Dynamic response of prestressed concrete frames

Dynamic response of prestressed concrete frames Y. L. Mo and W. L.

I-Iwang

Department of Civil Engineering, National Cheng Kung University, Tainan 701, Taiwan (Received July 1993; accepted October 1993)

Based on compatibility conditions, as well as a trilinear momentcurvature relationship, an analytical model was presented recently to predict the response of reinforced/prestressed concrete frames and continuous beams to static loads. The model was compared to the test specimens and was found to be applicable throughout the loading history. This paper shows that the inelastic response of prestressed concrete frames to dynamic loads can be predicted when the trilinear theory for the primary curve is combined with Takeda's rules for the hysteresis loops and with the linear acceleration step-by-step method for the dynamic model. The algorithm is described and a numerical example is given. Keywords: concrete frames, dynamic loads, algorithm

The nonlinear behaviour of prestressed concrete frames with sidesway is governed by two effects: firstly, the nonlinearity of materials due to cracking and the plastic behaviour of materials, and secondly, the nonlinearity of the geometry caused by the second-order deformation. These two effects may interact, and the whole phenomenon is known as the nonlinearity of goemetry and materialsL Prestressed concrete frames are frequently used in bridge design to resist wind or earthquake forces. These forces will accentuate the complexity of the frame behaviour because of the continuous change of the shape of the bending moment diagram. The change of moment diagram will in turn affect the magnitude of the cumulative plastic rotations 2. For prestressed concrete structures, the moments in some critical sections may be greater than those computed with a linear elastic analysis. Such sections could yield, and local failure could occur if these sections are not sufficiently ductile to redistribute the moments. In other words, the inelastic response of statically indeterminate prestressed concrete structures is difficult to determine. To predict ultimate strength, Walther 3 and Huber 4 utilized the theory of plasticity. However, the full moment redistribution with complete development of plastic hinges rarely occurs in prestressed concrete structures 5-7. Another method proposed by Guyon 8 can be applied, but the accuracy is doubtful 9,~°. Cohn and Frostig ~ have also used direct and incremental techniques for the analysis of the hyperstatic response beyond the elastic limit. In this analysis, the following assumptions are made. Firstly, the member stiffness was based on gross concrete sections for various loading stages; and secondly, prestressed concrete sections may exhibit sufficient ductility for a mechanism collapse to take place; otherwise, a local failure occurs at one of the critical sections that reaches its ultimate moment. It should, therefore, be noted that this method may not be

appropriate to investigate the inelastic response of prestressed concrete frames throughout the loading history. Recently Aalami ~2 studied the secondary actions of prestressed concrete continuous beams due to prestressing, however, this investigation is still based on elastic theory. The inelastic response of statically indeterminate reinforced concrete frames can be predicted by trilinear theory. 1'13 The trilinear theory has been further developed to include prestressed concrete continuous beams 14,15. It is shown in this paper that the recently reported trilinear theory can be applied to prestressed concrete frames for determining the load-deflection relationship as the primary curve for dynamic response analysis. In addition, the hysteretic rules proposed for dynamic analysis of reinforced concrete structures can be utilized for prestressed concrete frames under dynamic motions as long as the frames are designed for seismic loading ~6. Next, the primary curve and the hysteretic rules will be combined with the dynamic model ~7 to perform a dynamic response analysis. Finally, an algorithm and a numerical example are presented to illustrate the solution procedures.

Notation

Aps As

c, G cgs E

E~ L.

f,,~ f, I

area of prestressing strands area of mild steel compression contributed by concrete block compression contributed by compression steel centre of gravity of strands modulus of elasticity of concrete Young's modulus of steel ultimate strength of prestressing strands yield stress of prestressing strands compression strength of concrete moment of inertia of uncracked cross-section

0141-0296/94/080577-08 © 1994 Butterworth-Heinemann Ltd

Engng Struct. 1994, Volume 16, Number 8 577

Dynamic response of prestressed concrete frames: Y. L. Mo and W. L. Hwang

i i" ki

M M,.

Mi M u

M~ M~i~ m

P PI,P2

L

Tn~ t A, Yi as, Ay, Ay,

typical plastic hinge regions typical large deformation regions without hinge stiffness moment moment at cracking of concrete moment due to unit moment l, in primary statically, determinate system ultimate moment moment at yielding of steel bending moment of section i' due to unit moment mass seismic horizontal force at top of frame base shear of two columns induced by seismic loads tension contributed by mild steel tension contributed by prestressing strands time change in time displacement at time point i change in displacement at time point i change in velocity at time point i change in acceleration at time point i acceleration of ground at time point i incremental of ground acceleration

Ay.

Ayi + Ay~

Og

coefficient defined in equation (3) coefficient defined in equation (5) coefficient defined in equation (8) coefficient defined in equation (7) relative rotation of cross-sections adjacent to hinge i relative rotation of end cross-sections of plastic region i rotation in state I rotation in state II rotation in state III curvature curvature of cracking of concrete curvature at ultimate moment curvature at yielding of steel curvature in state I curvature in state II curvature in state III

/3 131 0

0, 0~ 0]' 4, 4',. q,. q,. ifjl

0 II i~/lll

Dynamic Response : Time - History Analysis

Hysteresis Loops : Takeda's Rules

Dynamic Model : SDOF System

Primary Curve : Trilinear Theory

Figure I Data structure for dynamic analysis of ~restressed concrete frame

is used (Figure 2). Since this continuous beam is statically indeterminate to the third degree, a primary statically determinate structure prior to collapse may include three plastic hinge regions and four regions of large deformation without hinges. A typical plastic hinge region is denoted i (1-3) and a typical region of large deformation without hinges i' ( 1'--4'). According to the principle of virtual work and the continuity of the deflection c u r v e 13'14, the compatibility condition for the plastic region i is as follows

f MiO' ds + f M, qJ" ds + f M, tplHds = O

(l)

where Mi is the bending moment due to the unit moment l i at the plastic hinge i in the primary statically determinate structure; qJ is the curvature caused by the external loads in the statically indeterminate structure; the curvature tp includes an elastic component, q~, a reduced-elastic component after cracking, q~t, and a plastic component which, after prestressing steel yields or concrete strain, is equal to 0.002, whichever reaches it first, q~n.

Moment-curvature relationship. Various types of analytical moment-curvature relationships have been proposed

O)

H

~

J,

4,

Dynamic modelling To perform a dynamic analysis of prestressed concrete frames, the primary curve and the theoretical hysteresis loops need to be combined with the dynamic modeP 7. The primary curve (also called the horizontal force-deflection curve) can be found using the trilinear theory ~'~3. Hysteresis loops which govern the relationship between the deflection and the horizontal force during repeated loading and unloading must be described. Finally, the motion of the prestressed concrete frame is computed using a dynamic model based on the linear acceleration method TM. Figure 1 indicates the data structure used in the dynamic analysis.

a

~1

2"

3'

4'

Primary curves The trilinear theory for reinforced/prestressed concrete structures ~.~3 consists of four parts and is briefly summarized below.

Compatibility condition. To illustrate the compatibility condition, a typical continuous prestressed concrete frame

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Engng Struct. 1994, Volume 16, Number 8

Figure2 Statically indeterminate frame to third degree. (a) vertical and horizontal loads; (b) plastic deformation regions

Dynamic response of prestressed concrete frames: Y. L. Mo and W. L. Hwang for concrete. Of these, the trilinear moment-curvature relationship has been shown to provide the best predictions for reinforced concrete frames 2. This trilinear moment-curvature relationship will, therefore, be employed for the behaviour of prestressed concrete frames (Figure 3). The first branch of the trilinear curve represents the behaviour of the prestressed concrete section up to flexural cracking (M~,qt,.). The second branch describes the behaviour from the cracking-up until the longitudinal steel yields or the concrete strain is equal to 0.002 (My, tpy). The third branch gives the postyield behaviour up to failure (M,,tp,). For a given cross-section the shape of the moment--curvature may be determined by using the principles of mechanics. The parabola-rectangle stress-strain curve of concrete specified in the CEB code t9 and the elastic-plastic stress-strain curve of steel are used in the computation.

/3 =

My

(5)

E l ( M y - Me)

When My < M -< M,, the third term of equation ( 1) can be substituted as follows

f Mil~ III =

0: II

=

f MiT(M- My) ds

E

M.

]

I~1u "l- /311//c- ~ y (l~//y + /31~//,.)

7=

M , - M,,

/3, = +yEI - M,,

Substituting equations (2), (4) and (6) into equation (1), the compatibility condition for prestressed concrete frames is as follows

r

(2)

aM ds + f Mi/3(M - 3/1,.)ds + f M i y ( M - M y ) ds=O

where

(9)

Solution procedures. The difficulties in solving equation (9) are avoided by making the following assumptions.

(3)

( 1) The plastic rotations concentrate on the critical sections (2) The moment-curvature curve of any section in the regions of the same sign of moments may be found through interpolation between the sections with the greatest and least eccentricities of prestressing steel

and E1 is the flexural rigidity of the uncracked prestressed concrete section. When Mc < M <- My, the second term of equation (1) can be substituted as follows

f M, t)n ds = 0]' = f Mi~(M - M,.) ds

(7) (8)

M,,- mc

in prestressed concrete frames the trilinear momentcurvature curves, as shown in Figure 3, will be used 13'14. The general equations are introduced below. When M <- M,, the first term of equation ( 1 ) can be substituted as follows

M, a=~-~

(6)

where

Determination of rotations. To determine the rotations

f M, q/ds = O]= f c~M ds

~E/-

Assumption ( 1 ) means that Mi in the critical sections has certain values which are mutually related. In addition, as shown in Figure 4, in the region i with plastic hinge, Mi is a unit moment (i.e., Mi = l). In the region i' without

(4)

where

Mtj Mv

I i

t11t

11/

MC p

-4 I ,

I

"

/

4

It-

q'c

~v

~' u

Figure 3 Trilinear moment-curvature relationship

Engng Struct. 1994, V o l u m e 16, N u m b e r 8

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Dynamic response of prestressed concrete frames: Y. L. Mo and W. L. Hwang

Mi -li-1

f

°i

cplII q)II

a

q)II

b

i'

i

Figure 4 Diagram of components of curvature. (a) in large deformation region /' without plastic hinge (e.g. 2' in Figure 2(b)); (b) in plastic hinge region i (e.g. 2 in Figure 2(b)) plastic hinge, M~ is a constant (i.e. Mi = M~~, where M~~ is the bending moment of the section i' due to a unit moment at plastic hinge i in the primary statically determinate structures). Therefore, equation (9) can be simplified to

f aMds+(M~ ~+ (f [3(M-M,)ds

1) + T(M-

M,)ds)=O

(10)

Equation (10) can be evaluated for every plastic hinge of the primary statically determinate structure. This should produce a set of n simultaneous compatibility equations in an n-degree statically indeterminate prestressed concrete frame. Assumption (2) means that/3 and ~/in equation (10) behave linearly along the member. In other words, this assumption is correct if the layout of the cables is linear and not curved. To predict the horizontal force-deflection curve the solution procedures are shown in Figure 5, and detailed explanations are reported in References 13 and 14.

Hysteresis loops A nonlinear dynamic analysis of prestressed concrete frames requires modelling for the horizontal force--deflec-

580

E n g n g Struct. 1994, V o l u m e 16, N u m b e r 8

tion relationship under load reversals. A qualitative model for reinforced concrete was developed by Clough 2° incorporating the stiffness degradation in the elastoplastic model. The response point during loading moved toward the previous maximum response point. The unloading slope remained parallel to the initial elastic slope. This modification improved the capability to simulate the flexural behaviour of the reinforced concrete. Compared with the elastoplastic model, less energy is absorbed per cycle beyond yielding by Clough's degrading model. A more refined and sophisticated hysteresis model was developed by Takeda et al.2~ for reinforced concrete on the basis of experimental observation. This model included stiffness changes at flexural cracking and yielding and also strain-hardening characteristics. The unloading stiffness was reduced by an exponential function of the previous maximum deformation. Takeda et al.2j also prepared a set of rules for load reversals within the outermost hysteresis loop. Frames incorporating precast prestressed concrete beam shells have been tested at the University of Canterbury ~6. These frames were designed for seismic loading, and exhibited very satisfactory strength and ductility characteristics. In addition, the hysteresis loops were not pinched, indicating satisfactory energy dissipation characteristics. In this study, it is also proved that extensive sliding shear

Dynamic response of prestressed concrete frames: Y. L. Me and W. L. Hwang

I

I

Find critical sections

I J

Determine mechanical properties of sections

r'----~

Select moment M1 for first critical section

Assume n moments for other n critical sections

Plot M-, a-,/3-, and y-diagrams

Calculate rotations

Are the n compatibility equations satisfied?

~ Yes I

A set of solutions

[ I

Is moment M1 larger than ultimate moment Mu?

Figure 5 Solution procedure for primary curve

displacements occurred along the vertical cracks in the beam at the face of the column as the test progressed, which was the main reason for the pinched hysteresis loops, if the frames were not designed for seismic loading, e.g. the potential plastic hinge region was not detailed with closely spaced stirrup ties for ductility. In other words, as long as prestressed concrete frames are designed to resist dynamic loads, the rules proposed by Takeda et al. 2~ can be employed to generate the hysteresis loops. Therefore, Takeda's rules will be used here. D y n a m i c analysis

A prestressed concrete frame is modelled as a single degree-of-freedom system for the purpose of the dynamic analysis. Figure 6 describes this system conceptually. The equation of motion is mfli 4- ci.~i q- kiy i = - m ~ g i

(11)

where m is the mass at the top of the frame, k is the stiffness, and c~ is the damping coefficient at time point i. They are assumed to remain constant during the time interval t~ to t~.l. The incremental quantities Ayi, m~i, and Ay~ are defined by Ayi = Yi÷l - Yi

(12)

Api = Pi+1 - Pi

(13)

Ayi = Yi+l - Yi

(14)

The ground is considered to be the point of reference even though it is in motion during the earthquake. The incremental acceleration of the ground is defined by ~Xy., = ?~.+,) - y~,

(15)

and obtained from the seismogram by linearly interpolating between the points. Subtracting equation ( 11 ) for i + 1 from the same equation with i produces the equation of equilibrium mAyi + ci Api + ki Ayi = - m A y g i

(16)

To solve equation (16), the linear acceleration method described in Reference 18 is used. In this method, the acceleration is expressed by a linear function of time during the time interval At. This approximation is valid when At = ti÷~ - ti is small. According to the NRC regulatory guide22, the damping coefficient ci is 2% of the critical damping for prestressed concrete structures under an operating basis earthquake. In general, sufficiently accurate results can be obtained if the time interval is no longer than one-tenth of the natural period of the structure.

Engng Struct. 1994, Volume 16, Number 8

581

Dynamic response of prestressed concrete frames: Y. L. Mo and W. L. Hwang

/

/

/

Mass

/

/

/

Form primary curve

/

I

Initialize values

]

Calculate displacement it

I Findlateralforce

I

Yg /

~

---..--~y

/

Spring

Calculate velocity and acceleration Mass

I Calculate effective stiffness /

Damper

/

I Calculate effective lateral force_l

b

Find i n c r e m e n t a l d i s p l a c e m e n t , I

velocity and acceleration k i AY i ~

mA~;it

I Output time, lateral force and displacement I c i AYi Figure 7 Algorithm for dynamic analysis of step-by-step sol-

e

A J)it=AS, i+zS~g i

Figure6 Prestressed concrete frame as a single degree-offreedom system

Algorithm The algorithm for numerically computing the solution is shown in Figure 7. (1) Form the primary curve (2) Initialize i, t, y, 29,Y, Ay, and A29to 0. This corresponds to the observation that the frame is stationary before the earthquake begins (3) Calculate the displacement y (4) Calculate the horizontal force (5) Calculate the velocity 29 and the acceleration y at the end of the time interval (6) Calculate the effective stiffness (7) Calculate the effective force (8) Calculate the incremental displacement, incremental velocity, and incremental acceleration at the end of the time interval (9) Output the time t, horizontal force, and displacement y (10) Repeat steps (2)-(10) for each time interval This algorithm has been implemented on a personal computer in the C language using the MSDOS operating system.

582 Engng Struct. 1994, Volume 16, Number 8

ution

Numerical example As an example, a prestressed concrete frame, shown in Figure 8, is subjected to the 1940 E1 Centro earthquake. The height is 20 m, the span is 40 m, the cross-section is 80 × 170 cm 2 for the beam, 120 × 210 cm 2 for the columns, and the mass supported by the frame is 251.5 kN s2/mm. The assumed physical constants are the yielding stress of steel fv = 412 MPa, the yielding stress of prestressing steel fpy 1578 MPa, the concrete strength f~' = 41.2 MPa for the beam and 34.3 MPa for the columns and the effective prestress f,e = 0.81 fpy. The seismic response of the frame is shown in Figure 9. It shows the computed hysteresis loops of the frame. For this frame, the cracking, yielding and ultimate loads are 216 kN, 785 kN and 1040 kN, respectively. Because of the cracking and yielding, the response is highly inelastic and when the horizontal force is close to the ultimate load where the frame fails, the energy absorbing characteristics are clearly demonstrated. The dynamic inelastic response calculated by the proposed method is plotted in Figures 10 and 11. Figure 10 indicates the deflection history while Figure 11 shows the horizontal force history. The linear elastic response obtained by a similar step-by-step analysis is also plotted as a comparison. The effect of the inelastic response on the deflection and horizontal force shows up clearly in this comparison. =

Dynamic response of prestressed concrete frames: Y. L. Me and W. L. Hwang

1.4 I-"

4000 cm

~1 "-I 50¢m 120cm

~-t

200 cm {

"i

/\ I~

4

II

210 cm

4

80 cm

w!

..t.": 10cm ~ 1 ~

I~10cm

~

S-32 i T (As = 65.36 cm 2)

~2o

Lit::

J}J

16-32 @ each side ( As = 130.72 cm2 )

I[ .... 10o *--,--~--,,-

Section l-I and 4-4

* 9 prestressing strands Aps = 9 * 11.845 =106.6 cmz

f'c f

pu

= 34.3 41.2 =

Es = f-py

fy

1862 Mpa 2.06"105

Mpa

= 1578 Mpa =

Section 2-2 and 3-3

Mpa for c o l u m n s Mpa for beam

412

• prestressing strands ( Aps ) • rebars(As)

•

Mpa

Figure 8 Dimensions and material properties of prestressed concrete frame - - - INELASTIC -- ELASTIC 1888.88 ..... ULTIMATE ..... ....

P~IMARY CURVE ~ SEISMIC RESPONSE / ;

YIELDING

8.oo~"i!

~ z

R / - C~ACK~N6

0.88

~

/

'i/

L Ld 0

CRACKINO

- "40.08

-500.80 ..-

.5-" ~08,80

---

ULTIMATE -~8.88

8.08 58.88 DEFLECTION (CM)

8 0 . 8 0 rl,,,Tir ,I,,,,,,,i,r,rr,,, 8.00 2.@8 4.80 188.88

Figure 9 Seismic response of prestressed concrete frame

,,,,,,,,rt,,,,,,,i,iil,,r,,,,

6.~@ B.80 TIME [ SEC)

I

18.00

12.80

Figure 10 Deflection history for El Centre records, scaled to O.5g

E n g n g Struct. 1994, V o l u m e

16, N u m b e r 8

583

Dynamic response of prestressed concrete frames: Y. L. Mo and W. L. Hwang 2@@e,@@~ INL_ ASTIC EL AST IC f;

2~z

,':,

5

1@@@.@@2

6

A

L~

:':: ~i~

i

4

:

ixx,

0£ 0 J_

, ; 7

\

"i i/,'

© N

;J

'~t_y,/

8

Q

9 10

11 %; Horizontal force h i s t o r y for El Centro r e c o r d s , l I M[

Figure 71

( %t

scaled

12

t o 0.5 g

13

Conclusions An algorithm for computing the inelastic response of prestressed concrete frames to dynamic loads is presented, and a numerical example is illustrated to demonstrate the applicability. Although the proposed theory appears to be quite promising, the absence of systematic tests of prestressed concrete frames subjected to dynamic loads precludes an experimental check of the calculated dynamic response. Systematic tests of such frames are therefore needed. References 1 Mo, Y. L. 'Zur Ermittlung der Momentenumlagemng in StahlbetonRahmen tragwerken', Dissertation, Institut ftir Massivbau, Universit~it Hannover, West Germany, 1982 2 Mo, Y. L. 'Moment redistribution in reinforced concrete frames', J. ACI 1986, 83 (4), 577-587 3 Walther, R. 'Behandlung yon Zwangsschnittgr6[3en beim Bruchsicherheits-nachweis yon Betonbauten, herausgegeben vom Deutschen Beton-Verein', e.V. Wiesbaden, West Germany, 1973

584

Engng Struct. 1994, Volume 16, Number 8

14

15 16 17 18 19

20

21

22

Huber, A. 'Zur Frage der Zwangschnittkr~ifte aus Vorspannung und Deren Einflu[3 auf die Sicherheit der Tragwerke', Beton-undStahlbetonbau, 1983, 78 (3), 69-73 Lin, T. Y. 'Strength of continuous prestressed concrete beams under static and repeated loads', ACI J. 1955, 52, 1037--1059 Mattock, A. H., Yamazaki, J. and Kattula, B. T. 'Comparative study of prestressed concrete beams with and without bond', ACI J. 1971, 68, 116-125 Morice, P. B. and Lewis, H. E. 'The ultimate strength of two-span continuous prestressed concrete beams as affected by tendon transtormation and untensioned steel', FIP Second Con eress, Amsterdam, 1955 Guyon, Y. Prestressed Concrete Vol 2, Contractors Record, London, 1960 Mallick, S. K. 'Redistribution of moments in two-span prestressed concrete beams', Mag. Concrete Res. 1962, 42, 171-183 Mallick, S. K. and Sastry, M. K. L. N. 'Redistribution of moments in prestressed concrete continuous beams', Mag. Concrete Res. 1966, 57, 207-220 Cohn, M. Z. and Frostig, Y. 'Inelastic behavior of continuous prestressed concrete beams', J. Struct. Div ASCE 1983, 109, 2292-2309 Aalami, B. O. 'Load balancing: a comprehensive solution to posttension', ACI Struct. J. 1990, 87, 662-670 Mo, Y. L. 'Investigation of reinforced concrete frame behavior theory and tests', Mag. Concrete Res. 1992, 44, 163-173 Mo, Y. L. and Chien, C. F. 'Response of prestressed concrete continuous beams to static loads', Mag. Concrete Res. 1993, 45 (163), 89-96 Mo, Y. L. 'Zur Ermittlung der Momentenumlagerung in SpannbetonTragwerken', Beton-und-Stahlbetonbau, 1993, 88 (10), 262-266 Park, R. and Bull, D. K. 'Seismic resistance of frames incorporating precast prestressed concrete beam shells', PCI J, 1986, 31, 54-77 Mo, Y. L. and Jost, S. D. 'Tool for dynamic analysis of reinforced concrete framed shearwalls', Comput. Struct. 1993, 46 (4), 659-667 Paz, M. Structural Dynamics-Theory and Computation, Van Nostrand Reinhold, New York, 1991 CEB-F1P Model Code for Concrete Structures, (3rd Edn) Comitd Euro-lnternational du B6ton/F6ddration Internationale de la Pr6contrainte, Paris, 1978 Clough, R. W. 'Effect of stiffness degradation on earthquake ductility requirements', Structural and Materials Research, Structural Engineering Laboratory, University of California, Berkeley, CA, Report 66-16, 1966 Takeda, T., Sozen, M. A. and Neilsen, N. N. 'Reintorced concrete response to simulated earthquakes', J. Struct. Div., ASCE 1970, 96 (ST12), 2557-2573 Regulatory Guide 1.61, Washington, DC, Nuclear Regulatory Commission, 1973