SMEARED CRACK MODEL FOR REINFORCED CON CRETE MASONRY SHEAR WALL A N ALYSIS

K . S . Ibrahim 1 and G. T . Suter 2

1. ABSTRACT Analytical models are needed to predict the behaviour of reinforced ma­

sonry buildings under different complex states of stress. Such models are useful to assess the behavioural seismic characteristics of masonry buildings undergo­ing progressive failure up to the ultimate stage. In this paper, an analytical model based on both a hypoelastic model for uncracked masonry and an orthog­onally anisotropic model for cracked masonry is presented and implemented in a general purpose finite element programo Cracks were modeled in a smeared fashion and reinforcement was modeled using smeared or discrete models. Due to local stress concentrations at a crack location, force transmitting mechanisms between masonry and reinforcement were modeled. Note that the analytical model presented in this paper will be referred to as the CUTTent model.

The performance of the model was verified using experimental data from rein­forced masonry shear wall panels. The load/ displacement relationship was traced up to the maximum lateralload resistance and beyond to study the softening be­haviour of the cracked reinforced masonry systems undergoing relatively ductile flexure-dominated and brittle shear-dominated progressive failures. Analytical results indicate good agreement with test evidence for the flexure-dominated panel. While the load/displacement relationship for the shear-dominated panel indicates some divergence from the experimental results, the model was success­fuI in capturing the softening behaviour of the system under increased controlled lateral displacement. AIso, the analytical results of the current model were com­pared with those of a plasticity model; good agreement was evident for both crack and yielding patterns.

Keywords: Reinforced Masonry, Shear Walls, Analytical Modeling, Smeared Crack, Finite Element.

1 Ph.D. Candidate, Civil Eng. Dept., Carleton Uni., Ottawa, Ont., Canada, lOS 5B6

2 Professor, Civil Eng. Dept. , Carleton Uni., Ottawa, Ont., Canada, lOS 5B6

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2. INTRODUCTION Analytical and experimental investigations of components, subassemblages,

and prototypes of reinforced masonry buildings are commonly used to provide a better understanding of principal behavioural characteristics and to help derive rational design models and codes. Since cost and laboratory facilities impose significant limitations on the extent of full-scale testing, current research efforts are mainly concentrated on studying experimentally the physical behaviour of masonry components and utilizing the experimental results to verify and validate sim pIe or com pIe x analytical models.

Typically, problems are highly nonlinear in nature and various mo deIs have been developed to trace associated behaviour. The finite element method offers a powerful tool for modeling complex structural systems under various loading patterns and tracing the complete nonlinear behaviour up to the ultimate stage. Using this technique, different physical characteristics of reinforced masonry such as post-cracking or post-crushing phenomena can be successfully modeled.

3. SCOPE AND OBJECTIVES A smeared crack mo deI approach has been used in finite element analyses of

concrete structures. While such a technique proved to be reliable and numerically efficient in many instances, its capability in predicting the mechanical behaviour for a variety of matrix-aggregate composites has not been thoroughly studied especially in the field of complex composite materiaIs such as reinforced masonry.

While numerous problems are associated with the smeared crack mo deI such as mesh-size dependency of numerical solution, failure to model elements with dominant shear failure, and spurious kinematic modes, it is still the simplest and most efficient approach to model the fracture behavior of reinforced ma­sonry shear walls exhibiting diffuse crack patterns. The main objective of the current study being undertaken at Carleton University is to evaluate the capabil­ity of smeared crack models in capturing various failure mechanisms of reinforced masonry shear walls undergoing different monotonic and cyclic loading patterns.

A smeared-crack finite element formulation adopting a hypoelastic model for uncracked masonry and nonlinear orthotropic constitutive mo deIs for cracked masonry has been utilized. In this numerical modeling, most of the current de­velopments in the field of reinforced masonry modeling, such as fixed, coaxial rotating, non-coaxial rotating crack representation, are incorporated into a finite element package. As part of this work, the Modified Compression Field Theory (MCFT) has been implemented; further, constitutive mo deIs for reinforced ma­sonry under tension and compression have been incorporated into the program and their significance on the numerical solution has been assessed. Post-tension cracking and post-compression crushing behaviour have also been modeled. Ad­ditionally, strain-softening behaviour and different tension-stiffening models are incorporated in the numerical solution. AIso, the influence of the anisotropy of partially grouted masonry has been considered.

4. ANALYTICAL MODEL ADINA is a general purpose finite element program for linear and nonlinear,

static and dynamic analysis which was originally developed at the Massachusetts Institute of Technology (Bathe, 1977). While the program has a lot of comput­ing capabilities and has been widely used in solving many practical problems,

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the program is limited when it deals with brittle material modeling such as concrete or masonry. Therefore, Vecchio and Collins's Modified Compression Field Theory was implemented in ADINA where a rotating smeared crack con­cept is employed. In the following subsections, different constitutive relations for uncracked/ cracked masonry and reinforcement along wi th clifferent strategies associated with the post-cracking and post-crushing behaviour modeling will be discussed.

4.1 Masonry Compressive Stress-Strain Relationships

A series of tests by Vecchio and Collins (Vecchio and Collins, 1986) have demonstrated that the concrete compressive strength is greatly reduced when coexisting principal tensile strains are high. This work resulted in the develop­ment of the relationship between the principal compressive stress and principal compressive strain for concrete as shown in Fig. 1. This model will be adapted for masonry where the principal compressive stress-strain law of grouted masonry (Im, é m ) was assumed as

(1)

where f' m and é o are the uniaxial peak compressive stress and corresponcling strain of grouted masonry and

f3 = 0.8 - 0.34( é m 1) 2: 1

é o

(2)

where éml is the tensile principal strain in the clirection normal to that being considered. In Eq. 2, the empirical constants were developed originally for concrete under biaxial states of stress. Since there are no detailed experimental data available for concrete masonry under biaxial states of stress and also since the masonry is fully grouted, it is reasonable to employ the same equation (Eq. 2) for concrete masonry.

4.2 Tension Stiffening Models for Masonry

Similar to concrete, as the masonry reaches its tensile strength, pnmary cracks form and their number and extent are controlled by the size and placement of the reinforcing steel. At the primary cracks, the masonry stress drops to zero and the steel carnes the full load. However, the masonry between cracks still maintains some tensile resistance which will drop as the load increases. The drop will be associated with a breakdown in bond as a secondary system of internaI cracks develops around the reinforcement.

Gopalaratnam and Shah (Gopalaratnam and Shah, 1985) have proposed an analytical model which assumes a unique stress-strain relationship only in the ascending part and a uni que relationship only between stress and crack width in the descending parto This was deduced from results obt ained from an experimental program that was carned out on different concrete , mortar, and paste specimens subjected to tensile stress.

For the ascending part , the following simple expression that has been used for defining compressive behaviour of concrete (Shah et al. , 1983) is recommended for the uniaxial tensile stress-strain curve up to the peak stress value:

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Em A 1m = Imer[l - (1 - -) 1 €mCT

where

1m = tensile stress

ImcT = peak value of 1m EmCT = value of strain, Em, at 1 mCT

A = E t EmcT /lmcT

Em = tensile strain Et = initial tangent modulus

(3)

For the descending part, no uni que stress-strain relationship existed due to the localized nature of the post-peak deformations. Instead, Gopalaratnam and Shah found a uni que relationslúp between average stress and average crack width in the form:

(4)

where w is the crack width, k and À are constants.

Now a new term, Iracture process zone strain, Ef' can be defined as the ratio between the sum of individual microcracks' openings, óf, and the crack band width W C :

and

Óf Ef=­

W c

where

W c = crack band width

TJ = constant (between 1.5 and 4)

da = maximum aggregate size.

(5)

(6)

Therefore, based on the results by Gopalaratnam and Shah and on the ex­isting relationslúp between the post-peak cracking stress and crack width, and also utilizing the crack band model of the fracture process zone, the total strain of a domain loaded in tension is equal to the sum of three main components as follows :

Et = Ee + Emp + Ef (7)

and

1m ImcT 111m!. Et = - + Emer - -- + -( --ln--)-"

Et Et W c k Imcr (8)

where the terms have been previously definedj the different strain components are illustrated in Fig. 2.

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Strain €m

Fig. 1 Stress-Strain Relationship for Cracked Masonry in Compression

f m •

mer

€.p/ Ef

r······························ ~~------------------------__ E

Fig. 2 Tension-Stiffening Model Based on Gopalaratnam and Shah Test Results

4.3 Transmitting Loads Across Cracks

Dealing with average stress in masonry or reinforcement results in ignoring local concentrations of stresses and, accordingly, force transmitting mechanisms between masonry and reinforcement. At a crack, the tensile stresses in rein­forcement are expected to be higher than average while midway between cracks the tensile stresses wiU be lower than average. On the other hand, the ma­sonry tensile stresses wiII be zero at a crack and higher than average midway between cracks. These local variations may govern the reinforcement's ability to transmit tension across the cracks and accordingly the ultimate capacity of a biaxially stressed element. Therefore, this load transmitting mechanism across the developing cracks was modeled and was implemented into the numerical solution.

4.4 Constitutive Relations

The biaxiaI constitutive equations for reinforced grouted masonry are based on Bathe's orthogonally anisotropic model (Bathe and Ramaswamy, 1979) and on Vecchio and Col!ins's Modified Compression Field Theory (Vecchio and CoUins, 1986). This model can be classified as a smeared crack model where fixed and rotating crack approaches are employed. Cracks are assumed to be weII dis­tributed across the elements. Both materiais, masonry and steel, are assumed to be perfectly bonded and subjected to the same strain field. The incrementaI stress-strain relation in the local n-s coordinates can be expressed as folIows:

dO" = D'déj DI = D'm + Dfs (9)

where DI m is the tangent constitutive matrix of grouted masonry, DI s is the tangent constitutive matrix of smeared reinforcement, O" = {O"n O"S O"ns}T and é = {én és éns } T.

4.4.1 Uncracked masonry

Let O"pl and O"p2 be the principal stresses with O"p2 ::::; O"pl and Epl and E p2 the corresponding uniaxial Young's moduli . The uncracked masonry is considered as isotropic with an equivalent multiaxial Young's modulus when subjected to tension or low compression j such an uncracked state is defi.ned by O" p2 2:: "-O" m

l .

The variable K, is typically 0.4 and the equivalent multiaxial Young's modulus , E, is then obtained using the folIowing weighting scheme:

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E = (Jp1Ep1 + (Jp2 E p2

(J pl + (J p2

and Poisson's ratio is assumed to be constant.

(la)

Then, the tangent stiffness matrix D for uncracked masonry is assumed to have the following form:

v 1 O ,~" 1 (ll )

If the masonry is under high compression, i.e., (Jp2 :::; K,(Jm', an orthotropic stress-strain matrix with the direction of orthotropy defined by the principal stress directions is employed. The stress-strain matrix corresponding to these directions is

l-V~E 1 -2- 12

(12)

where v is the constant Poisson's ratio, and the shear modulus in a coordinate plane is calculated from the weighted Young's modulus corresponding to that plane as

E 12

G12 = 2(1 + v)

4.4.2 Cracked masonry

1 (Jp1Ep1 + (Jp2Ep2

2(1 + v) (Jpl + (Jp2 (13)

Tensile failure occurs if the tensile stress in a principal stress direction ex­ceeds the tensile failure strength and a plane of failure develops perpendicular to the principal stress direction. This material failure will result in reductions in the normal and shear stiffnesses across the plane of failure and then, the cor­responding normal stress is released. Assuming that the principal stress (J pl is larger than the tensile failure stress, the new material stress-strain relation is

VT)n O 1 01 ?-V

T)s-2-

(14)

N ow, assuming that the principal stress (J p2 is larger than the tensile failure stress, the new material stress-strain relation that represents two perpendicular cracks is

VT)n

T)n

O (15)

where, typically, T)n = 0.001 and T)s = 0.5. The factor T)n is not set equal to zero in order to avoid the possibility of a singular stiffness matrix while the value employed for T)s depends on a number of physical factors.

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Rots and Blaauwendraad (Rots and Blaauwendraad, 1989) assumed for Tis a linear and nonlinear function of the tensile strain €l as follows:

_ ( _ €l - Ecr )P Tis - 1

Eu - Ecr (16)

in which €l is the crack normal strain at the beginning of the load increment , €cr is the strain at which a crack initiates, Eu is the tensile strain at wruch the shear retention factor is assumed to be zero, and p is a constant.

4.4.3 Reinforcement

The constitutive matrix for the reinforcement is given by:

PsE. (17)

in wruch Dls is the tangent constitutive matrix of smeared reinforcement, Pn and P. are the reinforcement ratios in the n, s directions, and En, E. are the tangent moduli of steel in the n, s directions obtained from a bilinear stress­strain relationsrup.

4.5 Calibration of the Finite Element Models

To evaluate the capability of smeared crack models , the in-plane resis t ance of two reinforced masonry shear wall panels tested by Srung et ai. (Srung et aI., 1988) is considered. The wall specimens were 6-ft long and 6-ft high , and they were constructed of fully-grauted 6 x 8 x 16-in. hollow concrete blocks , having five standard reinforcing bars uniformly distributed in the vertical as well as horizontal directions at a centre-to-centre spacing of 16 in. The lateral displacement was controlled by a horizontal actuator which applied symmetric cyclic reversals of gradually increasing amplitudes while the axialload was kept constant during each test.

Two test panels designated as specimens 5 and 12 in the aforementioned experimental study will be utilized to calibrate the finite element smeared crack models. Both wall panels were subjected to a constant axial stress of 100 psi ; specimen 12 failed by a ductile ftexural mode that displayed severe yielding of the vertical reinforcing bars accompanied by considerable toe crusrung. On the other hand, specimen 5 experienced a brittle shear failure dominated by diagonal tensile cracks.

The analysis is conducted under displacement contrai by specifying hori­zontal displacement increments at the top nodes while a constant axialload is applied at the beginning of the incremental analysis . Eight node quadrilaterals with 3 x 3 Gauss points are used in the current analysis. The material properties utilized in the numerical analyses were obtained from the test data published by Srung et aI. (Shing et aI., 1988).

5. NUMERICAL RESULTS The results for specimen 12 are shown in Fig. 3. As shown in Fig. 3a,

the numerical load-displacement curve dosely matches the experimental curve. Furthermore, the vertical steel yields substantially due to the widening of ftexural cracks, while no yielding is detected in the horizontal steel. This can be seen

737

in Figs. 3b and 3c. Toe crushing first occurs at a lateral load of 68 kips in the analysis versus 71 kips in the actual test. The extent of the crushing is very similar to that observed experimentally.

The same panel was modeled numerically using the plasticity Jz model devel­oped by Lotfi and Shing (Lotfi and Shing, 1991); the resultant load-displacement curve and crack pattern are shown in Figs. 3a and 3c, respectively. By com­paring Figs. 3b and 3c, close agreement between the current model and the plasticity mo deI is evident.

90

60

70

60 a. ~ 50

1l 40 o

..J

30

20

10

o

Currenl Modcl

(b)

o

__ Exporimenllll

__ '" _. Pt .... tic:r.y mod.t (lolll ~Shrf\g)

.. ........ . RoI.tlng ct-.;k mod.1

0.5 1.0 1.5 2.0 Displacement (in)

(a) PlasticHy Modcl

Fig. 3 Current Vs. Plasticity Model for Specimen 12, Flexure Dominated Failure. (a) Effect on Load-Displacement History, (b) and (c) Cracking Pattern.

(at 0.6 in. Displacement)

The results for specimen 5 are illustrated in Fig. 4. As shown in Fig. 4a, the numericalload-displacement curve agrees in general with the experimental curve. The predicted ultimate load by the analysis is slightly higher than that shown in the test and occurs at a slightly lower lateral displacement . Again, the same panel was modeled numerically using the plasticity Jz model developed by Lotfi and Shing; the resultant load-displacement relationship is shown in Fig. 4a. In both models, the lateral resistance continues to increase as the tensile strain at the diagonal cracks increases. While the plasticity model failed to capture the rapid drop in lateral load resistance once a major diagonal crack had occurred, the current model successfully captured this drop as a diagonal crack opened. This rapid drop is due to the relatively small amount of shear reinforcement provided in specimen 5.

By comparing Figs. 4b and 4c, close agreement between the current model and the plasticity madel in detecting the crack pattern was faund. AIso, it was

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observed from the numerical results that the post-peak load degradation in panel 5 is accompanied by compressive toe crushing and substantial flexural cracking.

130

120 úpclimtnl&1

110

100

90

Co 60 R 70 1l 60 -o

._ .... -. Pl •• Ucity modtl (lelll 6Shln;t) 01,'·7::,">,:. .. • ........ Rot.tl/"lQcroc:kmod.,

./1/ ':\.,." .. _.

.. . '-"

....I 50

40

30

20

10

o o 0.5 1.0 1.5 2.0

Displacoment (tn)

Curuml Idodel (a) Plaslicily Model

(b) (c)

Fig. 4 Current Vs. Plasticity Model for Specimen 5, Shear Dominated Failure. (a) Effect on Load-Displacement History, (b) and (c) Cracking Pattern.

(at 0.48 in. Displacement)

6. DISCUSSION OF RESULTS The analysis of the fiexure dominated waU indicated a very good match

with the experimental results. On the other hand, the analysis of the shear dominated wall indicated a slight increase in the walllateralload resistance over the experimental results; a sudden drop was experienced once a major diagonal crack occurred. As a diagonal crack opened, the aggregate-interlock forces and the residual shear at the crack interface were substantially reduced. Because of the small amount of shear reinforcement, this led to a rapid drop in the lateral load resistance. This, to a great extent, was predicted by the current model. AIso, it is observed from the numerical results that the post-peak load degradation is accompanied by a diagonal strut mechanism which failed quickly by crushing of the masonry in the diagonal strut direction.

Lotfi and Shing (Lotfi and Shing, 1991) analyzed the same shear wall (spec­imen 5) using a plasticity h model. The numerical solution of the aforemen­tioned shear wall was influenced by spurious shear strength which consequently prevented the realistic modeling of brittle behaviour. They claimed that this spurious lateral resistance is due to the diagonal compressive strain that is as­sociated with shear distortion which in turn is caused by the diagonal crack opemng.

Lotfi and Shing claimed that the shear capacity will be maintained as long

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as the diagonal compressive strength is present, regardless of the shear retention along the diagonal crack surface. While the last statement is true to a great extent, it could not explain the spurious shear strength. Utilizing the current model indicated the development of the strut mechanism. Concurrently, due to the increased compressive strains, a progressive diagonal compressive failure was initiated starting from the crushed toe and extending diagonally until complete failure of the modeI. Numerical results, using the current model, indicated no spurious shear strength for the shear dominated specimen 5 and the sudden drop due to the excessive opening of the diagonal crack has been properly captured.

7. CONCLUSIONS A rotating smeared crack mo deI has been developed and implemented in

the general purpose finite element package. The finite element model has been calibrated against experimental work where two reinforced masonry wall panels of flexural and shear dominated failure modes were modeled. Some conclusions can be drawn at this stage of the research.

• Good agreement was found between the experimental behaviour and the current mo deI behaviour regarding the flexural response of masonry shear walls.

• The response of masonry walls dominated by brittle shear failure accom­panied by the opening of diagonal cracks could not be accurately modeled especially in the pre-peak zone. However, it was possible to predict the ul­timate load quite accurately and to capture the sudden drop in the lateral load resistance.

• The analytical results of the current model were compared with those based on the plasticity mo deI developed by Lotfi and Shing and good agreement was found between both models.

8. REFERENCES • Bathe, K.J., "Static and Dynamic Geometric and Material Nonlinear Anal­

ysis Using ADINA", Massachusetts Institute of Technology, Report No. 82448-1, 1977.

• Vecchio, F.J., and Collins, M.P., "The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear", ACI Structural Jour­nal, 1986, pp. 219-23l.

• Gopalaratnam, V.S., and Shah, S.P., "Softening Response of Plain Concrete in Direct Tension", ACI Journal, May-June 1985, pp. 310-323.

• Bathe, K.J., and Ramaswamy, S., "On Three-Dimensional Nonlinear Analy­sis of Concrete Structures", Nuclear Engineering and Design, No. 52, 1979, pp. 385-409.

• Rots, J.G. and Blaauwendraad, J., "Crack Models for Concrete: Discrete or Smeared? Fixed, Multi-Directional or Rotating?", Heron, Vol. 34, No.l, 1989.

• Shing. P.B., Noland J.L., Klamerus, E., and Spaeh, H., "Inelastic Behaviour of Concrete Masonry Shear Walls", Journal of the Structural Engineering, ASCE, VoI. 115, No. 9, July 1988, pp. 2204-2225.

• Lotfi, H. R and Shing, P.B, "An Appraisal of Smeared Crack Models for Masonry Shear Wall Analysis", Computers and Structures, Vol. 41 , No. 3, 1991, pp. 413-425.

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