NONLINEAR MODELING AND SEISMIC ANALYSIS OF MASONRY SHEAR WALLS

By Parviz Soroushian,1 Associate Member, ASCE, Kienuwa Obaseki,2 and Ki-Bong Choi3

ABSTRACT: Empirical hysteretic models are developed for masonry shear walls using the results of cyclic tests performed by different investigators on 37 single-story walls. The models account for the deteriorating nature of the response of masonry shear walls to cyclic loads, and distinguish between the hysteretic characteristics of walls with shear or flexural modes of failure. These models are used to compute the nonlinear response of single-story reinforced masonry shear walls to 16 recorded earthquake ground motions with different intensities and frequency contents. The effects of the deteriorating nature of masonry shear wall hysteretic behavior on its nonlinear seismic response characteristics are investigated by comparing the computed responses of masonry walls and some hysteretically stable (e.g., elasto-plastic) systems. The effects of the wall dominant failure mode (shear or flexure) on its response to earthquakes are also evalu-ated. Suggestions are made for the required strength of masonry shear walls located in regions with different seismic risks.

INTRODUCTION

Among all of the major construction materials, masonry is one of the least understood. Considering that a major fraction of the earthquake fatalities worldwide result from the collapse of masonry buildings, it is apparent that more studies are needed in this area (1,12,14-16,18,22).

In load-bearing masonry construction, both gravity and lateral loads are usually carried by masonry walls. Under the lateral loads of earthquakes, masonry buildings may collapse due to the in-plane or out-of-plane failure of these walls (Fig. 1). This study is concerned with the in-plane behavior of single-story masonry walls under seismic loads. Only solid walls (with no opening) have been considered in this investigation.

The in-plane failure of masonry walls under lateral loads is dominated by either flexure or shear. If flexure dominates the behavior, some horizontal cracks will widen at the wall base [Fig. 2(a)], and the vertical steel crossing these cracks will generally yield prior to failure. Yielding of the vertical steel at the wall base is usually followed by a relatively long plateau on the lateral load-deformation (p-A) diagram of the wall [Fig. 2(b)], Failure in this case is finally due to the crushing of the compressive corner or rupture of the extreme tensile bars at the base.

'Asst. Prof, Dept. of Civ. and Envir. Engrg., Michigan State Univ., East Lansing, MI 48824.

2Doctoral Candidate, Dept. of Civ. and Envir. Engrg., Michigan State Univ., East Lansing, MI 48824.

3Doctoral Candidate, Dept. of Civ. and Envir. Engrg., Michigan State Univ., East Lansing, MI 48824.

Note. Discussion open until October 1, 1988. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on July 17, 1985. This paper is part of the Journal of Structural Engineering, Vol. 114, No. 5, May, 1988. ASCE, ISSN 0733-9445/88/0005-1106/$1.00 + $.15 per page. Paper No. 22445.

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Flexural Crack {out-of-Plane)

Shear Crack (In-Plane)

FIG. 1. Failure Modes of Load-Bearing Masonry Buildings Subjected to Seismic Loads

l l l l l l l l l I 1 I 1

I I I 1- 1 1 1

I I I I I I I

1 1 1 1 1 1 ' I

1 1 1 I I 1 1 1 1 "1 1 . I ' l l I 1 1 "- 1

I I

(a) ( b )

FIG. 2. Behavior of Walls with Flexural Failure: (a) Failure Mode; (b) Hysteretic Behavior

If the flexural strength of a masonry wall subjected to lateral loads exceeds its shear strength, then shear will dominate the failure. The tendency towards shear failure increases with increasing values of the gravity load, length-to-height ratio, and vertical reinforcement of the wall. Shear failure is more brittle than flexural failure, and it is marked by large diagonal cracks [Figs. 3(a and b)]. If the bond strength of the mortar joint is sufficiently large, then the diagonal crack indiscriminately crosses the masonry units and mortar [Fig. 3(a)], Otherwise, the crack results from sliding of the units over joints under shearing stresses [Fig. 3(b)]. A typical hysteretic behavior of reinforced walls with shear failure is shown in Fig. 3(c). This behavior, when compared with the flexural behavior, is more brittle, with a larger stiffness deterioration and a smaller energy-absorption capacity.

In this study, empirical hysteretic models were produced for typical reinforced-masonry shear walls with shear and flexural failure modes. These models were then used for the nonlinear seismic analysis of single-story load-bearing masonry buildings. The results of seismic analy-sis were used to derive conclusions regarding the effects of the distinct

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(a) (b)

FIG. 3. Behavior of Walls with Shear Failure: (a) Failure Mode with Tensile Cracking of Units; (b) Failure Mode with Sliding of Units; (c) Hysteretic Behavior

H / /-/ / U 1/ If, A ,? v-y 7

/;r). Angular deformation j> is defined as the ratio of the top lateral displace-ment A to the wall height h.

In the proposed model, the hysteretic envelope of reinforced-masonry

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FIG. 5. Hysteretic Model: (a) Envelope Curve; (h) Branch Curves (Loops)

shear walls is represented by the following equation, which is capable of simulating the behavior observed in tests [Fig. 5(a)]:

P = 1 f1

,/exp ( - - - ) + (! -DC/2 ~2f) exp 3/

2 V ^ T (f + 1) (1)

where p = \PlPuU\; f = l/

amplitude; as, a7 = variables deciding the deteriorations in area and stiffness of the loops, respectively, under repeated load cycles.

In the equations presented so far, lateral forces and angular defor-mations were generally normalized by their values at the ultimate lateral load. The angular deformation at ultimate load ,(/, is treated as one of the model variables to be derived empirically, and the ultimate lateral load corresponds to the smaller of the in-plane shear and flexural capacities of the wall.

The flexural and shear strengths of the reinforced-masonry shear walls were calculated following the conventional procedures used in masonry design (16). These procedures are comparable to those practiced in the strength method of reinforced-concrete design. The maximum error of these procedures in predicting test results was found to be below 25%.

Failure of the wall is assumed in this model to occur when an ultimate ductility factor is reached. Ductility factor/ is denned as the ratio of the angular deformation to its value at the ultimate lateral load. The maximum

TABLE 1. Properties of Walls with Shear Failure

Wall num-ber

(D 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Refer-ence

(2)

13

13

13

12

12

12

12

22

22

22

22

22

22

22

22

22

Height x length x thickness

(in.)

(3)

110 X 106 X 5.9

110 X 106 X 5.9

110 X.106 X 5.9

108 X 128 X 5.9

108 x 128 x 4.7

108 x 128 x 5.9

108 x 128 x 5.9

45 x 44 x 4.25

45 x 44 x 4.25

48 x 48 x 3.6

48 x 48 x 3.6

45 X 44 X 4.25

45 x 44 x 4.25

47 X 26 X 4.25

47 x 26 x 4.25

38 x 73 x 4.25

Compres-sive stress

(psi)

(4)

0.0

35.1

0.0

0.0

0.0

29.2

0.0

250.0

500.0

250.0

500.0

250.0

250.0

250.0

500.0

250.0

Units

(5)

Concrete block

Concrete block

Concrete block

Concrete block

Hollow brick

Concrete block

Concrete block

Hollow

brick Hollow

brick Concrete

block Concrete

block Hollow

brick Hollow

brick Hollow

brick Hollow

brick Hollow

brick

Masonry compres-

sive strength

(psi)

(6)

851

851

851

851

370

851

851

4,076

4,076

2,772

2,772

4,076

4,076

4,076

4,076

4,076

Grout compres-

sive

strength (psi)

(7)

2,559

' 2,559

2,559

3,768

2,872

3,527

3,449

2,800

2,800

2,800

2,800

2,800

2,800

2,800

2,800

2,800

Vertical reinforce-

ment ratio"

(8)

0.40

0.40

0.40

0.26

0.19

0.26

0.26

0.24

0.24

0.26

0.26

0.67

0.67

1.63

0.20

0.22

Horizonta reinforce-

ment ' ratio"

(9)

0.00

0.00

0.00

0.21

0.82

0.00

0.02

0.00

0.00

0.00

0.00

0.00

0.33

0.00

0.00

0.00

Steel yield

strength (psi) (10)

56,880

56,880

56,880

35,550

35,550

35,550

50,000

50,000

50,000

50,000

50,000

50,000

50,000

50,000

50,000

50,000

aValues on the gross cross-sectional area of the wall. NOTE: 1 in. = 2.54 cm.; 1 psi = 6.9 KN/m2.

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TABLE 2. Properties'of Walls with Flexurai Failure

Wall num-ber

(D 17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

Refer-ence

(2)

13

13

13

13

13

13

22

22

22

22

22

22

22

22

22

22

22

18

18

18

18.

Height x length x thickness (in.)

(3)

110 x 106 X 5.9

110 X 106 X 5.9

110 X 106 X 5.9

110 X 106 X 5.9

110 x 106 X 5.9

110 x 106 x 5.9

45 X 44 x 4.3

45 x 44 x 4.3

48 X 48 x 3.6

48 X 48 x 3.6

47 X 26 x 4.3

47 X 26 x 4.3

38 x 73 x 4.3

45 x 44 x 4.3

45 X 44 x 4.3

47 x 26 X 4.3

47 X 26 X 4.3

60 X 64 x 8.7

60 X 64 x 8.7

62 x 60 x 5.5

62 x 60 x 5.5

Compres-sive

stress (psi)

(4)

0.0

60.7

0.0

60.7

0.0

121.3

0.0

125.0

0.0

125.0

250.0

125.0

0.0

0.0

125.0

250.0

0.0

0.0

0.0

0.0

0.0

Units

(5)

Concrete block

Concrete block

Concrete block

Concrete block

Concrete block

Concrete block

Hollow brick

Hollow brick

Concrete block

Concrete block

Hollow brick

Hollow brick

Hollow brick

Hollow brick

Hollow brick

Hollow brick

Hollow brick

Solid brick

Solid brick

Hollow brick

Hollow brick

Masonry compres-

sive strength

(psi)8

(6)

851

851

851

851

851

851

4,076

4,076

2,772

2,772

4,076

4,076

4,076

4,076

4,076

4,076

4,076

2,129

2,314

1,835

1,815

Grout compres-

sive strength

(psi)

(7)

2,560

2,560

2,560

2,560

2,560

2,560

2,800

2,800

2,800

2,800

2,800

2,800

2,800

2,800

2,800

2,800

2,800

5,640

4,118

3,741

4,510

Vertical reinforce-

ment ratio"

(8)

0.07

0.07

0.07

0.07

0.25

0.25

0.24

0.24

0.26

0.26

0.20

0.20

0.22

0.24

0.24

0.20

0.20

0.63

0.22

0.07

0.54

Horizontal reinforce-

ment ratio8

(9)

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.41

0.15

0.77

0.29

Steel yield

strength (psi) (10)

56,880

56,880

56,880

56,880

56,880

56,880

50,000

50,000

50,000

50,000

50,000

50,000

50,000

60,000

60,000

60,000

60,000

64,380

43,790

61,625

42,340

aVaIues on the gross cross-sectional area of the wall. NOTE: 1 in. = 2.54 cm.; 1 psi = 6.9 KN/m2.

ductility factor at failure fmax is also treated as one of the model variables to be obtained from test results.

The model presented includes nine variables {a^-a-,, (j>, and/,,.), the values of which were found empirically using test results on 37 reinforced-masonry shear walls presented in Refs. 10, 12, 15, and 22. In 21 of these walls, failure was dominated by flexure, and in the remaining 16, shear dominated the failure. Tables 1 and 2 present the properties of these walls with shear and flexurai failure modes, respectively.

For each of the tested walls, parameters a,-a7, 4>, and fmax were

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- 1 6~\ , 1 , - , . 1 , 1 , 1 , 1 - 0 . 6 - 0 . 4 - 0 . 2 0.0 0.2 0.4 0.6

DEFLECTION (in) (b)

FIG. 6. Comparison of Proposed Model with Test Results: (a) Wall 15 with Shear Failure; (b) Wall 30 with Flexural Failure

derived by least-square fitting of the proposed equations to the experimen-tal hysteretic curves. Typical comparisons of the experimental and theo-retical hysteretic diagrams are shown in Fig. 6(a) for wall 15 of Table 1 (with a shear mode of failure) and in Fig. 6(b) for wall 30 of Table 2 (with a flexural failure mode).

Table 3 presents the means and coefficients of variation of the variables a,-fl7 , (j>, and fmax for walls with shear and flexural modes of failure (based on test results on 16 walls failing in shear shown in Table 1 and 21 walls failing in flexure shown in Table 2). The coefficients of variation of the parameters in each group of walls (with shear or flexural failure modes) are of the order of 65%. This means that for the number of tests considered, the standard error (defined as the coefficient of variation of the mean values) of each variable is about 15%.

In order to derive the hysteretic model of a single-story, reinforced-masonry shear wall using this formulation, first the dominant failure of the

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TABLE 3. Wall Hysteretic Parameters Derived from Test Results

Parameter

(1)

Mft, andfmax given in Table 3 for shear and flexural failure modes can then be used in the proposed formulations for hysteretic modeling of walls. Considering the observed variations in the

1.0

0 . 8

0 .6

0 .4

u 0 . 2

o,3 0 .0

* - 0 . 2

- 0 . 4

- 0 . 6

-0.8

- 1 . 0

1.0

0.8

0 .6

0 .4

0 .2

0 .0

- 0 . 2

- 0 . 4

- 0 . 6

- 0 . 8

0 .000

4 / h

(a)

0.000

A/h

FIG. 7. Typical Hysteretic Diagrams for Reinforced Masonry Shear Walls Built with 70-Percentile Values of Parameters Given in Table 3: (a) Walls with Shear Failure; (b) Walls with Flexural Failure

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values of hysteretic variables for different masonry shear walls, it was decided that instead of the mean values of the variables, their 70-percentile values [=mean - (standard deviation/2), assuming a normal distribution] should be used in hysteretic modeling. These smaller values of the variables are more conservative, in the sense that they make the hysteretic behavior more brittle with larger deteriorations and smaller energy-absorption capacities. The 70-percentile values of the variables are also shown in Table 3. Figs. l(a and b) show typical hysteretic diagrams built with the proposed models using the 70-percentile values of the variables for walls with shear and flexural modes of failure, respectively. The flexural wall, as expected, has higher ductility and energy-absorption capacity, and its deterioration under cyclic loads is also smaller when compared with the wall failing in shear.

SEISMIC ANALYSIS

In single-story, load-bearing masonry buildings, the seismic forces as well as the gravity loads are usually carried by the shear walls. In this study, the seismic responses of typical masonry shear walls with shear and flexural failure modes were computed and compared with those of struc-tural systems with simpler and less degrading hysteretic characteristics.

The single-story, load-bearing masonry building was idealized as a single-degree-of-freedom system, with its hysteretic characteristics corre-sponding to either the proposed masonry wall models [Figs. l{a and b)~\ or the more stable types of hysteretic behavior [elastoplastic, or the stiffness degrading type shown in Fig. 4(a)]. For the purpose of nonlinear dynamic analysis, a computer program was written that used the average acceler-ation version of the Newmark-p method (4,13) in a step-by-step numerical integration of the dynamic equilibrium equations. This program was used for finding the minimum in-plane lateral strength of a shear wall, with certain hysteretic characteristics and natural periods of vibration, required for survival under a specified earthquake ground motion. A wall is assumed to survive under an earthquake if its maximum ductility at failure is not exceeded. The natural period of vibration was based on the initial lateral stiffness of the wall.

16 earthquake ground-motion records (17) were chosen to represent the maximum-intensity earthquakes expected during the lifetime of structures in seismic zones 1-4 of the Uniform Building Code (5,6,20). Seismic zone 0 was excluded from this study because considerations other than the earthquakes govern the structural design in this zone. Four recorded earthquake ground motions were selected for each of the four seismic zones, such that the statistical variation in the earthquake characteristics were taken into account (21).

Fig. 8(a) shows typical response spectra of the masonry shear walls with shear mode failure and the corresponding linear response spectra of systems with 2% and 5% viscous damping ratios subjected to one of the earthquake ground motions in zone 3 (S70E component of the 1971 San Francisco earthquake recorded at Glendale, Calif. (17). Fig. 8(a) shows that the nonlinear hysteretic energy absorption substantially reduces the required lateral strength for survival. It may also be concluded from this figure that viscous damping seems to be less effective in the case of

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Nonlinear (Masonry Wall, Shear Failure)

GHT

"R

EN

GTH

1,2-

1.0-

0 .8-

0 . 6 -

0.4-^

A

0 0.2-j

0.0 0.5 I 1.0

NATURAL PERIOD (sec)

O 1 4.

5 1.2-1

o a: Lo-ts

O.B

0.6

0.4

0.2-1

<

ui O.O-tn 0,

Elaato Plastic Stiffnesa Degrading Masonry Wall With Shear Failure

- 1 0.5 1.0

NATURAL PERIOD (sec)

FIG. 9. Comparison of Acceleration Response Spectra Normalized with Respect to Peak Ground Accelerations

1.0-1 Wean + Std. Dev. Mean

- 0 . 2 5 0.00 0.25 0.50 0.75 LOO 1.25 1.50

NATURAL PERIOD (sec)

FIG. 10. Response Spectra of Walls with Shear Failure in Different Seismic Zones (5% Viscous Damping)

were compared: masonry walls with shear mode of failure, elastoplastic, and stiffness degrading [the elastoplastic and stiffness-degrading systems were assumed to fail at a displacement ductility factor of 3.0 (Fig. 3(a)]. The comparison (Fig. 9) was made using the acceleration response spectra of these systems normalized with respect to the peak ground acceleration. For the elastoplastic and stiffness-degrading systems, the normalized response spectra presented in Ref. 13 were used, while the shear wall response spectrum curve used in this comparison is the mean value of the normalized shear wall response to 16 earthquake ground motions con-sidered in this investigation. From Fig. 9, it may be concluded that for natural periods longer than about 0.2 sec, the acceleration spectral values for the elastoplastic system are larger than the corresponding values for the

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stiffness-degrading hysteretic model of shear walls, while at shorter periods, the spectral values of masonry shear walls are larger than those of the other systems. The peak acceleration of the deteriorating hysteretic model of shear walls is also observed to be reached at shorter natural periods when compared with the elastoplastic and stiffness-degrading systems.

These results can be justified considering the deteriorating nature of the shear wall hysteretic behavior which, on one hand, adversely influences the seismic response characteristics due to the lower inelastic energy-absorption capacity of the system and, on the other hand, increases the natural period of vibration that might have positive or negative effects on the response of the system to specific earthquake ground motions.

From this comparative study, one may also conclude that hysteretic modeling is a critical issue requiring careful consideration in nonlinear seismic analysis of the systems with high degrees of deterioration.

For each of the seismic zones, the values of mean and mean plus standard deviation of the minimum required lateral strengths of walls with a shear mode of failure and a 5% viscous damping ratio are shown in Fig. 10. This figure shows that, at shorter natural periods, the required strength for survival of typical masonry shear walls with shear failure in zones 1, 2, and 3 are about 20%, 35%, and 70% of that required in seismic zone 4.

SUMMARY AND CONCLUSION

Results of tests on 37 single-story, reinforced-masonry shear walls were used to derive typical hysteretic models for masonry shear walls failing in shear and in flexure. The proposed models consider the severe stiffness and strength degradations, as well as the low energy dissipation capacity and brittle failure of reinforced-masonry shear walls. From nonlinear seismic analysis of masonry shear walls, using the developed hysteretic models, it was concluded that: (1) The hysteretic behavior and energy absorption capacity of shear walls substantially reduced the seismic forces (when compared with the linear systems of similar initial periods of vibration); (2) viscous damping was less influential on the seismic response characteristics of the (nonlinear) shear walls, when compared with the linear systems; (3) under seismic excitations, the walls with shear mode of failure, have less ductility and energy dissipation capacity and larger deteriorations, required higher strengths than the comparable shear walls failing in flexure for survival under similar earthquakes; and (4) systems with a more stable hysteretic behavior than masonry shear walls did not necessarily perform better under earthquake ground motions. Large stiff-ness deteriorations of shear walls produced elongations in the natural period of vibration that positively influenced the response of certain shear walls to some earthquake ground motions.

The mean and mean plus standard deviation of the shear strength per unit weight of reinforced masonry shear walls with shear mode of failure, required for survival in different seismic zones (based on the computed responses under four earthquake ground motions in each zone), were also generated.

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ACKNOWLEDGMENTS

The writers wish to thank Drs. Floyd O. Slate and Peter Gregley for their contributions to this research project which was initiated at Cornell University and later completed at Michigan State University.

APPENDIX I. REFERENCES

1. Amrhein, J. E. (1983). Reinforced masonry engineering handbook. 4th ed., Masonry Institute of America, Los Angeles, Calif.

2. Anagnostopoulos, S. A., and Rosset, S. M. (1973). "Ductility requirements for some nonlinear systems subjected to earthquakes." Proc, 5th World Conf. on Earthquake Engineering, Rome, Italy, Vol. II, 1748-1751.

3. Bazan, E., and Meli, R. (1980). "Seismic analysis of structures with masonry walls." Proc, Seventh World Conf. on Earthquake Engineering, Istanbul, Turkey, Vol. V, 633-640.

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APPENDIX II. NOTATION

The following symbols are used in this paper:

A = area of hysteretic loop; a i - a 7 = parameters of hysteretic model;

h = height of wall; n = number of cycles repeated with constant amplitude; P = lateral load; R = tangent modulus of branch curve (loop) at zero lateral deflec-

tion;

1 = maJ$ult ! A = lateral deflection; p = LP/Pl;and (j) = angular deformation defined as ratio of lateral deflection to

wall height (A/h).

Subscripts max = value at maximum lateral deflection; and

ult = value at ultimate lateral load.

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