119891119909 119889119860119889119909

+ int119897int1198608119867

12057512057611987911990941205901199094 119889119860119889119909 + int119897120575120577119879120589 119889119909] 119889119905

(25)

By substituting (18)ndash(20) into (24) the SCCTB elementmass matrix is

M119890 = [[[[

M120585120585 M120585120579 M120585119908M120579120585 M120579120579 M120579119908M119908120585 M119908120579 M119908119908

]]]]

M120585120585 = int119897int119860119888

1205881198881198961198882120593120585119879120593120585 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199042120593120585119879120593120585 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199042120593120585119879120593120585 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199042120593120585119879120593120585 119889119860119889119909

Shock and Vibration 5

M120585120579 = int119897int119860119888

1205881198881198961198881199111120593120585119879120593120579 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199041199112120593120585119879120593120579 119889119860119889119909

M120579120579 = int119897int119860119888

12058811988811991121120593120579119879120593120579 119889119860119889119909+ int119897int119860119909119892

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int119860119890119902

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1cos12057312058811990411991122120593120579119879120593120579 119889119860119889119909

M120585119908 = 0M120579119908 = 0M119908119908 = 119898int

119897120593119908119879120593119908 119889119909

M120579120585 = M119879120585120579M119908120585 = M119879120585119908M119908120579 = M119879120579119908

(26)

where119898 = 1198981+1198982+1198983+1198984+1198985+1198986+1198987+11989881198981 = 12058811988811986011198982 = 1205881198881198602 1198983 = 1205881199041198603 1198984 = 1205881199041198604 1198985 = 120588119904119860511988751198874 1198986 =120588119904119887611986061198874 1198987 = 120588119904119887711986071198874 1198988 = 120588119904119860811988781198874 1198961 = 1198962 = 1198961198881198963 = 1198964 = 119896119904 1199111 = 119911119888 minus 119911 1199112 = 119911119904 minus 119911By substituting (18)ndash(20) into (25) the SCCTB element

stiffness matrix is

K119890 = [[[[

K120585120585 K120585120579 K120585119908K120579120585 K120579120579 K120579119908K119908120585 K119908120579 K119908119908

]]]]

K120585120585 = int119897int119860119888

119864119888119896119888212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int1198608

119864119904119896119904212059310158401205851198791205931015840120585 cos3120573119889119860119889119909+ int119897119896119904119897120593120585119879120593120585119889119909

K120585120579 = int119897int119860119888

119864119888119896119888119911112059310158401205851198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904119911212059310158401205851198791205931015840120579 119889119860119889119909+ int119897int1198608

119864119904119896119904119911212059310158401205851198791205931015840120579 cos3120573 119889119860119889119909+ int119897120593120585119879119896119904119897ℎ120593120579 119889119909

K120579120579 = int119897int119860119888

1199112111986411988812059310158401205791198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

(1199112211986411990412059310158401205791198791205931015840120579 + 119866119904120593120579119879120593120579) 119889119860119889119909

+ int119897int1198605

11988741198872611988735 119864119904120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1199112211986411990412059310158401205791198791205931015840120579 cos3120573119889119860119889119909+ int119897ℎ2119896119904119897120593120579119879120593120579 119889119909

K120579119908 = int119897int1198605

(minus11988741198872611988735 1198641199041205931205791198791205931015840119908)119889119860119889119909minus int119897int119860119909119892

1198661199041205931205791198791205931015840119908 119889119860119889119909

K119908119908 = int119897int1198605

11988741198872611988735 11986411990412059310158401199081198791205931015840119908 119889119860119889119909+ int119897int119860119909119892

11986611990412059310158401199081198791205931015840119908 119889119860119889119909K120585119908 = 0K120579120585 = K119879120585120579K119908120585 = K119879120585119908K119908120579 = K119879119908120579

(27)

32 Solving the Natural Vibration Frequency of SCCTBBased on the SCCTB element stiffness matrix K119890 elementmass matrix M119890 and element displacement vector q119890 =q119879120585 q119879120579 q119879119908119879 and using the ldquoseat by numberrdquo method theSCCTB overall stiffness matrix K overall mass matrix Mand overall degree of freedom vector q can be obtainedThe common boundary conditions for the SCCTB can beexpressed as follows [17 19]

1205851003816100381610038161003816119909=0119871 = 119908|119909=0119871 = 120579|119909=0119871 = 0 (28)

6 Shock and Vibration

The transformational relationship between the SCCTBoverall degree of freedom vector before and after applying theboundary constraint can be expressed as follows

q = Sq (29)

where S is the transformation matrix for the overall degreeof freedom vector which can be obtained for the givenboundary conditions and q is the SCCTB overall degree offreedom vector after applying the boundary constraint

The free vibration function for a structure with multipledegrees of freedom can be expressed as follows

Mu + Ku = 0 (30)

Let

u = q sin120596119905 (31)

where 120596 is the structurersquos natural vibration frequencySubstituting (31) into (30) gives

(minus1205962M + K) q = 0 (32)

Substituting (29) into (32) gives

(minus1205962M + K) q = 0 (33)

where

M = S119879MSK = S119879KS (34)

Using (33) the frequency equation of SCCTB can beobtained as follows 10038161003816100381610038161003816minus1205962M + K10038161003816100381610038161003816 = 0 (35)

The software MATLAB has been used to develop theFBEM program of the abovementioned SCCTB elementThen the natural vibration frequency of SCCTB has beensolved by including the effects of both the interface slip andshear deformation

33 Simplification of SCCTB Element Stiffness Matrix If theshear deformation is not considered then

120574119908 = 1199081015840 minus 120579 = 0 997904rArr1199081015840 = 120579 997904rArr

1205931015840119908q119908 = 120593120579q120579119908101584010038161003816100381610038161003816119909=0 = 1205931015840119908q11990810038161003816100381610038161003816119909=0 = 11990810158401119908101584010038161003816100381610038161003816119909=119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=119897 = 11990810158402

119908101584010038161003816100381610038161003816119909=05119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=05119897 = [minus 32119897 minus14 32119897 minus14] q119908120579|119909=0 = 120593120579q1205791003816100381610038161003816119909=0 = 1205791

120579|119909=05119897 = 120593120579q1205791003816100381610038161003816119909=05119897 = 1205792120579|119909=119897 = 120593120579q1205791003816100381610038161003816119909=119897 = 1205793

(36)

Using (36) the following expression can be obtained

1205791 = 119908101584011205792 = [minus 32119897 minus14 32119897 minus14] q1199081205793 = 11990810158402

(37)

Using (37) and excluding the effect of the shear deforma-tion the element stiffness matrix K119890 and mass matrix M119890 are

1006704K119890 = S119889K119890S119879119889

1006704M119890 = S119889M119890S119879119889

(38)

where S119889 is a 7times10 transformationmatrix and can be obtainedusing (37)

After obtaining the SCCTB element stiffness matrix andmass matrix without the effect of the shear deformation thesame method in Section 32 can be used to calculate thenatural vibration frequency of SCCTB without the effect ofthe shear deformation

4 Cases for Analyses

To verify the accuracy of the developed FBEM (Section 3)the natural frequencies of two groups of clamped supportedSCCTBs (ie SCCTB-1 and SCCTB-2) have been calculatedusing both the finite elementmethod and the FBEM For eachgroup there are cases with five degree of shear connection(119903 = 04 06 08 10 20) and two spans The mechanical andgeometrical parameters of the SCCTBs are as follows

119897119904 = 25mm119899119904 = 101198871 = 400mm1198872 = 200mm1198873 = 200mm1198874 = 200mm1198875 = 4472mm1198876 = 400mm1198877 = 760mm1198878 = 7859mm1199051 = 150mm1199052 = 150mm1198673 = 40mmℎ3 = 34mm1198674 = 40mm

Shock and Vibration 7

ℎ4 = 34mm1198675 = 43mmℎ5 = 37mm1198676 = 22mmℎ6 = 17mm1198677 = 22mmℎ7 = 17mm1198678 = 22mmℎ8 = 17mm1198679 = 22mmℎ9 = 17mm1198793 = 40mm1199053 = 34mm1198794 = 40mm1199054 = 34mm1198795 = 43mm1199055 = 37mm1198796 = 22mm1199056 = 17mm1198797 = 22mm1199057 = 17mm1198798 = 22mm1199058 = 17mm1198799 = 22mm1199059 = 17mm119864119904 = 21 times 105MPa119864119888 = 45 times 104MPa120588119904 = 7900 kgsdotmminus3120588119888 = 2400 kgsdotmminus3119891119910 = 400MPa120583119904 = 03120583119888 = 021198711 = 10m1198712 = 12m

(39)

119871 119894 (119894 = 1 2) are the calculation lengths of the two groupsof SCCTBs respectively Further 120583119904 and 120583119888 are Poissonrsquos ratioof steel and concrete respectively

The calculations by the finite element method have beencarried out using the finite element program ANSYS Theupper chord and the lower chord have been simulatedusing SHELL43 shell element The vertical web memberoblique web member lower horizontal connection memberand oblique bracing member have been simulated usingBEAM188 elements The concrete slab has been simulatedusing SOLID65 solid elementsThe studs have been simulatedusing COMBIN14 spring elements The elastic modulus 1198701of the spring element has been calculated using (14) In thefinite element models by coupling the degrees of freedomin the vertical direction of the nodes at the same positionthe interface between the concrete slab and the steel trussbeam is connected in the vertical that is there is no verticalseparation between the concrete slab and the steel truss beamIn order to simulate the clamped supported at the ends ofthe beams in the finite element models there are constrainsin the degrees of freedom in the vertical transverse andlongitudinal directions

Tables 1 and 2 show a comparison of the results of thecalculated natural frequencies by the FBEM and ANSYSmodel 119877AN is the calculation result from the ANSYS model119877FB is the calculation result from the FBEM including theeffects of the shear deformation and the interface slip 119877SDis the calculation result without including the effect of theshear deformation Further 119890FB = 100(119877FB minus119877AN)119877AN is thecalculation error in FBEM and 119890SD = 100(119877SD minus 119877FB)119877FBand 119862119904 = (119877FB|119903=20 minus 119877FB|119903=04)119877FB|119903=20 where 119890SD and 119862119904are the errors due to the effects of the shear deformation andthe interface slip respectively

From Tables 1 and 2 and Figures 3ndash5 the following can beseen

(1) By including the effects of the interface slip sheardeformation moment of inertia and many other influencingfactors on the SCCTB the results of the FBEMrsquos calculationsagree with those of the ANSYSrsquos finite element calculationsThe maximum calculation error 119890FB of the first six ordersof natural vibration frequency is less than 46 This is anindication that the SCCTB element stiffness matrix and massmatrix developed in this study are rational and effectiveAccording to the calculation results of ANSYS the localvibration of SCCTB-1 with small span is becoming obviousin the 6th mode while SCCTB-2 with larger span doesnot cause this kind of vibration in high modes Neglectinglocal vibration will result in generating additional restraintson the SCCTB which will overestimate the restrain rigidityof the SCCTB The proposed FBEM model in this papercan be used to describe the actual behavior of SCCTBs inflexural vibrations effectively but it does not take into accountthe local vibration Therefore the 6th natural frequency ofSCCTB-1 using FBEM is slightly larger than the ANSYSrsquoresults

(2) Without including the effect of the shear deformationon the SCCTBrsquos natural vibration frequency the FBEMrsquoscalculation results are greater than those of the ANSYSrsquoscalculations Further SCCTBrsquos shear deformation effect is

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 3

where 119911119904 is the 119911-coordinate of the centroid of the steel trussbeam 119911119888 is the 119911-coordinate of the centroid of the concreteslab 120579(119909) = 1199081015840(119909) minus 120574119908(119909) is the cross-sectional rotationangle 119908(119909) is the vertical deflection of the SCCTB 120585(119909)is the difference between the longitudinal displacement ofthe concrete slab centroid and that of the steel truss beamcentroid 119864119904 is the elastic modulus of the steel truss and 119864119888is the elastic modulus of the concrete slab Further 119899 =119864119904119864119888 119860119888 = 1198601 + 1198602 119860 119904 = 1198603 + 1198604 + 1198608119867 1198600 = 119860119888119899 +119860 119904 1198601 = 211988711199051 1198602 = 211988721199052 1198603 = 2(11986731198793 minus ℎ31199053) 1198604 =2(11986741198794 minus ℎ41199054) 1198605 = 2(11986751198795 minus ℎ51199055) and 1198608 = 11986781198798 minus ℎ81199058

The axial displacement of the oblique web member isgiven by

Δ 5 = 1205751 cos1205721205751 = 1198874 (1199081015840 minus 120579) (7)

where 120572 is the angle between the oblique and vertical webmembers

22 Strain Model of SCCTB According to the displacementmodel of SCCTB in Section 21 the longitudinal strain of thecross-section can be expressed as

120576119909119894 = 120597119906119894120597119909 = 1198961198881205851015840 minus (119911 minus 119911119888) 1205791015840 119894 = 1 21198961199041205851015840 minus (119911 minus 119911119904) 1205791015840 119894 = 3 4 (8)

120574119908 = 1199081015840 minus 120579 (9)

120576119891119909 = Δ 51198875 = 1198874 (1199081015840 minus 120579) cos1205721198875 = 1198874119887611988752 (1199081015840 minus 120579) (10)

where 120576119909119894 (119894 = 1 2 3 4) are the longitudinal strains ofthe top slab cantilever slab upper chord and lower chordrespectively 120576119891119909 is the normal strain of the oblique webmember

Derived from (5) the longitudinal relative slip 120577(119909 119905)between the concrete slab and the truss is

120577 (119909 119905) = 120585 + ℎ119888120579 + ℎ119904120579 = 120585 + ℎ120579 (11)

where ℎ = ℎ119888 + ℎ119904 ℎ119888 and ℎ119904 are the distances from thecentroids of the concrete slab and steel truss to the interfacerespectively

23 Stress Model of SCCTB Based on the strain models ofSCCTB in Section 22 the stresses of SCCTB are

120590119909119894 = 119864119888 [1198961198881205851015840 minus (119911 minus 119911119888) 1205791015840] 119894 = 1 2119864119904 [1198961199041205851015840 minus (119911 minus 119911119904) 1205791015840] 119894 = 3 4

120591119908 = 119866119904 (1199081015840 minus 120579) 120590119891119909 = 119864119904 1198874119887611988752 (1199081015840 minus 120579)

(12)

where 120590119909119894 (119894 = 1 2 3 4) are the longitudinal stresses ofthe top slab cantilever slab upper chord and lower chord

respectively 120591119908 is the shear stress of the upper chord and thelower chord respectively and 120590119891119909 is the normal stress of theoblique web member

The shearing force per unit length of the interface is givenby

120589 (119909 119905) = 119896119904119897120577 (119909 119905) = 119896119904119897 (120585 + ℎ120579) (13)

119896119904119897 = 11987011198971199041198701 = 066119899119904119881119906119881119906 = 119860 119904119891119904119903119897119904(119871119899119904)

(14)

where 119896119904119897 is the longitudinal shear stiffness in per unit lengthbetween the concrete slab and the steel truss beam andthe unit is Nmm2 1198701 is the longitudinal shear stiffnessof the shear connections and the unit is Nmm 119899119904 is thenumber of shear connections in a crosswise row 119897119904 is thelongitudinal spacing between two studs119871 is the effective spanof SCCTB 119903 is the degree of shear connection 119891119904 is the yieldstrength of steel material and 119881119906 is the shear strength of thestud

3 Finite Beam Element Method of SCCTB

31 Improved SCCTB Element Stiffness Matrix and MassMatrix Including the Effects of Interface Slip and Shear Defor-mation In order to satisfy the consistency requirementsfor the different displacement functions to contribute to thesame strain function the degree of polynomial has to bethe same after the finite element approximation Otherwisethere are unreal geometric constraint conditions which cansignificantly reduce the accuracy of the finite elementmethodand result in shear locking [5 18] The highest-order deriva-tives of each displacement function in (8)ndash(11) show that thehighest derivatives of 120585(119909) and 120579(119909) are first-order derivativesHence the corresponding shape function can only satisfythe continuity condition for 1198620 Further (9) shows that theshape function of the deflection function 119908(119909) must satisfythe continuity condition of 1198621 According to the principle ofnumerical calculation in order to satisfy the continuity con-dition of 1198621 the Hermite polynomial has to be at least cubicHowever (9) shows that the contribution of the deflectionfunction 119908(119909) to the cross-section rotation function 120579(119909) isa quadratic polynomial Hence in order to satisfy the consis-tency requirements the displacement functions 120585(119909) and 120579(119909)have to be approximated using the quadratic polynomialsFurther the elements satisfying the consistency requirementshave to have at least 10 degrees of freedom as shownin Figure 2 Assume that the nodal displacement vectoris

120585 (119909) = 120593120585q120585120579 (119909) = 120593120579q120579119908 (119909) = 120593119908q119908

(15)

4 Shock and Vibration

kc1 kc2 kc3

ks1 ks2 ks3

1 2 3

w1 w2

w1

w2

Figure 2 Nodal degrees of freedom of the SCCTB element

q120585 = [1205851 1205852 1205853]119879 q120579 = [1205791 1205792 1205793]119879 q119908 = [1199081 11990810158401 1199082 11990810158402]119879

(16)

The Hermite polynomial shape function to satisfy thecontinuity condition of 1198621 at the element boundary is

120593120585 = 120593120579 = [1 minus 3119896 + 21198962 4119896 minus 41198962 minus119896 + 21198962]120593119908 = [1 minus 31198962 + 21198963 (119896 minus 21198962 + 1198963) 119897 31198962

minus 21198963 119897 (minus1198962 + 1198963)] (17)

where 119896 = 119909119897 and 119897 is the element lengthSubstituting (15) into (8) gives

120576119909119894 = 1198961198881205931015840120585q120585 minus (119911 minus 119911119888) 1205931015840120579q120579 119894 = 1 21198961199041205931015840120585q120585 minus (119911 minus 119911119904) 1205931015840120579q120579 119894 = 3 4

120574119908 = 1205931015840119908q119908 minus 120593120579q120579120577 (119909) = 120593120585q120585 + ℎ120593120579q120579120576119891119909 = 1198874119887611988752 (1205931015840119908q119908 minus 120593120579q120579)

(18)

Substituting (15) into (12)-(13) gives

120590119909119894 = 119864119888 [1198961198881205931015840120585q120585 minus (119911 minus 119911119888) 1205931015840120579q120579] 119894 = 1 2119864119904 [1198961199041205931015840120585q120585 minus (119911 minus 119911119904) 1205931015840120579q120579] 119894 = 3 4 (19)

120591119908 = 119866119904 (1205931015840119908q119908 minus 120593120579q120579) 120589 (119909) = 119896119904119897 (120593120585q120585 + ℎ120593120579q120579) (20)

120590119891119909 = 119864119904 1198874119887611988752 (1205931015840119908q119908 minus 120593120579q120579) (21)

The kinetic energy of the SCCTB can be expressed as

119879 = 12 int1198971198982119889119909 + 12

4sum119894=1

int119897int119860119894

120588119894 2119894 119889119860119889119909

+ 12 int119897int1198605

120588119904 2119894 11988751198874 119889119860119889119909 + 12 int119897int1198608

1205881199042cos120573119889119860119889119909

(22)

where 119898 = 120588119888119860119888 + 120588119904119860 119904 120588119888 and 120588119904 are the densities of thesteel material and the concrete material respectively and1205881 = 1205882 = 120588119888 1205883 = 1205884 = 120588119904

The strain energy of SCCTB can be expressed as

119881 = 12 int119897( 4sum119894=1

int119860119894

120590119909119894120576119909119894 119889119860 + int119860119909119892

120591119908120574119908 119889119860

+ int119860119890119902

120590119891119909120576119891119909 119889119860 + int1198608119867

12059011990941205761199094 119889119860 + 120589120577)119889119909(23)

where 119860119890119902 = 1198605 sin120572 and 119860119909119892 = 1198603 + 1198604According to (22) and (23) the variation forms of the

strain energy and the kinetic energy can be expressed as

int11990521199051

120575119879119889119905 = minusint11990521199051

[119898int119897120575119908119889119909

+ int119897(int119860119888

120588119888119888120575119906119888 119889119860 + int119860119909119892

120588119904119904120575119906119904 119889119860)119889119909+ int119897int119860119890119902

120588119904119904120575119906119904 119889119860119889119909+ int119897int1198608

1cos120573120588119904119904120575119906119904 119889119860 119889119909]119889119905

(24)

int11990521199051

120575119881119889119905 = int11990521199051

[int119897int119860119888

120575120576119879119909120590119909 119889119860119889119909+ int119897int119860119909119892

(120575120576119879119909120590119909 + 120575120574119879119908120591119908) 119889119860119889119909+ int119897int1198605

1sin 120572120575120576119879119891119909120590119891119909 119889119860119889119909

+ int119897int1198608119867

12057512057611987911990941205901199094 119889119860119889119909 + int119897120575120577119879120589 119889119909] 119889119905

(25)

By substituting (18)ndash(20) into (24) the SCCTB elementmass matrix is

M119890 = [[[[

M120585120585 M120585120579 M120585119908M120579120585 M120579120579 M120579119908M119908120585 M119908120579 M119908119908

]]]]

M120585120585 = int119897int119860119888

1205881198881198961198882120593120585119879120593120585 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199042120593120585119879120593120585 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199042120593120585119879120593120585 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199042120593120585119879120593120585 119889119860119889119909

Shock and Vibration 5

M120585120579 = int119897int119860119888

1205881198881198961198881199111120593120585119879120593120579 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199041199112120593120585119879120593120579 119889119860119889119909

M120579120579 = int119897int119860119888

12058811988811991121120593120579119879120593120579 119889119860119889119909+ int119897int119860119909119892

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int119860119890119902

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1cos12057312058811990411991122120593120579119879120593120579 119889119860119889119909

M120585119908 = 0M120579119908 = 0M119908119908 = 119898int

119897120593119908119879120593119908 119889119909

M120579120585 = M119879120585120579M119908120585 = M119879120585119908M119908120579 = M119879120579119908

(26)

where119898 = 1198981+1198982+1198983+1198984+1198985+1198986+1198987+11989881198981 = 12058811988811986011198982 = 1205881198881198602 1198983 = 1205881199041198603 1198984 = 1205881199041198604 1198985 = 120588119904119860511988751198874 1198986 =120588119904119887611986061198874 1198987 = 120588119904119887711986071198874 1198988 = 120588119904119860811988781198874 1198961 = 1198962 = 1198961198881198963 = 1198964 = 119896119904 1199111 = 119911119888 minus 119911 1199112 = 119911119904 minus 119911By substituting (18)ndash(20) into (25) the SCCTB element

stiffness matrix is

K119890 = [[[[

K120585120585 K120585120579 K120585119908K120579120585 K120579120579 K120579119908K119908120585 K119908120579 K119908119908

]]]]

K120585120585 = int119897int119860119888

119864119888119896119888212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int1198608

119864119904119896119904212059310158401205851198791205931015840120585 cos3120573119889119860119889119909+ int119897119896119904119897120593120585119879120593120585119889119909

K120585120579 = int119897int119860119888

119864119888119896119888119911112059310158401205851198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904119911212059310158401205851198791205931015840120579 119889119860119889119909+ int119897int1198608

119864119904119896119904119911212059310158401205851198791205931015840120579 cos3120573 119889119860119889119909+ int119897120593120585119879119896119904119897ℎ120593120579 119889119909

K120579120579 = int119897int119860119888

1199112111986411988812059310158401205791198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

(1199112211986411990412059310158401205791198791205931015840120579 + 119866119904120593120579119879120593120579) 119889119860119889119909

+ int119897int1198605

11988741198872611988735 119864119904120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1199112211986411990412059310158401205791198791205931015840120579 cos3120573119889119860119889119909+ int119897ℎ2119896119904119897120593120579119879120593120579 119889119909

K120579119908 = int119897int1198605

(minus11988741198872611988735 1198641199041205931205791198791205931015840119908)119889119860119889119909minus int119897int119860119909119892

1198661199041205931205791198791205931015840119908 119889119860119889119909

K119908119908 = int119897int1198605

11988741198872611988735 11986411990412059310158401199081198791205931015840119908 119889119860119889119909+ int119897int119860119909119892

11986611990412059310158401199081198791205931015840119908 119889119860119889119909K120585119908 = 0K120579120585 = K119879120585120579K119908120585 = K119879120585119908K119908120579 = K119879119908120579

(27)

32 Solving the Natural Vibration Frequency of SCCTBBased on the SCCTB element stiffness matrix K119890 elementmass matrix M119890 and element displacement vector q119890 =q119879120585 q119879120579 q119879119908119879 and using the ldquoseat by numberrdquo method theSCCTB overall stiffness matrix K overall mass matrix Mand overall degree of freedom vector q can be obtainedThe common boundary conditions for the SCCTB can beexpressed as follows [17 19]

1205851003816100381610038161003816119909=0119871 = 119908|119909=0119871 = 120579|119909=0119871 = 0 (28)

6 Shock and Vibration

The transformational relationship between the SCCTBoverall degree of freedom vector before and after applying theboundary constraint can be expressed as follows

q = Sq (29)

where S is the transformation matrix for the overall degreeof freedom vector which can be obtained for the givenboundary conditions and q is the SCCTB overall degree offreedom vector after applying the boundary constraint

The free vibration function for a structure with multipledegrees of freedom can be expressed as follows

Mu + Ku = 0 (30)

Let

u = q sin120596119905 (31)

where 120596 is the structurersquos natural vibration frequencySubstituting (31) into (30) gives

(minus1205962M + K) q = 0 (32)

Substituting (29) into (32) gives

(minus1205962M + K) q = 0 (33)

where

M = S119879MSK = S119879KS (34)

Using (33) the frequency equation of SCCTB can beobtained as follows 10038161003816100381610038161003816minus1205962M + K10038161003816100381610038161003816 = 0 (35)

The software MATLAB has been used to develop theFBEM program of the abovementioned SCCTB elementThen the natural vibration frequency of SCCTB has beensolved by including the effects of both the interface slip andshear deformation

33 Simplification of SCCTB Element Stiffness Matrix If theshear deformation is not considered then

120574119908 = 1199081015840 minus 120579 = 0 997904rArr1199081015840 = 120579 997904rArr

1205931015840119908q119908 = 120593120579q120579119908101584010038161003816100381610038161003816119909=0 = 1205931015840119908q11990810038161003816100381610038161003816119909=0 = 11990810158401119908101584010038161003816100381610038161003816119909=119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=119897 = 11990810158402

119908101584010038161003816100381610038161003816119909=05119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=05119897 = [minus 32119897 minus14 32119897 minus14] q119908120579|119909=0 = 120593120579q1205791003816100381610038161003816119909=0 = 1205791

120579|119909=05119897 = 120593120579q1205791003816100381610038161003816119909=05119897 = 1205792120579|119909=119897 = 120593120579q1205791003816100381610038161003816119909=119897 = 1205793

(36)

Using (36) the following expression can be obtained

1205791 = 119908101584011205792 = [minus 32119897 minus14 32119897 minus14] q1199081205793 = 11990810158402

(37)

Using (37) and excluding the effect of the shear deforma-tion the element stiffness matrix K119890 and mass matrix M119890 are

1006704K119890 = S119889K119890S119879119889

1006704M119890 = S119889M119890S119879119889

(38)

where S119889 is a 7times10 transformationmatrix and can be obtainedusing (37)

After obtaining the SCCTB element stiffness matrix andmass matrix without the effect of the shear deformation thesame method in Section 32 can be used to calculate thenatural vibration frequency of SCCTB without the effect ofthe shear deformation

4 Cases for Analyses

To verify the accuracy of the developed FBEM (Section 3)the natural frequencies of two groups of clamped supportedSCCTBs (ie SCCTB-1 and SCCTB-2) have been calculatedusing both the finite elementmethod and the FBEM For eachgroup there are cases with five degree of shear connection(119903 = 04 06 08 10 20) and two spans The mechanical andgeometrical parameters of the SCCTBs are as follows

119897119904 = 25mm119899119904 = 101198871 = 400mm1198872 = 200mm1198873 = 200mm1198874 = 200mm1198875 = 4472mm1198876 = 400mm1198877 = 760mm1198878 = 7859mm1199051 = 150mm1199052 = 150mm1198673 = 40mmℎ3 = 34mm1198674 = 40mm

Shock and Vibration 7

ℎ4 = 34mm1198675 = 43mmℎ5 = 37mm1198676 = 22mmℎ6 = 17mm1198677 = 22mmℎ7 = 17mm1198678 = 22mmℎ8 = 17mm1198679 = 22mmℎ9 = 17mm1198793 = 40mm1199053 = 34mm1198794 = 40mm1199054 = 34mm1198795 = 43mm1199055 = 37mm1198796 = 22mm1199056 = 17mm1198797 = 22mm1199057 = 17mm1198798 = 22mm1199058 = 17mm1198799 = 22mm1199059 = 17mm119864119904 = 21 times 105MPa119864119888 = 45 times 104MPa120588119904 = 7900 kgsdotmminus3120588119888 = 2400 kgsdotmminus3119891119910 = 400MPa120583119904 = 03120583119888 = 021198711 = 10m1198712 = 12m

(39)

119871 119894 (119894 = 1 2) are the calculation lengths of the two groupsof SCCTBs respectively Further 120583119904 and 120583119888 are Poissonrsquos ratioof steel and concrete respectively

The calculations by the finite element method have beencarried out using the finite element program ANSYS Theupper chord and the lower chord have been simulatedusing SHELL43 shell element The vertical web memberoblique web member lower horizontal connection memberand oblique bracing member have been simulated usingBEAM188 elements The concrete slab has been simulatedusing SOLID65 solid elementsThe studs have been simulatedusing COMBIN14 spring elements The elastic modulus 1198701of the spring element has been calculated using (14) In thefinite element models by coupling the degrees of freedomin the vertical direction of the nodes at the same positionthe interface between the concrete slab and the steel trussbeam is connected in the vertical that is there is no verticalseparation between the concrete slab and the steel truss beamIn order to simulate the clamped supported at the ends ofthe beams in the finite element models there are constrainsin the degrees of freedom in the vertical transverse andlongitudinal directions

Tables 1 and 2 show a comparison of the results of thecalculated natural frequencies by the FBEM and ANSYSmodel 119877AN is the calculation result from the ANSYS model119877FB is the calculation result from the FBEM including theeffects of the shear deformation and the interface slip 119877SDis the calculation result without including the effect of theshear deformation Further 119890FB = 100(119877FB minus119877AN)119877AN is thecalculation error in FBEM and 119890SD = 100(119877SD minus 119877FB)119877FBand 119862119904 = (119877FB|119903=20 minus 119877FB|119903=04)119877FB|119903=20 where 119890SD and 119862119904are the errors due to the effects of the shear deformation andthe interface slip respectively

From Tables 1 and 2 and Figures 3ndash5 the following can beseen

(1) By including the effects of the interface slip sheardeformation moment of inertia and many other influencingfactors on the SCCTB the results of the FBEMrsquos calculationsagree with those of the ANSYSrsquos finite element calculationsThe maximum calculation error 119890FB of the first six ordersof natural vibration frequency is less than 46 This is anindication that the SCCTB element stiffness matrix and massmatrix developed in this study are rational and effectiveAccording to the calculation results of ANSYS the localvibration of SCCTB-1 with small span is becoming obviousin the 6th mode while SCCTB-2 with larger span doesnot cause this kind of vibration in high modes Neglectinglocal vibration will result in generating additional restraintson the SCCTB which will overestimate the restrain rigidityof the SCCTB The proposed FBEM model in this papercan be used to describe the actual behavior of SCCTBs inflexural vibrations effectively but it does not take into accountthe local vibration Therefore the 6th natural frequency ofSCCTB-1 using FBEM is slightly larger than the ANSYSrsquoresults

(2) Without including the effect of the shear deformationon the SCCTBrsquos natural vibration frequency the FBEMrsquoscalculation results are greater than those of the ANSYSrsquoscalculations Further SCCTBrsquos shear deformation effect is

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

kc1 kc2 kc3

ks1 ks2 ks3

1 2 3

w1 w2

w1

w2

Figure 2 Nodal degrees of freedom of the SCCTB element

q120585 = [1205851 1205852 1205853]119879 q120579 = [1205791 1205792 1205793]119879 q119908 = [1199081 11990810158401 1199082 11990810158402]119879

(16)

The Hermite polynomial shape function to satisfy thecontinuity condition of 1198621 at the element boundary is

120593120585 = 120593120579 = [1 minus 3119896 + 21198962 4119896 minus 41198962 minus119896 + 21198962]120593119908 = [1 minus 31198962 + 21198963 (119896 minus 21198962 + 1198963) 119897 31198962

minus 21198963 119897 (minus1198962 + 1198963)] (17)

where 119896 = 119909119897 and 119897 is the element lengthSubstituting (15) into (8) gives

120576119909119894 = 1198961198881205931015840120585q120585 minus (119911 minus 119911119888) 1205931015840120579q120579 119894 = 1 21198961199041205931015840120585q120585 minus (119911 minus 119911119904) 1205931015840120579q120579 119894 = 3 4

120574119908 = 1205931015840119908q119908 minus 120593120579q120579120577 (119909) = 120593120585q120585 + ℎ120593120579q120579120576119891119909 = 1198874119887611988752 (1205931015840119908q119908 minus 120593120579q120579)

(18)

Substituting (15) into (12)-(13) gives

120590119909119894 = 119864119888 [1198961198881205931015840120585q120585 minus (119911 minus 119911119888) 1205931015840120579q120579] 119894 = 1 2119864119904 [1198961199041205931015840120585q120585 minus (119911 minus 119911119904) 1205931015840120579q120579] 119894 = 3 4 (19)

120591119908 = 119866119904 (1205931015840119908q119908 minus 120593120579q120579) 120589 (119909) = 119896119904119897 (120593120585q120585 + ℎ120593120579q120579) (20)

120590119891119909 = 119864119904 1198874119887611988752 (1205931015840119908q119908 minus 120593120579q120579) (21)

The kinetic energy of the SCCTB can be expressed as

119879 = 12 int1198971198982119889119909 + 12

4sum119894=1

int119897int119860119894

120588119894 2119894 119889119860119889119909

+ 12 int119897int1198605

120588119904 2119894 11988751198874 119889119860119889119909 + 12 int119897int1198608

1205881199042cos120573119889119860119889119909

(22)

where 119898 = 120588119888119860119888 + 120588119904119860 119904 120588119888 and 120588119904 are the densities of thesteel material and the concrete material respectively and1205881 = 1205882 = 120588119888 1205883 = 1205884 = 120588119904

The strain energy of SCCTB can be expressed as

119881 = 12 int119897( 4sum119894=1

int119860119894

120590119909119894120576119909119894 119889119860 + int119860119909119892

120591119908120574119908 119889119860

+ int119860119890119902

120590119891119909120576119891119909 119889119860 + int1198608119867

12059011990941205761199094 119889119860 + 120589120577)119889119909(23)

where 119860119890119902 = 1198605 sin120572 and 119860119909119892 = 1198603 + 1198604According to (22) and (23) the variation forms of the

strain energy and the kinetic energy can be expressed as

int11990521199051

120575119879119889119905 = minusint11990521199051

[119898int119897120575119908119889119909

+ int119897(int119860119888

120588119888119888120575119906119888 119889119860 + int119860119909119892

120588119904119904120575119906119904 119889119860)119889119909+ int119897int119860119890119902

120588119904119904120575119906119904 119889119860119889119909+ int119897int1198608

1cos120573120588119904119904120575119906119904 119889119860 119889119909]119889119905

(24)

int11990521199051

120575119881119889119905 = int11990521199051

[int119897int119860119888

120575120576119879119909120590119909 119889119860119889119909+ int119897int119860119909119892

(120575120576119879119909120590119909 + 120575120574119879119908120591119908) 119889119860119889119909+ int119897int1198605

1sin 120572120575120576119879119891119909120590119891119909 119889119860119889119909

+ int119897int1198608119867

12057512057611987911990941205901199094 119889119860119889119909 + int119897120575120577119879120589 119889119909] 119889119905

(25)

By substituting (18)ndash(20) into (24) the SCCTB elementmass matrix is

M119890 = [[[[

M120585120585 M120585120579 M120585119908M120579120585 M120579120579 M120579119908M119908120585 M119908120579 M119908119908

]]]]

M120585120585 = int119897int119860119888

1205881198881198961198882120593120585119879120593120585 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199042120593120585119879120593120585 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199042120593120585119879120593120585 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199042120593120585119879120593120585 119889119860119889119909

Shock and Vibration 5

M120585120579 = int119897int119860119888

1205881198881198961198881199111120593120585119879120593120579 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199041199112120593120585119879120593120579 119889119860119889119909

M120579120579 = int119897int119860119888

12058811988811991121120593120579119879120593120579 119889119860119889119909+ int119897int119860119909119892

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int119860119890119902

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1cos12057312058811990411991122120593120579119879120593120579 119889119860119889119909

M120585119908 = 0M120579119908 = 0M119908119908 = 119898int

119897120593119908119879120593119908 119889119909

M120579120585 = M119879120585120579M119908120585 = M119879120585119908M119908120579 = M119879120579119908

(26)

where119898 = 1198981+1198982+1198983+1198984+1198985+1198986+1198987+11989881198981 = 12058811988811986011198982 = 1205881198881198602 1198983 = 1205881199041198603 1198984 = 1205881199041198604 1198985 = 120588119904119860511988751198874 1198986 =120588119904119887611986061198874 1198987 = 120588119904119887711986071198874 1198988 = 120588119904119860811988781198874 1198961 = 1198962 = 1198961198881198963 = 1198964 = 119896119904 1199111 = 119911119888 minus 119911 1199112 = 119911119904 minus 119911By substituting (18)ndash(20) into (25) the SCCTB element

stiffness matrix is

K119890 = [[[[

K120585120585 K120585120579 K120585119908K120579120585 K120579120579 K120579119908K119908120585 K119908120579 K119908119908

]]]]

K120585120585 = int119897int119860119888

119864119888119896119888212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int1198608

119864119904119896119904212059310158401205851198791205931015840120585 cos3120573119889119860119889119909+ int119897119896119904119897120593120585119879120593120585119889119909

K120585120579 = int119897int119860119888

119864119888119896119888119911112059310158401205851198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904119911212059310158401205851198791205931015840120579 119889119860119889119909+ int119897int1198608

119864119904119896119904119911212059310158401205851198791205931015840120579 cos3120573 119889119860119889119909+ int119897120593120585119879119896119904119897ℎ120593120579 119889119909

K120579120579 = int119897int119860119888

1199112111986411988812059310158401205791198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

(1199112211986411990412059310158401205791198791205931015840120579 + 119866119904120593120579119879120593120579) 119889119860119889119909

+ int119897int1198605

11988741198872611988735 119864119904120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1199112211986411990412059310158401205791198791205931015840120579 cos3120573119889119860119889119909+ int119897ℎ2119896119904119897120593120579119879120593120579 119889119909

K120579119908 = int119897int1198605

(minus11988741198872611988735 1198641199041205931205791198791205931015840119908)119889119860119889119909minus int119897int119860119909119892

1198661199041205931205791198791205931015840119908 119889119860119889119909

K119908119908 = int119897int1198605

11988741198872611988735 11986411990412059310158401199081198791205931015840119908 119889119860119889119909+ int119897int119860119909119892

11986611990412059310158401199081198791205931015840119908 119889119860119889119909K120585119908 = 0K120579120585 = K119879120585120579K119908120585 = K119879120585119908K119908120579 = K119879119908120579

(27)

32 Solving the Natural Vibration Frequency of SCCTBBased on the SCCTB element stiffness matrix K119890 elementmass matrix M119890 and element displacement vector q119890 =q119879120585 q119879120579 q119879119908119879 and using the ldquoseat by numberrdquo method theSCCTB overall stiffness matrix K overall mass matrix Mand overall degree of freedom vector q can be obtainedThe common boundary conditions for the SCCTB can beexpressed as follows [17 19]

1205851003816100381610038161003816119909=0119871 = 119908|119909=0119871 = 120579|119909=0119871 = 0 (28)

6 Shock and Vibration

The transformational relationship between the SCCTBoverall degree of freedom vector before and after applying theboundary constraint can be expressed as follows

q = Sq (29)

where S is the transformation matrix for the overall degreeof freedom vector which can be obtained for the givenboundary conditions and q is the SCCTB overall degree offreedom vector after applying the boundary constraint

The free vibration function for a structure with multipledegrees of freedom can be expressed as follows

Mu + Ku = 0 (30)

Let

u = q sin120596119905 (31)

where 120596 is the structurersquos natural vibration frequencySubstituting (31) into (30) gives

(minus1205962M + K) q = 0 (32)

Substituting (29) into (32) gives

(minus1205962M + K) q = 0 (33)

where

M = S119879MSK = S119879KS (34)

Using (33) the frequency equation of SCCTB can beobtained as follows 10038161003816100381610038161003816minus1205962M + K10038161003816100381610038161003816 = 0 (35)

The software MATLAB has been used to develop theFBEM program of the abovementioned SCCTB elementThen the natural vibration frequency of SCCTB has beensolved by including the effects of both the interface slip andshear deformation

33 Simplification of SCCTB Element Stiffness Matrix If theshear deformation is not considered then

120574119908 = 1199081015840 minus 120579 = 0 997904rArr1199081015840 = 120579 997904rArr

1205931015840119908q119908 = 120593120579q120579119908101584010038161003816100381610038161003816119909=0 = 1205931015840119908q11990810038161003816100381610038161003816119909=0 = 11990810158401119908101584010038161003816100381610038161003816119909=119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=119897 = 11990810158402

119908101584010038161003816100381610038161003816119909=05119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=05119897 = [minus 32119897 minus14 32119897 minus14] q119908120579|119909=0 = 120593120579q1205791003816100381610038161003816119909=0 = 1205791

120579|119909=05119897 = 120593120579q1205791003816100381610038161003816119909=05119897 = 1205792120579|119909=119897 = 120593120579q1205791003816100381610038161003816119909=119897 = 1205793

(36)

Using (36) the following expression can be obtained

1205791 = 119908101584011205792 = [minus 32119897 minus14 32119897 minus14] q1199081205793 = 11990810158402

(37)

Using (37) and excluding the effect of the shear deforma-tion the element stiffness matrix K119890 and mass matrix M119890 are

1006704K119890 = S119889K119890S119879119889

1006704M119890 = S119889M119890S119879119889

(38)

where S119889 is a 7times10 transformationmatrix and can be obtainedusing (37)

After obtaining the SCCTB element stiffness matrix andmass matrix without the effect of the shear deformation thesame method in Section 32 can be used to calculate thenatural vibration frequency of SCCTB without the effect ofthe shear deformation

4 Cases for Analyses

To verify the accuracy of the developed FBEM (Section 3)the natural frequencies of two groups of clamped supportedSCCTBs (ie SCCTB-1 and SCCTB-2) have been calculatedusing both the finite elementmethod and the FBEM For eachgroup there are cases with five degree of shear connection(119903 = 04 06 08 10 20) and two spans The mechanical andgeometrical parameters of the SCCTBs are as follows

119897119904 = 25mm119899119904 = 101198871 = 400mm1198872 = 200mm1198873 = 200mm1198874 = 200mm1198875 = 4472mm1198876 = 400mm1198877 = 760mm1198878 = 7859mm1199051 = 150mm1199052 = 150mm1198673 = 40mmℎ3 = 34mm1198674 = 40mm

Shock and Vibration 7

ℎ4 = 34mm1198675 = 43mmℎ5 = 37mm1198676 = 22mmℎ6 = 17mm1198677 = 22mmℎ7 = 17mm1198678 = 22mmℎ8 = 17mm1198679 = 22mmℎ9 = 17mm1198793 = 40mm1199053 = 34mm1198794 = 40mm1199054 = 34mm1198795 = 43mm1199055 = 37mm1198796 = 22mm1199056 = 17mm1198797 = 22mm1199057 = 17mm1198798 = 22mm1199058 = 17mm1198799 = 22mm1199059 = 17mm119864119904 = 21 times 105MPa119864119888 = 45 times 104MPa120588119904 = 7900 kgsdotmminus3120588119888 = 2400 kgsdotmminus3119891119910 = 400MPa120583119904 = 03120583119888 = 021198711 = 10m1198712 = 12m

(39)

119871 119894 (119894 = 1 2) are the calculation lengths of the two groupsof SCCTBs respectively Further 120583119904 and 120583119888 are Poissonrsquos ratioof steel and concrete respectively

The calculations by the finite element method have beencarried out using the finite element program ANSYS Theupper chord and the lower chord have been simulatedusing SHELL43 shell element The vertical web memberoblique web member lower horizontal connection memberand oblique bracing member have been simulated usingBEAM188 elements The concrete slab has been simulatedusing SOLID65 solid elementsThe studs have been simulatedusing COMBIN14 spring elements The elastic modulus 1198701of the spring element has been calculated using (14) In thefinite element models by coupling the degrees of freedomin the vertical direction of the nodes at the same positionthe interface between the concrete slab and the steel trussbeam is connected in the vertical that is there is no verticalseparation between the concrete slab and the steel truss beamIn order to simulate the clamped supported at the ends ofthe beams in the finite element models there are constrainsin the degrees of freedom in the vertical transverse andlongitudinal directions

Tables 1 and 2 show a comparison of the results of thecalculated natural frequencies by the FBEM and ANSYSmodel 119877AN is the calculation result from the ANSYS model119877FB is the calculation result from the FBEM including theeffects of the shear deformation and the interface slip 119877SDis the calculation result without including the effect of theshear deformation Further 119890FB = 100(119877FB minus119877AN)119877AN is thecalculation error in FBEM and 119890SD = 100(119877SD minus 119877FB)119877FBand 119862119904 = (119877FB|119903=20 minus 119877FB|119903=04)119877FB|119903=20 where 119890SD and 119862119904are the errors due to the effects of the shear deformation andthe interface slip respectively

From Tables 1 and 2 and Figures 3ndash5 the following can beseen

(1) By including the effects of the interface slip sheardeformation moment of inertia and many other influencingfactors on the SCCTB the results of the FBEMrsquos calculationsagree with those of the ANSYSrsquos finite element calculationsThe maximum calculation error 119890FB of the first six ordersof natural vibration frequency is less than 46 This is anindication that the SCCTB element stiffness matrix and massmatrix developed in this study are rational and effectiveAccording to the calculation results of ANSYS the localvibration of SCCTB-1 with small span is becoming obviousin the 6th mode while SCCTB-2 with larger span doesnot cause this kind of vibration in high modes Neglectinglocal vibration will result in generating additional restraintson the SCCTB which will overestimate the restrain rigidityof the SCCTB The proposed FBEM model in this papercan be used to describe the actual behavior of SCCTBs inflexural vibrations effectively but it does not take into accountthe local vibration Therefore the 6th natural frequency ofSCCTB-1 using FBEM is slightly larger than the ANSYSrsquoresults

(2) Without including the effect of the shear deformationon the SCCTBrsquos natural vibration frequency the FBEMrsquoscalculation results are greater than those of the ANSYSrsquoscalculations Further SCCTBrsquos shear deformation effect is

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

M120585120579 = int119897int119860119888

1205881198881198961198881199111120593120585119879120593120579 119889119860119889119909+ int119897int119860119909119892

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int119860119890119902

1205881199041198961199041199112120593120585119879120593120579 119889119860119889119909+ int119897int1198608

1cos1205731205881199041198961199041199112120593120585119879120593120579 119889119860119889119909

M120579120579 = int119897int119860119888

12058811988811991121120593120579119879120593120579 119889119860119889119909+ int119897int119860119909119892

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int119860119890119902

12058811990411991122120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1cos12057312058811990411991122120593120579119879120593120579 119889119860119889119909

M120585119908 = 0M120579119908 = 0M119908119908 = 119898int

119897120593119908119879120593119908 119889119909

M120579120585 = M119879120585120579M119908120585 = M119879120585119908M119908120579 = M119879120579119908

(26)

where119898 = 1198981+1198982+1198983+1198984+1198985+1198986+1198987+11989881198981 = 12058811988811986011198982 = 1205881198881198602 1198983 = 1205881199041198603 1198984 = 1205881199041198604 1198985 = 120588119904119860511988751198874 1198986 =120588119904119887611986061198874 1198987 = 120588119904119887711986071198874 1198988 = 120588119904119860811988781198874 1198961 = 1198962 = 1198961198881198963 = 1198964 = 119896119904 1199111 = 119911119888 minus 119911 1199112 = 119911119904 minus 119911By substituting (18)ndash(20) into (25) the SCCTB element

stiffness matrix is

K119890 = [[[[

K120585120585 K120585120579 K120585119908K120579120585 K120579120579 K120579119908K119908120585 K119908120579 K119908119908

]]]]

K120585120585 = int119897int119860119888

119864119888119896119888212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904212059310158401205851198791205931015840120585 119889119860119889119909+ int119897int1198608

119864119904119896119904212059310158401205851198791205931015840120585 cos3120573119889119860119889119909+ int119897119896119904119897120593120585119879120593120585119889119909

K120585120579 = int119897int119860119888

119864119888119896119888119911112059310158401205851198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

119864119904119896119904119911212059310158401205851198791205931015840120579 119889119860119889119909+ int119897int1198608

119864119904119896119904119911212059310158401205851198791205931015840120579 cos3120573 119889119860119889119909+ int119897120593120585119879119896119904119897ℎ120593120579 119889119909

K120579120579 = int119897int119860119888

1199112111986411988812059310158401205791198791205931015840120579 119889119860119889119909+ int119897int119860119909119892

(1199112211986411990412059310158401205791198791205931015840120579 + 119866119904120593120579119879120593120579) 119889119860119889119909

+ int119897int1198605

11988741198872611988735 119864119904120593120579119879120593120579 119889119860119889119909+ int119897int1198608

1199112211986411990412059310158401205791198791205931015840120579 cos3120573119889119860119889119909+ int119897ℎ2119896119904119897120593120579119879120593120579 119889119909

K120579119908 = int119897int1198605

(minus11988741198872611988735 1198641199041205931205791198791205931015840119908)119889119860119889119909minus int119897int119860119909119892

1198661199041205931205791198791205931015840119908 119889119860119889119909

K119908119908 = int119897int1198605

11988741198872611988735 11986411990412059310158401199081198791205931015840119908 119889119860119889119909+ int119897int119860119909119892

11986611990412059310158401199081198791205931015840119908 119889119860119889119909K120585119908 = 0K120579120585 = K119879120585120579K119908120585 = K119879120585119908K119908120579 = K119879119908120579

(27)

32 Solving the Natural Vibration Frequency of SCCTBBased on the SCCTB element stiffness matrix K119890 elementmass matrix M119890 and element displacement vector q119890 =q119879120585 q119879120579 q119879119908119879 and using the ldquoseat by numberrdquo method theSCCTB overall stiffness matrix K overall mass matrix Mand overall degree of freedom vector q can be obtainedThe common boundary conditions for the SCCTB can beexpressed as follows [17 19]

1205851003816100381610038161003816119909=0119871 = 119908|119909=0119871 = 120579|119909=0119871 = 0 (28)

6 Shock and Vibration

The transformational relationship between the SCCTBoverall degree of freedom vector before and after applying theboundary constraint can be expressed as follows

q = Sq (29)

where S is the transformation matrix for the overall degreeof freedom vector which can be obtained for the givenboundary conditions and q is the SCCTB overall degree offreedom vector after applying the boundary constraint

The free vibration function for a structure with multipledegrees of freedom can be expressed as follows

Mu + Ku = 0 (30)

Let

u = q sin120596119905 (31)

where 120596 is the structurersquos natural vibration frequencySubstituting (31) into (30) gives

(minus1205962M + K) q = 0 (32)

Substituting (29) into (32) gives

(minus1205962M + K) q = 0 (33)

where

M = S119879MSK = S119879KS (34)

Using (33) the frequency equation of SCCTB can beobtained as follows 10038161003816100381610038161003816minus1205962M + K10038161003816100381610038161003816 = 0 (35)

The software MATLAB has been used to develop theFBEM program of the abovementioned SCCTB elementThen the natural vibration frequency of SCCTB has beensolved by including the effects of both the interface slip andshear deformation

33 Simplification of SCCTB Element Stiffness Matrix If theshear deformation is not considered then

120574119908 = 1199081015840 minus 120579 = 0 997904rArr1199081015840 = 120579 997904rArr

1205931015840119908q119908 = 120593120579q120579119908101584010038161003816100381610038161003816119909=0 = 1205931015840119908q11990810038161003816100381610038161003816119909=0 = 11990810158401119908101584010038161003816100381610038161003816119909=119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=119897 = 11990810158402

119908101584010038161003816100381610038161003816119909=05119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=05119897 = [minus 32119897 minus14 32119897 minus14] q119908120579|119909=0 = 120593120579q1205791003816100381610038161003816119909=0 = 1205791

120579|119909=05119897 = 120593120579q1205791003816100381610038161003816119909=05119897 = 1205792120579|119909=119897 = 120593120579q1205791003816100381610038161003816119909=119897 = 1205793

(36)

Using (36) the following expression can be obtained

1205791 = 119908101584011205792 = [minus 32119897 minus14 32119897 minus14] q1199081205793 = 11990810158402

(37)

Using (37) and excluding the effect of the shear deforma-tion the element stiffness matrix K119890 and mass matrix M119890 are

1006704K119890 = S119889K119890S119879119889

1006704M119890 = S119889M119890S119879119889

(38)

where S119889 is a 7times10 transformationmatrix and can be obtainedusing (37)

After obtaining the SCCTB element stiffness matrix andmass matrix without the effect of the shear deformation thesame method in Section 32 can be used to calculate thenatural vibration frequency of SCCTB without the effect ofthe shear deformation

4 Cases for Analyses

To verify the accuracy of the developed FBEM (Section 3)the natural frequencies of two groups of clamped supportedSCCTBs (ie SCCTB-1 and SCCTB-2) have been calculatedusing both the finite elementmethod and the FBEM For eachgroup there are cases with five degree of shear connection(119903 = 04 06 08 10 20) and two spans The mechanical andgeometrical parameters of the SCCTBs are as follows

119897119904 = 25mm119899119904 = 101198871 = 400mm1198872 = 200mm1198873 = 200mm1198874 = 200mm1198875 = 4472mm1198876 = 400mm1198877 = 760mm1198878 = 7859mm1199051 = 150mm1199052 = 150mm1198673 = 40mmℎ3 = 34mm1198674 = 40mm

Shock and Vibration 7

ℎ4 = 34mm1198675 = 43mmℎ5 = 37mm1198676 = 22mmℎ6 = 17mm1198677 = 22mmℎ7 = 17mm1198678 = 22mmℎ8 = 17mm1198679 = 22mmℎ9 = 17mm1198793 = 40mm1199053 = 34mm1198794 = 40mm1199054 = 34mm1198795 = 43mm1199055 = 37mm1198796 = 22mm1199056 = 17mm1198797 = 22mm1199057 = 17mm1198798 = 22mm1199058 = 17mm1198799 = 22mm1199059 = 17mm119864119904 = 21 times 105MPa119864119888 = 45 times 104MPa120588119904 = 7900 kgsdotmminus3120588119888 = 2400 kgsdotmminus3119891119910 = 400MPa120583119904 = 03120583119888 = 021198711 = 10m1198712 = 12m

(39)

119871 119894 (119894 = 1 2) are the calculation lengths of the two groupsof SCCTBs respectively Further 120583119904 and 120583119888 are Poissonrsquos ratioof steel and concrete respectively

The calculations by the finite element method have beencarried out using the finite element program ANSYS Theupper chord and the lower chord have been simulatedusing SHELL43 shell element The vertical web memberoblique web member lower horizontal connection memberand oblique bracing member have been simulated usingBEAM188 elements The concrete slab has been simulatedusing SOLID65 solid elementsThe studs have been simulatedusing COMBIN14 spring elements The elastic modulus 1198701of the spring element has been calculated using (14) In thefinite element models by coupling the degrees of freedomin the vertical direction of the nodes at the same positionthe interface between the concrete slab and the steel trussbeam is connected in the vertical that is there is no verticalseparation between the concrete slab and the steel truss beamIn order to simulate the clamped supported at the ends ofthe beams in the finite element models there are constrainsin the degrees of freedom in the vertical transverse andlongitudinal directions

Tables 1 and 2 show a comparison of the results of thecalculated natural frequencies by the FBEM and ANSYSmodel 119877AN is the calculation result from the ANSYS model119877FB is the calculation result from the FBEM including theeffects of the shear deformation and the interface slip 119877SDis the calculation result without including the effect of theshear deformation Further 119890FB = 100(119877FB minus119877AN)119877AN is thecalculation error in FBEM and 119890SD = 100(119877SD minus 119877FB)119877FBand 119862119904 = (119877FB|119903=20 minus 119877FB|119903=04)119877FB|119903=20 where 119890SD and 119862119904are the errors due to the effects of the shear deformation andthe interface slip respectively

From Tables 1 and 2 and Figures 3ndash5 the following can beseen

(1) By including the effects of the interface slip sheardeformation moment of inertia and many other influencingfactors on the SCCTB the results of the FBEMrsquos calculationsagree with those of the ANSYSrsquos finite element calculationsThe maximum calculation error 119890FB of the first six ordersof natural vibration frequency is less than 46 This is anindication that the SCCTB element stiffness matrix and massmatrix developed in this study are rational and effectiveAccording to the calculation results of ANSYS the localvibration of SCCTB-1 with small span is becoming obviousin the 6th mode while SCCTB-2 with larger span doesnot cause this kind of vibration in high modes Neglectinglocal vibration will result in generating additional restraintson the SCCTB which will overestimate the restrain rigidityof the SCCTB The proposed FBEM model in this papercan be used to describe the actual behavior of SCCTBs inflexural vibrations effectively but it does not take into accountthe local vibration Therefore the 6th natural frequency ofSCCTB-1 using FBEM is slightly larger than the ANSYSrsquoresults

(2) Without including the effect of the shear deformationon the SCCTBrsquos natural vibration frequency the FBEMrsquoscalculation results are greater than those of the ANSYSrsquoscalculations Further SCCTBrsquos shear deformation effect is

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

The transformational relationship between the SCCTBoverall degree of freedom vector before and after applying theboundary constraint can be expressed as follows

q = Sq (29)

where S is the transformation matrix for the overall degreeof freedom vector which can be obtained for the givenboundary conditions and q is the SCCTB overall degree offreedom vector after applying the boundary constraint

The free vibration function for a structure with multipledegrees of freedom can be expressed as follows

Mu + Ku = 0 (30)

Let

u = q sin120596119905 (31)

where 120596 is the structurersquos natural vibration frequencySubstituting (31) into (30) gives

(minus1205962M + K) q = 0 (32)

Substituting (29) into (32) gives

(minus1205962M + K) q = 0 (33)

where

M = S119879MSK = S119879KS (34)

Using (33) the frequency equation of SCCTB can beobtained as follows 10038161003816100381610038161003816minus1205962M + K10038161003816100381610038161003816 = 0 (35)

The software MATLAB has been used to develop theFBEM program of the abovementioned SCCTB elementThen the natural vibration frequency of SCCTB has beensolved by including the effects of both the interface slip andshear deformation

33 Simplification of SCCTB Element Stiffness Matrix If theshear deformation is not considered then

120574119908 = 1199081015840 minus 120579 = 0 997904rArr1199081015840 = 120579 997904rArr

1205931015840119908q119908 = 120593120579q120579119908101584010038161003816100381610038161003816119909=0 = 1205931015840119908q11990810038161003816100381610038161003816119909=0 = 11990810158401119908101584010038161003816100381610038161003816119909=119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=119897 = 11990810158402

119908101584010038161003816100381610038161003816119909=05119897 = 1205931015840119908q11990810038161003816100381610038161003816119909=05119897 = [minus 32119897 minus14 32119897 minus14] q119908120579|119909=0 = 120593120579q1205791003816100381610038161003816119909=0 = 1205791

120579|119909=05119897 = 120593120579q1205791003816100381610038161003816119909=05119897 = 1205792120579|119909=119897 = 120593120579q1205791003816100381610038161003816119909=119897 = 1205793

(36)

Using (36) the following expression can be obtained

1205791 = 119908101584011205792 = [minus 32119897 minus14 32119897 minus14] q1199081205793 = 11990810158402

(37)

Using (37) and excluding the effect of the shear deforma-tion the element stiffness matrix K119890 and mass matrix M119890 are

1006704K119890 = S119889K119890S119879119889

1006704M119890 = S119889M119890S119879119889

(38)

where S119889 is a 7times10 transformationmatrix and can be obtainedusing (37)

After obtaining the SCCTB element stiffness matrix andmass matrix without the effect of the shear deformation thesame method in Section 32 can be used to calculate thenatural vibration frequency of SCCTB without the effect ofthe shear deformation

4 Cases for Analyses

To verify the accuracy of the developed FBEM (Section 3)the natural frequencies of two groups of clamped supportedSCCTBs (ie SCCTB-1 and SCCTB-2) have been calculatedusing both the finite elementmethod and the FBEM For eachgroup there are cases with five degree of shear connection(119903 = 04 06 08 10 20) and two spans The mechanical andgeometrical parameters of the SCCTBs are as follows

119897119904 = 25mm119899119904 = 101198871 = 400mm1198872 = 200mm1198873 = 200mm1198874 = 200mm1198875 = 4472mm1198876 = 400mm1198877 = 760mm1198878 = 7859mm1199051 = 150mm1199052 = 150mm1198673 = 40mmℎ3 = 34mm1198674 = 40mm

Shock and Vibration 7

ℎ4 = 34mm1198675 = 43mmℎ5 = 37mm1198676 = 22mmℎ6 = 17mm1198677 = 22mmℎ7 = 17mm1198678 = 22mmℎ8 = 17mm1198679 = 22mmℎ9 = 17mm1198793 = 40mm1199053 = 34mm1198794 = 40mm1199054 = 34mm1198795 = 43mm1199055 = 37mm1198796 = 22mm1199056 = 17mm1198797 = 22mm1199057 = 17mm1198798 = 22mm1199058 = 17mm1198799 = 22mm1199059 = 17mm119864119904 = 21 times 105MPa119864119888 = 45 times 104MPa120588119904 = 7900 kgsdotmminus3120588119888 = 2400 kgsdotmminus3119891119910 = 400MPa120583119904 = 03120583119888 = 021198711 = 10m1198712 = 12m

(39)

119871 119894 (119894 = 1 2) are the calculation lengths of the two groupsof SCCTBs respectively Further 120583119904 and 120583119888 are Poissonrsquos ratioof steel and concrete respectively

The calculations by the finite element method have beencarried out using the finite element program ANSYS Theupper chord and the lower chord have been simulatedusing SHELL43 shell element The vertical web memberoblique web member lower horizontal connection memberand oblique bracing member have been simulated usingBEAM188 elements The concrete slab has been simulatedusing SOLID65 solid elementsThe studs have been simulatedusing COMBIN14 spring elements The elastic modulus 1198701of the spring element has been calculated using (14) In thefinite element models by coupling the degrees of freedomin the vertical direction of the nodes at the same positionthe interface between the concrete slab and the steel trussbeam is connected in the vertical that is there is no verticalseparation between the concrete slab and the steel truss beamIn order to simulate the clamped supported at the ends ofthe beams in the finite element models there are constrainsin the degrees of freedom in the vertical transverse andlongitudinal directions

Tables 1 and 2 show a comparison of the results of thecalculated natural frequencies by the FBEM and ANSYSmodel 119877AN is the calculation result from the ANSYS model119877FB is the calculation result from the FBEM including theeffects of the shear deformation and the interface slip 119877SDis the calculation result without including the effect of theshear deformation Further 119890FB = 100(119877FB minus119877AN)119877AN is thecalculation error in FBEM and 119890SD = 100(119877SD minus 119877FB)119877FBand 119862119904 = (119877FB|119903=20 minus 119877FB|119903=04)119877FB|119903=20 where 119890SD and 119862119904are the errors due to the effects of the shear deformation andthe interface slip respectively

From Tables 1 and 2 and Figures 3ndash5 the following can beseen

(1) By including the effects of the interface slip sheardeformation moment of inertia and many other influencingfactors on the SCCTB the results of the FBEMrsquos calculationsagree with those of the ANSYSrsquos finite element calculationsThe maximum calculation error 119890FB of the first six ordersof natural vibration frequency is less than 46 This is anindication that the SCCTB element stiffness matrix and massmatrix developed in this study are rational and effectiveAccording to the calculation results of ANSYS the localvibration of SCCTB-1 with small span is becoming obviousin the 6th mode while SCCTB-2 with larger span doesnot cause this kind of vibration in high modes Neglectinglocal vibration will result in generating additional restraintson the SCCTB which will overestimate the restrain rigidityof the SCCTB The proposed FBEM model in this papercan be used to describe the actual behavior of SCCTBs inflexural vibrations effectively but it does not take into accountthe local vibration Therefore the 6th natural frequency ofSCCTB-1 using FBEM is slightly larger than the ANSYSrsquoresults

(2) Without including the effect of the shear deformationon the SCCTBrsquos natural vibration frequency the FBEMrsquoscalculation results are greater than those of the ANSYSrsquoscalculations Further SCCTBrsquos shear deformation effect is

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

ℎ4 = 34mm1198675 = 43mmℎ5 = 37mm1198676 = 22mmℎ6 = 17mm1198677 = 22mmℎ7 = 17mm1198678 = 22mmℎ8 = 17mm1198679 = 22mmℎ9 = 17mm1198793 = 40mm1199053 = 34mm1198794 = 40mm1199054 = 34mm1198795 = 43mm1199055 = 37mm1198796 = 22mm1199056 = 17mm1198797 = 22mm1199057 = 17mm1198798 = 22mm1199058 = 17mm1198799 = 22mm1199059 = 17mm119864119904 = 21 times 105MPa119864119888 = 45 times 104MPa120588119904 = 7900 kgsdotmminus3120588119888 = 2400 kgsdotmminus3119891119910 = 400MPa120583119904 = 03120583119888 = 021198711 = 10m1198712 = 12m

(39)

119871 119894 (119894 = 1 2) are the calculation lengths of the two groupsof SCCTBs respectively Further 120583119904 and 120583119888 are Poissonrsquos ratioof steel and concrete respectively

The calculations by the finite element method have beencarried out using the finite element program ANSYS Theupper chord and the lower chord have been simulatedusing SHELL43 shell element The vertical web memberoblique web member lower horizontal connection memberand oblique bracing member have been simulated usingBEAM188 elements The concrete slab has been simulatedusing SOLID65 solid elementsThe studs have been simulatedusing COMBIN14 spring elements The elastic modulus 1198701of the spring element has been calculated using (14) In thefinite element models by coupling the degrees of freedomin the vertical direction of the nodes at the same positionthe interface between the concrete slab and the steel trussbeam is connected in the vertical that is there is no verticalseparation between the concrete slab and the steel truss beamIn order to simulate the clamped supported at the ends ofthe beams in the finite element models there are constrainsin the degrees of freedom in the vertical transverse andlongitudinal directions

Tables 1 and 2 show a comparison of the results of thecalculated natural frequencies by the FBEM and ANSYSmodel 119877AN is the calculation result from the ANSYS model119877FB is the calculation result from the FBEM including theeffects of the shear deformation and the interface slip 119877SDis the calculation result without including the effect of theshear deformation Further 119890FB = 100(119877FB minus119877AN)119877AN is thecalculation error in FBEM and 119890SD = 100(119877SD minus 119877FB)119877FBand 119862119904 = (119877FB|119903=20 minus 119877FB|119903=04)119877FB|119903=20 where 119890SD and 119862119904are the errors due to the effects of the shear deformation andthe interface slip respectively

From Tables 1 and 2 and Figures 3ndash5 the following can beseen

(1) By including the effects of the interface slip sheardeformation moment of inertia and many other influencingfactors on the SCCTB the results of the FBEMrsquos calculationsagree with those of the ANSYSrsquos finite element calculationsThe maximum calculation error 119890FB of the first six ordersof natural vibration frequency is less than 46 This is anindication that the SCCTB element stiffness matrix and massmatrix developed in this study are rational and effectiveAccording to the calculation results of ANSYS the localvibration of SCCTB-1 with small span is becoming obviousin the 6th mode while SCCTB-2 with larger span doesnot cause this kind of vibration in high modes Neglectinglocal vibration will result in generating additional restraintson the SCCTB which will overestimate the restrain rigidityof the SCCTB The proposed FBEM model in this papercan be used to describe the actual behavior of SCCTBs inflexural vibrations effectively but it does not take into accountthe local vibration Therefore the 6th natural frequency ofSCCTB-1 using FBEM is slightly larger than the ANSYSrsquoresults

(2) Without including the effect of the shear deformationon the SCCTBrsquos natural vibration frequency the FBEMrsquoscalculation results are greater than those of the ANSYSrsquoscalculations Further SCCTBrsquos shear deformation effect is

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

Table 1 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-1)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 9443 24192 44764 70425 100160 122580119877FB 9313 24049 44384 68993 96811 126960119877SD 9706 26119 50630 83145 123639 172038

060119877AN 9580 24359 44930 70570 100280 122610119877FB 9480 24274 44629 69225 97022 127146119877SD 9895 26404 50977 83524 124037 172447

080119877AN 9701 24512 45087 70709 100390 122630119877FB 9628 24482 44861 69448 97227 127329119877SD 10064 26670 51311 83892 124427 172850

100119877AN 9809 24653 45236 70844 100510 122660119877FB 9760 24676 45083 69665 97428 127508119877SD 10216 26920 51631 84251 124810 173248

200119877AN 10214 25236 45884 71454 101020 122770119877FB 10257 25484 46056 70653 98362 128356119877SD 10790 27980 53066 85920 126630 175165119862119904 () 9206 5633 3629 2350 1577 1087

Table 2 Comparison of calculation results between the FBEM and ANSYS model (SCCTB-2)

119903 Computation methods Natural frequencies (Hz)1st 2nd 3rd 4th 5th 6th

040119877AN 6717 17314 32190 50824 72887 97708119877FB 6598 17177 32065 50470 71700 95122119877SD 6795 18224 35278 57903 86098 119830

060119877AN 6833 17465 32347 50966 73013 97813119877FB 6734 17370 32284 50686 71903 95307119877SD 6944 18454 35562 58214 86427 120170

080119877AN 6935 17602 32494 51103 73134 97915119877FB 6853 17548 32491 50893 72100 95488119877SD 7075 18666 35832 58516 86750 120504

100119877AN 7025 17729 32634 51235 73252 98014119877FB 6957 17712 32688 51094 72292 95665119877SD 7190 18864 36090 58810 87065 120834

200119877AN 7352 18242 33234 51826 73796 98481119877FB 7338 18384 33539 52001 73182 96501119877SD 7612 19683 37228 60157 88549 122409119862119904() 10079 6564 4396 2944 2026 1429

greater for SCCTB with the high-order natural vibrationfrequency If SCCTBrsquos natural vibration frequency orderis six the shear deformation effect is 365 Hence theshear deformation effect on the SCCTBrsquos high-order naturalvibration frequency cannot be ignored

(3) For SCCTBwith a shorter span the shear deformationeffect is greater

(4) The section rotation caused by the bending momentand the shear deformation caused by the shear force bothresult in a bending displacement of the structural memberThe shear deformation effect on SCCTB at the low-orderfrequency is insignificant This is an indication that the

SCCTBrsquos bending vibration-type of low-order is mainly dueto the bending deformation caused by the section rotationand the bending deformation caused by shear deformationis small

(5) For the shear deformation effect-natural vibrationfrequency order curves of SCCTB under different degrees ofshear connections they overlap with each other This is anindication that the effect of the degree of shear connectionson SCCTBrsquos shear deformation effect is not significant It isbecause the shear force of SCCTB is mainly carried by theweb member of the steel truss and has little relation to thedegree of shear connections

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

SCCTB-1

1 2 3 4 5 6 70

Order

minus6

minus4

minus2

0

2

4

6e

(

)

r = 04r = 06r = 08

r = 10r = 20

SCCTB-2

1 2 3 4 5 6 70

Order

r = 04r = 06r = 08

r = 10r = 20

minus6

minus4

minus2

0

2

4

6

e

()

Figure 3 Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency

SCCTB-1

0

5

10

15

20

25

30

35

40

e SD

()

1 2 3 4 5 6 70

Order

SCCTB-2

1 2 3 4 5 6 70

Order

0

5

10

15

20

25

30

35

40

e SD

()

r = 04r = 06r = 08

r = 10r = 20

r = 04r = 06r = 08

r = 10r = 20

Figure 4 Relationship between the shear deformation effect and the mode orders of natural vibration frequency

(6) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant the effecton the SCCTBrsquos low-order natural vibration frequency canbe as high as 101 Hence the effect of interface slip stiffnesson the SCCTBrsquos low-order natural vibration frequency cannotbe ignored According to the results of the analyses thisis because SCCTBrsquos section bending stiffness increases withincreasing interface slip stiffness while SCCTBrsquos bendingvibration-type of low-order is dominated by the bendingdeformation caused by the section rotation

Figures 6 and 7 show the comparison between the firstsix flexural mode shapes of ANSYS and FBEM modelsof SCCTB-1 and SCCTB-2 It can be seen that the firstsix flexural mode shapes of FBEMrsquos calculations agree well

with those of the ANSYSrsquos finite element calculations whenincluding the effects of the interface slip shear deformationmoment of inertia andmany other influencing factors on theSCCTB which shows that the proposed FBEMmodel can beused to describe the actual behavior of SCCTBs in flexuralvibrations effectively and accurately

5 Conclusions

By including the effects of the interface slip shear deforma-tion moment of inertia and many other influencing factorson SCCTB and based on Hamiltonrsquos principle the SCCTBelement stiffness matrix and mass matrix with cubic Hermitepolynomial shape function have been developed Based on

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

L = 10 mL = 12 m

SCCTB

0

2

4

6

8

10

12

14

Cs

()

1 2 3 4 5 6 70

Order

Figure 5 Relationship between the interface slip effect and the mode orders of natural vibration frequency

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

Mode 4

ANSYSFBEM

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

8 1062 40

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 100

(m)

ANSYSFBEM

Mode 6

2 4 6 8 100

(m)

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

Figure 6 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-1

the developed SCCTB element this study has also developeda FBEM program which can calculate the natural vibrationfrequency of SCCTBs that are common in the engineeringpractice The program has been used to calculate the naturalvibration frequencies of several SCCTBs with different spansand different degrees of shear connections The conclusionsare as follows

(1) The FBEMrsquos calculation results of the natural vibra-tion frequencies of several SCCTBs agree well with

the results ofANSYSrsquos finite element calculationsThisis an indication that the SCCTB element stiffnessmatrix and mass matrix developed in this studyare effective Hence this is a basis for the furtherapplications of the FBEM to the dynamic calculationsof SCCTB

(2) The SCCTBrsquos bending vibration-type of low-order isdominated by the bending deformation caused bythe section rotation and the shear deformation effect

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

ANSYSFBEM

Mode 1

00

02

04

06

08

10

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 2

minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

Disp

lace

men

t

ANSYSFBEM

Mode 4minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 5minus12

minus08

minus04

00

04

08

12D

ispla

cem

ent

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 6minus12

minus08

minus04

00

04

08

12

Disp

lace

men

t

2 4 6 8 10 120

(m)

ANSYSFBEM

Mode 3minus12

minus08

minus04

00

04

08

12

2 4 6 8 10 120

(m)

Figure 7 Comparison between the first six flexural modes of ANSYS and FBEMmodels of SCCTB-2

of SCCTBrsquos low-order natural vibration frequency isinsignificant

(3) For the SCCTB with high-order natural vibration fre-quency the effect of the shear deformation is greaterHence the shear deformation effect on the SCCTBrsquoshigh-order natural vibration frequency cannot beignored

(4) Under different degrees of shear connections thecurves for the shear deformation effect and naturalvibration frequency order of SCCTBs are overlappedwith each other This is an indication that the effectof the degree of shear connections on SCCTBrsquos sheardeformation is not significant

(5) While the interface slip effect on the SCCTBrsquos high-order natural vibration frequency is insignificant theeffect on the SCCTBrsquos low-order natural vibrationfrequency is dominated by the section rotation defor-mation which cannot be ignored

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51778630 51408449 and 51378502) the Special Fund ofStrategic Leader in Central South University of China (under

Grant 2016CSU001) and the Fundamental Research Fundsfor the Central Universities of Central South University ofChina (2016zzts078)

References

[1] F-X Ding J Liu X-M Liu F-Q Guo and L-Z JiangldquoFlexural stiffness of steel-concrete composite beam underpositive momentrdquo Steel and Composite Structures vol 20 no6 pp 1369ndash1389 2016

[2] J Liu F-X Ding X-M Liu and Z-W Yu ldquoStudy on flexuralcapacity of simply supported steel-concrete composite beamrdquoSteel and Composite Structures vol 21 no 4 pp 829ndash847 2016

[3] J G Nie C S Cai T R Zhou and Y Li ldquoExperimental andanalytical study of prestressed steel-concrete composite beamsconsidering slip effectrdquo Journal of Structural Engineering vol133 no 4 pp 530ndash540 2007

[4] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007

[5] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013

[6] W Zhou and W Yan ldquoRefined nonlinear finite element mod-elling towards ultimate bending moment calculation for con-crete composite beams under negative momentrdquo Thin-WalledStructures vol 116 pp 201ndash211 2017

[7] A Monaco ldquoNumerical prediction of the shear response ofsemi-prefabricated steel-concrete trussed beamsrdquo Constructionand Building Materials vol 124 pp 462ndash474 2016

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

[8] G Monti and F Petrone ldquoShear resisting mechanisms andcapacity equations for composite truss beamsrdquo Journal ofStructural Engineering vol 141 2015 UNSP 0401505212

[9] G Campione P Colajanni and A Monaco ldquoAnalytical evalua-tion of steelndashconcrete composite trussed beam shear capacityrdquoMaterials and StructuresMateriaux et Constructions vol 49 no8 pp 3159ndash3176 2016

[10] B Giltner and A Kassimali ldquoEquivalent beam method fortrussesrdquo Practice Periodical on Structural Design and Construc-tion vol 5 no 2 pp 70ndash77 2000

[11] J Machacek andM Cudejko ldquoLongitudinal shear in compositesteel and concrete trussesrdquo Engineering Structures vol 31 no 6pp 1313ndash1320 2009

[12] J Machacek and M Cudejko ldquoComposite steel and concretebridge trussesrdquo Engineering Structures vol 33 no 12 pp 3136ndash3142 2011

[13] S L Chan and M Fong ldquoExperimental and analytical investi-gations of steel and composite trussesrdquo 2011

[14] M Fong S L Chan and B Uy ldquoAdvanced design for trussesof steel and concrete-filled tubular sectionsrdquo Engineering Struc-tures vol 33 no 12 pp 3162ndash3171 2011

[15] J Bujnak and A Bouchair ldquoTheoretical and Experimentalresearch on Steel-Concrete Composite Trussrdquo 2014

[16] W Siekierski ldquoAnalysis of concrete shrinkage along trussbridge with steel-concrete composite deckrdquo Steel and CompositeStructures vol 20 no 6 pp 1237ndash1257 2016

[17] W Zhou L Jiang Z Huang and S Li ldquoFlexural natural vibra-tion characteristics of composite beam considering shear defor-mation and interface sliprdquo Steel and Composite Structures vol20 no 5 pp 1023ndash1042 2016

[18] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005

[19] ZWangbao L Shu-Jin J Lizhong and Q Shiqiang ldquoVibrationanalysis of steel-concrete composite box beams consideringshear lag and sliprdquo Mathematical Problems in Engineering vol2015 Article ID 601757 2015

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of