An Approach forCreep Analysis of Cracked

Partially PrestressedConcrete Members

7Shunji InomataDr. Eng Consulting EngineerJapan Bridge & Structure Institute Inc.Tokyo, JapanAlso, Professor of Civil EngineeringAichi Institute of TechnologyAichi, Japan

Some partially prestressed concretemembers may be designed without

following the zero tension requirementtinder dead Ioad. In these members,cracks caused by dead load or serviceloads most likely applied at an early stagewill remain open permanently underdead load. Since the cracked part of thesection cannot resist any tensile stresses,including the tensile stresses caused bytime-dependent internal stress redis-tribution, creep analysis of the sectionshould be performed by taking into ac-count the presence of cracks.

There are many well-known methods'for computing time-dependent effects ofuncracked sections. However, only afew authors- discuss these effects incracked sections.

This paper deals with a creep analysis

for cracked prestressed concrete sec-tions under dead load, based on the fol-lowing assumptions:

1. The section is designed withoutconsideration of a zero tension require-ment under dead load; cracking couldbe caused by dead Ioad or service loadswhich are most likely applied beforetime-dependent effects occur.

2. Plane sections remain plane afterbending.

3. The common practice of ignoringtensile stresses in concrete is assumed.

4. The stress-strain relations forbonded prestressed and nonprestressedsteel are linear.

5. The concrete strain response togradually varying stress is given by Baz-ant's F age-adjusted effective modulusmethod.

104

COMPRESSIVE STRESSDISTRIBUTION AT TIME t

A practical method for directly com-puting the concrete strain under a vary-ing stress was developed by Trost ? in1967 and later improved by Bazant whocalled his method the age-adjusted ef-fective modulus method. He introducedthe concept of an aging coefficientwhich expresses the aging effect oncreep of concrete loaded gradually.Neville et al B showed the aging coeffi-cient in the form of graphs, which werederived according to the procedure re-ported by Bazant fi by using the 1978CEB-FIPO creep function instead of theACI creep function.

The response of the concrete strain attime t is given by the age-adjusted ef-fective modulus formula as follows:

€.(t) =f̀ (t°[I + (t, to) I +f(t) –f(t) X

F

[ 1 + x (t, t„) i*p (t, t„) I + E, (t, t,,) (1)

where E. (t) is total concrete strain attime t, and Ec, (t, t o) is free shrinkagesince the time of prestressing t, (t, t)and x (t, t,) are creep coefficient andaging coefficient at time t for concreteloaded at age t, respectively, and fF (t)and fC (t o ) are concrete stresses at time tand t,, respectively.

From Eq. (1), the concrete stress attime t is written as:

f(t)= Er[EF(t)—E^a]— (1 —x) a f,, (to)1+x w

(2)

where E, = E F/(1 + X ip). The concretestress f, (t) is given by subtracting theconcrete stress f^. (t o ) multiplied by afactor (1 – x) col (1 + Xfp) from the con-crete stress,E; a [E F (t) — e}.

Assuming a linear distribution of thetotal strain E F (t) in the cracked section, itis necessary to differentiate two caseswhere the distribution of the concretestress at time t, fF (t), becomes linear and

SynopsisThis paper deals with the creep

analysis of cracked prestressed con-crete sections based on the relation-ship between concrete strain re-sponse and gradually varying stressgiven by Bazant's age-adjusted effec-tive modulus formula.

The computed results show thatthe long-term curvature of crackedconcrete sections is much largerthan that calculated for uncrackedsections. Therefore, it is apparent thatthe creep analysis for cracked sec-tions is important in order to esti-mate precisely the long-term deflec-tion of cracked prestressed con-crete members.

in addition to the detailed proce-dure, the author also proposes an ap-proximate calculation procedure. Theanalysis and proposed calculationprocedure will enable designers to ac-curately estimate long-term sectioncurvature of cracked prestressed sec-tions which could result in a precisecalculation of long-term deflection ofthe member.

bilinear in the compressed area of thecracked section.

(a)When the value of the second termof ft (t) exceeds that of the first term, theconcrete stress f,. (t) becomes negative,which means a tensile stress. Ignoringthe tensile stress in the concrete, alinear distribution off (t) is created asshown in Fig. I (a).

(b) When the value of the first term offF (t) exceeds that of the second termeverywhere in the compressed part ofthe section, a bilinear distribution off

(t) is obtained as shown in Fig. 1 (b). Thisoccurs because of the nonexistence of f,(to) below the tip of the crack caused attime to.

PCI JOURNAUJanuary- February 1987 105

! Thh

:)/E< I!

Strain at time t

PROPOSED ANALYTICALMETHOD

The proposed method is based onsatisfying equilibrium of forces andmoments and compatibility of strains.Typical equilibrium and strain compati-

hility equations for a general cross sec-tion with several layers of prestressedand nonprestressed steel and for thestrain diagrams shown in Fig, 2, aresummarized below, The symbols areexplained in the Appendix. Allmathematical derivations and other

h

fs,(ta/E,

0002.apA LJ

f0/ E0

`t fP{t^-'fPa}/EV

f2 (t/E,Cross-section Strain at time t,,

Fig. 2. Strain distribution diagram of cracked section.

106

supporting material are not published inthe PCI JOURNAL due to space limita-tions. However, they are kept on file atPCI Headquarters and are available onrequest.

1. At time t„

Equilibrium equations:

0 021^D 2 DJ (t)[,D te ^ + ^

F

FP2, LM,! (4)

where the following notations are intro-duced for the cracked section:

Pretensioning Post-tensioning"

D, Drn(xo) =E 5 A0 (x0) +E, A,+E. A I, Dtn (,xo) =E5 A11(x0 ) +Ea A$D,, D211 (x 0 ) = EeQcn(xe) + E,Q,+ EnQ P Dzu(xe) =EcQcn 0) + E,Q, (5)

D a Dare (xo) = E c i c a (x0 ) + E,I a + E,I, D311 (x o) =Eel ctt (xe ) + E1

FP 1 Y. A,n fPn 2^ Avn .fPn (tn) (6)FP2 S AP,, f,10!/r

1. A pn fpn(t o) yn

• Since there is no bond between tendons and concrete at transfer, the terms A, Q, and I,,

should be omitted andf_can be replaced by f,,, (t,)•

Equation for determining neutral axisdepth x,:

E co (t o ) + 4)(t o ) (yr —x0) =0 (7)

Concrete stress at level k:

fMM(to) =Ec[erg(t0) + `i'(to) yx] (8)

Stresses in nonprestressed reinforce-ment at level m and bonded pre-stressed steel (pretensioned memberonly) at level n:

fam( to) = E,[Eca(t,,) + 4)(t 1)i1 ] `9)

1 p,, `t0)=Jaa +E[€0(t0) + (t0)y,,1 (10)

Stress in tendon at level n for post-tensioned member:

.f". (ta) (11)

For post-tensioned members, sincethe bond between the tendons andducts is generally established just aftertransfer, at this state the constants Dsare the same as those for pretensionedmembers. The steel stresses f in ten-dons are given as:

f 5 —f 1 (ta) J N[ €rQ (to) + c11(t0)yn] (12)

and FP 1 and FP2 in Eq. (6) are also thesame as those for pretensioned rnem-hers.

By solving Eqs. (3) and (4), €r1 (to) and(P (ta) are obtained in terms of x n . Sub-stituting € ,, (to ) and da (t0 ) into Eq. (7), acubic equation in xa is found. Thus, xa,eco (to ) and (D (to) at time to are obtained.

2. At time tEquilibrium equations:

fD (x) a (t) 1 1 F1 0 (13)

1 15 n(t) Dsn(x ") t(t) + [FFP2} + ^FFR2j – [F2j = lMd ] (14)

where

D_

$

t

nn(

(r –Q 11 (x 1 )x,) =Re A cn(x t) +E,AR+EPAP

+Ee Q g +E Q (15) FR1 = 1A,8f,(t)D(16)ECCP Pi5(x) = Eclrn(xt) +E,I 1 +E,I FR2= IAx,ns.fnn(t)yn

PCI JOURNAL/January-February 1967 107

Ec [(1 – X) ,p {E A (to) A, n (x,) + c) (t,) Q, (x,) } + E,, A,ll (x =) I for x, > x,,Fl

IEc[(I— x)+P {Eca(t,)Acn(x1) +d)(t,.)Qcn(x()}+ €, A,,(x,)Iforx,<X,

(17)F2 E'I(1– x),p {€,.(t,)Q11(x,) + (to)I^n (xo)}+ E,R ,n(xe)Iforx, >xo

E, [( 1 — X) (P {€v(to) Qc,l (x,) + 1 (to ) I c rt (x,)} + EcaQc[1 (x e ) I fOrx, C xo

Equation for determining neutral axisdepthx,:

(a) Whenx,<xo:

€.g (t) + 1(t) (y1 — x ,) — Eca— (1 — x) tp {Eca (to) + 1I1 (to) (U1 — x t)} = 0

(18a)

(6) When x, > :xo A,, (x) _ (b – b) ht + b,n x – A,,

(t)+ (t)(y, x,)–,,=0 (18b) QCEJ(x)= (b–b)hf(y,–h1/2)+b,,x(y,–x /2) –A6,y„

A, = A ,1 + A,,Qr = An ye, – A,, y, = a1U

ze1 + A .Y1 t!"4

A, = A,.

Qa = – A ,.1ly

lJz1,. =AaP

(a) x > hr:

Concrete stress at level k:

(a) Decreasing neutral axis depth asshown in Fig. 1(a):

for(y,–xt)= yk-_y..fek (t) = E^ [ { e^9 (t) + 4(t)w k — Ecr}

–(1 – x) ^, ELF (to)1 (19a)

(b) Increasing neutral axis depth asshown in Fig. 1(b):

for (y, - x,) Y k < (y, – x,)fck (t ) = E,{E,.(t) + ((t ) y k — EC,)for (y1–x,.) = Yk -- Yi (19h). rk(t)=[{€,.(t) + 1(t)y--(1

Steelstresses in nonprestressed rein-forcement at level m and in bonded ten-don at level n:

f,,„(t) =E,Ie,,(t) + 1D(t)q} (20)

fP,(t)=f,.,, +E.,{€ ,(t) + 1(t)y,} +S .fY, (t1 (21)

Solving Eqs. (13) and (14), e,q (t) and1 (t) can be obtained in terms of x, andsubstituting them into Eq. (18), a fifthdegree equation in x, is found. Thus, x1,Eeg(t) and 1 (t) at time t can be com-puted.

For a cracked cross section shown inFig. 2, the constants in Eqs. (5) and (15)are given as:

I^„ (x) = (b – b) hr (h j112 + (y, –h,/2)'}+ bx {x 2/12 + (y 1 – x12) 2} –A,,ya,

(b)(y,–y,1)<x--hr:

An(x) = bx –A„QN „(x) = bx (y 1 –x12) –A.,y.1I(x) = bx {x2/12+(y1–x/2)'} —A ,1 y , a,

STRESS ANALYSISUNDER SERVICE LOADS

In the analysis of cracked sections, itis advantageous to bring the prestressedsection to a condition identical to that ofa conventionally reinforced sectionsubject to combined axial force andbending moment. The force required toachieve this state is called the fictitiousdecompression force by which thereinforcement, prestressed and nonpre-stressed, must be held externally inorder to eliminate stresses in the con-crete while no external loads are pres-ent. Tadros 10 presented a theoreticalprocedure for determining the fictitiousdecompression force.

The fictitious stresses in steel rein-forcement at the decompression stateare given as follows:

f,,,, =E„(e p+ cp*y,R ) (22)

f;, = .f,.,,+ s f,., (t) +E p (E a+ 4*y„) (23)

108

^g= 1 lxfrEllW + (1 — x) p .g(tn)+ Ece}1+xrp

(24)

1+xc

(a) When the neutral axis depth x under than x 1 , x can be obtained by solving theservice loads is larger than x 0, but less following equations:

D 1 (x) Dsn(x) AE CV + (I — x) cE rp (tq )1(I + xcp) 'PI

D2 11 (x) D3 « (x) ] [ A 4) + (1— x) c(t0 )/(l+ x.P) + FP2*

— 0 + (1 — x) w — FP 1 (25)

LM4 +M 1 ] 1 + xi° M,d—FP2 (26)

.^ Ecp + (1 ^- x)c^ Eca(t0) + f i1

+ (1-

1 + x) (P (tn) } ( i — x) = 0 `27)x x ca

where :] E,. Q and A cp are incremental service loads. The following equationsaxial strain and section cur vature due to may he written:

FP 1 * _ f + YEA, f,

FP2* = 4Aam f * ym('

+ 4Ayn J Payn

D i n(X) = E,A, n (x) f E,A,+EpAp (28)

Den (X ) =E.Qen(x ) +E8Q$+E,,Qn

D311(X ) =Ez lcn(x ) +EgIS+E,,I,,

By solving Eqs. (25) and (26), A Ell and cubic equation in x is obtained.

A4 can be expressed in terms of x and The stresses in the section can besubstituting A Ell and J (D into Eq. (27), a written as follows:

&1 a y, > (y 1 —x0), ffk = Ec(AEcg+ A4 jk)

( NI x0) }i!k } (y 1 - x), .fek — E. [( A€ ,+ AFL/k)

(1 — x) {Ec11(tp) + (to)Ykf J (29)+

1+xc

Inn = fsm + E (AEcp +A4)ym)

fP„ _ ,J-,,+E,(AEel+ A(by)

and the axial strain and section curvature are:

Ecv = E*g + AEco l

4 =d>*+A(P I(30;

(b) When the neutral axis depth x is the following equations, AE Ce , A4) and xsmaller than xo and x 1 , then by solving can be obtained:

PCI JOURNAUJanuary-February 1987 109

I

D i ri W D211(x) 1 [A,,,] + [ FP1* 1 [ 0 1 (31)

D2n(x ) D3ii (x) J A(1) LFP2* ] [ M,+M, (32)

A C' + A(P(Y l –x)=0 (33)

Finally, the stresses in the concrete section of the member are given as foI-and steel, and the deformations in the lows:

fnk =Ee( J E Cp+ A4)yk),farce = .fam +Es(AECgt 0ym).f f +E,(A +.](PY,1 (34)Eca = fc0 + QEcaID=(*+

DESIGN EXAMPLE

The following example illustrates theproposed calculation method and willshow readers how to quickly apply thebasic computation steps. A prestressedconcrete single T section is selected.The dimensions, material propertiesand moments due to loading are shownin Fig. 3.

The section is analyzed manuallyunder the following conditions:

Loading Case 1: Dead load plus pre-stressing force at transfer.

Loading Case 2: Application of liveload after establishing bond be-tween tendon and concrete.

Loading Case 3: Removal of live Ioad.Loading Case 4: Dead load plus com-

bined effects of creep and shrinkageof concrete and relaxation of pre-stressing steel.

Loading Case 5: Application of liveload after all losses have occurred.

Loading Case 1First, the section is assumed to be un-

cracked just after transfer. From thewell-known analysis of uncracked pre-stressed sections, the following stressesin the concrete and steel, and deforma-tion in the section are obtained:

J35x1030.06700 + 0.1701x0.326)x10''= 4.29 N/mm 2 (622 psi)

t',g= 35x103x(0.06700 – 0.1701x0.674)x1O-= –1.67 N/mm 2 ( -242 psi)

fs1=200xlO3x(0.06700 +0.1701x0.276)x10.3

–22.79 N/mm 2 (3.30 ksi)f,2=200x1O 3x(0,06700

0.1701x0.624)x10-3=-7.82 N/mm 2 ( - 1.13 ksi)

ff-fp (to)= –1140 Nlmm z (-165.3 ksi)

Since the concrete stress f« at the ex-treme tension fiber is in tension, andless than the tensile strength of con-crete, the assumption of an uncrackedsection used here is correct.

Loading Case 2Since full service loads are most likely

applied after establishing the bond be-tween the tendon and concrete, it is as-sumed that any time-dependent defor-mations have not occurred before thisloading. The section can be assumed tobe cracked under the service loads.

Assuming the neutral axis of themember is in the web, the section prop-erties of the effective concrete zone arewritten as:

C)

0Cz

m`

CCD

CD

Cw

CD

V

180

o^.

LZ

SECTION PROPERTIES

NET CONCRETE SECTION : A,=0.52947 m', I,=0.049158 m°

NON PRESTRESSED REINFORCEMENT: A,=0.004013 m',

Q,=-0.001986X 0.276-0.002027X 0624 = -0.0007167 I rn3

1, = 0.001986x 0.276'+0.002027X 0.624' = 0.00094055 m"

PRESTRESSED STEEL: AP =0.0011 I5 rn',

QP 0.001115X( 0.574)=-0.00064001 m3

I= 0.00!! 15x0574 2 =0.00036736 m4

DIMENSIONS cm

BENDING MOMENTS:

DEAD LOAD MOMENT: 1044.5 kN-nm

LIVE LOAD MOMENT : 337.5 kN-m

MATERIAL PROPERTIES:

CONCRETE : E,=35 kN/mm', TENSILE STRENGTH=3.3 N/mrn', 2.6, e1 , = 25X 10 "S

NON PRESTRESSED STEEL: E,=200 kN/mm'

PRESTRESSED STEEL : E P =200kN/mut', TENSILE STRESS AT TRANSFER =1140N/mm=,

LOSS DUE TO STEEL RELAXATION=5 PERCENT OF INITIAL STRESS

Fig. 3. Example: Dimensions, moments, section and material properties of prestressed concrete T beam.(Conversion factors: 1 cm = 0.3937 in., 1 N/mm 2 = 145 psi, I kNr'mm2 = 145 ksi.)

Arn (x) = 1.48x0.12 + 0.74x0.05+ 0.32x - 0.001986

0.2126 + 0.32x m2QCn (z) = 0.1776x(0.326 - 0.006)

+ 0.037x(0.206 -0.05/3) +0.32x(0.326 - x/2) -0.001986x0.276

= 0.053699 + 0.10432x-0.16x 2 m3

Ic „ (x) = 0.1776x{0.12 2!12 +(0.326 - 0.06) 2 } +0.037x{0.052118 +(0.206 - 0.0513) 2} +0.32x{x 2!12 + (0.326- xl2) 2J -0.001986x0.2762

= 0.013959 + 0.034008x- 0.10432X 2 +0.10667x 3 m 1

From Eq. (12), the initial steel stressf in the bonded tendon can he calcu-lated by referring to the section defor-mations of Loading Case 1:

_ - 1140 - 200x103x(O.067000.1701x0.574)xl(r3

_ - 1133.9 N/mm 2 (36)Substituting Eq. (35) into Eq. (5), the

constants D,,,, D 211 and D. , for thecracked section with bonded tendonare:

Substituting Eq. (36) into Eq. (6),FPIand FP2 for the bonded tendon are:

FP I = 1115x(-1133.9) = -1264.3 kNFP2 = ---1264.3x(-0.574)

=725.7kN•m

By replacing Md in Eq. (4) with (Ma +Me), the equilibrium equations for ser-vice loads are written as follows:

Diu(x) = E, A,. 11 (x) +E e A,+Ep A„(35) _ (8467 + 11200x)x103

D2[1 (x) =E^Q (x ) +E,Q, +E9Q_ (1608 + 3651x - 5600x 2)xI0 3 (3

Ds,, (x) = E eI,„ (x) + E,IF + E„I,_ (750.1 + 1190x - 3651x2

+ 3733x3)x10-"

D

I I Ecu

j

+ -1264.3 _ 0 (38)D2 Dan ¢' [ 725.7 - 1044.5 + 337..5 ]

Solving Eq. (38), e^.,and 4) are obtained as follows:

e,p - (12643D3i , - 0.65629D 2(1 )x1031(3,,, Dan -Ds ij l(39)

4) - (0.65629D 11 --I.2643 D2,t)x1031(Dtn Dail - D 211) JSubstituting Eq. (39) into Eq. (7), an

equation for determining the neutralaxis depth x can be written as:

1.2643 Da „ - 0.6563D2 + 0.326 -x = 0

0.6563D,,, - 1.2643D2R(40)

Substituting Eq. (37) into Eq. (40), acubic equation inx can he obtained as:

-107.0 - 891.6x - 940.7x 2 + 4720x'3524 + 2734x + 7080x2

+ 0.326 - x = 0

and the neutral axis depthx is calculatedto be 0.258 m.

Substitutingx = 0.258 m into Eqs. (37)and (39), E^,O and 4) are obtained as fol-lows:

= -0.06100x10-3, 4) = 0.898140-31mTherefore, the stresses in the section

in Loading Case 2 are:

= 8.12 N/mm 2 (1177 psi)fs , = 37.40 N/mm 2 (5.42 ksi)fRZ = -124.26 Nlmm 2 ( - 18.02 ksi)f,, - -1249.2 N/mm 2 (--181.1 ksi)

112

Loading Case 3Since cracking has been caused by the

application of live load as shown in theprevious calculation (Loading Case 2),and the zero tension requirement is notsatisfied under the dead load, the cracksremain open under the dead load afterremoval of live load. Hence, a crackedsection should be assumed in the cal-culation of the stresses under the deadload. The calculation procedure issimilar to that for Loading Case 2, butthe right hand term of Eq. (38) should bewritten in terms of the dead load mo-ment as follows:

L 1044.5 j

Solving a cubic equation, x a is ob-tained as 0.572 m and € 2 (t„) and 4) (to)

are:

E(t0) = 0.05631x10•."aD (t o) = 0.2286x1& /m (41)

Therefore, the stresses in the cracked

section under the dead load before thetime-dependent effects occur are found:

= 4.58 N/mm 2 (664 psi)fat = 23.88 N/mm 2 (3.46 psi)fg ^ = 17.26 Nimm I ( - 2.50 psi)f = - f 148.8 N/mm2 ( - 166,6 ksi)

Loading Case 4For this loading case, the creep

analysis of cracked sections is per-formed. By assuming the aging coeffi-cient xis equal to 0.80, 8 the age-adjustedeffective modulus E. can be obtainedfrom Eq. (2) as follows:

= 351(1 + 0.80x2.6)= 11.36 kNlmm 2 (1647 ksi)

The steel stress loss 8 f,,(t) due to therelaxation of prestressing steel is:

S f(t) = 0.05x1140 = 57 N/mm2

Substituting Eq. (35) into Eq. (15), theconstants for the cracked section D,,ll21 and D2R are obtained as follows:

D I ^ ^ Ec ArR (x t ) + E, A,+ E„ A i, = (3441 + 3636xt)x10'

Dn = E.Qcn (x) + EAQ2+ E np„ = (338.9 + 1185x,- 1$1$xt)x10s (42)

D 311 = Ecle, (x,) + E BIB i- E„I 2, = (420.2 + 386.4x, - 1158x' + 1212x

.

)x 10d

From Eq. (16), FRI andFR2 are:

FRI =A,8 f„(t) = 1115x57 = 63.56 kNFR2=A p 3 fp (t)y p =63. 6x(-0.574)

= -36.48 kN •m

By assuming a decreasing neutral axis

depth with time (x, < xe), F I and F2 areobtained from Eq. (17). BY substituting

€ CQ (t o) = 0.05631x10-3 and (D (t o ) =0.2286x1(0 3lm given by Eq. (41) into Eq.(17), the following expressions are ob-tained:

F1 = 11.36x10 sx[(1 - 0.80)x2.6x{0.05631x10•2x(0.2126 + 0.32x,)+ 0.2286x10- 3x(0.053699 + 0,10432x - 0.16x )}+ 0.25x10-x(0.2126 + 0.32x)]

= (0.7470 + 1.1548x: - 0.2144x2)x103

F2 = 11.36x101h [(1 - 0.80)x2.6x{0.05631x10-3x(0.053699 + 0.10432x- 0.16x)+ 0.2286xi00x(0.013959 + 0.034008x, - 0.10432x?+ 0.10667x?)}+ 0.25x10-ax(0.053699 + 0.10432x, - 0,16X2) ]

(0.1892 + 0.3764x 2 - 0.6473x+ 0.1429x; )x103

From Eqs. (13) and (14), the equilibrium equations are rewritten as:

PCI JOURNAL/January-February 1987 1 13

D_ ^ri D 2 l egg{t) 1264.3 _ 63.56 F1 DD 3 (t) i 725.7 -36.48

LF2

J[M

d J

1.948 + 1.1548x, - 0.21616xi0.5445 + 0.3770x, - 0.6487x^ + 0.1441x; {43)

Solving Eq. (43), e,„ (t) and cj) (t) are:

E^o (t) = [(1.948 + 1.1548x, - 0.21616x;') Ds1f - 1- (0.5445 + 0.3770x, - 0,6487x 2 + 0.1441x 3) DE , i/A(44)(P(t) = [(0.5445 + 0.3770x,- 0.6487x,2 + 0.1441x =)D1,t

- (1.948 + 1.1548x,- 0.21616xi)D E1 ]/A

where 4 = D 1 nD - D Qa.

Substituting Eqs. (42) and (44) intoEq. (18a), an equation for determiningthe neutral axis depth can be written as:

ErQ (t) - (I - X) € (t0 ) -- "sID (t) - (I - x) V O (ta ) + ( y i - x)

_ 262.2 - 109.9x, - 3950x? + 1147x^ - 0.1007x

1055 + 338.5x, + 1721x - 0.02470x3 + 0.038012; + 0,326 -x= 0 (45)

The above fifth degree equation in xrleads to:

x t = 0.478 m [<x, = 0.572 m (ok) I

Substituting x, = 0.478 m into Eqs.(42) and (44), E,,, (t) and cD (t) are:

E, W = 0.3921x10-',4) (t) = 0.8573x10•s/m (46)

and the stresses in the section are ob-tained by Eqs. (19a), (20) and (21) asfollows:

= 11,36x10sx{(0.3921+0.8573x0.326 - 0.25)x10-3- (I - 0.80)x2.6x(0.05631+ 0.2286x0.326)xIO-3 J

= 4.02 NImm 2 (583 psi)

f8 z = 125.77NImm2(18.24ksi)f2 = -28.55 N/mm 2 (-4.14 ksi)fp = 1097.2 N/mrn s (--I59.1 ksi)

Loading Case 5From Eq. (24), the axial strain Q and

and the section curvature 4)* at the de-compression state are obtained by refer-ring to Eqs. (41) and (46):

CCU+ 0.80x2.6 {U.80x2.6x0.39211

+ (1- 0.80)x2 .6x0.05631 + 0.25)x10.3= 0.3555x10-3

V = 1 {0.80x2.6x0,85731 + 0.80x2.6

+ (1-0.80)x2.6x022861x10.30,6176x10 3

The fictitious stresses in the rein-forcing steel are obtained by Eqs. (22)and (23) as follows:

f i = 200x10sx(0.3555+ 0.6176x0.276)x10-s

= 105.2 N/mm 2 (15.25 ksi)f,2 = 200x10c(0.3555

- 0.6176xO.624)x1O-_ - 5.97 N/mm 2 ( -0.866 ksi)

f = -1133.9+57+200x103x(0.3555 - 0.6176x0.574)x1U-3-1076.7 N/mm 2 (-156.12 ksi)

114

23.88 N/mm'

1

f ^r,

0.2286x10 /iii r

L L

125.77 N /mm' 138.72 N/mm'

IIII

/ E ^/

0.392/ 0.2621x10'

C,

0.8573x10Jm //

/ /

//A1

1, 1 .5628x 103/m

I.'

—17.26 N/mm' - 28.55 N/mm'

142.57 N/unm2(3) (4)

(5) Loading Case

4.29 N /mm2 8.12 Nlmm' 4.58 N /mm' 4.02 N/mm= 7.52 N/mm' Concrete Stresses

C)20C7]z

C-2)CN

N

Cv

J

N I I00

67 N /mm2

Stresses in non prestressed steel & strain distributions in section

22.79 N/mm' 37.40 N/mm'

—1 E --7/

II1 /

1 ++ j

1 0.06700x 10 -3 —0.06100x 10'E 1_ ^^

+ 0.1701x10'/m //r 1^ /^ 0.8982x 10 1/m

/

/r //

—7,82 N/mm2 —124.26 N /m,n=

(I) (2)

I, `I Fig. 4. Stresses in concrete and nonprestressed steel. (Creep analysis for cracked section.)(Conversion factor: 1 N /mm 2 = 145 psi.)

From Eqs. (31) and (32), the equilib-rium equations are given below, pro-vided thatx is less than x0.

0 = D ii c -A E ,+ D$„p$+(1986x105.2- 2027x5.97 - 1115x1076.7)x10-3

(1044.5 + 337.5)= D211 A ECn + D3 1, 4 F -} {208.9x0.276

-12.lOx(-0.624) -1200.5x(-0.574)}

(47)

Solving Eq. (47), A € 1 iD areobtained and substituting them into Eq.(33), the following equation for deter-mining the neutral axis depth is ob-tained:

1.0037D,,, - 0.6277D2R0.6277D,,, - 1.0037D2 + (0.326 - x)-256.5 - 1097x - 149.4x 2 + 3747x3

3701 + 3366x + 5621x2+ 0.326 - x = 0 (48)

Solving Eq. (48), x is 0.227 m (< xaand, therefore, the neutral axis is belowthe bottom of the top flange) andand a c are:

0 c,, = -0.0934x10-3A 4) = 0.9452x 10-3

Finally, the stresses and deformationsin the section under the service loadsare obtained by Eq. (34) as follows:

L, = 7.52 N/mm 2 (1089 psi)f„ = 138.7 Nlmm 2 (20.11 ksi)f,2 _ -142.6 N/mm 2 (-20.67 ksi)

= 1203.9 N/mm i (-174.56 ksi)c, = 0.2621 x 10-14 = 1.5628x10.3

The distribution of the stresses in thesection under various loading condi-tions is shown in Fig. 4.

In order to reduce the amount of in-terpolation in solving the equations fordetermining the neutral axis depth, acomputer program has been developed

for the creep analysis of cracked pre-stressed sections, based on the equa-tions presented in this paper. The com-puter program is available directlythrough the author.

DISCUSSION1. The effect of free shrinkage on the

depth of the neutral axis was investi-gated on two T sections (Fig. 3), namely,without any compression reinforcement(A s1) and with 1986 mm 2 of A,,. Therange of shrinkage considered variesfrom zero to 25x10-5.

The results show that the depth of theneutral axis always increases with timefor the free shrinkage range smaller than13x10-5 (Fig. 5). These results suggestthat the neutral axis depth increaseswith time are not usually expected undernormal environmental conditions. Thisis because the free shrinkage of concreteis usually much larger than this range.

Another result is that the use of com-pression reinforcement will reduce theaxial strain, the section curvature, in ad-dition to the concrete stress at the ex-treme compression fiber after the occur-rence of all time-dependent effects asshown in Table 1.

2. When the effect of free shrinkagelarger than 13x10'' is taken into accountin the creep analysis of cracked sections,the neutral axis will not coincide withthe zero strain axis as shown in Fig. 6a.Also, the depth of the zero strain axis in-creases with time while the depth of theneutral axis decreases with time. Twoaxes coincide only when the free shrin-kage is zero (Fig. 6b).

3. Since the stress-strain relation ofsteel remains elastic regardless of anytime-dependent deformations of con-crete, the zero strain axis of the sectioncoincides with the zero stress axis of thesteel section. Therefore, the height ofthe crack in the cracked section can bedetermined by the zero stress axis of thesteel section.

116

Table 1. Effect of compression reinforcement on x, f, eC,

and ch.

A. i (mm 2) 1986 0 Remarks

At time t x, (cm) 47.8 50.6 Creep analysisf,, (N/mm') 4.02 4.91 of crackedE eD x 10 3 0.3921 0,4350 sectioncp x 10"/m 0.8573 0.9777

Table 2. € t1) in cracked section.

Creep analysis

Service loads arc a lpph cd

Before all losses After all losses

Crackedsection

Uucrackedsection

Dead load e,,,x 104x 10 :;

0.39210.8573/m

0.39960.8192/ n

Service loads E ^p x 10 1its x i0l

0.2612l .5628/Ti

0.26011.54831m

70

60

50

40

30

20

10

00 5 10 15 20 25

e5 x 105

Fig. 5. Variations of neutral axis depth with free shrinkage E^5.

(Conversion factor: 1 cm = 0.3937 in.)

PCl JOURNALJJanuary-February 1987 117

70Asp =O : x 6 =56.9 cm

60

50A3j,= 1986 mm' : xo =57.2 cm

40 180 cm

30 All

0° Asp : Variable

20 A S2 A,2— 2027 mm2

=2.610

0

Strain distributions Concrete stresses4.58 N(mm'

0.6716x 10 3 4.02 NJmm'

' j 125.77 NJmm'

Eu u^ op(a)

fa =25x10 ' ^

fx2 =-28.55 NJmm=

I^ ^1-0.1857x 10'

4.55 N(mm2

0.4196x 10 4.00tIImi2fi' =7735 N/mm2

E uri

r y r,L^ c-1

t r-= p o vEc^

^n `

La

1.50 NJmm'

A2 =-4 1.1 7 Nfmmi'

—0.2389x 10 S

Fig. 6. Effect of free shrinkage on stress and strain in section given in Fig. 4.{Conversion factors: 1 cm = 0.3937 in., 1 N/mm 2 = 145 psi.)

4. The creep analysis may be per-formed on the basis of an uncrackedsection if no service loads are appliedbefore occurrence of all time-dependentdeformations. However, after occur-rence of all Iosses and application of ser-vice loads, cracking will result and anystress calculation afterwards should beperformed using a cracked section, evenwhen the live load is removed.

Except for the deformations of sec-tion, it can be concluded from Figs. 4and 7 that different creep analyses giverelatively close results in the strain dis-tributions for a cracked section underservice and dead Ioads after all time-dependent deformations have occurred.The axial strain and the curvature in thecracked section are given in Table 2.

Therefore, for a high value of dead

118

Concrete stresses

4.29 N1mm' 3.78 N /mm' 7.93 Nlmm' 4.49 Nlmm'

UIEO U

CJE 0-Ii

UO 0

t`

LtI .67 N/mm —2.47 N/mm'

Stresses in lion prestressed steel & strain distributions in section

22.79 N/mm' 122.96 N/mm =137.52 N/mm° 125.16 N/nm'

/ /

E j 3 0 4385 x 10' j 0.2601 x I0 0.3996x 10 //o ^; 0.06700x 10 ce a (;^ G^

i l / / /n / / / 0,8192xiO3fn

0.1701 x I O'/mq 7/ // °O

0,6385x10Jm /'u/

5483 x 10 3/iu

j /

7.82 Nlmm' 8.03 N/isim' 141.17 N/mm =-22.29 N/mm'

At transfer After all losses Under service loads Removal of L.L.

Fig. 7. Stresses in concrete and nonprestressed steel and strain distributions unaervarious loading conditions. (Creep analysis is performed on uncracked section.)(Conversion tactors: 1 cm = 0.3937 in., 1 N/mm 2 – 145 psi.)

load degree of prestress tc (=M d /M4),such as 0.8 in this example, the long-termdeflections under dead load may be cal-culated using the results of creepanalysis based on uncracked sectionsregardless of the existence of cracksduring time-dependent deformations.

5. Tn order to compare the long-termcurvatures and stresses in nonpre-stressed steel calculated by two differ-

ent creep analyses using uncracked andcracked sections, a rectangular sectionshown in Fig. 8 with various combina-tions of prestressed and nonprestressedsteel is analyzed. All the cross sectionshave the same ultimate strength corn-puted by ACI 318-83. Variations of long-term section curvature, stress in nonpre-stressed steel and fictitious decompres-sion force with K are shown in Fig. 9.

PCI JOURNALIJanuary-February 1987 119

roO

BENDING MOMENTS;

DEAD LOAD MOMENT: 975 kNm

LIVE LOAD MOMENT : 625 kNm

REQUIRED FLEXURAL STRENGTH:

M„ =1.4 M d + 1.7 M 1 = 2427.5 kNm

MATERIAL PROPERTIES:

CONCRETE: f40 N/mm E,=35 kN/mm? TENSILE STRENGTH=3.3 N/mm2

CREEP COEFFICIENT, a = 26, FREE SHRINKAGE, E,,= 25x10-`NONPRESTRESSEI) STEEL: fy =400 N/mrn 2, E s =200 kN/mm2

PRESTRESSED STEEL: f = 1900 N/mrn 2 (STRESS-RELIEVED STRAND)

E. P = 200 kN/mm 2, `I ENSILF, STRESS AT TRANSFER= 1 140 N/mm2

LOSS DUE "10 STEEL RELAXATION =5 PERCENT OF INITIAL STRESSDIMENSIONS cm

AREAS OF REINFORCING STEEL

Ap (mm 2 ) 2400 2057 1714 1371 1029 686A. (mm = ) 169 1288 2430 3592 4771 5971A P +As (mm') 2569 3345 7 4144 4963 5800 6657

Fig. 8. Example: Dimensions, moments, section and material properties of rectangular beam. (Conversion factors: 1 cm = 0.3937 in.,1 N/mm 2 = 145 psi, 1 kN/mm 2 = 145 ksi, 1 kN - m = 0.7375 kip-It.

C)

p 2.4CMZ 2.2nrC_

z.nC 5)R

1.8mQC

1.6

1.4

1.2

1.0

Cracked section analysis based on decompression force obtained fromcreep analysis of untracked section

2400

^ 1

1 ^\

2200

Creep analysis2000

1900

4 1600

1400

1200I-

- O 100C0)

d

0.8 E 801)

\ o

06-• 600

Creep analysis of cracked section0.4- 400

0,2 200

-140[!

-1200

of untracked section-1000

-800

E -6011

d 41111 ^

/200 -

l v ^

O `^

f

k200

0o y`

vl400 6_

fm

800

I l 11x1110

00 0.2 0,4 0.6 0.8 1.0 1.2 U - 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2

K = (M dec(M d)

k ' (MdPJMd) K =(Mdec/Md)

N

Fig. 9. Variations of long-term curvature, stress in nonprestressed steel and decompression force with dead load degree of prestress K.

(Conversion factor: 1 N?mm2 = 145 psi.)

It can be observed from Fig. 9 that fora low value of K the result of an un-cracked section is always much smallerthan that of a cracked section. However,this difference becomes considerablysmall when the section which was as-sumed to he uncracked is analyzed bythe elastic theory using a fictitious de-compression force derived from a creepanalysis for an uncracked section and as-suming that the section is now crackedunder dead load.

For practical use in routine designwork, it is suggested that even for lowervalues of K, the creep analysis and com-putation of fictitious decompressionforce may be carried out on the hasis ofuntracked sections. Then the long-ternsection curvature and steel stresses in acracked section under dead load can becomputed by elastic theory and apply-ing dead load moment to the section atthe decompression state.

CONCLUDING REMARKSIn this paper, the creep analysis for

cracked sections of partially prestressedconcrete members, which are designedwithout following the zero tension re-quirement in concrete under dead load,has been developed based on Bazant'sage-adjusted effective modulus method.The author has also introduced asimplified design method for routinework. The calculation steps of this

method are as follows:1. Creep analysis is performed based

on the assumption that the section is notcracked, and by using one of the well-known methods for estimating time-dependent effects in uncracked pre-stressed concrete sections.

2. Section deformations at the decom-pression state and the fictitious decom-pression force are calculated on thebasis of the creep analysis for uncrackedsections.

3. Section curvature and steel stress incracked sections under dead load arecomputed using elastic theory and thefictitious force.

This simple method results in a goodapproximation of the exact method de-veloped by the author according to thenumerical example carried out in thispaper.

Although a description of the long-term deflection analysis is not includedin this paper, the mean curvature' 2 canbe computed by taking into account thecontribution of concrete in tension fromthe two calculated curvatures of thesection in uncracked and cracked states.The long-term deflection can be ob-tained by double integration of themean curvature along the member,while taking into account the supportconditions.

The proposed approximate procedurecan be used to predict the long-term de-flections of cracked prestressed concretemembers.

122

REFERENCES

1. Dilger, W. H., "Creep Analysis of Pre-stressed Concrete Structures UsingCreep-Transformed Section Properties,"PCI JOURNAL, V. 27, No. 1, January-Fchruary 1982, pp. 98-118.

2. Madsen, K., "Matrix Formulation forPractical Solution of Concrete CreepProblems," Dialog 1-79, DenmarksIngenirakademi.

3. Frey, J., and Trost, H., "Zur Berechnungvon teilweise Vorgespannten BetonTragwerken in Gebrauchzustand,"Beton-und Stahlbetonbau, No. 11, 1983,No. 12, 1983.

4. Bazant, Z. P., and Oh, B. H., "Deforma-tion of Progressively Cracking Rein-forced Concrete Beams," ACI Journal,V. 81, No. 3, May-June, 1984, pp. 268-278.

5. Debernardi, P. C., "Analyse Syst2ma-tique de la Curbure Instantantse et aLong Term des Pieces en Baton Arme,"Annales de l'Institut Technique duBatiment et des Travaux Publics, No.419, November 1983.

6. Bazant, Z. P., "Prediction of ConcreteCreep Effects Using Age-Adjusted Ef-

fective Modulus Method," ACI Journal,V. 69, No. 4, April 1972, pp. 212-217.

7. Trost, H., "Auswirkungen des Superpo-sitionspringzips auf Kriech-undRelaxations-probleme bei Beton andSpann-beton," Beton -und Stahlbeton-bau, V. 62, No. 10, 1967; No. 11, 1967.

8. Neville, A. M., Dilger, W. H., andBrooks, J. J., Creep of Plain and Struc-tural Concrete, Longman, London, En-gland, 1982.

9. CEB/FIP Model Code for ConcreteStructures, Paris, France, 1978.

10. Tadros, M. K., "Expedient Service LoadAnalysis of Cracked Prestressed Con-crete Sections," PCI JOURNAL, V. 27,No. 6, November-December 1982, pp.86-111.

H. Bachmann, H., "Design of Partially Pre-stressed Concrete Structures Based onSwiss Experiences," PCI JOURNAL, V.29, No. 4, July-August, 1984, pp. 84-105.

12. Favre, R. Koprna, M., and Putallaz, J. C.,"Deformation of Concrete Stuctures,Theoretical Basis for the Calculation,"IABSE SURVEYS S-16/81, February1981.

NOTE: Supplemental information (including mathematicalderivations) on this article is available from PCI Headquarters.

PCI JOURNALIJanuary-February 1987 123

APPENDIX - NOTATIONA f = net area of untracked concrete (negative for tensile stress)

section f2,, (t„) = stress in prestressed rein-Ax) = net area of concrete hounded forcement at level n im-

by neutral axis of depth x and mediately alter transferextreme compression fiber f = stress in nonprestressed

A„ = area of prestressed reinforce- reinforcement at level mment (negative for tensile stress)

A, = area of nonprestressed rein- f(t) = stress in nonprestressedforcement reinforcement at time t (nega-

D, ic (x), D211 (x), D. 11 (x) = various elastic tive for tensile stress)stiffnesses of ft,, = fictitious stress in prestressedcracked sec- reinforcement at level n whention. Argu- stresses in concrete are elim.i-ment (x) may natedbe omitted. fg*,,, = fictitious stress in nonpre-See Eq. (5). stressed reinforcement at_ _ _

Din (x), DaR (x), D 311 (x) = various stiff- level m when stresses in con-nesses of creteareeliminatedcracked sec- I,n (x) = moment of inertia of concretetion when section bounded by neutraltime-depen- axis of depth x and extremedent effects compression fiber about cen-are consid- troidal axis of net concretee r e d . See sectionEq. (I5). I. = moment of inertia of pre-

Ee = modulus of elasticity of con- stressed reinforcement aboutCrete centroidal axis of net concrete_

E, = age-adjusted effective mod- sectionulus of concrete I8 = moment of inertia of nonpre-

E, = modulus of elasticity of pre- stressed reinforcement aboutstressed steel centroidal axis of net concrete

E 8 = modulus of elasticity of non- sectionprestressed steel Ma = moment due to dead load

fck = concrete stress at level k (po- M&, = decompression moment, pro-sitive fbr compressive stress) dotes zero stress at extreme

f(t) = concrete compressive stress at tension fibertime t M, = moment due to live load

fD„ = stress in prestressed rein- Q^ R (x)= first static moment of concreteforcement at level n (negative section bounded by neutralfor tensile stress) axis of depth x and extreme

f, (t) = stress in prestressed rein- compression fiber about cen-forcement at time t (negative troidal axis of net concretefor tensile stress) section

Fs f(t) = loss of tensile stress in pre- Q p = first static moment of pre-stressing steel at time t due to stressed reinforcement aboutrelaxation of steel centroidal axis of net concrete

frmo = stress in prestressed rein- sectionforcement at level n im- Q = first static moment of nonpre-mediately before transfer stressed reinforcement about

124

ttoxa

xt

Yk

Y1

centroidal axis of net concretesection

= time elapsed after transfer= time at transfer

depth of neutral axis of con-crete stress distribution attime to

= depth of neutral axis of con-crete stress distribution attime t

= distance from centroidal axisof net concrete section to afiber k (positive in upward di-rection)

= distance from centroidaI axisof net concrete section to ex-treme compression fiber (po-sitive)

Er (t) = concrete strain at time t (po-sitive sign indicates short-ening)

e,„(t) = axial strain at centroid of netconcrete section at time t

e^Q = axial strain when concretestresses are eliminated

D e^, = incremental axial strain due toservice loads from referencesection

E r , = free shrinkage of concretesince time of prestressing to

Sp = concrete creep coefficientsince time of prestressingt,

ci>(t0 ) = section curvature at time t,,I>W = section curvature at time tV = section curvature when con-

crete stresses are eliminatedA = incremental section curvature

due to service loads from thereference section

x = aging coefficient at time t forconcrete loaded at age t,

►c = dead load degree of prestress

NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by September 1, 1987.

PCI J0URNAUJanuary-February 1987 125

Unusually long precast prestressed bulb T beamswere used to build a three-span bridge to carrybicycle and pedestrian traffic over a busy interstatefreeway and railroad mainline.

1-205 Bikeway OvercrossingBanfield Freeway

Dhe problem confronted by the Bridge Design Section of the Oregonepartment of Transportation was to design a three-span bikeway structure

to be visually compatible with an existing parallel freeway bridge adjacent tothe bikeway while at the same time provide minimum disruption to freeway andrail traffic during construction.

The solution to the problem was to design a structure using six 6 ft 2 in. (1.87m) deep precast prestressed concrete bulb T beams ranging in length from 110to 166 ft (33.5 to 50.6 m) with the longest beams weighing 79 tons (72 t) each.

The resulting structure is a 433 ft (132 m) long three-span bridge with a l0 ft 9in. (3.28 m) wide deck having a total area of 4654 sq ft (433 m 2 ). The detailedplan, elevation and cross section are shown on p. 128.

126

The three long spans, column orientation and grade line were designed to beaesthetically compatible with the existing parallel freeway bridge adjacent tothe bikeway. In addition, the end abutments of the bikeway structure aresupported on the wingwall footings of the existing freeway bridge for economyof design as well as visual compatibility with this bridge.

The bikeway structure, which is located in East Portland at the SouthBanfield Interchange with 1-205, carries both bicycle and pedestrian trafficover a railroad mainline and six lanes of interstate freeway. The structureparallel and adjacent to the bikeway is a three-span post-tensioned concretebox girder bridge carrying seven lanes of freeway traffic.

The pile footing supported wing walls of the existing freeway bridge weremodified to serve as end abutments for the bikeway structure. This not onlyreduced construction costs but gives the bridge ends of the bikeway a pleasingcontinuous appearance in line with the ends of the existing structure. A singlecolumn at each interior bent was located to align with the four columns at eachinterior bent of the existing freeway structure.

Prestressed bulb T beams were chosen for the superstructure because theycould be precast, trucked to the job site and erected quickly over a busyfreeway and railroad line without the need of falsework support. An unusualfeature of this structure is the extremely long [up to 166 ft (50.6 m) ] beams.Each span contains two bulb T beams with 5 ft 4¼ in. (1.63 m) wide top flanges.The wide top flanges allowed the entire bridge deck to be poured directly on thebulb T beams without bottom deck forms.

Protective fencing completely encloses the deck surface providing safety forthe bikeway traffic as well as for freeway and rail traffic below.

The completed structure provides an attractive, safe crossing for bicycles andpedestrians at an intersection of freeways and a railroad line.

The total cost of the bikeway structure was $266.800 with the precastprestressed concrete portion of the work amounting to $112.000. Based on theabove figures, the unit cost was $57.33 per sq ft.

PCI JOURNAL/January-February 1987 127

IDECK SECTION

END BENT DETAIL

Z

yv RENT rBIKEWAYN

y ,ter EXIST. NT. I

-B V4"C-C END BENTS - PRECAST PRESTRESSED BULB -T BEAMS

III- SPAN 2 } 164'-5y2"ISPAN 3 )BENT 2

^11` BENT 3

I, P. R.R. BENT 4

RET WALLSF- ITYPI

EXISTBT- 4

I ST. STRUCTURE^, \ DE R

S 5151 BT p EXIST BT.3$MEDIAN _ e^

PLAN

CHAIN LINK ENCLOSURE

EXP UFRR PIN FREE FRF= PIN ED€ANTRACKS' FREEWAY FREEWAY

a ,1 I 11 _

^JLSPAN I ---^ SPAN 2

BENT I BENT2 BENT 3ELEVATION

%P ^I

un

SPAN 3

BENT

The bikeway structure was completed in January 1986 after a 9 monthconstriction period. During the past year it has become a popular facility forboth cyclists and pedestrians.

CreditsOwner: Oregon Department of Transportation — Highway Division, Salem.

Oregon.Architect/Engineer: Bridge Design Section, Oregon Department of

Transportation — Highway Division, Salem. Oregon.Contractor: Auburn Construction Company, Wilsonville, Oregon.Prestressed Concrete Manufacturer: Concrete Technology Corporation,

Tacoma, Washington.

PCI JOURNAL/January-February 1987 129