Energy dissipation in R.C. Beams under cyclic loadings

Engineering

Fraerure

Mechanics

Vol. 39, No. 2, pp. 177-184, 1991

0013-7944/91 $3.00 + 0.00 Pergamon Press pk.

Printed in Great Britain.

ENERGY

DISSIPATION IN R.C. BEAMS CYCLIC LOADINGS

UNDER

ANDREA CARPINTERI Istituto

di Scienza e Tecnica delle Costruzioni, Universitl di Padova, Via Marzolo 9, 35131 Padova, Italy

Abstract-A reinforced concrete beam cross section subjected to unidirectional cyclic bending moment is analysed. A fracture mechanics model explains the hysteretic phenomenon which occurs when the maximum bending moment is greater than twice the bending moment of slippage or yielding of the steel reinforcement. The critical relative crack depth &a above which the hysteretic phenomenon cannot occur can be determined for each value of the parameter N,, N, being a dimensionless number dependent on the mechanical and geometrical properties of the reinforced concrete beam cross section. A numerical procedure can follow the fatigue crack growth and calculate the total energy dissipation until an unstable concrete fracture phenomenon occurs.

INTRODUCTION SEVERAL authors have already studied the hysteretic behavior of reinforced concrete beams under cyclic loadings[l-71. A fracture mechanics model proposed in refs [8-lo] analyses the phenomena which occur in a reinforced concrete beam cross section subjected to unidirectional cyclic bending moment when the maximum bending moment M is greater than the bending moment Mp of slippage or yielding of the steel reinforcement[l l] and lower than the unstable concrete fracture value MF. It can be verified that an elastic shake-down due to slippage or plastic deformation of the steel bars occurs for Mp < M < Ms., with M,, = 2&f, = shake-down bending moment; when the phenomenon is plastic and dissipative, that is, the steel stress-strain curve shows M>M,,, hysteretic loops and energy dissipation. In the present paper the diagram to against Np is obtained, rcR being the critical relative crack depth, i.e. the relative crack depth r for which M,, = MF. The parameter Np is dependent on the mechanical and geometrical properties of the reinforced concrete beam cross section. The phenomenon of plastic shake-down cannot occur for 5 > &, because it is preceded by an unstable concrete fracture, MF being lower than or equal to Mso. It is possible to verify that the critical relative crack depth decreases by increasing the dimensionless number Np. Therefore, the plastic fatigue can occur only for very low values of the relative crack depth if the value Np is large. Moreover, it can be remarked that the critical bending moment MCR increases by increasing the dimensionless number Np, MCR being the value of the bending moments MS0 and MF for r = (CR. A numerical procedure can follow the fatigue crack growth and calculate the total energy dissipation. As a matter of fact, if an untracked cross section is subjected to a cyclic bending moment, fracturing of concrete can occur and the energy dissipated in the steel reinforcement can be computed for each loading cycle from the moment-rotation diagram. When the crack depth increases, the hardening line of the above-mentioned diagram becomes more and more inclined, as shown in the following sections. Finally, an unstable concrete fracture phenomenon occurs when the value & is reached, rF being the relative crack depth for which the maximum value of the cyclic bending moment is equal to MF.

ELASTIC AND PLASTIC

SHAKE-DOWN

A reinforced concrete beam cross section subjected to unidirectional cyclic bending moment is considered (Fig. 1). A through-thickness edge crack of depth a 2 h is assumed to exist in the stretched part, h being the steel’ reinforcement distance from the external surface. EFM ,9,2--A

177

A. CARPINTERI

178

TIME

0 Fig. 1. Cracked reinforced concrete beam cross section, subjected to unidirectional moment.

cyclic bending

Applying a rotation congruence condition[l 11, the bending moment of steel plastic flow due to yielding or slippage can be calculated:

where Fp can indicate either the force of yielding@, u, = steel yield strength; A, = steel area) or the force of pulling-out, when the latter is lower than the former, and r(5) is a function of the relative crack depth 5 = u/b. If the section is loaded with a maximum bending moment M greater than or equal to M, and then is unloaded, a shake-down phenomenon will occur. The unknown steel compression F after unloading can be obtained[&lO] and then the moment of plastic shake-down MS0 can be computed as the lowest maximum moment for which F = Fp, that is, for which the reinforcement yields even in compression:

and in dimensionless form: (3) where: Np=

p F K IC b”2t

_W2AS K IC

A

(4)

with A = bt = cross section area and K,c = concrete fracture toughness. The shake-down phenomenon is elastic for M, -=cM < MS, (Fig. 2); the shake-down is plastic, i.e. the steel stress-strain curve presents hysteretic loops, when MSD< M < MF. The unstable fracture bending moment MF can be calculated by equalling the stress-intensity factor expression[ 121 to the concrete fracture toughness and, consequently, the dimensionless fracture moment A?, is: (5) where YM(5) and Y&c) are functions of the relative crack depth[ll]. The dimensionless bending moments fiS. and I@, against the relative crack depth are shown in Fig. 3 for h/b = 0.1 and Np = 0.3. It can be remarked that as,, increases by increasing {, while fi, decreases by increasing <. Therefore, the two curves intersect in a critical point, the coordinates of which are tCR= 0.245 and ficR = 0.364. The phenomenon of plastic shake-down cannot occur for5 >LR, because it is preceded by the unstable concrete fracture, &#Fbeing lower than or equal For different h/b and Np values it is possible to obtain tiSs, and fi, curves qualitatively to &. analogous to those in Fig. 3. The diagram of the critical relative crack depth cCRagainst the dimensionless number NP is shown in Fig. 4, varying the parameter h/b. It can be pointed out that rCRdecreases by increasing

179

Energy dissipation in R. C. beams

ROTATION

Fig. 2. Elastic shake-down (for M, < A4 Q MS,) and plastic shake-down (for MS0 < M < MF).

Np. Therefore, the plastic shake-down can occur only for very low 5 if Np is large. It can lx seen from eq. (4) that the dimensionless number Np is large if the cross section depth b and/or the steel percentage ,4,/A are large. The same result can be reached in the case of high steel yield strength and/or low concrete fracture toughness. It can be noticed that the curve for h/b = 0.05 is lower than that for h/b = 0.10 (Fig. 4), but the two curves are very close. Finally, it is possible to verify that the critical dimensionless bending moment i@c, increases by increasing Np (Fig. 5), the curve for h/b = 0.05 being higher than that for h/b = 0.10. However, the two curves are very close, as occurs for the curves LjcRagainst Np (Fig. 4). ENERGY DISSIPATION On the basis of what has previously been stated, it is possible to follow the fatigue crack growth and calculate the total energy dissipation during the hysteretic phenomenon. As a matter of fact,

A

I

0.0 0.0

I

I

I

+

OS2 kR OS4 Oe6 RELATIVE

CRACK

DEPTH, 6

Fig. 3. Dimensionless bending moments tiSD and fiF against relative crack depth 5, for h/b = 0.1 and NP = 0.3.

180

A. CARPINTERI

w.F I; $j :: d u g F 4 E J $ F z *

A

0.6-

0.4-

0.2-

1 0.1

0.0 0.0

I 0.2

I 0.3

DIMENSIONLESS

I 0.4

NUMBER,

I 0.5

b

N,

Fig. 4. Critical relative crack depth <,-R against dimensionless number N,, varying parameter h/b.

when an untracked cross section is subjected to a cyclic bending moment, fracturing of concrete can occur. If the initial relative crack depth is assumed to be equal to l,, the slope of the hardening line in Fig. 6 is [l 11: 1 b2tE -= (6) 1MM 2 r ’ I”,(<> d5 ’ Jo where E is the Young modulus of concrete and 5 is equal to r, Therefore, the energy dissipated in each cycle is equal to the area l-2-3-4: work = ~wL.fM(M - Mso), cycle

(7)

__

where MS, and A,,,, are calculated from eqs (2) and (6) 5 being equal to 5,.

DIMENSIONLESS

NUMBER,

Np

Fig. 5. Critical dimensionless bending moment fi,-R against dimensionless number Np, varying parameter h/b.

Energy dissipation in R. C. beams

181

7

3 ROTATION

Fig. 6. Fatigue crack growth from (, to & and hysteretic loops in moment-rotation

diagram.

When 4 increases up to &, the bending moment of steel plastic flow increases very little and the hardening line becomes more inclined. The energy dissipated per cycle in the new configuration is equal to the area S-6-7-8 and can be obtained from eqs (2), (6) and (7), 5 being equal to <, . Then, r can still increase and the energy dissipated in the steel reinforcement can be computed at each step, the hardening line being more and more inclined. If the dimensionless maximum bending moment A? is greater than or equal to tiCR (Fig. 7), the fatigue crack growth occurs from the initial relative crack depth 4, to the unstable fracture value &, the shake-down phenomenon being plastic during the whole crack propagation, Therefore, a hysteretic loop is described for each loading cycle and the total energy dissipation can be computed by adding all the values of dissipated energy per cycle (see eq. (7)) from <, to {, (Fig. 7). If fi < aCCR(Fig. 8), the fatigue crack growth occurs from <, to & as in the previous case, but the shake-down is plastic only up to tSo. Therefore, the energy dissipation is calculated from 4, to &, because the loop described in each loading cycle degenerates into a segment for

h RELATIVE

CRACK

DEPTH

ROTATION

Fig. 7. Case I@ & ficR: plastic shake-down from 5, to &.

A. CARPINTERI

RELATIVE

Paris-Erdogan

ROTATION

CRACK DEPTH

Fig. 8. Case fi < w,,:

plastic shake-down from 5, to &,

law[l3] is utilized to follow the fatigue crack growth: da/dN = A(AK,)“,

(f-9

where N = number of cycles, AK, = stress-intensity factor range, A and m = material constants. Several authors have experimentally determined A and m for concrete[l4-161. In the present paper the values proposed in ref. [16] are considered: A = 7.71 x lO-25 m = 3.12, for crack propagation

(9)

rate da/dN in m cycle-’ and stress-intensity factor range AK, in N/m”‘. NUMERICAL

RESULTS

Consider a reinforced concrete beam cross section with t = 0.2 m, h = 0.3 m, a = h = 0.03 m (Fig. 1). The concrete properties E and K,= are assumed to be equal to 3.21 x 10’0N/m2 and 1.75 x lo6 N/m312respectively. If the steel reinforcement consists of two bars with diameter 4 = 10 mm and yield strength fY = 373 N/mm*, the parameter N, is equal to about 0.3. Different values Np can be obtained, for instance, by varying the steel reinforcement. As an example, three values of Np are considered: 0.1, 0.2 and 0.3. The critical bending moments acR related to these three values are 0.139, 0.255 and 0.364 respectively (see Fig. 5). Figure 9 shows fatigue life (continuous curves) against dimensionless maximum bending moment A?, the former being the number of loading cycles from the initial relative crack depth 5, = 0.1 to the unstable fracture value tF. It can be remarked that fatigue life obviously decreases by increasing i@, while, for a certain value of fi, it increases by increasing N,. The dashed curves in Fig. 9 represent the number of cycles for which the cross section behavior is hysteretic. These curves coincide with the continuous ones for fi 2 fiCRr according to what has been deduced in the previous section. The energy dissipated per cycle during the fatigue crack growth is shown in Fig. 10 for values fi. The total energy dissipated during the plastic shake-down can be obtained for each value A: as a matter of fact, it is the area under the curve related to the value fi considered (Fig. 10). The total dissipated energy against the dimensionless maximum bending moment is shown in Fig. 11. It can be remarked that the maximum value of energy dissipation decreases by increasing Np, while the value of h? for which the above-mentioned maximum occurs increases by increasing

Energy dissipation in R. C. beams

183

t \

N, =O.l I

IT,\\ I

-

10"

\

r

-

NP

=

0.2

’

I

I

I

0.1

0.2

0.3

0.4

DIMENSIONLESS BENDING

+

MAXIMUM

MOMENT,

&l

Fig. 9. Fatigue life (continuous curves) and number of hysteretic cycles (dashed curves) against dimensionless maximum bending moment a, for different numbers N,, in the case c, = 0.1.

NP. For each value NP considered in Fig. 11, the abscissa of the maximum is approximately to 0.16 for N, = 0.1, 0.28 for N, = 0.2 and 0.39 for N, = 0.3.

equal

CONCLUSIONS If a reinforced concrete beam cross section is subjected to unidirectional cyclic bending moment, a hysteretic phenomenon can occur if the maximum value of the cyclic loading is greater than iUS0 = 2 M,, M, being the bending moment of slippage or yielding of the steel reinforcement, and lower than the unstable concrete fracture bending moment MF. The critical relative crack depth tCRabove which the hysteretic phenomenon cannot occur can be determined for each value of the parameter N,,, N, being a dimensionless number dependent on the mechanical and geometrical properties of the reinforced concrete cross section. By increasing NP, tCR decreases while A?,-, increases, the latter being the value of the dimensionless bending moments fiSD and I@~for < = &. 4

-E 0.4 z ?If: 0.3 0 2 0.2 E 3

0.1 0.0

I

I

500

1000

NUMBER

OF CYCLES

Fig. 10. Dissipated energy per cycle against number of cycles, for different values I$#,in the case N, = 0.2 and t,=O.l.

184

A. CARPINTERI

L

1

0.1

I

0.2

DIMENSIONLESS Fig. 11. Total dissipated

I

energy against

1

0.3 MAXIMUM

I

0.4 BENDING

b

0.5 MOMENT,

dimensionless maximum bending Np, in the case l, = 0.1.

iii

moment,

for different

numbers

A numerical procedure can follow the fatigue crack growth under the considered cyclic loading and calculate the total energy dissipation during the hysteretic phenomenon. It can be noticed that the maximum value of energy dissipation decreases by increasing NP. The dimensionless maximum bending moment for which the above-mentioned maximum of energy dissipation occurs increases by increasing N, and, for each value N, considered, is slightly greater than fiCR. Acknowledgements-The author gratefully acknowledges the research support Technological and Scientific Research (M.U.R.S.T.) and the Italian National

of the Italian Ministry for University Research Council (C.N.R.).

and

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