the stability of a movable high-strength inverted

Research ArticleThe Stability of a Movable High-Strength Inverted-TriangularSteel Bridge

Lei Gao Linyue Bai Kebin Jiang QiangWang and Xiaohui He

Army Engineering University of PLA Field Engineering Institute Nanjing 210007 China

Correspondence should be addressed to Qiang Wang qwangjxsinacom

Received 15 June 2018 Revised 20 August 2018 Accepted 27 August 2018 Published 19 September 2018

Academic Editor Ricardo Branco

Copyright copy 2018 Lei Gao et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The overall stability of a movable high-strength inverted-triangular steel bridge is worth studying because of its new truss structureIn this study an approach was proposed based on the stiffness equivalence principle in which the inverted-triangle truss structurewasmodeled as a thin-walled triangular beamOn this basis the calculation of the critical load of elastic stability of a movable high-strength inverted-triangular steel bridge with variable rigidity at both ends and locally uniformly distributed load was carried outbased on the energy theory which was in good agreement with existing theories A material performance test at BS700 was carriedout to establish the material properties and then a finite element model of the bridge was established the results of which werecompared with those of the experimental load test in order to verify the accuracy of the finite elementmodel Considering materialnonlinearity and geometric nonlinearity nonlinear buckling analysis of the bridge was conducted and the factors influencing thebridgersquos ultimate bearing capacity were analyzed

1 Introduction

High-strength steel with a yield strength of 460ndash690MPa hasbeen commonly used in construction projects in Japan theUnited States and some other countries [1ndash3] In additionthe 1100MPa-level high-strength steel has been used in somemilitary bridges such as FB48 Over the past few yearsan increasing number of high-strength steel buildings havebeen built in China [4ndash6] with Q420 and Q460 as themain two types of high-strength steel being used Accordingto the research the application of high-strength steel caneffectively contribute to self-weight reduction and cost saving[6] which hasmade itmore suitable for the design ofmovableemergency bridgesThis is the reason why high-strength steelwith a yield strength greater than 700 MPa (12059002 ge 700 MPa)is used in some movable bridges [7] Movable high-strengthinverted-triangular steel bridges developed with BS700 high-strength steel (12059002= 700 MPa) are a new type of long-spanbridge whose length and deck load can reach 51 m and60 ton respectively This kind of bridge consists of twoside bridge segments with variable cross-sections and fouridentical central bridge segments

Apart from BS700 high-strength steel a new inverted-triangular truss structure was used in this bridge Two piecesof inverted-triangular truss structures made up two laneswhich were connected and integrated in between on the topusing cross beams Central bridge segments are shown inFigure 1 Normally a rectangular truss system is used in atruss bridge yet research findings suggested that a triangulartreadway bridge performs better in saving materials andreducing weight than a rectangular truss bridge [8] Mean-while the triangular truss was usually employed on a limitedbasis for a number of structures such as roof truss girderstransmission towers and crane booms [9] In some com-posite bridges the triangular trusses were also widely used[10]Thementioned triangular trusses were usually verticallytriangular but an inverted-triangular truss system emergedwith the development of long-span spatial structures [11]Today the inverted-triangular truss is more commonlyapplied in composite bridges [12 13] Inverted-triangulartrusses were used in the movable bridge designed in thisstudy To make it convenient for mechanized bridging there

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1568629 12 pageshttpsdoiorg10115520181568629

2 Mathematical Problems in Engineering

inverted trussstrengthentransverse beam

top chord Itop chord II

transverse beam

slanted web memberbottom chordupright web member

Figure 1 The members of a typical bridge segment

was no connection to the bottom chord and the top chordwas also weakly connected which offers a promising area forfurther studies on the bridgersquos overall stability

Considerable research on the stability of triangular orinverted-triangular truss structures has been doneThe eigen-value stability of truss structures was studied through criticalpoint theory [14 15] Some laboratory-scale tests and full-scale testing of old steel truss tests were performed to getthe load-carrying capacity of truss structures [16 17] Afull-scale triangular composite truss structure specimen wastested and compared with finite element analysis [13] and thefinite element model (FEM) was validated with experimentalresults Recently the truss bridgersquos linear buckling analysisand nonlinear stability analysis which can consider materialnonlinearity and initial geometric defects were both doneby the finite element method [18ndash20] The stability ultimateload influence factors of truss structures were also studied Itwas found that the joint rigid connection has more influenceon the ultimate bearing capacity of triangular space-trussstructure [21] Height-width ratio has more influence onthe integral stability and the proposed height-width ratiois 075sim0875 [22] In comparison with the paper that hasalready been published the main purpose of this work isto investigate the stability of a novel high-strength inverted-triangular steel truss bridge that has varied sections in itsside bridge segments In this study the elastic stability of themovable inverted-triangular steel bridge was analyzed theo-retically and a calculation equation was obtained to predictits elastic load-carrying capacity The material performancetest of BS700 and the field test of load-carrying capacitywere conducted The bridgersquos elastic stability load-carryingcapacity was calculated using the finite element methodand compared with the theoretical result Both materialnonlinearity and geometric nonlinearity were consideredin analyzing the nonlinear stability of the bridge by thefinite element method Moreover factors influencing thestability load-carrying capacity of the movable high-strengthinverted-triangular steel bridge were analyzed

A1 A2 A1

b

A3

ohx

Ｂ1

Ｂ2

Figure 2 Schematic diagram of the cross section of the movablehigh-strength inverted-triangular steel bridge

2 Analysis of Elastic StabilityLoad-Carrying Capacity

Analysis of the elastic stability load-carrying capacity laysthe foundation for analyzing the nonlinear stability load-carrying capacity To perform the theoretical analysis thetruss structure is usually modeled as a beam following theprinciple of equivalent stiffness [23] In terms of the trussstructure of amovable high-strength inverted-triangular steelbridge the stiffness parameters are mainly axial rigidity 119864119860in-plane flexural rigidity 119864119868119909 out-plane flexural rigidity 119864119868119910and torsional rigidity 11986611986811989621 Simplification to Thin-Walled Triangular Beam Basedon Equivalent Stiffness The simplified diagram of the crosssection of the movable high-strength inverted-triangularsteel bridge is shown in Figure 2

There are two different sections on the top chord whoseareas are 1198601 and 1198602 The area of the bottom chord is 1198603The position of a section centroid is determined according to119878119909=0 Supposing that point 119900 is the centroid and ℎ1 and ℎ2 arethe distances between the centroid and the top and bottomchords then

ℎ1 = 1198603ℎ21198601 + 1198602 + 1198603ℎ2 = (21198601 + 1198602) ℎ21198601 + 1198602 + 1198603

(1)

The equivalent area is

119860 = 21198601 + 1198602 + 1198603 (2)

Neglecting the contribution of the componentrsquos cross sectionto the overall moment of inertia moment of inertia theequivalent moment of inertia is

119868119909 = (21198601 + 1198602) ℎ21 + 1198603ℎ22119868119910 = 1211986011198872

(3)

Therefore the axial rigidity 119864119860 in-plane equivalent flexuralrigidity 119864119868119909 and out-plane flexural rigidity 119864119868119910 could becalculated respectively In terms of torsional rigidity 119866119868119896since the web member connecting the chord primarily bears

Mathematical Problems in Engineering 3

A3

A1+05A2 A1+05A2

bi

d d d d

Figure 3 Schematic diagram of the movable high-strength inverted-triangular steel bridge

＜Ｃ

ＰＣＰＣ

Ｃ

ＥＣ

Ｃ

Figure 4 One segment of the movable high-strength inverted-triangular steel bridge

the shear force its contribution to torsion could not beignored There are two types of web members used in themovable high-strength inverted-triangular steel bridge oneis connected to the bottom and top chord I and the other isconnected to the bottom chord and top chord II (Figure 1)For simplification the middle chord (top chord II) and webmember are evenly distributed to both sides forming twosections of trusses on the left and right sides (Figure 3)

One section of the movable high-strength inverted-triangular steel bridge is taken as a calculation unit andconverted into a thin plate whose thickness is te at each sideThe whole section to be calculated is converted into a thin-walled triangular beam with its torsion strength calculatedIf the intersegment length is 119889 and height is 119887119894 according tothe equation of strain energy density the strain energy of eachpiece of thin plate 119894 is

119880119890119894 = 119902211988921198663sum119894=1

119887119894119905119890119894 (4)

The thickness 119905119890119894 can be determined based on the conditionthat strain energy 119880119894 and 119880119890119894 of a truss with a length of d areequal Suppose the total shear force that results from shearflow 119902 acting upon the truss plane is119881119894 = 119902119887119894 A segment of thebridge is taken as the research object as shown in Figure 4

The internal force generated by a slanted web memberwith the action of shear force is

119863 = 119881119894sin 120572119894 = 119881119894 119896119894119887119894 = 119902119887119894 119896119894119887119894 = 119902119896119894 (5)

If the length and area of the web member are 119897 and 119860 and itbears a constant force 119875 its strain energy is

Δ119906 = 11987521198972119864119860 (6)

The chordrsquos axial force varies linearly within the length of 119897Assuming the maximum force 119878 = 12119902119889 the chordrsquos strainenergy is

Δ119906 = 12119864119860 int1198970(119878 minus 2119897 119878119909)

2 119889119909 = 11987821198976119864119860 (7)

The strain energy of each piece of truss unit with a length ofd can be obtained according to (6) and (7)

Top chord

Δ1199060 = 11990221198893241198641198600 (8)

where 1198600 = 11986012 + 11986024Slanted web member

Δ119906119889 = 119902211989632119864119860119889 (9)

where 119860119889 = 11986011988912 and 1198601198891 is the area of central slanted webmember

Bottom chord

Δ119906119906 = 1199022119889324119864119860119906 (10)

where 119860119906 = 11986032Vertical web member

Δ119906V = 119902211988932119864119860V(11)

where 119860V is the area of the slanted web memberThe equation for the 119894119905ℎ piece of truss unit could be calcu-

lated by summing up the strain energy of the aforementionedfour chords and web members119906119905119894 = Δ119906119900119894 + Δ119906119906119894 + Δ119906V119894

= 11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] (12)

The following equation is obtained according to the principleof equal strain energy


11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] = 11990221198892119866 119887119894119905119890119894 (13)

Then the thickness of equivalent thin-walled beam is

119905119890119894 = 119864119866 119887119894119889(119889312) (1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894 119860V119894 (14)

At this point the converted thicknesses of the trianglersquos threesides 1199051198901 1199051198902 and 1199051198903 are calculated The equivalent torsionmoment of inertia is obtained according to the constantequation of free torsion of a thin-walled closed section ofarbitrary shape [24]

119868119896 = 411986020∮ (119889119904119905) = 41198602011987811199051198901 + 11987821199051198902 + 11987831199051198903 (15)

where 1198600 is the area of the closed section formed by themedians of the thin-walled triangle 1198781 1198782 and 1198783 are thewidths of three thin-walled plates ie lengths of the threesides of the trianglersquos cross section and 1199051198901 1199051198902 and 1199051198903 arethe converted thicknesses of three sides 119866119868119896 the torsionalrigidity of the thin-walled triangular beam simplified fromthe movable high-strength inverted-triangular steel bridgecan be obtained based on (14) and (15)

The equivalent rigidity can be calculated using MATLABaccording to this theoretical derivation [25] Due to the factthat the rigidities of both end units of the movable high-strength inverted-triangular steel bridge are variable whilethe central rigidity remains unchanged two programs arecompiled to calculate their respective equivalent rigiditieswhich can be calculated by entering the bridgersquos height andwidth and the sizes of various chords and bars into theprograms With this calculation the moments of area of thesections where central rigidity remained unchanged were119868119909 = 000961198984 119868119910 = 000181198984 and 119868119896 = 0000163119898422 Critical Load of Linear-Elastic Stability It is difficult tofind the solution to the torsional rigidity if the movable high-strength inverted-triangle steel bridge is simplified into athin-walled triangular beam since the bridgersquos rigidity variesat the two ends while the central rigidity stays the sameTo this end it is solved using the energy method [24] Theaxial compressive strain energy and shear strain energy aretoo low to be taken into account In addition a study madeby Liu [23] suggests that the warping strain energy of athin-walled triangular beam is negligible This strain energycontains lateral bending strain energy 1198801 in-plane bendingstrain energy 1198802 and pure torsion strain energy 1198803

Lateral bending strain energy can be obtained accordingto the strain energy equation [24]

1198801 = 12int119897

0119864119868119910 (11990610158401015840)2119889119911 (16)

In-plane bending strain energy is

1198802 = 12int119897

0119864119868119909 (V10158401015840)2119889119911 (17)

Free torsion strain energy is

1198803 = 12int119897

0119866119868119896 (1206011015840)2119889119911 (18)

where 119906 is the lateral displacement V is the vertical displace-ment and 120593 is the torsion angle

Therefore total strain energy is

119880 = 1198801 + 1198802 + 1198803= 12 int

119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 + 119864119868119909 (V10158401015840)2 119889119911 (19)

Chen [24] reported that the flexural barsrsquo elastic flexural-tensional buckling critical flexural moment was increasedto some extent by the effect of large counter-arch due toflexural deformation Therefore ignoring the impact of in-plane flexure would result in a slightly smaller critical loadand a more conservative result Equation (19) is thus reducedto

119880 = 1198801 + 1198802 + 1198803 asymp 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1205931015840)2 119889119911 (20)

Without considering residual stress the potential energy ofexternal force can be calculated with the following equationwhen any section is subjected to the combined action of axialforce and flexural moment [24]

119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 (21)

The potential energy of the external force should be (21)combined with the strain energy generated by the changesof the loadrsquos action spot Supposing the distance between theloadrsquos action spot and shearing center is 119886 (Figure 5) thepotential generated is

Δ119882 = minus12 int119897

01199021199111198861205932119889119911 (22)

Hence the total potential energy of the external force is

119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 minus 1199021199111198861205932119889119911 (23)

where 120573119910 = int119860119910(1199092 + 1199102)1198891198602119868119909 minus 1199100 is the section

asymmetry coefficient and 120573119910 = minus1199100 because the section issymmetrical 1199100 is the distance between the shearing centerand the centroid According to our calculation120573119910 = minus02 119886 =017

Total potential energy is

Π = 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 119889119911 + 2120573119910119872119909 (1206011015840)2

+ 211987211990911990610158401015840120601119889119911 minus 1199021199111198861206012119889119911(24)

The integral of the total potential energy should be taken insegments as the height of themovable high-strength inverted-triangular steel bridge varies linearly at both ends and the


a

section centroidshearing center

Ｓ0

Figure 5 Position of the shearing center of the movable high-strength inverted-triangular steel bridge

q

Bm m

L

Figure 6 Thin-walled triangular beam under locally uniformly distributed load

uniformly distributed load acts locally upon the mid-span(Figure 6) Due to the linear variation of height supposingthe flexural rigidity and torsional rigidity are also subjectto linear variations the maximum rigidities are 1198681199102 and 1198681198962respectively and the minimum rigidities are 1198681199101 and 1198681198961respectively Assume the width of the local uniformly dis-tributed load the length of the segment with variable rigiditythe intensity of uniformly distributed load and the span ofthe simply supported bridge are 119861 119898 119902 and 119871 respective-ly

The expressions of segmented load and rigidity are shownas follows

Uniformly-distributed load 119902z

=

0 0 le 119911 le (119871 minus 119861)2119902 (119871 minus 119861)2 lt 119911 le (119871 + 119861)20 (119871 + 119861)2 lt 119911 le 119871

(25)

Flexural moment of inertia Iy

=

I1199101 + 1198681199102 minus 1198681199101119898 0 le 119911 le 1198981198681199102 119898 lt 119911 le 119871 minus 1198981198681199101 minus 1198681199102119898 119911 minus 119871 (1198681199101 minus 1198681199102) minus 1198981198681199101119898 119871 minus 119898 lt 119911 le 119871

(26)

Torsion moment of inertia Ik

=


(27)

Flexural moment 119872119909

=

(119902119861) 1199112 0 le 119911 le (119871 minus 119861)21199021198611199112 minus 119902 [119911 minus (119871 minus 119861) 2]22 (119871 minus 119861)2 lt 119911 le (119871 + 119861)21199021198611199112 minus 119902119861 (119911 minus 1198712) (119871 + 119861)2 lt 119911 le 119871(28)

Therefore the integral of total potential energy is performedon five intervals along the beamrsquos lengthwise directionnamely

[0119898] [119898 (119871 minus 119861)2 ] [ (119871 minus 119861)2 (119871 + 119861)2 ] [ (119871 + 119861)2 119871 minus 119898] [119871 minus 119898 119871] (29)

TheRayleighndashRitzmethod is used to solve the critical flexuralmoment [21] Suppose the shape function that matches theboundary conditions is


119906 = 1198881 sin 120587119911119871 120593 = 1198882 sin 120587119911119871 (30)


Π = 12 int119897

0

12058741198641198681199101198714 11988821 sin2120587119911119871 + 11986611986811989612058721198712 11988822+ 212057311991011987211990912058721198712 11988822 cos2120587119911119871 minus 211987211990912058721198712 11988811198882sin2120587119911119871minus 11990211991111988611988822 sin2120587119911119871 119889119911

(31)

After the segment integral total potential energy is

Π = 11989111198681199101 + 1198912 (1198681199102 minus 1198681199101) + 11989171198681199102 + 119891111198681199102 + 119891161198681199102+ 11989122 (1198681199101 minus 1198681199102) minus 11989123 [119871 (1198681199101 minus 1198681199102) minus 1198981198681199101] 11988821+ 11989131198681198961 + (1198681198962 minus 1198681198961) 1198914 + 1199021205731199101198915 + 11986811989621198918 + 1199021205731199101198919+ 119891121198681198962 + 11990212057311991011989113 minus 11990211988611989115 + 119891171198681198962 + 11990212057311991011989118minus 11990212057311991011989119 + 11989124 (1198681198961 minus 1198681198962)minus 11989125 [119871 (1198681198961 minus 1198681198962) minus 1198981198681198961] + 11989126119902120573119910 minus 11989127119902120573119910 11988822+ 119902 (minus1198916 minus 11989110 minus 11989114 minus 11989120 + 11989121 minus 11989128 + 11989129) 11988811198882

(32)

where 1198911 sim 11989129 are the parameters related to 119871 119861 and119898According to the minimal potential energy principle we

have

120597Π1205971198881 = 0120597Π1205971198882 = 0 (33)

Three parameters are introduced

1198701 = 11989111198681199101 + 1198912 (1198681199102 minus 1198681199101) + 11989171198681199102 + 119891111198681199102 + 119891161198681199102+ 11989122 (1198681199101 minus 1198681199102)minus 11989123 [119871 (1198681199101 minus 1198681199102) minus 1198981198681199101]

(34)

1198702 = 11989131198681198961 + (1198681198962 minus 1198681198961) 1198914 + 1199021205731199101198915 + 11986811989621198918 + 1199021205731199101198919+ 119891121198681198962 + 11990212057311991011989113 minus 11990211988611989115 + 119891171198681198962 + 11990212057311991011989118minus 11990212057311991011989119 + 11989124 (1198681198961 minus 1198681198962)minus 11989125 [119871 (1198681198961 minus 1198681198962) minus 1198981198681198961] + 11989126119902120573119910minus 11989127119902120573119910

(35)

1198703 = 119902 (minus1198916 minus 11989110 minus 11989114 minus 11989120 + 11989121 minus 11989128 + 11989129) (36)

The buckling condition of the equivalent thin-walled trian-gular beam is 1003816100381610038161003816100381610038161003816100381610038161003816

21198701 11987031198703 211987021003816100381610038161003816100381610038161003816100381610038161003816 = 0 (37)

To separate the load 119902 from the equation above severalcoefficients ie 1199111 1199112 1199113 and 1199114 are introduced1199111 = 11989131198681198961 + (1198681198962 minus 1198681198961) 1198914 + 11986811989621198918 + 119891121198681198962 + 119891171198681198962

+ 11989124 (1198681198961 minus 1198681198962) minus 11989125 [119871 (1198681198961 minus 1198681198962) minus 1198981198681198961] (38)

1199112 = 1198915 + 1198919 + 11989113 + 11989118 minus 11989119 + 11989126 minus 11989127 (39)

1199113 = 11989115 (40)

1199114 = minus1198916 minus 11989110 minus 11989114 minus 11989120 + 11989121 minus 11989128 + 11989129 (41)The critical load is obtained119902119888119903

= 211987011205731199101199112 minus 211987011198861199113 + radic(211987011205731199101199112 minus 211987011198861199113)2 + 4119870111991111199112411991124 (42)

To verify the correctness of this expression a special cir-cumstance where load is uniformly distributed along the fullspan and rigidity remained unchanged along the beamrsquos axialdirection was modeled In this case the critical load is asfollows according to (42)

119902119888119903 = 1212058721205872 + 3 12058721198641198681199101198714

minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910

+ radic(minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910)2 + 11986611986811989611987121205872119864119868119910

(43)

According to Chen [24] the elastic critical flexural momentwith full-span uniform load distribution is

119872119888119903 = 312058722 (1205872 + 3) 12058721198641198681199101198712 minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910

+ radic(minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910)2 + 119868120596119868119910 (1 + 11986611986811989611987121205872119864119868120596)

(44)

Without considering warping the moment of inertia theresult of (43) is consistent with that of (44) Not consideringthe self-weight the elastic critical load of the movable high-strength inverted-triangular steel bridge was 23006 kNwhich was calculated by substituting the data obtained basedon the strain energy principle into (43)

3 Finite Element Analysis of Elastic StabilityLoad-Carrying Capacity

To further verify the accuracy of the equation for the elasticstability ultimate load a model was built using the finiteelement method to perform elastic buckling analysis


Table 1 MISO data used for ANSYS models

Strain 0001 000178 000203 0003 00037 000385 000426 000451 0008Stress[MPa] 209 370 419 592 689 704 734 754 790

(a) (b)

Figure 7 Material mechanical property test (a) Test procedure (b) The membersrsquo failure

23500

4000

2000 2000

D1 D3D2

steel plate

Figure 8 Diagram of test points and loading

31 Establishment andVerification of the Finite ElementModelThe beam element 188 of the ANSYS software which canimport a userrsquos section data was adopted in the finite elementmodel In order to analyze many modes used for studyingthe influence factors on stability ultimate load capacity themodel uses many parameters which can be easily changed inan APDL document (Analysis Parameter Design Language)[15] The whole finite element model contains a basic modeleigenvalue buckling analysis and the results of picking-upThe material model made use of the multilinear hardeningmodel

In order to get the material model data uniaxial tensiontests were carried out as shown in Figure 7(a) No obviousplastic flow or necking was observed in the tests The failureplanes were nearly placed along the 45∘ cross section of thespecimens as shown in Figure 7(b) Shear failure is dominantin the material

As shown in Figure 7 the complete stress-strain curve ofthe BS700 steel is gently strain-hardening According to thetest data the elastic modulus of the material is 2109 GPaThenominal yield stress 12059002 is 734 MPa and the nominal yieldstrain 12057602 is 000426 For the material model we chose themultilinear isotropic hardening (MISO) model The MISOmodel needs points as shown in Table 1

32 Finite Element Model In order to verify the basic finiteelement model validity the load capacity test of the bridgewas performed A 51m-long bridge where the bearing lengthat both ends was 2m and clearance height under the span was06 m was constructed Two sections of planks were placedat the middle of bridge span to simulate two decks Steelplates weighing 2542 t997888rarr4897 t997888rarr5842 t997888rarr6315 t997888rarr6787t997888rarr7117 t997888rarr7456 t997888rarr8084 t (1 tasymp10 kN)were loaded to andpiled up on the planks There were three deformation pointsat L6 5L6 and L2 (L is 51 m) Measuring points and theloading diagram of the test and field test images are shown inFigures 8 and 9 respectively

The test results compared with finite element analysis(FEA) results are shown in Table 2

According to Table 2 it can be declared that the finiteelement model can simulate the bridge accurately the errorswere below 5 excepting a few points

33 Eigenvalue Buckling Analysis The typical equation foreigenvalue buckling analysis was [26]([1198700] + 120582 [119870120590]) 119863 = 0 (45)

where [1198700] is the matrix of elastic rigidity [119870120590] is thematrix of stress rigidity 119863 is the characteristic displacementvector and 120582 is the eigenvalue


Table 2 Comparison of the displacement results of the test points and FEM calculation

119901[119896119873] 1199061 [119898119898] 1199061199041 [119898119898] 1199061199041 minus 11990611199061 1199062 [119898119898] 1199061199042 [119898119898] 1199061199042 minus 11990621199062 1199061199043 [119898119898] 1199061199043 [119898119898] 1199061199043 minus 11990631199063

2542 38 40 53 97 105 82 40 42 504897 73 78 68 202 211 45 80 83 365842 89 93 45 247 262 57 95 100 506315 95 100 53 272 282 35 105 108 286787 102 107 49 292 304 39 115 116 097117 107 112 47 312 319 22 120 121 087456 112 117 45 327 334 21 130 127 -248084 122 127 41 342 362 55 135 138 22Note p is load 119906i is test deformation u119904i is FEA calculation deformation (u119904i-119906i)119906i is ratio of FEA results to test results and i=1 2 3 represents the test pointsD1 D2 and D3

Figure 9 The load test scene

Without considering self-weight the eigenvalue 120582obtained from eigenvalue buckling analysis multiplied by theload applied is the eigenvalue buckling load In fact the loadkept changing so that1003816100381610038161003816[1198700] + 120582 [119870120590]1003816100381610038161003816 = 0 (46)

When the self-weight was not considered the first-order ei-genvalue buckling mode was obtained as shown in Figure 10The first-order eigenvalue buckling load was 24054 kNwhich was 46 larger than the 23006 kN load that wasderived by the thin-walled beam equivalent theory in thisstudy This suggests a highly accurate value of the eigen-value buckling load of the movable high-strength inverted-triangular steel bridge was derived in this study

4 Nonlinear Ultimate BearingCapacity of the Bridge

41 Theories on Nonlinear Analysis Values of bridgersquos elasticultimate bearing capacity obtained by theoretical derivationand eigenvalue buckling analysis were usually so large thatthey could only be used for qualitative analysis of the struc-turersquos load-carrying capacity instead of genuinely reflecting itTherefore nonlinear buckling analysis was required Consid-ering factors such as the structurersquos geometric defect and thenonlinearity of materials a nonlinear buckling analysis was

able to more truthfully present the structurersquos load-carryingcapacity In consideration of the material-geometry dualnonlinearity the basic buckling equation could be convertedinto

([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)

where [1198700] is the small-displacement elastic stiffness matrix[119870120590] is the initial stress stiffness matrix [119870120576] is the largedisplacement elastic stiffness matrix also known as the initialstrain matrix and 119880 and 119875 are nodal displacement andload vector respectively Solutions were found by meansof the iteration method by gradually applying load incre-ments Structure stiffness changed as load roseThe structurereached itsmaximum load-carrying capacitywhen det([1198700 ]+[119870120590]+[119870120576]) = 0 Ultimate bearing capacity was tracked usingthe arc-length method

42 Ultimate Bearing Capacity of the Bridge The geometricdeflection was introduced to the analysis model by theconsistent-deflection-mode method which simulates deflec-tion distribution by the lowest buckling mode shape Themaximal displacement of buckling analysis was multiplied bya coefficient to amplify to the 11000 span of bridge

Based on the above finite element model and consideringmaterial and geometric dual nonlinearity factors the stabilityultimate load of the bridge can be obtained by using the arc-length method [16] Firstly an elastic buckling analysis wasperformed when the load was 600 kN and then the elasticbuckling load which was the top limit load of bridge wasfound Secondly setting the elastic buckling load as the initialload the accumulated loads were then introduced graduallyuntil the solution was emanative The load-displacementcurve can be traced through time post-process (post 26)The highest point of the load-displacement curve reveals thestability ultimate load capacity of a movable high-strengthinverted-triangular steel truss bridge The load-displacementcurves are shown in Figure 11

From Figure 11 we can obtain the stability ultimate loadcapacity (119875119906) of the bridge which is 16376 kN The load-displacement curve is approximately linear and the verticaldisplacement is bigger than the lateral displacement beforethe load arrives at the ultimate load capacity Therefore the


Figure 10 First-order eigenvalue buckling mode of the movable high-strength inverted-triangular steel bridge

0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)

Vertical displacement at middle span (mm)

(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)

Lateral displacement at middle span (mm)

(b)

Figure 11 Load-displacement at middle span (a) Load-vertical (b) Load-lateral

bridgersquos deformation is mainly in the moment plane it isa kind of bending buckling when the load is small Whenthe load arrives at the stability ultimate load capacity ofthe bridge the lateral deformation becomes bigger and thebridge exhibits true bending-torsion buckling

5 Stability Ultimate LoadInfluence Factors Analysis

There are many factors which can affect the stability ultimateload capacity of a movable high-strength inverted-triangularsteel truss bridge such as bridge span bridge height trackwidth and memberrsquos section In order to study the influenceof these factors on the stability ultimate capacity of the bridgemany FEA (finite element analysis) models with differentparameters were analyzed

51 Influence ofHeight and TrackWidth Theheight and trackwidth are two important parameters for design When theanalysis was performed one parameter was changed eachtimeThrough FEA analysis the stability ultimate load capac-ity119875119906 was obtained Based on the actual bridgersquos ultimate loadcapacity 119875119906 (which is 16376 kN) the change percentage can

be determined (119875119887 minus 119875119906)119875119906 times 100 The results are shown inFigure 12

From Figure 12 a conclusion can be made that the heighthasmore influence on the stability ultimate load capacity thanthe track width The reason for this is that the bridgersquos heighthas more influence on section bending stiffness than trackwidth does Meanwhile the track width changes but thewhole bridgersquos width does not change the lateral torsion stiff-ness changes a little Thus track width has less influence onthe stability ultimate load capacity of the bridge

52 Influence of Chord The top and bottom chord are the im-portant members in the bridge How a chord influences thestability ultimate load was also studied in this paper Thetop chord has two different box sections which were labeledchord I and chord II The top chord area changed at thesame time of the width and height changes of the sectionThe multiplied area of the top chord can be found throughadjusting the width and height of section gradually Thebottom chord is a kind of complex section The multipliedarea of the bottom chord can be found through adjustingthe main dimensions of the section gradually The results areshown in Figure 13


minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age

09 10 11 12 13Height or track width change ratio

Height changeTrack width change

Figure 12 Change percentage of the bridgersquos ultimate load capacitywhen track width varies

minus20minus15minus10

minus505

101520253035

Top chord I changeTop chord II changeBottom chord change

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age

09 10 11 12 13Chord section area change multiplied

Figure 13 Change percentage of the bridgersquos ultimate load capacitywhen chord area varies

From Figure 13 it can be concluded that the top chord hasmore influence on the stability load capacity than the bottomchord The top chord I has more influence than chord IIThe bottom chord has a little influence on the stability loadcapacity that changes below 5 Even top chord I has moreinfluence than height when both two parameters change bythe same percentage as shown by comparing Figure 12 withFigure 13

53 Influence of Web Member The web member mainlybears shear force in the bridge structure but also providessome bending stiffness There are two different webmembersshown in Figure 1 the sections of which are box shapesThus

minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age

09 10 11 12 13Web member section area change ratio

Slanted web member changeUpright web member change

Figure 14 Change percentage of the bridgersquos ultimate load capacitywhen web member area varies

the influence of thewebmember on the stability ultimate loadwas studied using the same method as the chord The resultsare shown in Figure 14

According to Figure 14 the web member has a little influ-ence on the stability ultimate load capacity of the bridge Theinfluence of the upright webmember ismore than the slantedweb member

6 Conclusions

This paper investigated the stability of a movable high-strength inverted-triangular steel bridge by the theoreticalexperimental and finite element methods From this studythe following conclusions can be drawn(1) The movable high-strength inverted-triangular steelbridge can be modeled as equivalent to a thin-walled trian-gular beam according to the stiffness equivalence principleThe equivalent rigidities of the end units which are variableand the central units rigidity which remains unchanged werecalculated(2) The equation which can calculate the elastic stabilitycritical load of the inverted-triangular steel bridge withvariable rigidities at both endswas put forwardThis equationwas well agreed with existing theoretical calculation Thecalculated elastic stability load of the bridge was 23006 kNby substituting equivalent rigidity into the equation(3) The material performance test of BS700 (12059002= 700MPa) high-strength steel was carried out and the materialproperties used for finite element analysis were obtainedThe load capacity test of the movable high-strength inverted-triangular steel bridge whose length is 51m was performed(4)Afinite elementmodel for the elastic buckling analysisof the movable high-strength inverted-triangular steel bridgewas established The finite element simulated results wellagreed with field test results The elastic buckling critical loadof the bridge is 24054 kN which further verifies the accuracy


of the equation of the bridgersquos elastic stability critical loadproposed in this study(5) The nonlinear buckling analysis model was put for-ward with consideration of material and geometric nonline-arityThematerial modelmade use of themultilinear harden-ing model The geometric deflection was introduced to theanalysis model by the consistent-deflection-mode methodAccording to calculations the bridgersquos ultimate bearing ca-pacity is 16376 kN which is lower than the elastic stabilitycritical load(6) Analysis of factors influencing the bridgersquos ultimatebearing capacity suggests that top chord and bridge height aretwo major factors of which top chord 1 exerts the dominantinfluence Track width bottom chord and web member haveless influence on the bridgersquos ultimate bearing capacity

Nomenclature

119864119860 Axial rigidity119864119868119909 In-plane flexural rigidity119864119868119910 Out-plane flexural rigidity119866119868119896 Torsional rigidity1198601 The top chord I area1198602 The top chord II area1198603 The bottom chord areaℎ1 The distances between the centroid andthe top chordℎ2 The distances between the centroid andthe bottom chord119905119890119894 The thickness of equivalent thin-walledbeam119889 The intersegment length119887119894 The intersegment height119880119890119894 The strain energy of each piece of thinplate 119894119881119894 The total shear force that results fromshear flow 119902 acting upon the truss planeΔ1199060 The top chord strain energy of each pieceof truss unitΔ119906119906 The bottom chord strain energy of eachpiece of truss unitΔ119906V The vertical web member strain energy ofeach piece of truss unit119906119905119894 The total strain energy of the 119894119905ℎ truss unit1198801 Lateral bending strain energy1198802 In-plane bending strain energy1198803 Pure torsion strain energy

U The total strain energyΠ Total potential energy119906 The lateral displacementV The vertical displacement120593 The torsion angle119882 The potential energy of external force120573119910 The section asymmetry coefficient119886 The distance between the shearing center

and the top side of equivalent thin-walledbeam119861 The width of the local uniformlydistributed load

119898 The length of the segment with variablerigidity119902 The intensity of uniformly distributed load119871 The span of the simply supported bridge1198911 sim 11989129 The parameters related to 119871 119861 and 1198981198881 1198882 The shape function parameters of thelateral displacement and the torsion angle1198701 1198702 1198703 The parameters related to 119891 119871 119902120573119910119898119868119910and 1198681198961199111 1199111 sim 1199114 The parameters related to 119891 119871119898 and 119868119896119902119888119903 The critical load[1198700] The small-displacement elastic stiffnessmatrix[119870120590] The initial stress stiffness matrix[119870120576] The initial strain matrix119880 Nodal displacement vector119880 Nodal load vector119875119906 The stability ultimate load capacity of thebridge

Data Availability

(1) The bridgersquos height width sizes of various chords andbars used to support the moments of area of the sections ofthis study are available from the corresponding author uponrequest (2) The parameters related to L B and m used tosupport the elastic critical flexural moment of this study areavailable from the corresponding author upon request (3)Thematerial model used to support the stress-strain curve ofBS700 steel of this study is currently under embargo while theresearch findings are commercialized Requests for data [12months] after publication of this article will be considered bythe corresponding author (4) 11000 span of bridge is the dataused to supportmaximal displacement of buckling analysis ofthis study and is included within the article It can be foundin [13 17]

Conflicts of Interest

The authors declare that there are no conflicts of interest re-garding the publication of this paper

Acknowledgments

The authors wish to acknowledge the financial support fromthe project of the Major State Basic Research Developmentof China (973 Program No 2014CB046801) and ChinaPostdoctoral Science Foundation (Grant No 2017M623403)

References

[1] Y Fukumoto ldquoNew constructional steels and structural stabil-ityrdquo Engineering Structures vol 18 no 10 pp 786ndash791 1996

[2] G A Rosier and J E Groll ldquoHigh strength quenched and tem-pered steels in structuresrdquo Steel Construction vol 21 no 3 p 131987

[3] G Pocock ldquoHigh strength steel use in Australia Japan and theUSrdquo Structural Engineering International vol 84 no 21 pp 27ndash30 2006


[4] Z M Chen Y L Zhang and M X Peng ldquoApplication of high-strength steel and thick steel plates to CCTVnew site buildingrdquoSteel Construction vol 24 no 2 pp 34ndash38 2009

[5] S H Zhou Z Y Zhu and W H Qi ldquoStructural design onthe project of Phoenix International Media Centerrdquo BuildingStructure vol 41 pp 56ndash62 2001

[6] G Shi H Y Ban and Y J Shi ldquoOverview of research progressfor high strength steel structuresrdquo Engineering Mechanics vol30 no 1 pp 1ndash13 2013

[7] M K Gou and L Tao ldquoThe application of high strength steel(=700 MPa) to the movable bridgerdquo Steel Structure vol 17 no5 p 6 2002

[8] R H Durfee ldquoDesign of a triangular cross section bridge trussrdquoJournal of Structural Engineering vol 113 no 5 pp 2399ndash24141987

[9] R H Durfee ldquoReview of triangular cross section truss systemsrdquoJournal of Structural Engineering vol 112 no 5 pp 1088ndash10961986

[10] H G Dauner A Oribasi and D Wery ldquoThe Lully viaduct acomposite bridge with steel tube trussrdquo Journal of Construc-tional Steel Research vol 46 no 1ndash3 pp 67-68 1998

[11] G E Blandford ldquoProgressive failure analysis of inelastic spacetruss structuresrdquo Computer Structure vol 458 no 4 pp 981ndash990 1996

[12] A Reis and J J Oliveira Pedro ldquoComposite Truss Bridges newtrends design and researchrdquo Steel Construction vol 4 no 3 pp176ndash182 2011

[13] D Zhang Q Zhao F Li and Y Huang ldquoExperimental andnumerical study of the torsional response of a modular hybridFRP-aluminum triangular deck-truss beamrdquo Engineering Struc-tures vol 133 pp 172ndash185 2017

[14] G H Li Bridge Stability and Vibration Chinese Railway PressBeijing China 1st edition 2002

[15] H Sun Y Wang and W Zhao ldquoComparison of theories forstability of truss structures Part 1 Computation of criticalloadrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1700ndash1710 2009

[16] G Brunell andY J Kim ldquoEffect of local damage on the behaviorof a laboratory-scale steel truss bridgerdquo Engineering Structuresvol 48 pp 281ndash291 2013

[17] AAzizinamini ldquoFull scale testing of old steel truss bridgerdquo Jour-nal of Constructional Steel Research vol 58 no 5-8 pp 843ndash8582002

[18] Y P Tan and S Q Wang ldquoBuckling analysis of a long-spanfast launching simple-supported bridgerdquo Journal of ShenzhenUniversity science amp engineering vol 18 no 1 pp 64ndash70 2001

[19] M Tong F Mao and H Qiu ldquoStructural stability analysis fortruss bridgerdquo in Proceedings of the International Workshop onAutomobile Power and Energy Engineering APEE 2011 pp 546ndash553 China April 2011

[20] B Cheng Q Qian and H Sun ldquoSteel truss bridges withwelded box-section members and bowknot integral joints PartI Linear andnon-linear analysisrdquo Journal of Constructional SteelResearch vol 80 pp 475ndash482 2013

[21] A Fulop and M Ivanyi ldquoExperimentally analyzed stability andductility behaviour of a space-truss roof systemrdquo Thin-WalledStructures vol 42 no 2 pp 309ndash320 2004

[22] FN SunResearch on the integral stability of large span truss withinverted triangle section Gui Zhou University Guiyang 2008

[23] Y S Liu Analysis for out-plane global stability of spatiallylatticed rigid frame with triangular section Harbin Institute ofTechnology Harbin 2000

[24] J Chen Stability of Steel Structures Theory and Design SciencePress Beijing China 5th edition 2011

[25] XM ZhangMatlab Language andApplicationCase South-eastUniversity Press Nanjing China 1st edition 2010

[26] C H Zhang ANSYS 161 for the Analysis of Structure Engineer-ing China Machinery Press Beijing China 4th edition 2016

Hindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences


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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


inverted trussstrengthentransverse beam

top chord Itop chord II

transverse beam

slanted web memberbottom chordupright web member

Figure 1 The members of a typical bridge segment

was no connection to the bottom chord and the top chordwas also weakly connected which offers a promising area forfurther studies on the bridgersquos overall stability

Considerable research on the stability of triangular orinverted-triangular truss structures has been doneThe eigen-value stability of truss structures was studied through criticalpoint theory [14 15] Some laboratory-scale tests and full-scale testing of old steel truss tests were performed to getthe load-carrying capacity of truss structures [16 17] Afull-scale triangular composite truss structure specimen wastested and compared with finite element analysis [13] and thefinite element model (FEM) was validated with experimentalresults Recently the truss bridgersquos linear buckling analysisand nonlinear stability analysis which can consider materialnonlinearity and initial geometric defects were both doneby the finite element method [18ndash20] The stability ultimateload influence factors of truss structures were also studied Itwas found that the joint rigid connection has more influenceon the ultimate bearing capacity of triangular space-trussstructure [21] Height-width ratio has more influence onthe integral stability and the proposed height-width ratiois 075sim0875 [22] In comparison with the paper that hasalready been published the main purpose of this work isto investigate the stability of a novel high-strength inverted-triangular steel truss bridge that has varied sections in itsside bridge segments In this study the elastic stability of themovable inverted-triangular steel bridge was analyzed theo-retically and a calculation equation was obtained to predictits elastic load-carrying capacity The material performancetest of BS700 and the field test of load-carrying capacitywere conducted The bridgersquos elastic stability load-carryingcapacity was calculated using the finite element methodand compared with the theoretical result Both materialnonlinearity and geometric nonlinearity were consideredin analyzing the nonlinear stability of the bridge by thefinite element method Moreover factors influencing thestability load-carrying capacity of the movable high-strengthinverted-triangular steel bridge were analyzed

A1 A2 A1

b

A3

ohx

Ｂ1

Ｂ2

Figure 2 Schematic diagram of the cross section of the movablehigh-strength inverted-triangular steel bridge

2 Analysis of Elastic StabilityLoad-Carrying Capacity

Analysis of the elastic stability load-carrying capacity laysthe foundation for analyzing the nonlinear stability load-carrying capacity To perform the theoretical analysis thetruss structure is usually modeled as a beam following theprinciple of equivalent stiffness [23] In terms of the trussstructure of amovable high-strength inverted-triangular steelbridge the stiffness parameters are mainly axial rigidity 119864119860in-plane flexural rigidity 119864119868119909 out-plane flexural rigidity 119864119868119910and torsional rigidity 11986611986811989621 Simplification to Thin-Walled Triangular Beam Basedon Equivalent Stiffness The simplified diagram of the crosssection of the movable high-strength inverted-triangularsteel bridge is shown in Figure 2

There are two different sections on the top chord whoseareas are 1198601 and 1198602 The area of the bottom chord is 1198603The position of a section centroid is determined according to119878119909=0 Supposing that point 119900 is the centroid and ℎ1 and ℎ2 arethe distances between the centroid and the top and bottomchords then

ℎ1 = 1198603ℎ21198601 + 1198602 + 1198603ℎ2 = (21198601 + 1198602) ℎ21198601 + 1198602 + 1198603

(1)

The equivalent area is

119860 = 21198601 + 1198602 + 1198603 (2)

Neglecting the contribution of the componentrsquos cross sectionto the overall moment of inertia moment of inertia theequivalent moment of inertia is

119868119909 = (21198601 + 1198602) ℎ21 + 1198603ℎ22119868119910 = 1211986011198872

(3)

Therefore the axial rigidity 119864119860 in-plane equivalent flexuralrigidity 119864119868119909 and out-plane flexural rigidity 119864119868119910 could becalculated respectively In terms of torsional rigidity 119866119868119896since the web member connecting the chord primarily bears


A3

A1+05A2 A1+05A2

bi

d d d d


＜Ｃ

ＰＣＰＣ

Ｃ

ＥＣ

Ｃ




119880119890119894 = 119902211988921198663sum119894=1

119887119894119905119890119894 (4)



119863 = 119881119894sin 120572119894 = 119881119894 119896119894119887119894 = 119902119887119894 119896119894119887119894 = 119902119896119894 (5)


Δ119906 = 11987521198972119864119860 (6)


Δ119906 = 12119864119860 int1198970(119878 minus 2119897 119878119909)

2 119889119909 = 11987821198976119864119860 (7)


Top chord

Δ1199060 = 11990221198893241198641198600 (8)


Δ119906119889 = 119902211989632119864119860119889 (9)


Bottom chord

Δ119906119906 = 1199022119889324119864119860119906 (10)


Δ119906V = 119902211988932119864119860V(11)



= 11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] (12)



11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] = 11990221198892119866 119887119894119905119890119894 (13)


119905119890119894 = 119864119866 119887119894119889(119889312) (1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894 119860V119894 (14)


119868119896 = 411986020∮ (119889119904119905) = 41198602011987811199051198901 + 11987821199051198902 + 11987831199051198903 (15)




1198801 = 12int119897

0119864119868119910 (11990610158401015840)2119889119911 (16)


1198802 = 12int119897

0119864119868119909 (V10158401015840)2119889119911 (17)


1198803 = 12int119897

0119866119868119896 (1206011015840)2119889119911 (18)



119880 = 1198801 + 1198802 + 1198803= 12 int

119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 + 119864119868119909 (V10158401015840)2 119889119911 (19)


119880 = 1198801 + 1198802 + 1198803 asymp 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1205931015840)2 119889119911 (20)


119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 (21)


Δ119882 = minus12 int119897

01199021199111198861205932119889119911 (22)


119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 minus 1199021199111198861205932119889119911 (23)




Π = 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 119889119911 + 2120573119910119872119909 (1206011015840)2

+ 211987211990911990610158401015840120601119889119911 minus 1199021199111198861206012119889119911(24)



a


Ｓ0


q

Bm m

L





=


(25)


=


(26)


=


(27)


=






119906 = 1198881 sin 120587119911119871 120593 = 1198882 sin 120587119911119871 (30)


Π = 12 int119897

0


(31)



(32)


have

120597Π1205971198881 = 0120597Π1205971198882 = 0 (33)



(34)


(35)



21198701 11987031198703 211987021003816100381610038161003816100381610038161003816100381610038161003816 = 0 (37)



1199112 = 1198915 + 1198919 + 11989113 + 11989118 minus 11989119 + 11989126 minus 11989127 (39)

1199113 = 11989115 (40)




119902119888119903 = 1212058721205872 + 3 12058721198641198681199101198714

minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(43)


119872119888119903 = 312058722 (1205872 + 3) 12058721198641198681199101198712 minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(44)







(a) (b)


23500

4000

2000 2000

D1 D3D2

steel plate




















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









A3

A1+05A2 A1+05A2

bi

d d d d


＜Ｃ

ＰＣＰＣ

Ｃ

ＥＣ

Ｃ




119880119890119894 = 119902211988921198663sum119894=1

119887119894119905119890119894 (4)



119863 = 119881119894sin 120572119894 = 119881119894 119896119894119887119894 = 119902119887119894 119896119894119887119894 = 119902119896119894 (5)


Δ119906 = 11987521198972119864119860 (6)


Δ119906 = 12119864119860 int1198970(119878 minus 2119897 119878119909)

2 119889119909 = 11987821198976119864119860 (7)


Top chord

Δ1199060 = 11990221198893241198641198600 (8)


Δ119906119889 = 119902211989632119864119860119889 (9)


Bottom chord

Δ119906119906 = 1199022119889324119864119860119906 (10)


Δ119906V = 119902211988932119864119860V(11)



= 11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] (12)



11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] = 11990221198892119866 119887119894119905119890119894 (13)


119905119890119894 = 119864119866 119887119894119889(119889312) (1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894 119860V119894 (14)


119868119896 = 411986020∮ (119889119904119905) = 41198602011987811199051198901 + 11987821199051198902 + 11987831199051198903 (15)




1198801 = 12int119897

0119864119868119910 (11990610158401015840)2119889119911 (16)


1198802 = 12int119897

0119864119868119909 (V10158401015840)2119889119911 (17)


1198803 = 12int119897

0119866119868119896 (1206011015840)2119889119911 (18)



119880 = 1198801 + 1198802 + 1198803= 12 int

119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 + 119864119868119909 (V10158401015840)2 119889119911 (19)


119880 = 1198801 + 1198802 + 1198803 asymp 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1205931015840)2 119889119911 (20)


119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 (21)


Δ119882 = minus12 int119897

01199021199111198861205932119889119911 (22)


119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 minus 1199021199111198861205932119889119911 (23)




Π = 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 119889119911 + 2120573119910119872119909 (1206011015840)2

+ 211987211990911990610158401015840120601119889119911 minus 1199021199111198861206012119889119911(24)



a


Ｓ0


q

Bm m

L





=


(25)


=


(26)


=


(27)


=






119906 = 1198881 sin 120587119911119871 120593 = 1198882 sin 120587119911119871 (30)


Π = 12 int119897

0


(31)



(32)


have

120597Π1205971198881 = 0120597Π1205971198882 = 0 (33)



(34)


(35)



21198701 11987031198703 211987021003816100381610038161003816100381610038161003816100381610038161003816 = 0 (37)



1199112 = 1198915 + 1198919 + 11989113 + 11989118 minus 11989119 + 11989126 minus 11989127 (39)

1199113 = 11989115 (40)




119902119888119903 = 1212058721205872 + 3 12058721198641198681199101198714

minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(43)


119872119888119903 = 312058722 (1205872 + 3) 12058721198641198681199101198712 minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(44)







(a) (b)


23500

4000

2000 2000

D1 D3D2

steel plate




















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









11990222119864 [119889312 ( 1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894119860V119894] = 11990221198892119866 119887119894119905119890119894 (13)


119905119890119894 = 119864119866 119887119894119889(119889312) (1119860119900119894 + 1119860119906119894) + 1198963119860119889119894 + 1198873119894 119860V119894 (14)


119868119896 = 411986020∮ (119889119904119905) = 41198602011987811199051198901 + 11987821199051198902 + 11987831199051198903 (15)




1198801 = 12int119897

0119864119868119910 (11990610158401015840)2119889119911 (16)


1198802 = 12int119897

0119864119868119909 (V10158401015840)2119889119911 (17)


1198803 = 12int119897

0119866119868119896 (1206011015840)2119889119911 (18)



119880 = 1198801 + 1198802 + 1198803= 12 int

119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 + 119864119868119909 (V10158401015840)2 119889119911 (19)


119880 = 1198801 + 1198802 + 1198803 asymp 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1205931015840)2 119889119911 (20)


119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 (21)


Δ119882 = minus12 int119897

01199021199111198861205932119889119911 (22)


119882 = 12 int119897

02120573119910119872119909 (1205931015840)2 + 211987211990911990610158401015840120593119889119911 minus 1199021199111198861205932119889119911 (23)




Π = 12 int119897

0119864119868119910 (11990610158401015840)2 + 119866119868119896 (1206011015840)2 119889119911 + 2120573119910119872119909 (1206011015840)2

+ 211987211990911990610158401015840120601119889119911 minus 1199021199111198861206012119889119911(24)



a


Ｓ0


q

Bm m

L





=


(25)


=


(26)


=


(27)


=






119906 = 1198881 sin 120587119911119871 120593 = 1198882 sin 120587119911119871 (30)


Π = 12 int119897

0


(31)



(32)


have

120597Π1205971198881 = 0120597Π1205971198882 = 0 (33)



(34)


(35)



21198701 11987031198703 211987021003816100381610038161003816100381610038161003816100381610038161003816 = 0 (37)



1199112 = 1198915 + 1198919 + 11989113 + 11989118 minus 11989119 + 11989126 minus 11989127 (39)

1199113 = 11989115 (40)




119902119888119903 = 1212058721205872 + 3 12058721198641198681199101198714

minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(43)


119872119888119903 = 312058722 (1205872 + 3) 12058721198641198681199101198712 minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(44)







(a) (b)


23500

4000

2000 2000

D1 D3D2

steel plate




















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









a


Ｓ0


q

Bm m

L





=


(25)


=


(26)


=


(27)


=






119906 = 1198881 sin 120587119911119871 120593 = 1198882 sin 120587119911119871 (30)


Π = 12 int119897

0


(31)



(32)


have

120597Π1205971198881 = 0120597Π1205971198882 = 0 (33)



(34)


(35)



21198701 11987031198703 211987021003816100381610038161003816100381610038161003816100381610038161003816 = 0 (37)



1199112 = 1198915 + 1198919 + 11989113 + 11989118 minus 11989119 + 11989126 minus 11989127 (39)

1199113 = 11989115 (40)




119902119888119903 = 1212058721205872 + 3 12058721198641198681199101198714

minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(43)


119872119888119903 = 312058722 (1205872 + 3) 12058721198641198681199101198712 minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(44)







(a) (b)


23500

4000

2000 2000

D1 D3D2

steel plate




















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









119906 = 1198881 sin 120587119911119871 120593 = 1198882 sin 120587119911119871 (30)


Π = 12 int119897

0


(31)



(32)


have

120597Π1205971198881 = 0120597Π1205971198882 = 0 (33)



(34)


(35)



21198701 11987031198703 211987021003816100381610038161003816100381610038161003816100381610038161003816 = 0 (37)



1199112 = 1198915 + 1198919 + 11989113 + 11989118 minus 11989119 + 11989126 minus 11989127 (39)

1199113 = 11989115 (40)




119902119888119903 = 1212058721205872 + 3 12058721198641198681199101198714

minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(43)


119872119888119903 = 312058722 (1205872 + 3) 12058721198641198681199101198712 minus 61198861205872 + 3 + 1205872 minus 31205872 + 3120573119910


(44)







(a) (b)


23500

4000

2000 2000

D1 D3D2

steel plate




















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018











(a) (b)


23500

4000

2000 2000

D1 D3D2

steel plate




















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018


















([1198700] + [119870120590] + [119870120576]) 119880 = 119875 (47)







0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018










0 200 400 600 800 1000 12000

300

600

900

1200

1500

1800

Load

(kN

)


(a)

0 5 10 15 20 25 30 35 400

300

600

900

1200

1500

1800

Load

(kN

)


(b)










minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









minus10

minus5

0

5

10

15

20

Ulti

mat

e loa

d ca

paci

ty d

iffer

ence

per

cent

age





minus505

101520253035


Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age





minus5

0

5

10

Ulti

mat

e loa

d ca

paci

ty ch

ange

per

cent

age






6 Conclusions




Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018










Nomenclature





Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018







































Journal of












Journal of







Volume 2018






Volume 2018








the stability of a movable high-strength inverted

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