COMPUTER PROGRAM FOR ANALYSIS AND DESIGN OF SIMPLE …

COMPUTER PROGRAM FORANALYSIS AND DESIGNOF SIMPLE-SPAN PRECASTPRESTRESSED HIGHWAY ORRAILWAY BRIDGES

Clifford L. FreyermuthDesign Research SectionPortland Cement AssociationSkokie, Illinois

The Portland Cement Associationis developing a comprehensive se-ries of computer programs for analy-sis and design of concrete structuresand pavements. As these programsare developed, they will be madeavailable to the engineering profes-sion. In addition to this program,others are presently available onconcrete airport pavement design,design of reinforced concrete col-umns, and analysis and design offlat plates and continuous frames.

The computer program for analy-sis and design of simple-span, com-posite and non-composite highwayor railway bridges described in thispaper is based on conventionaldesign procedures(') and on appro-priate sections of the current high-way and railway standard specifica-tions d2 '3 ). The program provides ameans of reducing design costs andtime for the subject type of high-way or railway bridge.

DESCRIPTION OF PROGRAM

The program is written in FortranIV language for the IBM 1130 Com-puter. Output may be by 1132 print-er or console printer.

The program performs the analy-sis and design of simple-span, pre-

cast, prestressed highway or railwaybridges. The program will accom-modate the composite and non-com-posite sections included in Fig. 1,and will compute the following: sec-tion properties, dead load and liveload shears and moments, stressesfor various loading conditions, ulti-mate design moments and resistingmoments, spacing of shear rein-forcement, horizontal shear stressbetween the composite slab andprecast member, midspan elasticdeflections for various loading con-ditions, and the number and centerof gravity of prestressing strands re-quired.

The program is operated in oneof the following modes:

Mode 1. Analysis and design ofstandard sections with acomposite deck slab(Sections 1 through 5,Fig. 1).

Mode 2. Analysis and design ofnon-composite standardsections (Sections 6, 7and 8, Fig. 1).

Mode 3. Analysis and design ofnon-composite single-and double-celled boxbeams (Sections 9 and10, Fig. 1).

28 PCI Journal

The computer program applies to either highway or railroad bridgeswith various types of prestressed units on simple spans, eitherwith or without a composite deck. Based on conventional designprocedures, the program mechanizes all the routine calculationsand provides a sizeable reduction in design time and cost.

Mode 4. Analysis and design ofall sections in Fig. 1when the number andlocation of prestressingstrands are included inthe input data.

The detailed capabilities of theprogram are as follows:

Section Properties. The programcalculates section properties for non-composite single- or double-celledbox beams (Sections 9 and 10, Fig.1). The other precast sections inFig. 1 are all standards with sectionproperties available. For designswith these sections, it is requiredthat the section properties of theprecast section be included in theinput data. When the standard sec-tions are used in a composite de-sign, the program computes thecomposite section properties.

Loads. The standard AASHO load-ings are used to calculate shearsand moments as described below.The magnitude of highway live load(HS20-44, HS15-44, HS10-44, H20-44, H15-44, or H10-44) is indicatedby an input code number.

For railway bridges, the programuses Cooper's E-Loading. Live loadbending moments are required in-

put data. The magnitude of railwaylive load (E-72, E-60, etc.) is indi-cated by an input code number.

Shear and Moment due to Loads.For all standard AASHO loadingsthe program computes shears atthree locations (reaction, 1/4 span,1/3 span) and moments at up tofour specified locations. Truck load-ing and equivalent lane loading areconsidered, and the program selectsthe type that governs the design.Dead load shears and moments arecalculated at the sections specifiedfor live load. The program computesultimate moments and shears due toloads for highway bridges by multi-plying service load moments andshears by specified load factors.

For railway bridges, design mo-ments due to Cooper's loading atone to four locations are requiredinput data. The values are availablefrom moment tables. The programcomputes shears due to the speci-fied Cooper's loading at the 1/4-and 1/3-span points. Dead loadshears and moments are calculatedat the same sections as for live load.The program computes ultimate mo-ments and shears due to loads forrailway bridges by multiplying theservice load values by specified load

June 1968 29

factors.

Prestressing Force. In Modes 1 2and 3, the program calculates thenumber and center of gravity ofprestressing strands necessary to sat-isfy stress limitations, and to pro-vide an ultimate internal resistingmoment equal to, or greater than,the ultimate moment due to loads.The input data specifies the stress

limitations and the size and ulti-mate strength of the individual pre-stressing strand.

In Mode 4, the number and loca-tion of the prestressing strands areincluded in the input data, and theportions of the program used to cal-culate these values in the first threemodes are bypassed.

In any mode, the profile of thecenter of gravity of the prestressing

[i]

SPREADBOX

6. CHANNEL

2. 11 BOX 7. SLAB

8. Q 0 0 VOIDED

AASHO-PCI• TYPES I -IY

0 - PCI3YSYI

9. BO

[ LI I DOUBLE10' BOX

5. TEE

COMPOSITE SECTIONS NON-COMPOSITE SECTIONS

Fig. 1. Sections analyzed in computer program30 PCI Journal

force may be specified as straightparallel strands with constant eccen-tricity, or as depressed strands witha maximum of two hold-downpoints. The program will also ac-commodate post-tensioned designsand, for this case, the center of grav-ity of the prestressing force is con-sidered to be parabolic.

For pretensioned designs, part ofthe prestress losses occur due tostrand relaxation and elastic mem-ber shortening prior to developingthe stress in the section under the"initial" prestressing force. The pre-liminary loss due to strand relaxa-tion prior to release can be eval-uated from the formula^r):

; 1 1bt — 0.55(jf,

for fj 0.55

where fs = the remaining stress atany time, t, after pre-stressing

f the initial stressf, = 0.1 percent offset stresst = time in hours

For 270k strand stressed initially to0.70 f;, the ratio of f/ f Ri is about0.975 after 48 hours. In other words,the strand relaxation in this periodof time is about 21/2 percent.

The loss due to elastic shorteningand bending of the member can beconsidered to have occurred prior tocalculating stresses due to the initialprestressing force, because the mem-ber shortens as it is stressed. Thisloss can be calculated by the fol-lowing procedure:

1. Calculate concrete stress at re-lease at the level of the pre-stressing steel.

June 1968

f^ = Pti (1.0 — elastic loss) 1 + e—(_A I

where f, — concrete stress at levelof prestressing steel.

Pi = initial prestressingforce.

A = area of precast section.e = eccentricity of strands

(a weighted averagevalue may be used fordesigns with depressedstrands).

1= moment of inertia ofprecast section.

Elastic loss is a value to bedetermined. A value fromthe table in the operating in-structions can be used forthis calculation.

2. Percentage loss due to elastic

Es x f,,shortening = E°

fs^

where Eb = modulus of elasticity ofthe prestressing steel(usually 28 X 106 psi).

E, = modulus of elasticity ofthe concrete in the pre-cast section at the timeof prestressing (E=w1 X 33V7, wherew = weight of the con-crete and f,a = compres-sive strength of the con-crete at release).

fs = initial prestressing steelstress.

Table 1 contains approximate valuesfor the elastic shortening loss forthe 10 sections in Fig. 1.

Some economy can usually begained by considering the twolosses discussed above to have oc-curred prior to the initial stress dueto prestress. In the subject program,

31

this is achieved by an input vari-able (designated in the program asESR) representing the ratio ofwhere f $ is the stress immediatelyafter release and member shorten-ing. In general, a satisfactory valuefor this ratio is obtained by assum-ing a 2½ percent relaxation loss andan elastic shortening loss as indi-cated in Table 1. When the strandmanufacturer supplies relaxationloss data, it should be used in lieuof values in Table 1 or values com-puted from the above relaxation lossformula.

Tab e

For post-tensioned designs, wherethe steel elements are not tensionedsimultaneously, the value of theelastic shortening loss varies de-pending on the number of tendonsand the tensioning procedure. Therelaxation loss for post-tensioned de-signs before calculating stresses dueto the initial prestressing force isessentially zero. For most post-ten-sioned designs, it will be sufficientlyaccurate (and conservative) to entera value of 1.0 for the input variablerepresenting elastic shortening andpreliminary strand relaxation loss.

,.

1 2 3 4 5 6Approxi-

Sec- Type of Approxi- mate Strand Approxi-tion Section Approxi- mate Elas- Relaxation mate Value

Num- mate Value tic Shorten- Loss @ of ESRber for EMIN ing Loss 48 hr 0 _ (4 + 5)

(Fig. 1) in. percent percent 100

1 Compositespread box 2.0 5.0 2.5 0.925

2 Composite box 2.0 5.0 2.5 0.9253 Composite AASHO-

PCI standardsections

Type I 30-ft. span 2.0 6.0 2.5 0.915Type 145-ft. span 3.0 9.5 2.5 0.880Type II 40-ft. span 2.5 6.0 2.5 0.915Type II 60-ft. span 3.5 9.5 2.5 0.880Type III 55-ft. span 3.5 6.0 2.5 0.915Type III 80-ft. span 4.0 9.5 2.5 0.880Type IV 70-ft. span 3.5 6.0 2.5 0.915Type IV 100-ft. span 4.5 9.5 2.5 0.880

4 Type V 90-ft. span 4.5 7.0 2.5 0.905Type V 120-ft. span 5.0 9.5 2.5 0.880Type VI 110-ft. span 5.0 7.0 2.5 0.905Type VI 140-ft. span 5.5 9.5 2.5 0.880

5 Composite tee40-ft. span 4.550-ft. span 5.0 7.0 2.5 0.90560-ft. span 6.0

6 Channel20-ft. span 3.030-ft. span 4.0 7.0 2.5 0.90540-ft. span 5.0

7 Solid slab 2.0 4.0 2.5 0.9358 Voided slab 2.0 4.5 2.5 0.9309 Single box 2.0 5.0 2.5 0.925

10 Double box 2.0 5.0 2.5 0.925

32 PCI Journal

Use of the program requires thatthe designer enter a value for theminimum eccentricity of the pre-stressing force from the bottom ofthe precast section (designated inthe program as EMIN). Suggestedvalues for this input item are alsoincluded in Table 1.

Stresses. The program calculates topand bottom fiber stresses due toloads (beam weight, non-compositedead load, composite dead load, andlive load) and then calculates thestresses due to three combinationsof loads and prestress force (beamweight plus initial prestress, beamweight plus non-composite deadload plus final prestress, and allloads plus final prestress). Thesestresses are calculated at the loca-tions specified in the input data.

Ultimate Moments. The ultimate re-sisting moment provided is com-puted and compared to the ultimatemoment due to loads which is calcu-Iated as described above. If ultimatemoment governs the required areaof steel, the comparable number ofstrands are used by the program tocalculate stresses, etc.

Shear. The program computes thestirrup spacing required to meetspecifications at the 1/4- and 1/3-span points, which generally are thecritical locations for shear. Themaximum spacing permitted by spe-cifications is also calculated andcompared to the spacing determinedby stress considerations.

For composite highway bridgedesigns, the ultimate horizontalshear stress between slab and pre-cast member is calculated at thesupport.

Reaction. For highway bridge de-

signs, the program calculates thereaction due to live load plus im-pact, and the total reaction due todead load plus live load and impact.

Deflections. Instantaneous midspandeflections are calculated for threeloading conditions (beam weightplus prestressing force, total deadload plus prestressing force, and liveload).

INPUT

Standard input data sheets are in-cluded with the operating instruc-tions that may be reproduced fordesign office use. Fig. 2 presents thedesign data for one of the sampleproblems forwarded with the ob-ject or source deck and Fig. 3 is thecompleted input data sheet for thisdesign. Fig. 5 presents the designdata for the second sample prob-lem, and Fig. 6 is the completed in-put data sheet.

OUTPUT

Output may be specified in twolevels. The first level includes nearlyall of the significant design valuescomputed. The second level con-tains the minimum data required toprovide an adequate description ofthe design.

Figs. 4 and 7 contain sample out-put sheets. Fig. 4 is the output (firstlevel) for the sample problem andinput data presented in Figs. 2 and3. Fig. 7 is the output (second level)for the sample problem and inputdata presented in Figs. 5 and 6.

GENERAL COMMENTS ON USE OFTHE PROGRAM

The subject computer program isessentially a mechanization of con-ventional design procedures. Thedesigner selects and inputs the mainstructural parameters and the corn-

June 1968 33

puter makes the routine calcula-tions. In some instances, the pro-gram may not find a mutuallysatisfactory solution to the variousdesign requirements. This usuallymeans that the section is beyond itscapacity for the input design cri-teria. In these cases, an overstressindicator is printed in the output.If the overstress is significant andcannot be accommodated by in-creasing the design concretestrengths, some revision of the de-

sign criteria is necessary to enablethe program to find an acceptablesolution. In the initial design proc-ess, it may be desirable to inputseveral decks, varying some of thecontrolling design criteria. Usuallyone of these designs will be satis-factory. If not, it will generally bepossible to inspect the output andselect final design criteria for a sec-ond computer run.

After a satisfactory design is ob-tained, the designer must check to

140' C.TO C. BEARINGS

ELEVATION

31'– 6"

14'–O" 14'–O"

7" SLAB TYPE = AASHOGIRDER

3'— 4 EQUAL SPACES c 6'-3" • 25'–O"

SECTION

Loading: HS20-44

Prestressing tendons: 1/2"C/ 270k strands

Concrete strength:

Girders at release f. t = 4000 psiGirders at Z8 days fa = 5000 psiComposite slab f, = 4500 psiE composite slab/E girder = 0.8

Fig. 2. Design data for sample problem No. 134 PCI Journal

MODES I and 2 — Standard sections with or without a composite slob. PORTLAND CEMENT ASSOCIATIONINPUT FORM FOR PRESTRESSED BRIDGE PROGRAM

%%'110111111=^11111^i^iGIQ11^:Q0% IIIIIIIIIIIIQiilu1101^ICi611^11F^Q1110^ ^i1116^:1111

IIIIIIIIIIIIIIIIII^^a1111111Q ►̂ 5E1%Fi^Q1111111Q6111111Q^1i111Q Qi111111Q^Q11111 111

1115i111111^Q111111^Q1111111QQ^1111Q9^►11^1a1111111Q1 ^IIIIQQQIII^IIIIillllilll

®mo

.DIIIIQ^i11111Q111 1111111i1111Q1111111Q 5i1111QIN1111Q111N11Q111111111111111

1s^tspsuimnmmtcaraFprIcsssn .. , ..... . •,. ...

Fig. 3. Input data sheet for sample problem No. 1wN

5-

CDm

1007

2

3

4

5

6

7

BEAM WT +INITIAL PRESTRESS

TOP SOT0

.

836 1.6220.650 1.8130.662 1.8000.253 2.2189673. FT.KIPS

BEAM WT + SDL +FINAL PRESTRESS

TOP BOT1.602 0.5291.338 0.7991.232 0.9080.222.1.942

ALL LOADS +FINAL PRESTRESS

TOP SOT1.977 -0.190OVERSTRESS1.674 0.1551.520 0.3550.222 1.942

PORTLAND CEMENT ASSOC., JOB NO. 1 DATE 5 1 68DESIGN SECTION STRUCTURE TYPEVI AASHO

SUBJECT • 140FT SPAN 6FT3IN CTRSPRESTRESSED GIRDER ANALYSIS DATA BY CLF CHECKED BY ATD

ASECT • 1086.00 SO.IN. SECTI • 733400. IN4 YT • 35.580 IN. YB • 36.420 INe

WTS • 42.00 IN. BB • 8.00 IN. SB • 20137.2 IN3 ST • 20612.7 IN3

WCS • 75.00 IN. ICS • 7.00 IN. XNCS • 0.80

ACMP • 1506.00 SO.IN. CMPI • 1197669. IN4

YTC • 24.681 IN. YBC • 47.318 IN. YTSC • 31.681 IN.

STC • 48525.6 IN3 SBC • 25310.6 IN3 STSC • 37803.8 1N3 OTSC • 11836.0 IN3

MAX LL + I MOMENT • 1516.9 FT. KIPSLL + I SHEAR AT 1/4 PT • 34.0 KIPSLL + I SHEAR AT 1/3 PT • 30.2 KIPSLL +1 REACTION • 47.8 KIPS

STRESSES IN EXTREME FIBERS DUE TO EXTERNAL LOADS - KIPS PER SO.IN.DIST BEAM WT 50L CDL LIVE LOAD TOTAL

TOP SOT TOP 801 TOP SOT TOP BOT TOP BOT0.500 L 1.613 -1.651 0.668 -0.684 0.000 0.000 0.375 -0.719 2.657 -3.0550.330 L 1.426 -1.460 0.591 -0.605 0.000 0.000 0.335 -0.643 2.354 -2.7090.250 L 1.210 -1.238 0.501 -0.513 0.000 0.000 0.288 -0.552 1.999 -2.3040.000 L 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000COMPOSITE STRESS IN SLAB • 0.481 KIPS PER SO.IN.

MINIMUM STRANDS • 49.462 YM • 5.400 IN. YE • 21.370 IN.

STRESSES DUE TO EXTERNAL LOADS AND PRESTRESS - KIPS PER SQ. IN.

DIST INITIAL PRESTRESS

TOP SOT0.500 1. -0.776 3.2730.330 L -0.776 3.2730.250 L -0.547 3.0390.000 L 0.253 2.218ULTIMATE MOMENT REQUIREDULTIMATE MOMENT PROVIDED

DEPTH OF COMP BLOCK MORE THAN TCS+ ITS. UMP CALC APPROXIMATEFOR FCPC • 3.000 PSI AND 50. STRANDS UMP • 10985. FT.KIPS

CASE 1 STRESSES 50. STRANDS YM • 5.400 IN. YE • 21.370 IN.

DIST INITIAL PRESTRESS BEAM WT + BEAM WT + SDL + ALL LOADS +

INITIAL PRESTRESS FINAL PRESTRESS FINAL PRESTRESSTOP BOT TOP BOT TOP BOT TOP BOT

0.500 L -0.785 3.309 0.828 1.657 1.595 0.560 1.970 •0.1580.330 L -0.785 3.309 0.641 1.848 1.331 0.830 1.666 0.1860.250 L -0.553 3.072 0.656 1.833 1.227 0.937 1.515 0.3840.000 L 0.256 2.243 0.256 2.243 0.224 1.963 0.224 1.963

DEPTH OF COMP BLOCK MORE THAN TCS+TTS,UMP CALC APPROX

ULTIMATE MOMENT REQUIRED • 9673. FT.KIPS ULTIMATE MOMENT PROVIDED • 10985. FT.KIPS

AT 1/4 PT.REOUIRED STIRRUP SPACING • 30.99 IN.AT 1/3 PT.REOUIRED STIRRUP SPACING • 30.99 IN.

OL+LL+I REACTION PER BEAM • - 159.90 KIPS

ULT SHEAR STRESS BETWEEN SLAB AND BEAM AT REACTION • 28.169 PSI

DEFLECTIONSBEAM WEIGHT + PRESTRESS • -1.456 IN.TOTAL DEAD LOAD + PRESTRESS • 0.537 IN.LIVE LOAD + IMPACT - 1.038 IN.

Fig. 4. Output sheet for sample problem No. 1

36 PCI Journal

ELE VAT IC)N

141-0..7.—O"

5.. osyp

s„ 7„

size

SECTION

Loading: Cooper's E72Prestressing tendons: 1/2"Q 270k strandsConcrete strength:

Girders at release fit = 4000 psiGirders at 28 days f' = 5000 psi

Fig. 5. Design data for sample problem No. 2

see that a strand pattern can be pro-vided with the number and centerof gravity of strands indicated inthe output. Use of the EMIN valuesin Table 1 generally assures that asuitable strand pattern can beachieved. However, the EMIN val-ues in Table 1 were developed spe-cificially for %-in dia. 270k strand.Use of another size or strengthstrand may require some adjustmentof EMIN. If an adjustment is indi-cated, this can be accomplished byre-punching EMIN in card number

4 in the input deck, and making asecond computer run.

Values of EMIN and ESR aregiven for the extremes of the usualspan range for various sections inTable 1. For intermediate span val-ues, EMIN and ESR can be inter-polated from the values given.

ACKNOWLEDGMENT

The computer program describedhere is an enlarged and revised ver-sion of a program for non-compositesingle- and double-celled box beams

June 1968 37

MODE 3 – Non-composite single-ond double celled box beams PORTLAND CEMENT ASSOCIATION

INPUT FORM FOR PRESTRESSED BRIDGE PROGRAM

'^E^I^^O'JG0^0% rIF1^n1111^^11111n^^111rn1111nQ^ 'llnl^tJl^^^^01i^

0

1111%n^11^1^1nn^l^lrrll^llir1111^1111161'7n1r1^3^^111111111111111111n111111^

^rrrarlrrrrlolrlrrrror^^r^rraQr^l^r^olll non!.rrrrro^a^^rr^o^o^lllll^^^r1r111

uI 1111 ^uhIIulll _Inn^a 1uuuI1E In I ziu llluznuumumumflfld^Olrlln^ll , ,Erlll Ir - -^rllirâ ill^^1131111rr^rr rr^r^lir rrn

Fig. 6. Input data sheet for sample problem No. 2

PORTLAND CEMENT ASSOC., JOB NO. 2 DATE 5 1 68DESIGN SECTION STRUCTURE 48IN DOUBLE BOX 7FT WIDE

SUBJECT 45FT SPAN E72 LOADINGPRESTRESSED GIRDER ANALYSIS DATA BY CLF CHECKED BY ATD

BOX WIDTH • 7.000FT. BOX DEPTH • 4.000FT. TOP SLAB • 5.5001N. BUT SLAB • 5.5001N.SIDE WALL • 5.0001N. CEN. WALL • 7.0001N. FILLET • 3.0001N. DELTA • 0.0001N.

SECTION PROPERTIESAREA • 1589.00 YB • 24.000 YT • 24.000

1 • 502374. SB • 20932.2 ST • 20932.2 0 •, 13041.6

CASE 1 STRESSES 44. STRANDS TM • 5. 199 IN. YE • 5.199 IN.

DIST INITIAL PRESTRESS BEAM WT • B EAM WT + SOL + ALL LOADS +INITIAL PRESTRESS

FINAL PRESTRESS FINAL PRESTRESS

TOP BOT TOP BOT

TOP SOT TOP SOT0.500 L -0.318 1.807 -0.077 1.567

0.138 1.164 1.221 0.082

0.330 L -0.318 1.807 -0.105 1.595

0.090 1.213 1.085 0.2180.250 L -0.318 1.807 -0.137 1.627

0.034 1.269 0.853 0.449

0.000 L -0.318 1.807 -0.318 1.807 -0.278 1.581 -0.278 1.581

CHECK CONVENTIONAL TENSILE REINF. AT TOP SLAB.

ULTIMATE MOMENT REOIIRED • 5666. FT.KIPS

ULTIMATE MOMENT PROVIDED • 5781. FT.KIPSa

AT 1/4 PT,REOUIREO STIRRUP SPACING • 19.19 IN.AT 1/3 PT,REOUIRED STIRRUP SPACING • 29.17 IN.

DEFLECTIONSBEAM WEIGHT + PRESTRESS • -0.359 IN.TOTAL DEAD LOAD + PRESTRESS • -0.249 IN.LIVE LOAD + IMPACT • 0.336 IN. .

Fig. 7. Output sheet for sample problem No. 2

developed by the Southern PacificCompany. The cooperation and as-sistance of the Southern PacificCompany in making the originalprogram available is gratefully ac-knowledged.

REFERENCES

1. "Design of Highway Bridges in Pre-stressed Concrete," Portland CementAssociation, Old Orchard Road, Sko-kie, Ill. 60076.

2. "Standard Specifications for Highway

Bridges," Ninth Edition, American As-sociation of State Highway Officials,341 National Press Building, Washing-ton, D.C. 20004.

3. "Manual of Recommended Practice,"American Railway Engineering Associ-ation, 59 East Van Buren St., Chicago,.I11. 60605.

4. Magura, Donald D., Sozen, Mete A.and Siess, Chester P., "A Study ofStress Relaxation in Prestressing Rein-forcement," Journal of the PrestressedConcrete Institute, Vol. 9, No. 2, April1964, pp. 13-57.

Discussion of this paper is invited. Please forward your discussion to PCI Headquarters.by September 1 to permit publication in the December 1968 issue of the PCI JOURNAL.

June 1968 39,