Deformation Controlled Nonlinear Analysis of Prestressed Concrete Continuous Beams

T. I. Campbell Professor Department of Civil Engineering Queen's University Kingston, Ontario, Canada

Venkatesh Kumar R. Kodur Graduate Student Department of Civil Engineering Queen's University Kingston, Ontario, Canada

42

A finite element model, based on a curvature increment tech­nique, is applied to the nonlinear analysis of prestressed con­crete continuous beams. The computer model is capable of predicting the response of such beams over the entire loading range from precracking to the collapse stage. Nonlinearity is included in the model by means of a stepwise linear analysis technique, which makes use of the relevant se­cant stiffnesses obtained from the moment-curvature relation­ships for segments along the length of the beam. The use of curvature, as opposed to load, as the incrementing parameter facilitates the analysis of a beam with limited ductility, in which critical regions soften before failure occurs. A parametric study carried out to determine the optimum lay­out of the segments for the model is described. The optimum mesh is used in the analysis of a number of beams and the re­sults are compared with test data and other analytical predic­tions. It is concluded that the macroscopic finite element model gives an adequate prediction of the behavior of a con­tinuous prestressed concrete beam over the entire range of loading up to failure.

The fi nite element method of analysis affords the closest ap­proximation to the real behav­

ior of a structure. Based on the level of idealization of the structure, the fi­nite element method uti lizes either a macroscopic or a microscopic model. In a macroscopic model, the struc­ture is divided into segments of fi nite length and nonlinear material behav­ior is introduced through pregener­ated moment-curvature relationships for the segments. In the case of a mi-

croscopic model, the structure is di­vided into segments, which are sub­sequently divided into layers, and nonlinearity is introduced through the stress-strain relationships for the constituent materials in each layer. As compared to a microscopic finite element model, a macroscopic finite element model usually requires less effort for the analysis and at the same time yields good results.

A computer program for the Non­linear Analysis of Prestressed Con-

PCI JOURNAL

crete Continuous Beams (NAPCCB), which has been developed' utilizing a macroscopic finite element proce­dure, is outlined in this paper. The validity of the procedure is estab­lished by means of comparison with test data and other analytical predic­tions.

THE COMPUTER PROGRAM

The NAPCCB computer program is based on the stiffness method and was developed using the numerical procedure suggested and used by Warner and Yeo. 2 The program uses a curvature incrementing technique and is capable of tracing the response of a bonded prestressed concrete beam of rectangular, I or T section, subjected to either concentrated or uniformly distributed loading, over the entire range of behavior from prestressing to failure. A flow chart of the computer program is given in Appendix A.

The beam is idealized in the analy­sis by dividing it into segments, where a segment is defined as the length between two consecutive joints, with each joint having two de­grees of freedom, namely rotation and displacement. A linear elastic analysis is carried out to determine the effects of prestress and then the moment-curvature relationships for all the segments are established. A key segment, which is one located at a region where failure is likely to occur in the beam, is selected and the curvature in this segment is incre­mented in steps.

For each increment of curvature, a linear analysis, using the relevant se­cant bending stiffness for each seg­ment, is carried out to determine the moment and curvature in each of the segments. At each increment of cur­vature, the cycle of calculation is re­peated until equilibrium, compatibil­ity and convergence requirements are satisfied. At this stage, the actual load on the beam corresponding to the assumed curvature is determined from equilibrium considerations and the related deflections, reactions and segment end actions are calculated. The incrementing of curvature con-

Seotember-October 1990

Stress (fc)

I I I I I I I I I I

__ __.l¢Lo.15 f(:

Parabolic fc = f(; [ 2Ec - (~)

2 J 0.002 0.002

fJf~ 1.4

1.2

1.0

0.002 0.004 Strain (Ec)

(a) Unconfined Concrete

Linear fc = Kf fb[1 - Zm (Ec - 0.002 Kf)] ~ 0.2 Kf f(: 0.8

~ ( )2 J K f' 2Ec - Ec

f c 0.002Kf 0.002Kf

0.6 Parabolic fc =

0.4

0.2

0 0.002Kt 0.01 0.02

(b) Confined Concrete

~ ~ Emax (Eq. {3}) I I I I

0.03 0.04 Strain (Ec)

Fig. 1. Stress-strain relationships for concrete.

tinues until one of the segments reaches its ultimate curvature capac­ity, at which stage failure of the beam is said to occur. Further details of the steps involved in the analytical procedure have been given by Warner and Yeo2 and Kodur. 1

The moment-curvature relation­ship is generated for the central sec­tion of a segment using the strain compatibility technique.3 This rela­tionship is assumed to be valid throughout the length of the segment. For each segment the moment-curva­ture relationship is generated for both positive and negative bending, since some segments switch from positive to negative bending and vice-versa, due to redistribution of

moment in the inelastic range. The ultimate moment capacity, Mu, of a section is taken as that corresponding to the ultimate curvature, Ku, which is the curvature corresponding to a concrete strain, Emax , in the extreme compression fiber or to a steel strain equal to ultimate strain of the tension reinforcement at the section.

The two options in the model for selecting the stress-strain relation­ship for concrete in compression, one for unconfined concrete4 and the other for confined concrete,5 are shown in Fig. 1. The factors, K1, and Zm, used in Fig. l(b), are defined as:

K = 1 + p,[yh f t: (1)

43

and

Zm=[ ] 0.5 (3 + 0.29 J: h"

(l 4SJ: -lOOO +0.75ps.ff -0.002Kf

(2)

where Ps = ratio of volume of hoop re­

inforcement to volume of core measured to outside of the hoops

/yh = yield strength of the hoop reinforcement (MPa)

1: = compressive cylinder strength of concrete (MPa)

h" = width of concrete core mea­sured to outside of the pe­ripheral hoops (mm)

sh = center-to-center spacing of hoop sets (mm)

The maximum usable compressive strain of concrete, Emax , depends on the confinement resulting from hoop reinforcement present in the section and is given by:

Emax = 0.004 + 0.9ps [!b~) (3)

For unconfined concrete, where Ps is zero, the maximum usable strain is 0.004.

For concrete in tension, the stress­strain relationship is assumed to be linear with a slope equal to that of the curve for compression at zero stress. The contribution of concrete in the tension zone, after cracking, is not taken into account. An elastic­plastic stress-strain relationship is used for nonprestressed reinforce­ment, both in compression and ten­sion. A multilinear stress-strain rela­tionship is used for prestressed rein­forcement in tension.

namely 0, D and D' in Fig. 2, for the moment-curvature relationship was investigated.

If Point 0 is selected as the origin, computational problems such as non­convergence are encountered since the secant stiffness is negative for applied moments in the range from zero to M0 and becomes infinite at moment M 0 • When Point D is se­lected as the origin, it is found that the effects of prestress are not in­cluded in the analysis, since this choice implies zero curvature and zero moment at zero applied load. However, selection of Point D' as the origin, where D' is defined by the magnitude, Msec , of the secondary moment at the section due to pre­stress, accounts for prestressing ef­fects at all stages of loading and computational problems such as non­convergence are avoided. Hence, the moment-curvature relationship re­ferred to the M'- K' axes is used in

M

0

the analysis. This change of axis is a computational device only and the total moments are evaluated at each stage of loading.

The use of curvature as the incre­menting parameter has certain ad­vantages over the load incrementing technique.7

· 8 Curvature incrementing facilitates the analysis of beams where one or more critical sections undergo local softening (moment shedding) before failure. Also, the complete response of the beam can be determined in one computer run since the moment-curvature relation­ships are generated before the start of the nonlinear analysis, as opposed to several runs required in computer programs such as PCFRAME,9 based on a load incrementing technique.

FAILURE LOAD OF A CONTINUOUS BEAM

Subsequent to cracking of the con­crete, the bending moment diagram for a continuous prestressed concrete beam under load deviates from that given by a linear elastic analysis and, with increasing load, approaches that obtained from a plastic collapse anal-

-------------· K'

K

As shown in Fig. 2, the moment­curvature relationship for a pre­stressed concrete section does not usually pass through an origin corre­sponding to zero moment and zero curvature. Selection of a suitable ori­gin is an important step in the analy­sis since nonlinearity is accommo­dated by updating the secant stiff­ness, defined as the slope of the line from the origin to the relevant point on the moment-curvature diagram. The suitability of using the three pos­sible origins suggested by Arenas,6 Fig. 2. Axes system for moment-curvature relationship.

44 PCIJOURNAL

ysis. The failure load, based on a plastic collapse mechanism ap­proach, can be attained only if the sections at which plastic hinges form have sufficient rotational capacity to allow full development of the re­quired number of plastic hinges for formation of a mechanism in the beam. The rotational capacity avail­able at a plastic hinge can be as­sessed from the moment-curvature relationship of the cross section of the beam and increases with ductil­ity.

Determination of the required rota­tion at a hinge location necessitates an analysis of the beam under a load level corresponding to the plastic collapse load. If the required rotation at any of the plastic hinge locations is greater than that available, the beam will fail prior to reaching the plastic collapse load since the plastic moment capacity cannot be main­tained at this location and the col­lapse mechanism cannot develop fully. Partial redistribution of mo­ment occus in this case as opposed to full redistribution which accompan­ies the development of a plastic col­lapse mechanism.

In computing the required rota­tion at a plastic hinge location in a structure, inelastic deformations must be considered. This consider­ation has stifled the application of plastic methods of design to concrete framed structures. As a result, North American Codes 10

• 11 use a lower

bound approach to a failure load by utilizing elastic bending moments, which may be adjusted to account for redistribution of moment, to deter­mine the load at which the ultimate moment capacity is attained at a sin­gle section. The permitted amount of redistribution of moment is based on a measure of the ductility of a spe­cific cross section in a member. However, it has been suggested that redistribution of moment should be based on member ductility rather than on sectional ductility.2

Redistribution of moment in stati­cally indeterminate structures can be studied using nonlinear methods of analysis. The NAPCCB computer program has been developed as part of a study of redistribution of mo­ments in continuous prestressed con-

September-October 1990

~-----------;::;-------------~~~-:-::::,~:~_·:~:_::~--jjl_ (a) Beam and loading

I"" 1so ,. I Material Properties

"" {ij-:--:---.- f'c = 49.5 MPa

Ec = 30100 MPa ft = 3.30 MPa

Emax = 0.007

fy = 310 MPa fse = 1215 MPa

Es = 200000 MPa Eps = 200000 MPa

fpu = 1860 MPa Aps = 98.7 mm2

(b) Cross Section w All dimensions In mm

(c) Model 06SEGSB

r 8@250 LLl

(d) Model 12SEGSB w

(e) Model 14SEGSB w

Fig. 3. Idealized models for simply supported beam.

crete beams. 1•12 This program com­

putes the failure load of a beam by determining the load level at which the ultimate curvature is exceeded at a hinge location. Utilization of the program is incorporated in the analy­sis of the strength of a two-span beam presented in Appendix B.

OPTIMUM IDEALIZATION OF BEAM

Proper modeling of the regions in the beam where failure occurs is an important aspect in the NAPCCB

computer model since the failure load is governed by the strength of such critical sections. Warner and Y eo2 suggested the use of an aspect ratio (ratio of length to depth of the segment) of 1 to 2.

In order to assess an optimum ele­ment size, two prestressed concrete beams, one simply supported and the other two-span continuous, were an­alyzed using different idealizations in which the aspect ratio was reduced gradually in the critical regions, as shown in Figs. 3 and 4, respectively. The aspect ratio of the segments in

45

the critical regions for each idealiza­tion is given in Tables 1 and 2. In the case of the simply supported beam, the critical region is at the load point, while the load point and the central support are the critical regions in the continuous beam.

Both beams have a rectangular cross section, 250 mm (9.84 in.) deep and 150 mm (5.91 in.) wide, and the amounts of reinforcement and the material properties are as given in Figs. 3 and 4, respectively. The model designation, in Figs. 3 and 4 and in Tables 1 and 2, consists of two digits at the beginning identi­fying the number of segments in the beam, followed by letters SEG for SEGments and SB or CB for Simply supported or Continuous Beam.

1691.6

Prestressing tendon, ,W \ 2133.3 -·· 1066.7

~~==========~~~======~======~~~SYM ~ cgc

(a) Beam and Loading

H 8~ f'c=40MPa ~ Ec = 40000 MPa

8 .., 11 = 3.79 MPa

~ ~ 'max = 0.004

35.61

(d) Model 12SEGCB2

I

Material Properties

fy = 310 MPa fse = 1266 MPa

Es = 200000 MPa Eps = 200000 MPa

fpu = 1860 MPa

Aps = 98.7 mm2

All dimensions in mm

r 193.5 187

31

Results from the analyses of the idealized simply supported and con­tinuous beams are presented in Ta­bles 1 and 2, respectively. From Ta­ble 1 it can be seen that the failure load predicted by NAPCCB for the simply supported beam is the same for all the idealizations and is equal to the plastic collapse load, based on the ultimate moment capacity at the critical section. This is expected since the plastic collapse load is de­termined by equilibrium considera­tions alone for a statically determi­nate structure. However, the deflec­tion under the load point at the failure load decreases with the aspect ratio of the segment in the critical re­gion.

Fig. 4. Idealized models for continuous beam.

The deflection of the beam de­pends on the area of the curvature di­agram and, consequently, on the plastic hinge length, lP, as shown for the idealized curvature diagram13 in Fig. 5. The formulation in the model

is such that lP is equal to the length of the segment adjacent to the load point section since the entire critical segment is assumed to be fully plas­tic at failure. Thus, a model with a

higher aspect ratio, and related larger plastic hinge length, will predict a larger deflection for the same value of ultimate curvature at the critical section.

Table 2. Results from continuous beam analysis.

Table 1. Results from simply supported beam analysis. Model Aspect Failure Curvature Curvature Load designa- ratio load under at point

Model Aspect Failure Load point Load point designa- ratio load curvature deflection

tion (kN) load support deflection (xl0-6/mm) (xl0'6/mm) (mm)

tion (kN) (xl0'6/mm) (mm) 12SEGCB1 1.0 186.71 27.96 -55.09 10.53

06SEGSB 2.0 47.23 136.57 78.51 12SEGCB2 0.75 182.29 16.70 -55.09 9.00

12SEGSB 1.0 47.23 136.57 50.17 15SEGCB 0.50 158.71 12.23 -55.09 7.12

14SEGSB 0.5 47.23 136.57 31.93 18SEGCB 0.25 131.55 8.66 -55.08 5.16

16SEGSB 0.25 47.23 136.57 22.16 21SEGCB 0.125 119.20 7.23 -55.08 4.26

18SEGSB 0.125 47.23 136.57 17.49 24SEGCB 0.0625 113.77 6.63 -55.08 3.85

Plastic Plastic

analysis - 47.23 - - analysis - 186.28 - - -

Metric(SI)conversionfactors: I kip=4.448kN; I in.=25.4mm. Metric (SI) conversion factors: I kip= 4.448 kN; I in.= 25.4 mm.

46 PCIJOURNAL

L/2 L/2

(a) Beam and Loading

(b) Bending Moment Diagram at Failure

(c) Idealized Curvature Diagram at Failute

~ (d) Plastic Collapse Mechanism

Fig. 5. Failure conditions in a simply supported beam.

200 Plastic Analysis -----------------------------------------------------------------------------------

160

z ~ Q 120 ~ 0 ........ ...J

w a: 80 :::> ...J

< u.. 40 t:. NAPCCB

0 0.0 0.2 0.4 0.6 0.8 1.0

ASPECT RATIO

Fig. 6. Load vs. aspect ratio for continuous beam.

September-October 1990

Table 2 and Fig. 6 show that, for the continuous beam, the failure load decreases with aspect ratio. From Table 2 it can be seen that, while the support segment reaches its ultimate capacity, as characterized by the con­stant value of curvature at the sup­port for all the idealizations, the load point segment reaches a level of cur­vature dependent upon the aspect ratio. This can be seen also from the moment-curvature relationship in Fig. 7, where the numbers refer to the six different idealizations as listed. Fig. 6 shows that the failure load ap­proaches a value of 108 kN (24.3 kips) at zero aspect ratio, a value which could be interpreted as the ac­tual failure load of the beam. Since failure loads lower than that given by a plastic analysis were obtained from the NAPCCB analyses, it may be concluded that only partial redistri­bution of moment occurred in the beam at failure.

The variation in the magnitude of the failure load and the related de­flection of the continuous beam for the different idealizations can be ex­plained with reference to Fig. 8. At failure of the beam, the rotation, <1>., in the plastic hinge at the central sup­port can be determined using the free hinge approach. 13 The rotation at failure is given by the relation:

where <j> 1 and <j>2 , which are the rota­tions at the end of a simply sup­ported span subjected to a bending moment, M. [Fig. 8 (c)], and a load, W. [Fig. 8 (d)], respectively, can be computed using the relationships:

M,L 5W.U <l>t = 3EI and <1>2 = 81£/ (5)

where EI is the flexural stiffness of the span and L is the span length. The rotation, <1>., is also dependent on the product of plastic hinge length, !P, and ultimate curvature, K., at the central support and can be written as:

From Eq. (6), it can be seen that, for a specific moment capacity, M.,

47

120

100

'E 80 z ~ 1- 60 z UJ :E 0 :E 40

20

0 0 10 20 30

Ultimate Strain = 0.004

1 -12SEGCB1

40

2 -12SEGCB2

3-15SEGCB

4 -18SEGCB

5- 21SEGCB

6 • 24SEGCB

50

CURVATURE x 10·6 (/mm)

Fig. 7. Moment-curvature relationship for load segment in continuous beam.

t.---c -2U-3 -!-L--:Wu -U-3 -.~ (a) Beam and Loading

60

Plastic hinge

(b) Rotation at Failure

---------------------- \ ~~hlngo

fi 6_/Mu

(c) Rotation due to Mu

[i------

~'""'"9'

------- D

(d) Rotation due to Wu

Fig. 8. Rotation at center support under failure conditions.

48

and related curvature, Ku, an increase in plastic hinge length, lP, as a result of increased aspect ratio of the criti­cal segment, leads to a higher value of the failure load, Wu. In addition, the resulting increase in the area of the curvature diagram, resulting from an increased plastic hinge length, leads to a higher rotation in the plas­tic hinge length and hence a larger deflection at the load point. Thus, an increase in the aspect ratio of the segments in the critical regions leads to a higher failure load and a larger deflection in the model NAPCCB.

From the results of analyses of a number of additional two-span and three-span beams 1 under various load­ing conditions, it was found that an aspect ratio in the range of 0.2 to 0.4 in the critical regions gives a good estimate of the failure load and the de­flection in a beam in which partial redistribution of moment occurs. Use of an aspect ratio less than 0.1 in­duces round-off errors and also in­volves higher computer costs.

An investigation1 of the effect of refinement of mesh in two-span beams where redistribution of mo­ment is virtually complete at failure indicated that the aspect ratio has no significant effect on failure load. In this case significant plastic deforma­tion will occur in the region of the load point as well as at the central support. Since the rotation, <l>u , is re­lated to the area of the curvature dia­gram, the influence of an increase in the plastic hinge length at the support (negative curvature) will be counter­acted by a similar increase at the load point (positive moment).

Consequently, from Eq. (4), if <l>u and <1> 1 (which is dependent on Mu) are invariant, so will <1>2 be invariant, with the result that Wu will not be de­pendent on lP. However, the deflec­tion decreases with the aspect ratio as in the case of beams with partial redistribution of moment. This can be attributed to the fact that the de­flection at the load point will be in­fluenced primarily by plastic defor­mation in the load point region. Thus, an increased aspect ratio and related plastic hinge length will lead to increased deflection.

It is recommended that, since the

PCIJOURNAL

80 0

70 0 0 PLASTIC COLLAPSE LOAD ------------------------------f:[j·zs.··---t.-----A------------------------------------------

60 !::.

~

50 !::. BEAM 54 I z 4::1

~ 0!::.

== 40 !::.

c ~ tw .r 1600

<C 0 30 ~ ...I t .~. .t !::. 1600 •I• 1600 1600

20 t;l

10 l"ii

0 Test Data

!::. NAPCCB 0

0 5 10 15 20 25 30 DEFLECTION (mm)

(a) BEAM 54

100 0 0

0 PLASTIC COLLAPSE LOAD ---------------------------0--------------------------------------------------------------80

z c 60

== c <C g 40 !;1

0

20

.ro

t::.!::.t::.!::. Of::. I::.

!::. 0!::. BEAM 352 I

!::.

1. 15oo ,f, 16oo !w 16oo .~ '""' J

0 Test Data

!::. NAPCCB

OL-------~------~------~------~------~ 0 10 20 30 40 50

DEFLECTION (mm)

(b) BEAM 352

Fig_ 9. Load vs. load point deflection for Beams 84 and 382.

Failure load(kN)

Beam Test NAPCCB Plastic PC FRAME designation data analysis analysis analysis

S4 73.70 63.52 63.25 -

3S2 94.23 80.41 83.50 -Rectangular - 131.57 186.28 136.02

I - 68.65 68.70 71.80

September-October 1990

extent of redistribution of moment is not known before the analysis is car­ried out, an aspect ratio of 0.2 to 0.4 in the critical regions, together with an aspect ratio of 1 to 2 in other re­gions, should be used in all beams to obtain a reasonable estimate of both the failure load and the deflection.

EVALUATION OF THE MODEL

The validity of the computer model was further evaluated by com­paring its predictions with test data and with other analytical predictions. The analyses were carried out using an aspect ratio of 0.25 for segments in the critical regions. Confinement present in the section was taken into account by using the relevant stress­strain relationship for concrete in compression.

Test data12• 14 in the form of load­deflection and load-moment plots, for a two-span beam (S4) and a three­span beam (3S2), are compared with the results from the NAPCCB com­puter model in Figs. 9 and 10, re­spectively. In the load-moment plots the positive moment corresponds to that at the load point, while the nega­tive moment is that at the interior support. Good agreement is obtained between the test data and the pre­dicted responses over the entire load history for both beams.

The failure loads from tests, NAPCCB analyses and plastic anal­yses for Beams S4 and 3S2 are given in Table 3. The failure loads pre­dicted by the nonlinear analysis and by the plastic analysis are close, indi­cating that full redistribution of mo­ment occurred in both beams. The slightly higher failure load from the plastic analysis for the three-span beam is due to the use of maximum moments, as opposed to ultimate moments, at the critical sections which exhibited softening moment­curvature relationships.

The test failure load is approxi­mately 15 percent higher than the load predicted from plastic analysis for each beam. The higher failure load has been attributed to the pres­ence of the prestressing duct acting as extra reinforcement and neglect of

49

80

70 0

0 0

0 0 PLASTIC COLLAPSE LOAD 0 _____________ '& __________________________________________________________________ !# ___ _

60 !::. !::.

z 50 ~

3:: 40 c ct 0 30 ...1

20

~ ~ A !::. m .fiJ

'b ~ !::. !::.

Ot::. JSI cr. A

A !::.

0!::. ,fl

1 0 o Test Data m ~

100

90

80

70 z ~ 60

3:: 50 c ct 40 0 ...1

30

20

10

A NAPCCB [#. ~ n n

-70 -60 -50 -40 -30 -20 -10 0 10 20 30

MOMENT (kN.m)

(a) BEAM S4

0 0 0 0

-- -----------_I;:)------------------ ~~~§."!"!9.. ~Q~!:~!:?_~_ -~Q~I?---- ---------- .o. __ --- --

~0 I

o Test Data

A NAPCCB

t::.o ~

!::. !::. !::.0 !::.0

t::.o

0 !::.

0

0

LCI

0

0 A

0

0~--~---L--~----~~~---L--~----k---~--~ -50 -40 -30 -20 -10 0 10 20 30 40 50

MOMENT (kN.m) (b) BEAM 3S2

Fig. 10. Load vs. moment for Beams S4 and 3S2.

stmin hardening of the nonprestressed steel, two effects which are not ac · counted for in the analysis. 12 The smaller deflections predicted by NAPCCB at failure, as shown in Fig. 9, are due to the lower predicted fail­ure loads.

Predictions from the NAPCCB computer model were also compared with results from a microscopic fi­nite element model, PCFRAME,9 for two two-span beams, one of rectangu­lar and the other of I cross section. The use of similar stress-strain curves

50

for concrete and steel in both pro­grams facilitate a direct comparison of results. An aspect ratio of 0.25 was chosen for the mesh size in the critical regions for both analyses. Since confinement of concrete is not accounted for in PCFRAME, the analyses were carried out using the unconfined stress-strain curve for concrete in compression. Good agree­ment between the results from two programs over the entire loading range can be seen from the load-de­flection and the load-moment plots

in Figs. 11 and 12, respectively. In the case of the rectangular beam,

as can be seen in Fig. 12(a), the sup­port segment reached its ultimate ca­pacity of -50.67 kN.m (-37.37 kip­ft), but the load point segment did not reach its ultimate capacity of +98.71 kN.m (+72.79 kip-ft), indi­cating that the degree of redistribu­tion of moment was partial in this beam at failure. This is reflected in the large difference between the fail­ure load from the plastic analysis and the NAPCCB analysis [Table 3 and Fig. 11(a)]. The failure load pre­dicted by PCFRAME for this beam is slightly higher than that from NAPCCB because PCFRAME over­estimates the strength of the critical sections.

It has been shown1 that this over­estimation is due to the assumption in PCFRAME that the strain and the related stress is constant over the depth of a layer in an element. For the I beam, the predicted failure load from the NAPCCB analysis is very close to that from the plastic analy­sis, indicating that the redistribution of moment is almost complete. Again, it can be seen in Table 3 that PCFRAME overestimates the failure load. The slightly larger deflections at failure predicted by PCFRAME for both beams, as can be seen in Fig. 11, is due to the higher predicted failure loads.

As a result of the above evalua­tion, it is felt that the nonlinear math­ematical model, NAPCCB, may be used with confidence to examine the effects of the various parameters which may influence the redistribu­tion of moments in prestressed con­crete continuous beams. Such a study is being undertaken in order to refine and extend the approach pro­posed by Moucessian and Campbell8

for the determination of the failure load of a continuous prestressed con­crete beam.

CONCLUSIONS

1. An aspect ratio in the range of 0.2 to 0.4 for the segments in the critical regions should be used in order to yield acceptable results from the nonlinear analysis.

PCI JOURNAL

200 200 PLASTIC COLLAPSE LOAD ------------------- ---------------- ------------ ------------------- ..... P.~~?_TI~-~()LL_~_P.S_~_W!\1? ...... ---------------------------------------------

180 180

160 160

140 0 Ot. ~

0 ,..

z 120 ,..

Ill ~ ot. 3: 100 o"' Q ot. c(

140 §~ r# Ot. ct. z 120 ct Iii"'

~ rn rn 3: 100 ci' r:P> Q

19 r:f> c( 0 r::f ....1 80

0

0 ~,.. & ....1 80

,.. 0

!:70~1070r 2130

60 .fl

0 I. 2130 .. I 40 &I .... Jr"' '.

0 ,.. !>

60 0 0

q, p 0 0

40 lA IZil

0 0 0

20 Ill 0 PCFRAME 20 0 PCFRAME II! !!

# ,.. NAPCCB 0

!> NAPCCB q, ~ 0 0 1 2 3 4 5 6 -70 -50 -30 -10 10 30 50 70

DEFLECTION (mm) MOMENT (kN.m)

(a) RECTANGULAR BEAM (a) RECTANGULAR BEAM

80 80

PLASTIC COLLAPSE LOAD 0 0 70 -----------------·····---------0·;··B·····-7s.··---------n-----t.----------------------- 70 .,.P.~~~J!9..9.()~~-... P.~-~--~Q~I? ..•.• ······-····-···················1··········

60 ,..a:.

0 60 ~!> ff

0 0

z o"'

~ 50 0

3: !!

40 ,.. Q 0 w w c(

1 137o ~ 1370 t mo ! 1370 1

0 30 ....1 !! .. .. • • "' J' .. )I'

z s ~

~ 50 0 0

.[] ,(1]

3: 40 !> !> Q 0 0 c( 0 30 ....1 a. Iii

20 0 20 0 0

,.. 10 0 PCFRAME

0

!> ,.. 10 0 PCFRAME 0 0 ,.. NAPCCB !> NAPCCB

0 -2 2 6 10 14 18 22 26

0 -40 -30 -20 -10 0 10 20 30 40

DEFLECTION (mm) MOMENT (kN.m)

(b) I BEAM (b) I BEAM

Fig. 11. Load vs. load point deflection for rectangular and I beams.

Fig. 12. Load vs. moment for rectangular and I beams.

2. Predicted failure loads are sensi­tive to the aspect ratio of the seg­ments at the critical regions in beams where incomplete redistribution of moment occurs at failure, but not in beams where redistribution of mo­ment is complete at failure. Predicted deflections are sensitive to aspect ra­tio in all cases.

3. Comparison of results from the model contained in the computer pro­gram, NAPCCB, with results from laboratory tests and from other ana-

September-October 1990

lytical predictions showed that the model is capable of predicting the be­havior of a prestressed concrete con­tinuous beam at all stages of loading.

METRIC (SI) CONVERSION FACTORS

1 in. 1 kip 1 ksi 1 kip-ft =

25.4mm 4.448kN 6.895 MPa 1.356 kN.m

REFERENCES

1. Kodur, V. K. R., "Deformation Con­trolled Nonlinear Analysis of Pre­stressed Concrete Continuous Beams," MSc Thesis, Department of Civil Engi­neering, Queen's University, Kingston, Ontario, Canada, 1988.

2. Warner, R. F., and Yeo, M. F., "Ductil­ity Requirements for Partially Pre­stressed Concrete," Partial Prestress­ing, From Theory to Practice, V. II, Martinus Nijhoff Publishers, Dor­drecht, The Netherlands, 1986, pp. 315-326.

51

3. Priestley, M. J. N., Park, R., and Lu, F. P. S., "Moment-Curvature Relation­ships for Prestressed Concrete in Con­stant Moment Zones," Magazine of Concrete Research (London), V. 23, No. 75-76, January-February 1971, pp. 69-78.

4. Hognestad, E., "A Study of Combined Bending and Axial Load in Reinforced Concrete Members," Bulletin No. 399, University of Illinois Engineering Ex­periment Station, Urbana, IL, Novem­ber 1951, 128pp.

5. Park, R., Priestley, M. J. N., and Scott, B. D., "Stress-Strain Behaviour of Con­crete Confined by Overlapping Hoops at Low and High Strain Rates," ACI

Journal, Proceedings, V. 79,No. !,Jan­uary-February 1982,pp.13-27.

6. Arenas, J. J., "Continuous Partially Pre­stressed Structures: European Perspec­tive," Partial Prestressing, From The­ory to Practice, V. I, Martinus Nijhoff Publishers, Dordrecht, The Nether­lands, 1986,pp.257-287.

7. Scordelis, A. C., "Computer Models for Nonlinear Analysis of Reinforced and Prestressed Concrete Structures," PCI JOURNAL, V. 29, No. 6, November­December 1984,pp. 116-135.

8. Moucessian, A., and Campbell, T. 1., "Prediction of the Load Capacity of Two-Span Continuous Prestressed Concrete Beams," PCI JOURNAL, V. 33, No.2, March-April 1988, pp. 130-151.

9. Kang, Y. J., "Nonlinear Geometric Ma­terial and Time Dependent Analysis of Reinforced and Prestressed Concrete Frames," Report No. UC SESM. 77-1, College of Engineering, University of California, Berkely, CA, 1977.

10. Canadian Standards Association, "Code for the Design of Concrete Struc­tures for Buildings," National Standard of Canada, CAN3-A23.3-M84, Rex­dale, Canada, 1984.

11. ACI Committee 318, "Building Code Requirement for Reinforced Concrete (ACI 318-89) and Commentary (ACI 318R-89)," American Concrete Insti­tute, Detroit, Ml, 1989.

12. Moucessian, A., "Nonlinearity and Continuity in Prestressed Concrete Beams," PhD Thesis, Department of Civil Engineering, Queen's University, Kingston, Ontario, Canada, 1986.

13. Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley and Sons, New York, NY, 1975.

14. Bhatia, S., "Continuous Prestressed Concrete Beams in the Inelastic Range," MSc Thesis, Department of Civil Engineering, Queen's University, Kingston, Ontario, Canada, 1984.

52

APPENDIX A-FLOW CHART FOR NAPCCB PROGRAM

Analyse for prestressing effects

Generate M-K relationship

Choose a Target Curvature (TC) in Key Segment (KS)

Unit Load Analysis - ULM

ULM(KS) ULC(KS) = E/(KS)

TC SF = --------ULC(KS)

ULC(J) = ULM(J) E/(J)

Find BM(J) for K(J) from M-K relationship

NO

Find reactions and deflections

YES

PC! JOURNAL

APPENDIX B-ANAL YSIS OF TWO-SPAN BEAM

Fig. B 1 shows a two-span continu­ous fully prestressed concrete beam of T cross section. The beam is sub­jected to uniformly distributed load and has an idealized parabolic profile of the prestressing tendon. The pre­stressing force is assumed to be con­stant over the length of the beam. This beam has been analyzed pre­viously by Lin and Thornton (PCI JOURNAL, V. 17, No. 1, January­February 1972, pp. 9-20).

A linear elastic analysis of the beam gives the secondary moment at the central support, due to prestress, as 1897 kN.m (1399 kip-ft).

The beam is symmetric about the central support and has critical sec­tions, namely, the locations of poten­tial plastic hinges, one at the support and one in each span. For a uni­formly distributed load, the critical section in the span will be at a dis­tance of approximately 0.4L from the end support. A distance of 0.3961£, which corresponds to the location of the central section of a segment in the NAPCCB analysis, has been as­sumed here.

Properties of the two critical sec­tions, obtained using the strain com­patibility approach, are presented in Table Bl.

Plastic Collapse Load

Assuming a collapse mechanism to form in each span (full redistribu­tion of moment), where the plastic hinge in the span forms at a distance of 0.3961£ from the end support, the plastic collapse load, wP'' is given by:

Wpt= 2[MsL- M, a]

a (L- a)L (Bl)

2[4765x24.384- (-3646)9.659] 9.659(24.384- 9.659)24.384

= 87 .3lkN/m (5.97kips/ft)

Failure Load Based on Elastic Analysis

Assuming no redistribution of mo­ment, the failure load, w1e, based on elastic analysis, can be computed ac-

September-October 1990

I[>

ic 12192 ~I ... 12192 ~lc 12192 ~I ... 12192 ~I

3048

f' = 34.5 MPa (5000 psi) c

=l203.2 fpu = 1728 MPa (250 ksi)

p = 2675 kN (601 kip)

Ecu = 0.004 1014.4

Aps = 2580 mm2 (4 in2)

Note: Length in mm Aps

.:L

1 in =25.4 mm u 304.8

6

Fig. 81. Two-span beam.

cording to the CAN3-A23.3-M.8410

and ACI 318-89/318R-89 11 Codes by determining the load at which one of

The central support section will at­tain its moment capacity, M,, when M = M,- Mseo giving:

the critical sections reaches its ulti­mate capacity. M = -3646- 1897 = -5543 kN.m

Elastic analysis gives the moment at the central support as:

wU

Using this value of Min Eq. (B2), the failure load, w1e , based on elastic analysis, is given by:

M=--8- (B2) w1e = 74.58 kN/m (5.10 kips/ft)

Span critical Support critical Property section section

Moment(Mu) 4765kN•m 3646kNom (3513 kip-ft) (2688 kip-ft)

Curvature ( Ku) 37.21 x 10"6/mm 8.05 x 10"6/mm (945 x 10"6/in.) (204 X 10·6 /in.)

xldP 0.049 0.455

(l)p 0.0378 0.362

Msec 751 kN•m 1897kN•m (554 kip-ft) (1399 kip-ft)

53

Applying a similar procedure to the section at a distance 0.3961£ from the end support gives a failure load, w1,, equal to 96.44 kN/m (6.59 kips/ft).

Since the analysis based on the support section gives the lower load, the failure load, w1,, is taken as 74.58 kN/m (5.10 kips/ft).

Failure Load from NAPCCB

The beam was analyzed by means of the program NAPCCB using the idealization based on the recommen­dations given in this paper. This analysis indicated failure of the beam by crushing of the concrete at the central support at a load, wfl, of 76.79 kN/m (5.25 kips/ft).

As indicated in the moment-curva­ture relationships for the two sec­tions (Fig. B2), at failure, the central support section reached its ultimate capacity, while the critical span sec­tion reached a level of moment (and curvature) well below the ultimate capacity.

Failure Load Incorporating Redistribution of Moment

(a) CAN3-A23.3-M8410

This Code allows a percentage of redistribution of moment, for (x /d,) values in the range 0.2 to 0.6, of [30 - 50 (x/d,)] with an upper limit of 20 percent.

Hence, for the support section, where x/d, = 0.455, the redistribu­tion of moment is 7.25 percent. The central support section will attain its moment capacity, Mn when:

wV M = - -

8- ( 1 - 0.0725) =Me - Msec

= - 3646 - 1897 = -5543 kN.m

giving:

w =WeAN= 80.44 kN/m (5.51 kips/ft)

(B3)

The moment, allowing for redistri­bution of support moment, at the span critical section due to this load is 4271 kN.m (3150 kip-ft), which is less than the moment capacity (4765 kN.m) of the section. Since WeAN is

54

5000

c c c 4500 '-- c c c

c 4000 - ~

D 3500 -- c

~ 3000 - il ~

®CONDITION AT I 2500 1-- FAILURE r::: Cl) 2000 f-E 0

:a!! 1500 f-

1000 f-

500 f-

0 .I I I I I I I I -2 0 5 10 15 20 25 30 35 40

Curvature- K (x1o-6 I mm) (a) SPAN SECTION

4000

c c c ® 3500 - c D

c c

3000 - D c -E c

z 2500 - c r!)p

~ ® CONDITION AT -:a!! FAILURE

I 2000 1--r::: Cl)

E 1500 f-0

:a!! 1000 r-

500 f-

0 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Curvature - K (x1 o-6 I mm) (b) SUPPORT SECTION

Fig. 82. Moment-curvature relationships for critical sections.

higher than w11 , this Code overesti­mates the load carrying capacity of the beam.

(b) ACI 318-89/318R-89li This Code allows a percentage of

redistribution of moment of:

[ roP + 1<w- w')]

20 1- 0.36 ~~ (B4)

Application of Eq. (B4) to the cen­tral support section results in a nega­tive value, indicating that redistribu-

tion of moment is not permitted in this beam. Hence, the failure load ac­cording to this Code is the same as that computed previously neglecting redistribution of moment:

WAC!= W1e =74.58 kN/m (5.10 kips/ft)

ACKNOWLEDGMENT

The authors wish to acknowledge the financial assistance provided by the Natural Sciences and Engineer­ing Research Council of Canada under Grant No. A8255.

PCIJOURNAL

APPENDIX C- NOTATION

Aps = area of prestressed reinforce- fr = modulus of rupture of con- ULM= moment in a segment due to ment crete unit load intensity

a = distance from end support to /y = yield stress of nonprestressed w = concentrated load span critical section reinforcement wu = failure load

BM = bending moment in a seg- hh = yield stress of hoop rein- w = uniformly distributed load ment forcement X = distance from extreme com-

d = distance from extreme com- h" = width of concrete core mea- pression fiber to neutral axis pression fiber to centroid of sured to outside of the pe- zm = factor defined by Eq. (2) nonprestressed reinforce- ripheral hoops Ec longitudinal strain in con-ment K curvature crete

de distance from extreme com- Kt = factor defined by Eq. (1) £max = maximum usable compres-pression fiber to centroid of Ko = initial curvature due to pre- sive strain in concrete tension reinforcement stress <l>u = rotation at failure

dp = distance from extreme com- Ku = curvature at ultimate <j>, rotation in a simply sup-pression fiber to centroid of KY = curvature at yielding of steel ported beam with moment prestressed reinforcement L = length of a span M" atoneend

Ec modulus of elasticity of con- IP = plastic hinge length <l>z = rotation due to load W" in a crete M = moment simply supported beam

Eps = modulus of elasticity of pre- Me ultimate moment of resis- Ps = ratio of volume of hoop rein-stressed reinforcement tance at central support sec- forcement to volume of core

Es = modulus of elasticity of non- tion measured to outside of hoops prestressed reinforcement Mer = cracking moment ~I = factor defined in Section

e eccentricity of prestressing Ms ultimate moment of resis- 10.2.7.3 of ACI 318-89/ tendon tance at span critical section 318R-89

EI = flexural rigidity Msec = secondary moment (J) = reinforcement index of non-

Jc = longitudinal stress in con- Mu = ultimate moment of resis- prestressed tensile reinforce-crete at strain Ec tance ment

t: = compressive cylinder strength My moment at yielding of steel (!)' = reinforcement index of non-of concrete NEI = new flexural rigidity prestressed compressive re-

fru = ultimate stress in prestressed sh = center-to-center spacing of inforcement reinforcement hoop sets (J)p = reinforcement index of pre-

he = effective stress in prestressed SF = scaling factor stressed reinforcement reinforcement ULC= curvature corresponding to

ULM

September -October 1990 55