Serviceability Design ofContinuous Prestressed

Concrete Structures

Amin GhaliProfessor of Civil Engineering

The University of CalgaryCalgary, Alberta, Canada

Mamdouh M. ElbadryPost-Doctoral FellowDepartment of Civil EngineeringThe University of CalgaryCalgary, Alberta, Canada

C onerete structures reinforced with orwithout prestressing are designed

to satisfy the requirements of safetyagainst failure and serviceability. Safetyagainst failure can be assessed by esti-mating the ultimate load that can be car-ried by the structure. This check is rela-tively simple and is beyond the scope ofthis paper. But to ensure the service-ability requirements, it is essential topredict the stresses and deformations ofthe structure under service load condi-tions.

Stresses and deformations vary con-tinuously with time due to the effects ofcreep and shrinkage of concrete and re-laxation of prestressed steel. These ef-fects lead to a redistribution of stressesbetween various materials within a crosssection and to a change in reactions, andhence, a change in the internal forces ifthe structure is statically indeterminate,The importance of these tiine-depen-dent effects is much more pronounced

in structures built in stages than in thoseconstructed in one operation. Examplesof such structures are continuousbridges built span by span; segmentalconstruction; and bridges built of pre-cast prestressed concrete members con-nected and made continuous by cast-in-place concrete deck or joints and asubsequent prestressing.

Under increasing service loads,cracking occurs when the tensilestrength of concrete is exceeded, re-sulting in further redistribution ofstresses in individual sections, a consid-erable reduction in stiffness of differentmembers and important changes in de-formations. In statically indeterminatestructures, the changes in member stiff-nesses can result in changes in reactionsand internal forces.

In current practice, the initial pre-stressing forces are treated as externalforces applied on a plain concretestructure, The time-dependent change

54

in prestressing force, commonly re-ferred to as the prestress loss, is esti-mated and its effect is treated in thy•same way as the initial prestressingforce. The variation of the prestress lossfrom section to section is commonly ig-nored. Prestressed structures generallycontain a considerable amount of non-prestressed steel. The presence of thissteel, although frequently ignored, has asignificant effect on the time-dependentredistribution of stresses between con-crete and steel. Therefore, it is impor-tant to account for the time-dependentstress changes in both prestressed andnonprestressed steels.

Several methods and computer pro-grams are available in the literature forthe time-dependent analysis of seg-mental constructions and structuresbuilt in stages.'- 9 However, none ofthese methods and programs includesthe effect of cracking under increasingservice loads.

In this paper, a numerical procedureis presented and a computer program isdescribed for the analysis of reinforcedconcrete plane frames with or withoutprestressing. The analysis gives the in-stantaneous and time-dependentchanges in the displacements, in sup-port reactions and in statically indeter-minate internal forces. It also gives thecorresponding changes in stress andstrain at various sections of the structure.

The analysis accounts for the effects ofcreep and shrinkage of concrete and re-laxation of prestressed steel, for the ef-fects of sequence of construction andchange of geometry and support condi-tions, for the effects of temperature vari-ations and movement of supports, andfor the effects of cracking. At estimate ofthe average crack width is also made.With segmental construction and othermulti-stage casting and prestressingprocedures, the analysis gives the his-tory of stresses and deformations.

Cracking drastically reduces thestresses and internal forces induced bytemperature variations or support

SynopsisAn efficient numerical procedure is

presented and reference is made toan available computer program for theanalysis necessary in the design forserviceability of reinforced concreteplane frames with or without pre-stressing. Applications include contin-uous bridges and building frames.

The procedure accounts for the ef-fects of creep and shrinkage of con-crete and relaxation of prestressedsteel. The effects of cracking, partic-ularly on the deflections, the reactionsand the internal forces in statically in-determinate structures, are also con-sidered.

A frame member can be made up ofconcrete parts of different propertiesconstructed in different stages or ofconcrete and structural steel. Materialproperties and ages can vary alsofrom one member to another, as in thecase of segmental construction.

Instantaneous and time-dependentchanges in stress and strain in indi-vidual sections are calculated usingone set of equations applicable toboth cracked and noncracked states.The computer program is simple andcan be routinely employed in checkingthe design of reinforced and pre-stressed concrete structures for ser-viceability requirements using a mi-crocomputer. Two bridge examplesare presented to demonstrate the ap-plicability of the program.

movements and thus should not be ig-nored. This is discussed in a separatepaper. 10

In the analysis presented herein, theapproximate estimate of the time-de-pendent prestress loss is avoided. In-stead, the conditions of equilibrium offorces and compatibility of strains in the

PCI JOURNALJJanuary-February 1989 55

U,m

Fig. 1. Typical reinforced or prestressed concrete plane frame.

A

AXIS OF — NONPRESTRESSED AXIS OF —\ l NONPRESTRESSED AXIS OF -., NONPRESTRESSEDISYMMETRY STEEL SYMMETRY STEEL SYMMETRY ^^ STEEL

CAST-IN-SITU PRECAST 'CONCRETE PRESTRESSEDDECK 1K/-BEAM DECK STEELI i

j PRESTRESSED rSTCTuLPRESTRESSED STEEL STEEL GIRDERSTEEL

NON-

NON- PRESTRESSED

PRESTRESSED STEEL

gyp STEEL) b) (c)

Fig. 2. Typical cross sections treated in the present analysis.

concrete and steel in any section areemployed to determine the changes instrain and in forces in each of thesecomponents.

The input data for prestressing is sim-ply the magnitude of the initial pre-stressing force and the locations of thetendons at various sections. Whenpost-tensioning is employed, the loss inthe jacking force clue to friction and an-chorage slip is taken into account. Theso-called balancing forces exerted onconcrete wherever a prestressing ten-don changes direction are automati-cally included and need not be calcu-lated by the analyst.

The assumptions adopted in the anal-ysis concerning the structural discret-ization and the stress-strain relations aregiven in the following sections, Twonumerical examples are presented toillustrate the applicability and the prac-ticality of the proposed method.

STRUCTURAL ANDTEMPORAL

DISCRETIZATIONThe analysis is based on the dis-

placement method" in which a planeframe is idealized as an assemblage ofstraight beam elements connected at thejoints (nodes). The axes of the beams liein one plane and the external appliedloads act in the same plane, at the nodesor on the axes of the members. The cen-troid of a transformed cross section of amember changes position with time dueto varying concrete properties and dueto cracking. For this reason, a referenceaxis is arbitrarily chosen for each ele-ment and is kept unchanged through allsteps of the analysis. The nodes of theframe are located at the intersection ofthe' reference axes of individual ele-ments.

A member of the frame can be of con-stant or variable depth (Fig. 1). Themember cross section can consist of sev-eral concrete parts of different types or

of concrete and structural steel (Fig. 2).A concrete part can be divided into a setof rectangles or trapeziums for whichthe dimensions are specified. When asection has a structural steel part or astandard precast element, the area prop-erties and height of this part are enteredas data instead of its detailed dimen-sions. A cross section can also containmore than one layer of prestressed ornonprestressed steel reinforcements. Aprestressed tendon can be pretensionedor post-tensioned and is represented bya series of straight line and parabolic seg-ments. A prestressed or nonprestressedsteel layer can extend over a portion orover the full length of the member.

The time is divided into intervals, theinstant t, at the start of interval i coin-cides with the addition of new membersor new parts of a member, with the ap-plication of load or prestressing, or withthe change in support conditions. Foreach time interval, the analysis gives theinstantaneous and time-dependentchanges in three nodal displacementcomponents: two translations and a ro-tation (Fig. 3a), three forces at the twoends of individual members (Fig. 3h)and the reactions at the supports. Thecorresponding changes in stress andstrain in individual cross sections arecalculated using methods 'Y .' a , k4 whichare reviewed here.

Deformations due to shear are ig-nored, while those due to bending andaxial force are taken into account. Exter-nal Ioads can be in the form of forces orcouples applied at the nodes (Fig. 4a),concentrated loads or couples at anypoint on the axis of the member or a dis-tributed load of any variation covering apart or the full length of the member(Fig. 4b).

INITIAL PRESTRESSINGFORCE

In pretensioned members, the inputmust include the tension in the tendons

PCI JOURNAL(January- February 1989 57

Z X

NODEY

(a} GLOBAL AXES AND DISPLACEMENTSAT A TYPICAL NODE

3*

OR ` DI } r' i ^* REFERENCEAXIS

FI 2*/ r7f ^Y/i \ w

(b1LOCAL AXES AND POSITIVE DIRECTIONS OF ^ {D2}

MEMBER END FORCES AND DISPLACEMENTS ^S* ^* OR^Fx

ID } —^OR 0 r ,

{F} f~\.02

(c) MEMBER FIXED AT NODE 0 2 TOELIMINATE RIGID BODY MOTION

Fig. 3. Coordinate systems for plane frame analysis.

immediately before transfer. In case ofpost-tensioning, the jacking force is re-girired as input data. Instantaneouslosses due to friction and anchor set arecalculated by: ' 5' 1°

Pj Pre -tae,<r;.tli (1)

Area (ABC) = SA,^F, m (2)

where P i and Pt are the prestressingforces at two consecutive sections, withsection i closer to the jacking end; se aridB. are, respectively, the length of thetendon and the change in its slope, inradians, between sections i and j; 1, and

k are the curvature and wobble frictioncoefficients, respectively. Values for f.cand k are suggested in Refs. 17 and 18for different types of tendons. Succes-sive application of Eq. (1) starting fromthe jacking end gives the variation of theprestressing force along the tendonlength as shown in Fig. 5.

When the anchor sets a distance fi, thejacking force drops and the friction forcereverses direction over a length L. Theshortening of the tendon over L R is equalto a

This leads to Eq. (2) in which A y, andE „x are the cross-sectional area of the

58

O, -

Py Mz

Px NODE

{a) LOADS APPLIED AT A (bILOADS APPLIED ON ANODE MEMBER

Fig. 4. External loads on a typical node or a member.

END I Bij END 2i 1

Ls ' ^LsUJ A P.=P.e e ij +ks ij }C C ^ i I IA.G'

-` -v^ -- --W JACKING FROM JACKING FROM0 END I END 2wa

DISTANCE ALONG TENDON ,s

Fig. 5. Typical variation of prestressing force along a post-tensioned tendonafter losses due to friction and anchor set (jacking from both ends).

tendon and its modulus of elasticity, Inthe computer program developed for thepresent study, Point C in Fig. 5 is de-termined by trial such that Eq. (2) issatisfied.

When jacking takes place at both

ends, Eqs. (1) and (•?) are applied mea-suring the parameters BU and .sv fromeach end, giving two values of P i at eachsection; only the larger of the two valuesis of significance (curve BCEC'B inFig. 5).

PCI JOURNAL/January-February 1989 59

ASSUMPTIONS ANDCONSTITUTIVE RELATIONSIt is assumed that, under service

loads, instantaneous strains and creep ofconcrete are linearly proportional to theapplied stress. Steel reinforcements arealso assumed to be within the elasticrange. Plane cross sections before de-fonnation are assumed to remain planeafter deformation. Further, compatibil-ity of strains is assumed between con-crete and steel and between parts ofcomposite cross sections.

Shrinkage of Concrete

The symbol A€ t °) represents thefree (unrestrained) shrinkage of con-crete during a period to to t. In compos-ite sections the value E,, can vary frompart to part, but it is assumed to be con-stant over the cross-sectional area of anypart.

Creep of ConcreteA stress increment Arr, (to ) introduced

at time to and sustained without changein magnitude up to time t produces in-stantaneous strain and creep of totalmagnitude:

= Ao-c(t0) [1+0 (t,t0 )1 (3)E^ (to)

where E,. (ta) is the modulus of elasticityof concrete at age to and 0(t, t°) is thecreep coefficient.

When a stress increment r (t, t„) isintroduced gradually from zero at t„ to itsfull value att, the total strain att will be:

(t, t0 ) _ Aac(t, to) (4)to)

where E, is the age-adjusted modulus ofelasticity of concrete:

E^ (t, to) = Ec (t° ) (5)I + x 0(t, t0)

in which x = X(t, to), the aging coefl'i-

cient,'° ° is usually between 0.6 and 0.9.Suggested values of €C8 , 6, x and E,,,

which are dependent upon the relativehumidity, the size and shape of crosssection and the age of concrete, aregiven in Refs. 12 and 20 to 23.

Step-by-Step Analysis

Let t„ t 2 , ... represent instants atwhich external loads or prestressing areapplied. The symbol ,o-, (t;) will beused to represent a stress increment in-troduced at time t;. In reinforced andprestressed concrete, the reinforce-ments restrain the deformations due tocreep and shrinkage. The restraint sub-jects the concrete to stress incrementswhich develop gradually. Let wr (t+1,t)t i ) represent the stress increment gradu-ally developed between t; and t j+l . In alater section, the analysis will be donestep-by-step; for any inter val i, the stressincrements during earlier intervals willbe known from the preceding calcula-tions.

Assume that both Arr, (t i) and(tj + 1 , t ) ) are known forj – I, 2, ..., i-1. Itis required to calculate the hypotheticalfree strain which would occur in the ab-sence of the reinforcement during aninterval t t to t; +„ with i > j. For thispurpose, consider that u. (t i) and w,(t, + ,, t;) are lumped together as if thetwo increments occurred at t j . Thus, thelumped stress increment producescreep during the interval consideredequal to:

re(i3) + ©ac(t l+ ta t;) XE(t)

[0 (ti + 1 , tt) — d^ (t,, t,)]

If at t, external loads are applied pro-ducing a stress Arr, (t,), the correspond-ing creep during the time t; to t f+; willbe:

Ao-^(t{) ^(.t i+ late)E(t1)

Shrinkage during the same interval is

60

Aee, (t it1 , t i ). Thus, the total hypotheticalfree strain which would occur betweent i and t 1+1 is given by:

AEc ( t i+1, t t)Jie —

r=i

\ 1rXo (tj) + i^a (t j+ 1, ti)

t=i Er (ti)

[(t + 1 ,t) — W (t1,t1)]} +

AQ^( t;)(t+1,t) + De / trat F+1 t)

E,(ti)

(6)

Eq. (6) will later be used to de-rive the stresses developed in anytime interval when the free strain inconcrete is not free to occur due tothe presence of the reinforcement ordue to the attachment to other con-crete parts having different creep orshrinkage parameters.

Relaxation of Prestressed Steel

The intrinsic relaxation, JQp., is thereduction with time in the stress of aprestressed tendon when it is stretchedand held at a constant length betweentwo fixed points. 24 The amount of intrin-sic relaxation occuring during a givenperiod of time depends to a great extenton the stress level in the steel.

In a prestressed concrete member, theprestressed steel commonly experiencesa constantly dropping level of stress dueto the effects of creep and shrinkage ofconcrete. Thus, the actual relaxation isexpected to be smaller than the intrinsicvalue. Therefore, a reduced relaxationvalue . should be used in design.This reduced value equals the intrinsicrelaxation multiplied by a reductionfactor x,. given by:z5

Xr = e(-6.7; 5.3.11!1 (7)

= — ,Ilpr(8)

rr^

where A is the ratio of the tensile stress

a- in the tendon at the start of theperiod considered to the tensilestrength; is the change in stress inprestressed steel during the period con-sidered due to the combined effects ofcreep, shrinkage and relaxation; andOver is the intrinsic relaxation in thesame period. The value Ao-„ is generallynot known a priori because it dependsupon the reduced relaxation. Iteration istherefore necessary; first an assumedvalue Xr = 0.7 is used to calculate chap$and later adjusted by Eq. (7).

Cracking of Concrete

The analysis is linear before crackingoccurs. Nonlinearity occurs only whenthe stress in concrete exceeds its tensilestrength, producing cracking. Aftercracking, concrete in tension is ignoredand no tensile stresses can exist acrossthe crack face. Only compressivestresses can develop across the crackface and a load reversal will cause re-opening of the crack without any resis-tance.

In reality, tensile stresses exist in con-crete between the root of the crack andthe neutral axis. Also, between cracks,tensile stresses are transferred from thesteel to the surrounding concrete bymeans of bond stresses. This enablesconcrete in the tension zone to contri-bute, to some extent, to the stiffness ofthe member, an effect which is usuallyreferred to as the tension stiffening ef-fect of concrete. This effect can be sig-nificant in members subjected to serviceload levels, and ignoring it can result inunderestimation of stiffness and henceoverestimation of displacements. Con-sideration of tension stiffening is dis-cussed in a later section.

SIGN CONVENTIONFigs. 3 and 4 show the positive direc-

tions of the nodal displacements, of themember end forces and of the externallyapplied loads.

A tensile force, N, a tensile stress, u,

PCI JOURNAL/January-February 1989 61

CONCRETEPART 2 Ans I

REFEF^NCEAns 2 AEO IIPOINT p!AM'

AN(TENSION)^ A^' AYCONCRETE

y/

PART I --/ Apsf

-'

Aps2

'ans3(a) MEMBER CROSS SECTION (b) CHANGE IN (c) CHANGE IN

STRAIN STRESS

Fig. 6. Symbols and their positive sign convention.

and the corresponding strain, e, are po-sitive. A bending moment, M, is re-garded as positive when producing ten-sion at the bottom fiber. Positive curva-ture, qr, and slope of stress diagram, y,are associated with a positive moment(Fig. 6).

The reference axis 0102 of a member(Fig. 3b) intersects any section at a ref-erence point O. Any fiber below 0 has apositive y coordinate. The symbol A in-dicates a change in value; a positive Arepresents an increase. Thus, the sym-hols ae,, and Acr, are always negativequantities.

INSTANTANEOUSSTRESS AND STRAIN

Consider a composite cross sectionmade up of concrete parts of differentproperties and reinforced with severalprestressed and nonprestressed steellayers (Fig. 6a). At the start of any inter-val i, the cross section is subjected toincrements of an axial force N at anarbitrary reference point 0 and a bend-ing moment M.

The changes in stress and strain im-mediately after application of AN and

A M can he determined by the equationsgiven in Appendix A assuming that thecomposite section is replaced by atransformed section composed of thearea of concrete in each part plus thearea of the reinforcements, each multi-plied by its modulus of elasticity and di-vided by an arbitrary reference value,Ere,.

The distribution of the strain changeis assumed linear and is defined by thevalue De, at the reference point 0 andthe slope , (the curvature); see Fig. 6b.The two parameters can be determinedfrom Eq. (A6). The stress distributionI Eq. (A2)1 is in general represented by aseparate straight line for each concretepart of the section; each line can he de-fined by two parameters: the stressat 0, and Ay, the slope (Fig. 6c).

When construction is performed instages, some concrete parts may notexist at a particular instant. Moreover,particularly in segmental construction,grouting of the prestressing ducts is car-ried out in stages or at the end of pre-stressing. To simplify input data, the se-quence of grouting is ignored and it isassumed that grouting of an individualtendon is done shortly after its pre-stressing. When calculating the instan-

62

taneous changes in stress and strain atany stage, the properties of the trans-formed section exclude the areas of thenonexistent concrete parts and theirreinforcement layers, and also the areaof the ducts arid the tendons which areprestressed at the stage considered or atlater stages.

The equations in Appendix A are ap-plicable to cracked and noncracked sec-tions. The analysis of stresses andstrains in a cracked section is discussednext.

STRESS AND STRAIN IN ACRACKED SECTION

Consider the strain and stress changesclue to the effect of live load applied attime t producing at a composite crosssection (Fig. 6a) a normal force AN at areference point 0 and a bending mo-ment A M. Assume that at time t prior tothe application of AN and AM, thestress distribution is known and that it isdefined by two values, o- (t) and y(t), foreach concrete part in the composite sec-tion (Fig. 6c). Assume that the mag-nitudes of AN and AM are high enoughto produce cracking in concrete Part 1.

For the analysis of stresses and strainsafter cracking, partition AN and AMsuch thatl2,E4

L1 N = A Nrtecnmpraxrsnn + 4 Nn,,, crru'ks'd (9)AM = J.k1 ,frrnrnpw-mOn + A ^'^JMLly erarAf d

The pair (AN, A 4f },,QCOmPreSSiOn, re-ferred to as the decompression forces,represents the forces which, whenapplied on the noncracked compositesection, will bring the stresses in theconcrete Part 1 to zero. The values (AN,A M ^,^e.rnnjprrAR, are given by [Eq. (A5)]:

ANdernmvrr.Yinn =A ( — °n) i + B(–y)i }(10)^1 ArrnnvPrr.vrinn =B(— ro)1+t( —y}1

where the subscript 1 refers to concretePart 1; A, B and I are the area, and itsfirst and second moments about an axis

through the reference point 0 of thenoncracked transformed section forwhich E,T, is E,,(t), the elasticity mod-uhis of concrete of Part 1_

Under the effects of (AN, A M } ,rom _vreegion no cracking occurs and thechanges in strain and stress at this stagecan be determined by Eqs. (A6), (Al)and (A2) using the properties of the non-cracked transformed section.

The forces {AN, AIll } fn f^v rrarkre^

which represent the portions of AN andAM in excess of the decompressionforces, are applied on a transformedfrilly cracked section for which concretein tension is ignored. Eqs. (A6), (Al) and(A2) can again be applied to determinethe changes in strain and stress due to{AN, AMF .nav raked• The transformedsection properties A, B and Ito he usedin this stage must include only the areaof concrete in the compression zoneplus the area of reinforcements. Thus,the depth c of the compression zone(Fig. Blb) must be determined beforeA,B and I can be calculated. Determina-tion of the depth c is discussed in Ap-pendix B.

The total change in strain and stressdue to AN and AM is the sum of thevalues calculated for the decompressionand the cracking stages.

In the composite section consideredin Fig, 6a, it is unlikely that, under ser-vice conditions, cracking will extendbeyond the full height of concrete Part1. For this reason, AN and AM are par-titioned in Eq. (9) into two portionsonly. In a more general case, whencracking of the two concrete parts oc-curs, an additional portion of AN andAM necessary for decompression ofconcrete in Part 2 must be determinedbefore application of {AN, AM}

on the fully cracked section. Forfurther details of cracking in compositesections, the reader can see Ref. 14.

In the next section, time-dependentstresses and strains will be consideredfirst for noncracked sections and then forcracked sections.

PCI JOURNALIJarivary-February 1989 63

A 70 RESTRAINT )I

AN C ,

(ATRESTRAINT)I

(c) ARTIFICIAL RESTRAINT OF CREEPAND SHRINKAGE

0)

CONCRETEPART 2 —.

REFERENCE AnsiPOINT

CONCRETE APS

PARTAns2

(a) CROSS SECTION

[A''(ti+I,tirFREE12 [Leo(tj+j,ti)FREE12

REFERENCE—. ! CA£Q(ti+j,fi1FREE^IAXIS

^A*01 +l, ti) FREE j

SHRINKAGE

(b) UNRESTRAINED CREEP AND SHRINKAGEDURING THE PERIOD t i TO tj+I

Fig. 7. Analysis of changes in strain and stress due to creep, shrinkage and relaxation.

TIME-DEPENDENTSTRESSES AND STRAINS

Noncracked Sections

Consider a composite section (Fig. 7a)for which the distribution of thehypothetical free strain due to creep andshrinkage during a period t to t 1+1 hasbeen determined [Eq. (6)]. Thus, two pa-rameters are known defining the straindistribution over each concrete part(Fig. 7b): IJeo(tj+,, tf), **(t1+1, t1)]j q.The curvature lr(t r+,, t,}M.e can he de-termined by Eq. (6) by replacing a- withy (= dhidy), the slope of stress diagram,and setting ec„ = 0.

It is required to determine the time-dependent changes in stress and strainoccurring in each concrete part betweentime tr and t,+,, assuming the materialparameters 0, x, e c, and ^,,. are knownfor the time interval considered.

In Fig. 7c, the hypothetical free straincan be prevented by introducing an ar-tificial stress whose distribution overthe jth concrete part is defined by astress value at 0 and the slope:

(AO'O rertrulnt )! _ — [.Er AEo( t f+1, t1)frpJ)(11)

(AYrrstralnr)1 = -E. u1I (t f+1, t!)Iree]!

(12)

where Ej _ [Ec (t f+ 1 , t f )] 1 is the age-ad-justed modulus of elasticity of concretePart j [Eq . (5)).

The forces AN,,, and 0 Mme,, shown inFig. 7c represent the resultants of theartificial stress. The values of theseforces can be determined by [Eq. (A5)):

'R

A Nt,s = (A m rrnu^t + B c Ayreatraent )11 =1

(13)

Mc,s = (Bc Duo n- r,-' f*u + Ie ^Ym^rrsent)f^=1

(14)

The summations in these equationsare performed for the concrete parts. Foreach part A,, B, and 1, are the concretecross-section area, and its first and sec-ond moments about an axis through 0.

The strain in concrete due to relaxa-tion of prestressed steel can he artifi-cially prevented by applying the forces:

AN,,= Y,(Arp,A) k (15)k=1

AMp = ( prAp,1Jp,) k (16)k=1

where the summations are performedfor the prestressed steel layers ten-sioned before or at tj. A and yp.k arethe cross-sectional area and the y coor-dinate of the kth prestressed steel layer.

Summing up the forces in Eqs. (13) to(16) gives {AN, AM} reetrcinl, the totalforces which would artificially preventcreep, shrinkage and relaxation.

The artificial restraint is eliminatedby the application of{ A N, A M inreversed directions on the age-adjustedtransformed section (Fig. 7d). This pro-duces the change in strain Ae(t f + 1 , t1)

defined by the value at 0, Aeo (ti+l, ti),and the slope of the strain diagram,A'(ta + 1 , tf); see Fig. 7d. These two pa-rameters can be calculated by Eq. (A6)using the properties of the age-adjustedtransformed section: E r,t, A, B and 1.The age-adjusted transformed section iscomposed of the area of concrete in eachpart, multiplied by Ec 1 E,.1, plus the areaof reinforcements, multiplied by E,IE ref.

Multiplication of the strain shown inFig. 7d by E, of each concrete part givesthe corresponding stress change. Thesum of this stress and the stress in Fig.7c gives the total stress increment,tae(t f+l, t1).

Cracked Sections

The analysis of the time-dependentstrain and stress increments presentedearlier applies also to cracked sections

PCI JOURNAL/January-February 1989 65

in which the concrete in tension is ig-nored. The transformed section is com-posed only of the concrete in compres-sion and the reinforcements. AppendixB gives the equations from which thedepth c(t,) of the compression zone isdetermined due to any specified combi-nation of normal force and momentapplied at time t i . The nonnal force andmoment used here should be equal tothe increments introduced at t i minusthe decompression forces discussed inthe preceding section [Eq. (9)].

Due to creep and shrinkage duringthe period t I to t 1+1 , the depth c gradu-ally changes and the new value c(tr+i)can be determined by iteration:

1. Perform the time-dependentanalysis presented earlier, ignoring thechange in c. The depth of the area ofconcrete taken into account in deter-mining the properties of the transformedsection is c(t;). The analysis gives thestress distribution at time t f+ , and hencea new value e(t j fI).

2. Repeat the calculations using c(t,)when determining the artificial forcesnecessary to restrain creep, while usingc(t,+,) for the remainder of the calcula-tions. This will give a new stress dis-tribution and a new c(t,). If this valueis substantiall y different from the valuepreviously determined, this step is re-peated. Usually one or two iterations aresufficient.

It can he shown, however, that if thedepth c of the effective area of concreteis assumed constant between t and t^+„while allowing the neutral axis positionto change with time, a small error resultsin the time-dependent changes in strain(see discussion of Ref. 13). The error issmall because an area of concrete closeto the neutral axis is ignored, although itis subjected to compressive stresses.

The iteration procedure discusser) inSteps 1 and 3 is included in the coin-puter program CPF; t, hence, the effectof change in c on the stiffness of mem-bers and on the statically indeterminateinternal forces is not ignored.

MEAN STRAIN ANDCRACK WIDTH

Displacements of a member can becalculated by integration of the axialstrain and the curvature determined at anumber of sections along the memberlength. For a cracked member, dis-placements can be evaluated more accu-rately if the concrete in tension is notcompletely ignored (i.e., if the tensionstiffening effect of concrete is taken intoaccount). One way of doing this is to usemean values of axial strain and curvaturedetermined by interpolation betweentwo limiting states: the state with con-crete area assumed fully effective (non-cracked) and the state in which concretein tension is ignored. The followingempirical interpolation equation" z•29 isadopted here:

E (l I can = ( 1 — b EO Nu,,t,Fwke4 ^ E O hllucinc rd

inrnN — U — S1 h1,wru•nukrd + ^fhfl r,wb i

(17)

The interpolation coefficient C isgiven by:

= I — j3 $2 ( fie ) 2 (with s; 0.4)tJ

(18)

where f, is the tensile strength of con-crete; v,„Qr is the hypothetical stress atthe extreme tension fiber that wouldexist after application of the load as-suming no cracking; $, = 1 or 0.5 forhigh bond or plain reinforcing bars, re-spectively; $2 = 1 for the first loadingand equals 0.5 for loading applied in asustained manner or in large number ofcycles.

Assuming that cracks are spaced at adistance s, the mean value of crackwidth at the level of a steel layer is:

[li = 4S (19)

where €,r„u w rru. i,r,1 is the change in steelstrain calculated for a fully cracked see-

66

- f,

xdx;u

(20)

tion. Empirical equations are availablento predict the crack spacings. Here it isassumed thats is given as input data.

The parameter C represents the extentof cracking and the damage of bond afteroccurrence of cracks. The value of C ap-proaches unity as the internal forces in-crease above the values causing firstcracking. Once cracking has occurred ata section, it will remain cracked For anysubsequent loading even when theinternal forces drop below the valueswhich produced the first cracking. Also,the parameter C will continue to assumethe highest value reached under earlierloadings.

STIFFNESS MATRIXOF A MEMBER

In the preceding sections, equationswere presented to calculate the changesin axial strain and curvature in non-cracked and cracked sections due toforces applied on the section or due tothe effects of creep, shrinkage and relax-ation. In the present and followingsections, the changes in axial strain andcurvature will be used in the analysis ofthe corresponding changes in internalforces of statically indeterminate planeframes.

The typical plane frame membershown in Fig. 3b has six degrees of free-dom located at the two end nodes O, and02. Fix the member at end 0 2 (Fig. 3c)and generate a flexibility matrix [fl cor-responding to the three coordinates atend O. The elements in any columnj ofthe matrix [fl are:

f11 = -

= fo Iidx

where eoj and 4s; are the strain at the ref-erence axis 0,0 2 and the curvature pro-duced at any section at distance x from0, by a unit force applied at coordinate,

j, withj = 1, 2 or3.The integrals in Eq. (20) are evaluated

numerically employing values of €, and^i determined at a number of sectionsusing Eq. (A6). For analysis of instan-taneous effects, use the modulus ofelasticity of concrete and the trans-formed section properties at the time ofapplication of the load. When theanalysis is for the time-dependentchanges during a period t i to t j+E , theage-adjusted elasticity modulus E^(t,+„t i) and the properties of the age-adjustedtransformed section are to he used togive the age-adjusted flexibility, [T].

After cracking, the flexibility of acracked member is obtained by replac-ing E. and 4, in Eq. (20) with mean valueseo ,^^aA and ^i,,,ef1, determined from Eq.(17). This requires that the depthc oithecompression zone and the interpolationcoefficient be known a priori. Aniterative procedure will therefore benecessary (to be discussed in a separatesection).

Inversion of [/l gives a 3 x 3 stiffnessmatrix corresponding to the coordinatesatO, (Fig. 3c). The forces at end 0 2 (Fig.3h) are obtained by equilibrium andthus the stiffness matrix for the six coor-dinates is generated:

[s) = [H] T [f] -1 [111 (21)

where

1 O 0 -1 0 0[H]

I 0 1 0 0 -1 10 0 1 0 0 -1

with I being the length of the member.

FIXED-END FORCESFor external loads applied at any po-

sition between the two ends of amember (Fig. 4b), the fixed-end forcesat the three coordinates at end O, (Fig.3c) are:

{tF} = -[.f]-'{iD} (22)

where { A D} represents the three dis-

PCI JOURNAUJanuary-February 1989 67

placements at end 0, with the membertreated as a cantilever (Fig. 3c) andsubjected to the given loads. Corre-sponding axial strain and curvature de-termined by Eq. (A6) are to be used tocalculate { a D} :

AD S = – I Aeodx;0

flAD2– xdx;

U

tD,= AOdx (23)u

For a cracked section, A]eo meun andA mean are to he used in Eq. (23) insteadof Jeo and :]er,, and the values of c and Cmust be known from earlier steps ofanalysis. The forces at end 0 2 can hedetermined by equilibrium, and thus,the six fixed-end forces (Fig. 3b) can beexpressed as:

{ A F *} _ [M" { A F} + { A R} (24)

The first three elements in vector{ A R} are zero, while the last three arethe three reactions due to external loadsapplied on a cantilever fixed at end 0(Fig. 3c).

To determine the time-dependentchanges in the fixed-end forces duringan interval t f to t,+l , calculate for thecantilever the increments JEQ (t ++1 , t1)and 4ip (t {+1 , t i ). These values are sub-stituted in Eq. (23) to give the time-de-pendent displacement increments { A D(tip.1 , t, )} .

Substitution in Eq. (22), replacing[ f]with [f], gives the changes in thefixed-end forces { A F (t l , t i )} at end0 1 . These are substituted in Eq. (24),with {AR} = {0}, to givethe increments{ x'" (ts+l , t E)} of the six fixed-end forcesat the two ends.

ANALYSIS PROCEDUREFor each time interval, the conven-

tional displacement method of analysis"

is employed to determine the changes indisplacements, reactions and internalforces which occur instantaneously atthe beginning of the interval due toapplication of loads or prestressingand to calculate the time-dependentchanges due to creep, shrinkage andrelaxation.

The displacements at the nodes, theinternal forces, the stresses and thestrains at various sections existing be-fore introduction of new loads, or at thebeginning of any interval, are assumedknown. If cracking has occurred at anysection, the values c and i; are alsoknown. At the beginning of the analysis,before application of any loads, all ofthese variables are zero except c, whichequals the full depth of the section.

For each construction stage, load ap-plication or time interval, the analysis isperformed in steps:

1. Generate the stiffness matrix forindividual members. Calculate the rela-tive end displacements { D } and thefixed-end forces { F *} I see Figs. 3band c and Eqs. (22) to (24) ]. The valuesof c and , needed in the calculation ofaxial strain and curvature at any section,are those existing prior to application ofthe new loads. Assemble the fixed-endforces and apply in a reversed directionon the structure, and then determine bya conventional linear analysis the in-crements of nodal displacements andinternal forces. When the analysis is fortime-dependent changes, the stiffness tohe used in this step is the age-adjustedstiffness.

2. Add the increments of nodal dis-placements and internal forces to theexisting values. Compute the changes inaxial strain, curvature, and stresses, ac-counting for any cracking, at all sections.Add the changes in strain and stress tothe existing values. Update c and .

3. Use the current axial strain andcurvature to calculate the relative enddisplacements { D} of individual rnem-hers using Eq. (23). The same displace-ments can he calculated by:

68

{D} = [N] {D*} (25)

where {D*} are the nodal displacementsat the member ends (Fig. 3b). Whencracking does not occur, the relative enddisplacements by the two methods willbe equal. When this is not the case,calculate the difference in displace-ments and substitute in Eqs. (22) and(24) to obtain a vector of residual fixed-end forces. Note that for these calcula-tions {z R} = {O} in Eq. (24) and [f] isbased on the updated c and s; values.

4. The residual forces calculated inStep 3 for the individual members areassembled and applied in a reversed di-rection to the structure with its stiffnessupdated. Determine by a conventionalanalysis the increments in nodal dis-placements and in internal forces.

5. Go back to Step 2 and terminatethe analysis if the residual forces calcu-lated in Step 3 are smaller than pre-scribed values or when the incrementsin nodal displacements are less than aspecified percentage of the current totalvalues.

It is worth noting that this analysis hasan advantage over the standard finiteelement techniques, particularly whennonprismatic members are involved.The essential feature of the presentanalysis is that the actual deflectedshape of a member is obtained by inte-gration of the actual strains and curva-tures.

In the finite element method, the de-flected shape of a member is usuaIIy as-sumed as a function of the displace-inents at the nodes, and equilibriumbetween the external and the internalforces is satisfied only at the nodes. Alarger number of elements is usuallyneeded to overcome this drawback,especially in those places where amarkedly nonlinear behavior is ex-pected.

A computer program will greatly fa-cilitate the preceding analytical stepsand evaluation of the equations. This isdescribed next.

COMPUTER PROGRAMA computer program, CPF' (Cracked

Plane Frames in Prestressed Con-crete), L6 has been developed to performthe analysis presented here. The pro-gram gives the instantaneous values andthe time-dependent changes in jointdisplacements, support reactions andinternal forces, stresses and strains inconcrete and steel, and the crack widthat selected sections.

CPF is suitable for the analysis ofstructures composed of precast or cast-in-place segments or of members castand erected at different ages. The dif-ference in the time-dependent defor-mations of the parts is accounted forwhen the members have different agesor when the cross sections of individualmembers are composed of concreteparts of different ages.

The logic of the CPF program is illus-trated in the flow chart in Fig. 8. Theprogram requires a small core storageand can be used on a microcomputer.

To demonstrate the applicability ofthe present method of analysis, the CPFprogram is employed for the analysis oftwo bridge examples presented in thefollowing section. Further details andresults on these and other examples aregiven in Ref. 27.

APPLICATIONS

EXAMPLE 1Fig. 9 shows a three-span symmetrical

bridge made of a steel box and concretedeck. The deck is made of precast rec-tangular segments; each segment has thefull width of the deck and covers a shortpart of the span. The segments arepost-tensioned longitudinally as shownin Fig. 9a. Dimensions and area proper-ties of the cross section are given in Fig.9h and Table 1. The example borrowsmost of its dimensions from the designof Arvid Grant and Associates of theWallace Viaduct in Idaho.

PCI JCURNAUJanuary-February 19B9 69

Start.

Read number of ronatrucl ion and loading stagesor time intervals, geometry, material properties,boundary conditions at different stages, maximumnomher of iterations. tolerances.

Calculate prestressing forces in post: tensionedtendons after friction and anchor set losses.

luit ialiae nodal dispisermcnts, {D} = {L}. memberend forces, {A) _ {Ii}, strain, e = 0, streaa. a.. = 0,

a,,,, = 0, n^„ - ^„ r = full depth, r = 0.

Loop over construction and loadingstages or time intervals.

Read external loads, forces caused by removal ofsupports. Calculate self weight of newly constructedelements and concrete parts.

Generate attffneee usalrix [Eq. (21)] forthe completed part of the structure,

For each element, calculate relative end displace-menta, {D) [Ewq. (23)j and fixed-end forces (F'} [Eqs.122) and {24)[. Assemble fixed-end forces.

Fig. 8. Flow chart for computer program CPF.

70

Solve fnr incrr, ruts of nodal displaretnents andinternal forces, l Add increments to existing values.

For all sections, calculate :Xc, .:]t1 and .X' and thetolsI values. Calculate e„,,, and e,,, for crackedsections cif any). Update c and {.

Ear cracking occurred?

Yes

Are displacement increments less than / ]t tSme-dependent Analysis required?tolerable percentage of total values ? / I \

Yes

Is maximum number of 1er f Read d, x, Ac.-. and A^,for current interval.

iterations exceeded?

For earl, elemegt, raculatr relative end displace-ments, {L) [Eq. (23)1 using c,,, and W,,,• Calculatethe same displacement. from Eq. (2S).

Vu I,; uumi.er of loadingr' stages exceeded?

Calculate the difference in displacements from the two

equations and tine corresponding vector of residual yesfixed-end forces {Eqs. (22) and (u)].

Stop

I Update el iffncrs matrix- [

Fig. 8 (cont). Flow chart for computer program CPF.

PCI JOURNAL/January-February 1989 71

Table 1. Variation of cross section properties over bridge length (Example 1).

Section Region A B C

Top flange thickness (in.) 2'/4 1'/s Ya

Bottom flange thickness (in.) 2 1'/4 7/s

Web plate thickness (in.) 9'3s 911r 'lieSteel box Cross-sectional area (in. 2 ) 279 190 124

Centroid above bottom (in.) 32 29.5 25Moment of inertia aboutcentroid (in.') 248,000 146,000 81,500

Gross cross-sectionalarea (in, 2 ) 4,372 4,500 4,543

Concrete Centroid abovedeck bottom (in.) 9.5 10.34 10.61

Gross moment of inertiaaboutccntroicl (in.) 85,500 98,100 102,800

Nute: 1 in. = 25.4 mm; 1 in. 2 — 645.2 inrn 2 , I in.^ = 416231 mm.

AXIS OFA SYMM. PRESTRESSING

^--. .--^-- 4 TENDONS

A B C D

B B B BC A C A C

87ft I I 74ft^ I I I 87f1123ft 146tt 123ft

^A

(a) SPAN ARRANGEMENT

26 fg in. 9.5f1 -rj

19in. L ; + :. : a . ++. +. + + + . .

TOTAL Ans PRECAST CONCRETE

10.8 in2 DECK SEGMENTS64in POST-TENSIONED

STEEL STEEL, Aps=O.868in2/TENDONBOX- GIRDER ADUCT =3.142 in2

7ft —+

(b) CROSS SECTION A-A

Fig. 9. Three-span composite concrete-steel bridge (Example 1). Note: 1 in. = 25.4 mm;1 in.' = 645.2 mm 2 ; 1 ft = 0.3048 m.

72

The construction is performed in thefollowing sequence: The steel girder isplaced in position without shoring tocarry a load of 4.3 kips/ft (62.7 kN/rn),representing its own weight and theweight of the precast concrete segmentswhich is introduced in steps. First, seg-ments are placed in Region A (Fig. 9a)and post-tensioned with four tendons.Segments are then added in Region Band post-tensioned from end to endusing seven tendons. The bridge deck iscompleted by placing segments in Re-gion C and post-tensioning eight ten-dons throughout the bridge length.

The precast segments are of age 60days at the time of post-tensioning andthe prestressing force per tendon is 164kips (730 kN). Shortly after prestress-ing, the bridge is made composite bycasting concrete to fill in pockets in theprecast segments at the location of studswelded to the top flanges of the steelgirder. Finally, 30 days after erection ofthe steel girder, a superimposed deadload of 0.4 kip/ft (5.8 kN/m), represent-ing the surface cover, is applied and thebridge is opened to traffic.

Because of the advanced age of theprecast segments and the short period ofconstruction, the time-dependentchanges in stress and strain are calcu-lated for the time interval t = 60 to =00, and the self weight and the superim-posed dead load are applied at t = 60and sustained thereafter.

Other data are: E, = E,,, = 29,000 ksi(200 GPa); E. = 27,000 ksi (186 GPa);E, (60) = 3200 ksi (22 GPa); c(x, 60) =2.28; X = 0.788; Ac,, (x, 60) _ —230 x10-; pr(, 60) = —13 ksi (-90 MPa.)Friction is ignored here for simplicitybut is considered in Example 2.

For comparison, the analysis is per-formed with the steel girder unshored(as described above) and repeated forshored construction. The shores are as-sumed closely spaced and removed im-mediately after the structure becomescomposite, Some results for the un-shored and shored constructions are

given in Figs. 10 through 13. The resultsrepresent the effects of self weight,superimposed dead load and prestress-ing,

The restraint provided by the steelgirder and the prestressed and nonpre-stressed steels to the time-dependentdeformations of the concrete deck pro-duces important changes in internalforces. Fig. 10a shows the variation overhalf the bridge length of the bendingmoments immediately after completionof construction and at time t = oo. Thebending moment at any section is hereconsidered as the resultant of thestresses on all components: the steelbox, the nonprestressed and prestressedreinforcements, and the concrete.

The variations of the tensile force inthe prestressed tendons at the time ofprestressing and at time infinity areplotted in Fig. 10b.

The deflected shapes of the bridge atcompletion of construction and at t = =are depicted in Fig. 11. As expected,shoring during construction reducesdeflection considerably, and the time-dependent changes in deflection arelarger in shored than in unshored con-structions.

The stress distribution at two criticalsections are given at completion of con-stniction and at t = - in Figs. 12 and 13for unshored and shored constructions,respectively. A substantial reduction inthe compressive stresses produced byprestressing in the deck slab occurs dueto time-dependent effects. For example,the average stress in the slab is changedover the interior support from —675 to—175 psi (-4.7 to —1.2 MPa) (see Fig.12).

It can be noted that the small loss intension in the tendons (Fig. 10b) has nopractical significance because it doesnot represent loss of compression in theconcrete. It can also be seen from Figs.12 and 13 that the time-dependentchange in stress in the steel box ismainly compression; an increase of6,000 to 12,000 psi (41.4 to 82.8 MPa)

PCI JOURNAL/January-February 1989 73

-20,000 UN SHORED--SHORED AXIS OF-15,000 SYMM,I ^ r AT TIMEAT COMPLETION I I=M

a - 10,000 OF CONSTRUCTION fr ^,2 -5000 / ^/ \•^ \^^W •\00 5000 a)BENDING MOMENTS DUE TO SELF WEIGHT PLUSSUPERIMPOSED DEAD LOAD4000 --- AT TIME OF PRESTRESSING AXIS OF3000 AT TIME 1=ro SYMM.(UNSHORED) U 2000—fr0 1000 --- IO 0 20 40 60 80 100 120 140 160 180DISTANCE FROM SUPPORT A (ft)b) FORCE IN PRESTRESSING TENDONS

Fig. 10. Variation of bending moments and force in prestressing tendons in the bridge ofExample 1. Note: 1 ft = 0.3048 m; 1 kip = 4.448 kN; 1 kip-ft = 1.356 kN •m.AXIS OFI UNSHORED j SYMMSHORED0

zQ 2'VAT MEwU_ 3 ETIONw

4 OF CONSTRUCTIONFig. 11. Deflected shapes of half the length of the bridge of Example 1 due to self weightplus superimposed dead load. Note: 1 in. — 25.4 mm.

74

C-)C-0C33z

I-A)

C

T1roCA)

CDCD

-650-700 16590

-15660AT INTERIOR SUPPORT

-330

- 240-11550

7650AT MIDDLE OF INTERIORS PAN

-180-170 8,370

- 20630AT INTERIOR SUPPORT

30psi

- 17550 -20^

-4160AT MIDDLE OF INTERIORSPAN

q 1 IMMEDIATELY AFTER COMPLETION OF CONSTRUCTION bt AT TIME t = - AFTER OCCURRENCE OF TIME -DEPENDENT EFFECTS

Fig. 12. Stress distributions at critical sections — unshored construction. Note: 1 psi = 6.895 x 10- 3 MPa.

- 230

-575a 1210

- IlyaAT INTERIOR SUPPORT

-510-235 70

6570AT MIDDLE OF INTERIORSPAN

60 psi-2A-3220

-16840AT INTERIOR SUPPORT

-7380

-4655

AT MIDDLE OF INTERIORSPAN

a1 IMMEDIATELY AFTER COMPLETION OF CONSTRUCTION b) AT TIME t = co AFTER OCCURRENCE OF TIME --DEPENDENT EFFECTS

Vcn I Fig. 13. Stress distributions at critical sections - shored construction. Note: i psi = 6.895 x 10- 3 MPa.

36 ,1 4 tt

w ww/4 LIVE LOAD

B C p

75 ft 100 ft 75 ft

(a) SPAN ARRANGEMENT AND POSITION OF LIVE LOAD

CENTROID OF 16PRESTRESS TENDONS AXIS OFAp5 = 1.836in2 / TENDON SYMM

AREA OF DUCT

REFERENCE =43W/ TENDON 25 ft 22 5 ftAXIS

0.16% 0.50%6yn 0.16% TOP

173in 01 31 in Ihn^ Il in p38 42in

A B ^,

_^ -- 0.13 % 0.20 % F

30ft 36.25f18.75 7.9 14.6 ft

n5ft80T OM

75 ft 50ft ^{

(b} TENDON PROFILE

ft6in

35 PTOP

t in42 IOinin 40 in 2 in

CASTINGJOINT

PBOTTOM----^

L9.5ft ISft — F+^-9.5ftO 5ft 05ff

(c) CROSS-SECTION DIMENSIONS

Fig. 14. Three-span concrete bridge cast and prestressed in stages (Example 2).Note: 1 in. = 25.4 mm: 1 in.' = 645.2 mm 2 : 1 ft = 0.3048 m.

can be seen at various locations. Recallthat the analysis assumes no slip be-tween the concrete and the steel; the in-crease in compression on the steelwould be smaller if slip occurs.

EXAMPLE 2The computer program CPF is em-

ployed also for the analysis of the par-tially prestressed three-span continuousbridge shown in Fig. 14. The bridge

76

20ft75 ft

A ~P48 E

(a)PART CAST AND PRESTRESSED INSTAGE I

loftBOft

A B C t= t

(b) PART CAST IN STAGES I AND 2 AND PRESTRESSING APPLIED IN STAGE

-.. 55tt ---►,

(c) PART CAST IN STAGES I TD 3 AND PRESTRESSING APPLIED IN STAGE 3

P2

A B C

(d) CASTING OF CANTILEVERS AND PRESTRESSING APPLIED 1N STAGE 4

Fig. 15. Construction and prestressing stages of the bridge of Example 2.Note: 1 ft = 0.3048 m.

P1

cross section is made of a solid slab of3.5 ft (1.05 m) depth, with two side can-tilevers providing a two-lane roadway(Fig. 14c). The bridge is post-tensionedwith 16 tendons having the profileshown in Fig. 14b; each tendon consistsof twelve '/z in. strands. The span ar-rangement and the concrete dimensionsof the cross section are adopted fromthose reported in Ref. 28.

Here the bridge is assmned to be builtin the following stages (Fig. 15). At timet = 0, the thicker part of the deck ofwidth 16 ft (4.8 m) (referred to as thespine) is cast over a length AE coveringSpan AB and 20 ft (6 m) overhang (Fig.15a). At t = 14 days, the same part AF isprestressed with 11 longitudinal ten-dons (P i ) and its forms are removed.

Twenty-eight days later, the spine iscompleted for the remainder of Span BCand 20 ft (6 m) of Span CD. Prestressing

F i is applied on this part at t = 56 days(Fig. 15b). The remaining portion of thespine in Span CD is cast at t = 84 daysand prestressed with Pl at t = 98 days(Fig. 15c).

The completed spine serves as a trackcarrying a moving carriage for formingand casting of the two cantilevered partsof the deck in the period t = 98 to 126days. During the same period, five ad-ditional longitudinal tendons (P 2 ) areprestressed (Fig. 15d). To simplify theanalysis, the application of the weight ofthe cantilevers and the prestressing P 2 arelumped as if they occurred in one in-stant at t = 126 days. Transverse ten-dons are required to join the cantileversto the spine. However, their effects arenot included in the present analysis.The superimposed dead load of thewearing surface, curbs, etc., is intro-duced at t = 154 days.

PCI JOURNALJJanuary-February 1989 77

Table 2. Time-dependent properties of concrete in the bridge of Example 2.

(.f4nstruCtiof and

'lime Spine in Part ABE Cantilevers

E, (t ) Era (t j, tf) Ee (t l) Ee, (tl, tf)loading stages t, tj ksi d,(tt^) x(t i , t.) 10-° ksi CA(ti,t^) X(tj,t 4 ) 10-"

Stage 1 14 56 4500 0.88 0.85 -1314 98 1.1914 126 1.3214 154 1.4314 10000 3.21

Stage 2 56 98 5017 0.57 0.90 -1256 126 0.7756 154 0.91.5ti 10000 2.93

Stage 3 98 126 5106 0.36 0.91 -798 154 0.5798 1(0)00 2.70

Stage 4 126 154 5134 0.33 0.95 -7 450(1 0.91 0.83 -42126 10000 2.58 3.59

Superimposed dead load. 154 10000 5151 2.48 0.78 -300 4950 3.25 0.79 -371

Live load 10000 10000 5232 5232

Note: t = U at the day of casting of the spine in part ABE; I ksi = 6.895 111a.

6000 TOTAL INITIAL

.............

FORCE

5500 ........t ....

-x NONPRESTRESSED 1 AT TIMEa STEEL IGNORED t-10,000 DAYa 5000

0 ++-L NONPRE "4500STEEL CONSIDERED

4000'0 25 50 75 100 125

DISTANCE FROM SUPPORT A (ft)

(a) FORCES IN PRESTRESSING TENDONS

-5000NONPRESTRESSEDSTEEL IGNORED

-4500 -- ---_--ns

4000

au NONPRESTRESSEDUj STEEL

W -3500 CONSIDERED01.

H , I-3000

0 25 50 75 100 125

DISTANCE FROM SUPPORT A (ft)

(b) FORCE IN CONCRETE AT TIME t t 10,000 DAYS

Fig. 16. Forces in prestressing tendons and in concrete after time-dependent losses.Note: i ft = 0.3048 m; 1 kip = 4.448 kN,

The dead loads are: self weight ofspine = 8.2 kips/ft (120 kN/m); weight ofcantilevers = 2.6 kits/ft (38 kN/in); su-perimposed dead load = 1.65 kips/ft (24kN/m). The prestressing force at thetime of jacking = 390 kips (1735 kN) per

tendon. A live load representing a truckis applied at the position shown in Fig.14a at time t = 10,000 days.

'I'he spine and the cantilevers are as-siimed to be made of the same concrete;however, the parameters , 0, x and

PCI JOURNAL/January-February 1989 70

c.

A B

0.25 NONPRESTRESSED TOP OFSTEEL CONSIDERED CANTILEVER

0.00 f`

'u^ 0.25 `-- ---------- -----_--' -- __,

—0.50N

"]TDP–0.75 - -- --NDNPRESTRESSED SP NFSTEEL IGNORED

(a) TOP FIBER

0.00

r 0.50 ----------------- _______-----'+NONPRESTRESSED ^W NONPRESTRESSED ,'075 STEEL CONSIDERED < STEEL IGNORED<

(b) BOTTOM FIBER

Fig. 17. Concrete stresses at top and bottom fibers at t 10,000 days just beforeapplication of live load. Note: 1 ksi = 6.895 MPa.

differ according to the age. These aretaken according to the CEB ModelCode; 22 for brevity, Table 2 gives the pa-rameters only for the spine in Part ABEand for the cantilevers.

Other material properties are: fit0.35 ksi (2.4 MPa); E,,, = 29,000 ksi (200GPa); E, = 27,500 ksi (190 GPa); A = 1and / = 0.5; curvature friction coeffi-cient, u = 0.15/radian; wobble coeffi-cient, k = 7.5 X 10- 4/ft (2.5 x 10-3/m);anchor slip, S = 0.2 in. (5 mm). The in-trinsic relaxation at time infinity, aav,,,= –25 ksi (-172 MPa); values for

shorter periods are calculated from Eqs.(4) to (6) of Ref. 25; the relaxation re-duction factor, x,. = 0.7. The nonpre-stressed reinforcements near the top andbottom fibers are indicated in Fig. 14b-is percentages of the cross section area.

Two analyses are performed, one withthe nonprestressed steel considered andthe other with the presence of this steelignored. Figs. 16 through 21 presentsome results of the two analyses. Al-though the bridge is not symmetricalunder the effects of prestressing and liveload, results are presented only for the

80

c.

A B

0.5

CANTILEVER

CRACKEDZONE

0.0

Y –0.5 I^♦

\TOP OF NONPRESTRESSED–1.0 SPINE STEEL IGNORED

NONPRESTRESSEDSTEEL CONSIDERED

–1.5(a1 TOP FIBER

CRACKEDZONE r^-I

0.5

0.nNONPRESTRESSEDSTEEL CONSIDERED

-0.5

N ^NONPRESTRESSED–1'0 STEEL IGNORED

–1.5[b) BOTTOM FIBER

Fig. 18. Concrete stresses at top and bottom fibers at t = 10,000 days just afterapplication of live load, W = 130 kips (578 kN). Note: 1 ksi – 6.895 MPa.

half of the bridge length which experi-ences larger stresses and deformations.

The variations of the total prestressingforce in all tendons after the frictionlosses and after the time-dependentlosses are shown in Fig. 16a. As shown,presence of nonprestressed steel resultsin a small reduction in the loss of ten-

sion in the prestressed steel. However,the loss in tension is not equal in abso-lute value to the loss of compression inthe concrete. In fact, in the presence ofnonprestressed steel, a large compres-sive force is gradually transmitted fromthe concrete to this reinforcement, re-sulting in a much larger loss in compres-

PCI JOURNAL/January-February 1989 81

250 CRACKING ,'

CONSIDERED SECTION SECTION

----- CRACKING ,' 8 G

200 IGNOREDNQ /'

150 ,'

O

o 10o w ww A B+ 1 wC4 D

3 50 G

07 8 9 10 11 12 13 14 15 16

ABSOLUTE VALUE OF MOMENT M ( kip ft )

Fig. 19. Variation of bending moments at critical sections with increasing live load.Note: 1 kip = 4,448 kN; 1 kip-ft = 1.356 kN•m.

250 CRACKING

CONSIDERED .' ,'

----- CRACKING ,'

200 IGNORED ;

0. NONPRESTRESSED ' ,4— NONPRESTRESSEDY STEEL IGNORED / f STEEL CONSIDERED

150 ;

W cr = 121 kip iI

START OFp , CRACKING AT B

100 Wcr = 94 kip

W w wA B i i WC 4 D

J 50

00 0.5 1 1.5 2 2.5 3

DEFLECTION (in )

Fig. 20. Variation of deflection at middle of interior span with increasing live load.Note: 1 in. = 25.4 mm; 1 kip - 4.448 kN.

82

w wW/4

18A B C D

175 EXTENT OF CRACKING CONSIDERED

Y

TO BOTTOM OF FLANGES

MQb 170

cn START OF CRACKINGw OF SPINE

165 'CRACKINGIGNORED

1600 50 100 150 200 250

LIVE LOAD W (kips)

(a) PRESTRESSED STEEL

ww

5 A 8 t WCA D

025050 100 150 200

y LIVE LOAD W (kips )

--5 EXTENT OF,, CRACKING CONSIDERED CRACKING TOC BOTTOM OFb FLANGESSTART OF CRACKINGin ` 10 OF SPINEcn Ui

_15 CRACKINGIGNORED

—20

(b) NONPRESTRESSED STEEL

Fig. 21. Steel stresses at Section G, middle of interior span: variation with increasing liveload. Note: 1 kip = 4.448 kN; 1 ksi = 6.895 MPa.

PCI JOURNAL`January-February 1989 83

sion on the concrete. This is shown inFig. 16b, which represents the variationof the resultant compressive force inconcrete at t = 10,000 days. The differ-ence in the ordinates between the twocurves in Fig. 16b represents the com-pression in the nonprestressed steel.The two steps in the continuous curve atH and I in Fig. 16b are due to the cur-tailments of the nonprestressed steel(see Fig. 14b).

The variation of concrete stresses atthe top and bottom fibers just before andafter application of live load with W =130 kips (578 kN) (Fig. 14a) is depictedin Figs. 17 and 18, respectively. Fig. 18shows the zones in which the tensilestrength of concrete (f,r = 0.35 ksi) is ex-ceeded, indicating cracking. Wherecracking occurs, Program CPF recalcu-lates the stress distribution over thesection, ignoring the concrete in ten-sion, and determines the mean values ofaxial strain and curvature accounting forthe tension stiffening [Eqs. (17) and(18)]. Fig. 18 represents the stress varia-tion before the concrete in tension is ig-nored.

The dashed lines in Figs. 17 and 18represent the stresses when the pres-ence of nonprestressed steel is ignored.The analysis based on this assumptionwould indicate no cracking while, infact, cracking occurs over almost 30 per-cent of the length of the interior span.

The graphs in Figs. 19 through 21 rep-resent, respectively, the variation with

increasing live load, W, of the bendingmoment at two critical sections, and thedeflection and the stresses in the pre-stressed and nonprestressed steels at themiddle of the interior span.

The dashed lines in Figs. 19 through21 illustrate the case of a linear analysiswhich ignores cracking, while the can-tinuous lines are for the analysis whichaccounts for cracking. As shown, there isa substantial difference between the re-stilts of the two analyses.

Fig. 20 indicates that ignoring thepresence of nonprestressed steel resultsin an underestimation of the deflectionat the middle of the interior span and anoverestimation of the live load level atwhich cracking occurs [Wcr = 121 kips(538 kN) instead of94 kips (418 kN)].

For the stresses in the prestressed andnonprestressed steels, Figs. 21a and bshow that when the live load producescracking a large increase in stress oc-curs. This increase in stress can be im-portant if fatigue is of concern. Also, theincrement in stress in the nonpre-stressed steel at and after cracking canbe used to predict the width of cracks.

For the cracked section at the middleof the interior span and for a live load W= 250 kips (1112 kN), the analysis givesJEA roily craned = 735 x 10- ti for the bottomnonprestressed steel and i; = 0.93 [Eq.(18)1. Assuming a mean crack spacing, s= 1 ft (0.3 m), Eq. (19) gives a meanvalue for the crack width, w = 0.008 in.(0.2 min).

84

SUMMARY AND CONCLUSION

A numerical procedure based on thedisplacement method is presented forthe serviceability analysis of reinforcedconcrete plane frames with or withoutprestressing. The analysis accounts forthe effects of friction and anchor setting,creep and shrinkage of concrete and re-laxation of prestressed steel, and for theeffects of cracking on the stresses anddeformations.

Variation of concrete propertieswithin individual cross sections andfrom one member to another is takeninto account. External loads and pre-stressing can be applied in stages. Pre-stressing can be of any magnitude vary-ing from zero, allowing cracking, to fullprestressing, eliminating cracks. Theeffect of cracking on the reactions andinternal forces in statically indetermi-nate frames is analyzed by an iterativeprocedure_ Equilibrium of forces andcompatibility of strains in the pre-stressed and nonprestressed steels andin the concrete are used to calculatetime-dependent variations of the forcesin the three components. The need foruse of empirical equations for predictionof prestress losses is eliminated.

The procedure is implemented in anavailable computer program which is

suitable for the analysis of a wide rangeof frames including Continuous bridgesbuilt span by span, segmental construc-tion, and structures built of precast pre-stressed concrete members connectedand made continuous by cast-in-placeconcrete deck or joints and a secondstage prestressing. The program can alsobe used for the analysis of multistorystructures which are generally con-structed in several stages,

The program gives the instantaneousand time-dependent changes in the dis-placements, the reactions and staticallyindeterminate forces and the corre-sponding stresses and strains, and thecrack widths at various sections. Theprogram requires a small core storageand can be used on a microcomputer.*

Two bridge examples are presented toshow the significance of the time-de-pendent deformations and cracking onthe serviceability of a composite steelbridge and a partially prestressed con-crete bridge built in stages.

*A version of CPF on diskette, for use onIBM microcomputers, is available from theCivil Engineering Department, The Univer-sity of Calgary, 2500 University Drive N.W.,Calgary, Alberta, Canada T2N 1N4.

ACKNOWLEDGMENT

The research work reported in thispaper was financially supported by anIzaak Walton Killam Memorial Scholar-ship and a grant from the Natural Sci-ences and Engineering Research Coun-

cil of Canada which are greatly appreci-ated. Grateful appreciation is also due toArvid Grant and Associates, Olympia,Washington, for providing the data forthe bridge in Example 1.

PCI JOURNAL'January-February 1989 85

REFERENCES

1. Brown, R. C., Jr., and Burns, N. H.,"Computer Analysis of SegmentallyErected Bridges," Journal of the Struc-tural Division, American Society of CivilEngineers, V 101, No. ST4, April 1975,pp. 761-778.

2. Danon, J. R., and Gamble, W. L,, Time-Dependent Deformations and Losses inConcrete Bridges Built by the CantileverMethod, Structural Research Series No.437, Civil Engineering Studies, Univer-sity of Illinois at Urbana-Champaign,Urbana, Illinois, 1977, 169 pp.

3. EEC, B.C. Bridge Construction Com-puter Program Manual, Europe EtudesGecti, Paris, France, 1977.

4. Tadros, M. K., Ghali, A., and Dilger,W. H., "Long-Term Stresses and Defor-mations of Segmental Bridges," PCIJOURNAL, V. 24, No. 4, July-AugustI979, pp. 66-87.

5. Khalil, M. S. A., "Time-DependentNon-Linear Analysis of Prestressed Con-crete Cable-Stayed Girders and OtherConcrete Structures," PhD Thesis, De-partment of Civil Engineering, The Uni-versity of Calgary, Calgary, Alberta,Canada, April 1979, 246 pp.

6. Van Zyl, S. F., and Scordelis, A. C.,"Analysis of Curved Prestressed Seg-mental Bridges," Journal of the Struc-tural Division, American Society of CivilEngineers, V. 105, No. ST11, November1979, pp. 2399-2417.

7. Marshall, V., and Gamble, W. L., Time-Dependent Deformations In SegmentalPrestressed Concrete Bridges, StructuralResearch Series No. 495, Civil Engi-neering Studies, University of Illinois atUrbana-Champaign, Urbana, Illinois,1981.

8. Shushkewich, K. W., "Time-DependentAnalysis of Segmental Bridges," Com-puter and Structures, V. 23, No. 1, 1986,pp. 95-118.

9. Ketchum, M. A., "Redistribution ofStresses in Segmentally Erected Pre-stressed Concrete Bridges," Report No.UCBISESM-86/07, Structural Engi-neering and Structural Mechanics, Uni-versity of California at Berkeley, Berke-ley, California, May 1986, 238 pp.

10. Elbadry, M. M., and Ghali, A., "Designof Concrete Bridges for Temperature

Variations," presented at ACI Sympo-sium on "Serviceability Criteria andConcern for Bridges" at the ACI AnnualConvention in Seattle, Washington, No-vemher 1987.

11. Ghali, A., and Neville, A. M., StructuralAnalysis: A Unified Classical and MatrixApproach, Second Edition, Chapmanand Hall, London, England, 1978, 779pp -

12. Ghali, A., and Favre, R., ConcreteStructures: Stresses and Deformations.Chapman and Hall, London and NewYork, 1986, 352 pp.

13. Ghali, A., "A Unified Approach for Ser-viceability Design of Prestressed andNonprestressed Reinforced ConcreteStructures," PCI JOURNAL, V. 31, No,2, March-April, 1986, pp. 118-137. Seealso discussion, V. 32, No. 1, January-February 1987, pp. 133-140.

14. Ghali, A., and Elbadry, M. M., "Crackingof Composite Prestressed Concrete Sec-tions," Canadian Journal of Cicil Engi-neering, V. 14, No. 3, J tine 1987, pp.314-319.

15. Naaman, A. E., Prestressed ConcreteAnalysis and Design – Fundamentals,McGraw Hill Book Co., New York, N.Y.,1982, 670 pp.

16. Huang, T„ "Anchorage Take-up Loss inPost-tensioned Members," PCI JOUR-NAL, V. 14, No. 4, August 1969, pp.30-35,

17. ACI Committee 343, Analysis and De-sign of Reinforced Concrete BridgeStructures, Report 343-77, AmericanConcrete Institute, Detroit, Michigan,1977, pp. 73-80.

18. PCI Committee on Prestress Losses,"Recommendations for Estimating Pre-stress Losses," PCI JOURNAL, V. 20,No. 4, July-August 1975, pp. 43-75.

19. Trost, H., "Auswirkungen des Superpo-sitionsprinzips auf Kriech- and Relaxa-tionsprohleme bei Beton and Spann-beton," Beton- and Stahlbetonbau, V.62, No. 10, 1967, pp. 230 -238 and No. 11,1967, pp. 261-269.

20. Ba2ant, Z. P., "Prediction of ConcreteCreep Effects Using Age-Adjusted Ef-fective Modulus Method," ACI Journal,V. 69, No.4, April 1972, pp. 212-217.

21. ACI Committee 209, "Prediction of

86

Creep, Shrinkage and Temperature Ef-fects in Concrete Structures," ACI Spe-cial Publication SP-76, American Con-crete Institute, Detroit, Michigan, 1982,pp. 193-300.

22. Comite Euro-International du Beton(CEB) — Federation Internationale de laPr+ contrainte (FIP), Model Code forConcrete Structures, CEB, Paris,France, 1978, 348 pp.

23. Favre, R., Beeby, A. W., Falkner, H.,Koprna, M., and Schiessel, P., CEB De-sign Manual on Cracking and Deforrna-tions, Ecole Polytechnique Federale deLausanne, Switzerland, 1985.

24. Magura, D. D., Sozen, M. A., and Siess,C. P., "A Study of Stress Relaxation inPrestressing Reinforcement," PCIJOURNAL, V. 9, No. 2, April 1964, pp.13-57.

25. Chali, A., and Trevino, J., "Relaxation ofSteel in Prestressed Concrete," PCIJOURNAL, V. 30, No. 5, September-Oc-tober 1985, pp. 82-94.

26. Elbadry, M. M., and Chali, A., User's

Manual and Corn puter Program CPF:Cracked Plane Frames in PrestressedConcrete, Research Report.No. CE85-2,Department of Civil Engineering, TheUniversity of Calgary, Calgary, Alberta,Canada, January 1985. Program is avail-able on diskettes for IBM microcomput-ers.

27. Elbadry, M. M., "Serviceability of Rein-forced Concrete Structures," PhD The-sis, Department of Civil Engineerng,The University of Calgary, Calgary, Al-berta, Canada, November 1988, 294 pp.

28. Aparicio, A. C., Arenas, J. J., and Alon-so, C., "Examples of Moment Redistri-bution in Continuous Partially Pre-stressed Bridges," International Sympo-sium on Nonlinearity and Continuity inPrestressed Concrete, Proceedings V. 2,University of Waterloo, Waterloo, Can-ada, July 4-6, 1983, pp. 185-204.

29. Carnahan, B., Luther, H. A., and Wilkes,J. 0., Applied Numerical Methods, JohnWiley and Sons, Inc., New York, N.Y.,1969, 604 pp.

METRIC (SI) CONVERSION FACTORS

in, –25.4 mm 1 kip = 4.448 kN1 in. 2 = 645,2 mm2 1 psi = 6.895 x 10 - ' MPa1 in.3 = 0.0000164 m3 1 ksi = 6.895 MPa1 in" = 416231 mm" 1 kip-ft = 1.356 kN-m1 ft = 0.3048 m 1 kip/ft = 14.594 kNlm

NOTE: Discussion of this article is invited. Please submityour comments to PCI Headquarters by October 1, 1989.

PCI JOURNAL/January-February 1989 87

APPENDIX A - STRESS AND STRAININ A CROSS SECTION

Consider a cross section made of ahomogeneous elastic material subjectedto a normal force N at an arbitrary refer-ence point 0 and a bending moment M.Assuming that plane cross sections re-main plane, the strain and stress at anyfiber at a distance y below 0 can be ex-pressed as:

E Ep+ Lu1; Q = E(EO + qJ,)

(Al and 2)

where E is the modulus of elasticity, Eois the strain at 0 and i (= de/dy) is thecurvature (the slope of the strain dia-gram); see Fig. 6b. Equilibrium re-quires:

N=JadA; M= f QydA

(A3 and 4)

Substitution of Eq. (A2) into (A3) and(A4) gives:

N =A r +B y; M — Boo+l y

(A5)

where A, B and I are the area of the cross

section, and its first and second mo-ments about an axis through 0; o (= E€o) is the stress at 0 and y (- do-/dy =E -) is the slope of the stress diagram.Eq. (A5) can be used to determine thestress resultants when the stress (orstrain) distribution is known, For givenN and M, the strain at 0 and the curva-ture can he determined by:

_ IN—BM _ —BN+AMEQ E(AI —B E)' q E(AI —B2)

(A6)

When the equations of this appendixare applied to a composite reinforcedconcrete section with or without pre-stressing, the symbols A, B and I re-present the properties of a transformedsection composed of the area of concretein each part plus the area of reinforce-ments, each multiplied by its modulusof elasticity and divided by an arbitraryreference modulus, E,er. The referenceelasticity modulus can be convenientlytaken as the modulus of one of the con-crete parts.

88

APPENDIX B -- DEPTH OF COMPRESSION ZONEIN A FULLY-CRACKED SECTION

Figs. Bla-c show the strain and stressdistributions in a composite section dueto forces (AN, 0 M} a,,,y er kea producingcracking. Prior to applying these forces,the concrete stresses are assumed to bezero in the lower part of the section. Theresultant of (AN, AM } fultu eraeke.d is lo-cated at an eccentricity e from a refer-ence point 0 (e is positive when the re-sultant is situated below 0):

DMe = (Bl)AN Aran krod

The strain in any fiber can he ex-pressed by Eq. (Al), but because con-crete in tension is ignored, the stressesin concrete and steel are expressed by(go to top of right column):

Acrc = E e, (1 ----)A€0 when y < y„YU

Avc = 0 when y } y„

(B2)

Acr,= E, I – y8 Aeo (B3)Ya

where y„ (= c — do) is the y coordinate ofthe neutral axis, with do being the dis-tance from the top fiber to 0 (positivewhen 0 is below top fiber).

The depth c of the compression zonecan be determined by taking momentsabout an axis through the point of appli-cation of the resultant of IAN, AM)er,ji-krd and equating to zero. This leads tothe following equation:

UI U ill

e IA,ac(d(.—c)+

[A,(a,— ac)(d.—c}]^ = Apac[(d,c—d;)+dc(2d,_J1

do–c)+ 16^ l^ +1b2 + I [A.(a.–ac)(d,–c)(d.–do)1J} (B4)

For the special case when AN,,,sacked = 0, substitution of Eqs. (B2) and(B3) into (A3) gives:

m

^ A, ac(dc – c) +

[A,(a, – a)(14 – c)le = {1 (B5)1 = 1 ji

In the above two equations, subscriptsc and s refer to concrete and steel, re-spectively. An additional subscript p orn can be used with s to indicate pre-stressed or nonprestressed steel. Thesubscripts i and j refer to a concretetrapezium (Fig. Bid) and a steel layer;m is the total number of trapeziums andn is the number of steel layers includedin the ith trapeziurn;A, = E cf/E„erand a,(= E, /E,.ef. Note that E, = 0 for concretein the tension zone. Symbols A.1 and d,,

are the gross area of concrete trapeziumi and the distance from its centroid tothe extreme compression fiber; for atrapezium of widths h, and h 2 and heighth (Fig. Bid):

2 (b, + h2 )l + (B6)AQ^ L J=

h( b

1 + 2b2\1 (B7)- ______do{= [d1+ 3 b,+b t

Solution of Eq. (B4) or (B5) to deter-mine the value of c is best obtained bytrial using Newton's iterative tech-nique.29

Once c is known, the properties of thecross section A, B and I can be deter-mined leaving out the concrete belowthe neutral axis and Eqs. (Al), (A2) and(A6) can he used to find the stress andstrain changes in a fully cracked section.

PCJ JOURNAL/January-February 1989 89

CD0

REFERENCEPOINT

d-Arte_ 0 0 AFO

ANc

YT dDS dns

NEUTRAL Y' Yn iAXIS

Ps 0i

^- A ns

(a) FULLY-CRACKED COMPOSITE (b) STRAINSECTION

TYPICALTRAPEZIUM

p lGROSS AREA AgiTOP FIBER

dli

hi ty,ciCENTROID OFy. yn AREA Agi

VBOTTO M FIBER

i c) STRESS (d) DIVISION OF CONCRETE AREA INTOTRAPEZIUMS

Fig. B1. Concrete strain and stress in a fully cracked section due to { N, ,X M} 1a1,,, zfc1fed.

APPENDIX C - NOTATION

A andA = area of transformedand of age-adjustedtransformed sections

B and R = first moment of area oftransformed and ofage-adjusted trans-formed sections

= depth of compressionzone in a fully crackedsection

{U} and {D} = displacement vectorsE and E = modulus of elasticity

and age-adjusted elas-ticity modulus

{F} and {i~ *} = vectors of fixed-endforces_

[f 1 and [ f ] = flexibility and age-ad-justed flexibility ma-trices

f.t = tensile strength of con-crete

JH] = transformation matrixI andI = moment of inertia of

transformed and ofage-adjusted trans-formed sections

k = wobble friction coeffi-cient

M = bending momentN = normal forceP = absolute value of pre-

stressing forceISI = stiffness matrix

= length of prestressingtendon between sec-tions i andj

s = average crack spacingt = timew = mean crack widthy = coordinate of any sec-

tion fiber, measured

downwards from a reference point 0

a = modular ratiofi, and Qz = coefficients, 0.5 or 1 as

specified below Eq.(18)

y = slope of stress diagram= anchor set= increment or decre-

mente = normal strain

= interpolation coeffi-cient

= change in slope of aprestressing tendon, inradians, between sec-tionsi andj

A = ratio of the initial ten-sile stress in a tendonto its tensile strength

= curvature friction co-efficient

= stressi o and ]gi, = intrinsic and reduced

relaxation of pre-stressed steel

= creep coefficientX = aging coefficientX r = relaxation reduction

factor1u = curvature (slope of

strain diagram)

Subscriptsc, ps, its = concrete, prestressed

and nonprestressedsteel

Cs = shrinkage of concreteO = reference pointo = initial times = steel

PCI JOURNAL/January-February 1989 91