12TH INTERNATIONAL

BRICK/BLOCK Masonry c O N F E R E N C E

Sut

LlMIT ANAL YSIS OF ANISOTROPIC OR JOINTED MEDIA - APPLlCATION TO UNREINFORCED

MASONRY SHEAR WAllS

D.]. Sutcliffe1, H.S.Yu 2

, and A.W. Page3

1 Postgraduate student, Department of Civil Surveying and Environmental Engineering,

University of Newcastle, Australia

2Associate Professor, Department of Civil Surveying and Environmental Engineering, University

of Newcastle, Australia

3Professor and Dean, Faculty of Engineering, University of Newcastle, Australia

ABSTRACT

This paper presents a simplified numerical method for calculating the lower bound li­mit load in anisotropic media, with particular emphasis on the in-plane behaviour of unreinforced masonry walls. Results from this new technique are compared to those of an earlier lower bound formulation by the same authors. 80th techniques are ba­sed on the limit analysis of jointed rock but differ in the assumptions about the ma­terial properties. In the previously presented method the masonry material is assumed to be anisotropic and inhomogeneous due to the presence of the discontinuous joint sets in the masonry composite. The new, simpler technique, however, assumes ho­mogenous material properties. An example is given to illustrate the accuracy of the new technique.

Key words: Shear wall, In-plane, Limit analysis, Lower bound.

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1. INTRODUCTION

Masonry is a composite material composed of bricks or units set in a mortar ma­trix. The strength of this composite is affected by many factors including the strength of the unit and mortar constituents, the thickness and orientation of the joints, the size and aspect ratio of the unit and the bonding pattern. Another im­portant factor is the bond strength between the unit and the mortar, since the in­terface between the unit and the mortar has long been recognised as a plane of weakness along which many of the modes of failure displayed by masonry initia­te and propagate.

Due to the large number of factors which affect the strength of masonry, the nu­merical simulation of this material is difficult. The displacement finite element method has been used in the past, however, its application has been limited due to the need for extensive testing to provide an accurate material description. Furt­hermore, experience has shown that the displacement finite element method of­ten tends to overestimate the observed collapse load. An alternative technique which has been successfully applied to masonry (Ganz and Thurlimann, 1985; Sutcliffe, Yu and Page, 1999a and 1999b) and to geotechnical problems involving jointed rock (Yu and Sloan, 1994) is the application of the lower bound limit the­orem. A major advantage of the lower bound limit theorem from a design pers­pective is that the calculated load never exceeds the true collapse load resulting in conservative designs. Also, since the lower bound theorem deals solely with the strength properties of the material and does not deal with displacements, the ex­tent of testing required to provide a useable material description is significantly reduced.

The purpose of this paper is to present a new and simplified analytical method for determining the collapse load of unreinforced masonry shear walls which is ba­sed on an anisotropic but homogenous material description. In order to achieve this, several simplifying assumptions are made regarding the mode of failure as well as the preferred orientation of the failure plane. Common to both the pre­vious and present formulations for masonry is the assumption of a Mohr-Cou-10mb failure criterion for ali components.

2. CURRENT DESIGN PROCEDURES FOR SHEAR WAllS

Shear walls under monotonic loading display three basic modes of failure namely: (1) tensile failure at the heel of the wall due to overturning, (2) crushing failure at the toe of the wall, and (3) diagonal tensile failure of the panel near mid height. At a localised levei failure mode (3) can be broken down into: a) shear failure of the joint, b) tensile failure of the joint, c) diagonal tensile splitting of the unit, and d) direct tensile splitting of the unit. With the exception of the Swiss Masonry Standard (SIA V1 77/2 - 1989) current design provisions for masonry shear walls are empirical, and are typically a 'Iower bound' on a set of highly variable test re­sults displaying combinations of the above failure modes.

2.1 Australian Masonry (ode Provisions AS3700(1998)

For unreinforced masonry members the Australian masonry code uses equation (1) below which is based on the Mohr-Coulomb failure criterion .

where Vd = design shear force; Vo = shear bond strength; V, = shear friction strength = k, f d Adw;

k, = friction factor;

(1 )

f d = compressive stress on the bed joint under consideration; and, Adw = bedded area of the joint under consideration .

This method, which is similar to most other codes (page, 1989), has several short­comings, the greatest being its empirical nature. Equation (1) is based on a cru­de 'Iower bound' of test data from wall racking tests (refer Fig . 1). As a result, alt­hough the test data includes tests displaying ali the modes of failure Iisted above, the failure described by equation (1) represents only shear sliding along a single plane, with cohesion Vo and friction factor k,. Consequently, the cohesion and fric­tion factor from equation (1) are not well defined or even easily measured para­meters as they represent some average value related to numerous complex failu­re modes.

2.2 Swiss Masonry Standard SIA Vl77 /2 (1989)

The Swiss masonry design standard is the only standard which is based on a mo­re rational design approach as it considers the actual distribution of stresses wit­hin the material. The provisions are based on the work of Ganz and Thurliman (1985) who developed a lower bound Iimit approach for the design of unrein-

Points representing test results involving one or more 01 the p~ssible lailure mechanisms

• I Vnsafe design I

I Safe design I

(Compression)

orizontal oad

Vertical load

(Tension) Figure 1. Empirical/y based design of unreinforced masonry shear wal/s.

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+ + + + + + + + + + +Un Vs

I Figure 2. Paral/el compression field (Ganz and Thurliman, 1985).

forced masonry shear walls. The method is based on the manual formulation of a statically admissible stress field (see section 4.) as shown in Fig. 2. Due to the ne­ed to manually construct the discontinuous stress field, this method is difficult to apply to problems involving complex geometry and/or boundary conditions. The method is further limited by the failure criterion applied to the units which is highly dependant upon their geometry. As such the method is only relevant to the typically highly perforated Swiss masonry unit.

3. MOHR-COUlOMB CRITERION

3.1 Application to Anisotropic media

In terms of the Mohr-Coulomb failure criterion, an anisotropic material is one which displays a variation of cohesion and/or friction angle with varying orienta­tion within the material. Due to the anisotropic nature of the cohesion and fric­tional strengths, the conventional Mohr-Coulomb failure criterion is no longer va­lid . Assuming tensile stresses are positive, the Mohr-Coulomb failure criterion for shear strength that can develop on a plane at angle from the horizontal can be represented as:

I s.1 = c. + a n tan<)l. (2)

where Se represents the shear strength and ca, <)la represent the cohesion and fric­tion angle of the material at angle e respectively.

The normal and shear stress components on the plane can be described as:

(3)

(4)

In order to deal with a variation of both cohesion and friction angle with direc­tion within the bounds of linear programming, a series of weak planes are intro­duced into the material. Each plane has a value of cohesion (ce) and friction an­gle U e) associated with it. By solving equations (2) and (4) for I 'te - sei = O the failure surface on each plane of weakness can be defined by the yield surface:

F; = ~ I sin 28; (cr, - cry) + 2 cos 28; 't'y I

- C; + (sin' 8,cr, + COS' 8;cry - sin 28;'t,y) tan 4>; = O (5)

3.2 Application to unreinforced masonry

As stated previously, the presence of joints in the masonry composite create pla­nes of weakness and result in unreinforced masonry displaying anisotropic pro­perties. The five basic failure modes displayed by unreinforced masonry under in­plane loading are listed in section (2) above. It should be noted that failure modes 1), 2), 3a), 3b) and 3c) are ali influenced by the strength of the joint, with only failure mode 3d) occurring solely in the unit. If masonry material is to be mode­lIed as a series of units and joints in their true location as in the formulation of Sut­cliffe, Yu and Page (1999a, 1999b), the material must be assumed to be inho­mogeneous as well as anisotropic. Within the framework of a simplified micro-model, the strength of the units can be represented by a Mohr-Coulomb failure surface. This accounts for direct tensile failure of the units (mode 4). Failu­re modes 1), 2), 3), and 5) are ali assumed to be confined to the relatively weak joints. These joints are assumed to obey a linearised cap model which is based on the failure surface of Lourenco (1996) as shown in Fig.3.

In the simplified formulation presented here it is assumed that failure only occurs on continuous planes of weakness rather than occurring in both the units and the joints at random orientations. This simple assumption overcomes the need to tre-

/ /

/

/

/ r /

Figure 3. Spherical cop model (Lourenco, 7996) and linear approximation to cop model.

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at the material as inhomogeneous. Furthermore, it is also assumed that failure will occur on only two planes, namely failure parallel to the bed joint direction and failure parallel to the head joint direction. This assumption is necessary because there is a lack of relevant test data, but it also has the advantage of ensuring sim­plicity in the formulation .

Whilst it may seem crude to assume that fa ilure will only occur on the plane pa­rallel to the bed joint and/or the plane perpendicular to the bed joint, investiga­tion of experimental data on biaxial tests (Dhanasekar, 1985) indicates that this assumption produces results similar to the experimental f indings.

Several plots of normal stress versus shear stress are shown below in Fig . 4, whe­re e represents the angle between the vertical and bed joint direction. Com pari­son of these plots shows the similarity of the failure envelope for e = 0° and e =

22.5°, suggesting that the fa ilure plane is the same for these two panel orienta­tions. Consistencies can also be seen between the plots for e = 90° and e = 67.5". As such, it is reasonable to accept the assumption of two orthogonal fa ilure pla­nes w ith linearised failure criteria as demonstrated in Fig . 3. At the same time, the fa ilure envelope for e = 45° lies between the failure surfaces for the two orthogo­nal failure planes. For this reason, failure on the 45° plane may be assumed to in­volve interaction of the two fai lure planes parallel and perpendicular to the bed joint direction.

In the direction parallel to the bed joints it is valid to assume a continuous plane of failure (refer plane ao Figure 5). This is not the case, however, in the direction parallel to the head joint where three possible failure planes exist (Fig. 5). Section bb represents a continuous plane of failure through the units only, which is unli­kely given the strength of a typical unit as compared to the unit/mortar interface. Section cc is a continuous plane that incorporates fa ilure in both the head joints and the units. Section dd, while not continuous, does describe a mode of failure parallel to the head joints. The strength of the failure pattern represented by sur­face dd is considered to be a combination of the tensile strength of the head joints and the shear strength of the bed joints along path dd. By adopting the weakest of surfaces cc or dd for both direct tensile and shear failure modes it is possible to define the material strength by approximating it as a continuous plane.

Whereas the linearised cap failure criterion was used solely to represent the failure properties of the joints in the earlier formulation of the lower bound limit theorem, it is now used to define some average material properties. In the direction parallel to the bed joints the tensile and friction branches of the failure surface represent the strength values of the joint only. However, in the direction parallel to the he­ad joints these two branches describe some average strength of either surface cc (which involves the failure of both unit and joint) or surface dd (which involves the failure of joints alone but must also include the interlocking effect of the units). Compressive failure of the masonry is a combined failure of both the joints and the units and so the linearised cap of the failure surface represents the compressive strength of the composite material perpendicular to the two joint sets.

Figure 4. Failure envelopes for biaxial/y stressed panels.

THETA=O THETA=22.5

6

~I íi' ..... íi' t 4

.lf~~······~~.~ ... 11.

2 ~ c~·~a p ,!f . "x • •• -:~ p

o , -10 -8 ~ -4 -2 o 2 -12 -10 -8 ~ -4 -2 o 2

O'n(MPa) O'n(MPa)

THETA=67.5 lHETA=90

íi'

~ I íi' ! I ~ x'iS'::~~ 11.

1'~::E~hiL ~ p p ,

( ,~ , -10 -8 ~ -4 -2 o 2 -10 -8 ~ -4 -2 o 2

O'n(MPa) O'n(MPa)

THETA=45

11 íi' 11.

~~ ~ p ,

-10 -8 ~ -4 -2 o 2

O'n(MPa)

Figure 5. Planes of failure paral/el and perpendicular to the bed joint direction.

a a ,

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4. THE lOWER BOUND THEOREM

The lower bound theorem states that the collapse load obtained from any statically admissible stress field will give a lower bound of the true collapse load. A statically admissible stress field is one which a) satisfies the equations of equilibrium, b) sa­tisfies the stress boundary conditions and c) does not violate the yield criterion.

In order to model the stress field in the masonry material, a three noded triangu­lar stress element is used (Sloan, 1988). Each node is associated with three unk­nown stresses, a x ay, 'txy and with the variation of stresses throughout each ele­ment assumed to be linear. Unlike the elements used in displacement finite element analysis, several nodes may share the same coordinate with each node associated with only one element. Statically admissible stress discontinuities are allowed to occur at ali edges between adjacent triangles.

4.1 Yield Condition

For the masonry composite, the complete failure conditions are expressed by equation (5). Imposing the constraints Fi :s; O ensures that the yield conditions are satisfied.

4.2 Objective Function

The problem of finding a statically admissible stress f ield which maximizes the co­lIapse load may be expressed as:

Minimize -CTX

Subject to (6)

where A,. 8, represent the coefficients due to the equilibrium and stress boundary conditions; A2.82 represent the coefficients for the yield condit ions; eis the vector of objective function coefficients; and X is the global vector of unknown stresses. An active set algorithm is used to solve the above linear programming problem, details of which can be found in Sloan (1988).

5. APPlICATION OF THE lOWER BOUND lIMIT THEOREM TO UNREINFORCED MASONRY SHEAR WAllS

In order to demonstrate the applicability and accuracy of the proposed method to shear walls, the in-plane capacity of an unreinforced masonry panel has been analysed. The proposed method is directly compared with previous analyses con­ducted by Lourenco (1996) and Sutcliffe, Yu and Page (1999a, 1999b) using ex-

perimental findings reported by Raijmakers and Vermeltfoot (1992) and Vermelt­foot and Raijmakers (1993). A schematic diagram of the test configuration sho­wing boundary and loading conditions as well as a sample finite element mesh is shown in Fig. 6.

The material properties used in the analyses by Lourenco (displacement finite ele­ment method) and Sutcliffe, Yu and Page (lower bound limit formulation) are shown in Table 1. It should be noted that the tensile strength value for the joints has been modified to zero so that the assumption of perfect plasticity is valid . Af­ter tensile separation of the joint, the residual strength is equal to zero and as such the assumption of perfect plasticity requires that the initial tensile strength also be zero.

The properties reported by Lourenco are the material properties of the joint and unit. While the parameters still hold true for the failure plane parallel to the bed joint (where failure is confined to the relatively weak joints), the parameters for the failure in the orthogonal direction must be derived from these basic values. For tensile failure to occur along a plane perpendicular to the bed joint, either fai­lure pattern cc or dd (Fig. 5) must eventuate. Given that the tensile strength of the joint is assumed to be zero, failure mode cc should prevail and so the tensile strength of the masonry composite will simply be the shear strength of that por­tion of bed joint contained on the plane. The value of cohesion for the shear fai­lure mode on this plane however is not as easily determined. It is unclear from the data available whether failure would involve failure of the unit in shear or simply rotation of the interlocking components. It may be assumed however that the va­lue of cohesion should lie somewhere below that of an intact unit. The inferred failure parameters used in this investigation are shown in Table 2.

Figure 6. Shear wall test configuration with boundary conditions used in lower bound calculations and finite element mesh bound mesh.

Normal Load Fixed

Shear Load to be Determined ++++++++++.

0;/ = O 1"= O

+-- +-- +--I

I I

I I I I I I

I I I I I I

I I I I I

I I I I I I

I I I I I

>< ><

-- Element Boundary

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Table7. Masonry shear wall basie parameters

Tension Shear Cap Unit

J, ~, (z ~z J, ~J ( ~ 0.0 89.0 0.224 36.87 11.5 150.0 2.0 45.0

{MPa] {deg] {MPa] {deg] {MPa] {deg] {MPa] {deg]

Table2. Masonry shear wall parameters inferred for homogenous formulation

Tension Shear Cap

j , ~1 c, ~, J, ~J

Plane parallel to bed ioint 0.0 89.0 0.224 36.87 11.5 150.0 direction {MPa] {deg] MPa] {deg] {Mpa] {deg]

Plane parallel to head 0.4 89.0 2.0 36.87 11.5 150.0 joint direction {MPa] {deg] {MPa] {deg] {MPa] {deg]

The analysis was performed using a vertical prestress of 2.12MPa. A value of 80.7 kN was achieved for the shear load using the parameters shown in Table 2. This compares well with the result of 82.9kN from the previous lower bound formula­tion which assumed inhomogeneous material properties. The result also compa­res favourably with the experimental result of 97kN. The result is particularly go­od given the difficulty in interpreting the published material properties and boundary conditions for the tests.

Due to the indefinite nature of the cohesive strength on the plane orthogonal to the bed joint direction as discussed above, a parametric investigation on this pa­rameter has been undertaken. Results can be seen in Figure 7 below. These plots clearly demonstrate that, as expected, the value of cohesion on the plane parallel to the head joint direction is significant with decreases in cohesive strength re­sulting in reduced overall panel strength. This result is consistent with previous parametric studies (Sutcliffe Yu and Page 1999a, 1999b) which showed that by reducing the unit strength, failure is no longer contained solely within the weak joints, but is instead a combined failure of both unit and joint.

90

Z 80 ~ L -g 70 o I -: 60

I lO

~ 50

i UI 'O 40 t c 5 30 m J âJ 20 ~ o 10

..J

o o 1000 2000 3000 4000

c2 (kPa)

I-+-PHI1=89.0 desl

5000 Figure 7. Parametrie analysis - effeet of variation of parameter e2.

6. POTENTIAl APPlICATION TO SHEAR WAll DESIGN

The next step in the investigation will be the application of the lower bound limit theorem to the analysis of a wide range of shear walls. Variation of both the ge­ometry and boundary conditions, as well as the fundamental strength properties of the components will be studied . It is hoped that through extensive parametric analysis of this nature a simplified analytical design procedure based on the lower bound method, capable of handling complex geometry and boundary condi­tions, will be developed.

As with ali numerical modelling however, the use of the lower bound method re­quires representative failure criteria for ali masonry components. Shear wall test data reported in the literature have two major shortcomings from a modelling point of view: firstly, poorly reported loading and boundary conditions mean that assumptions must be made regarding support and loading points; and secondly, insufficient data on the strength of individual components makes accurate detai­led modelling impossible. Given this lack of usable data, a testing program is cu­rrently underway at the University of Newcastle using panels subjected to com­bined shear and normal loading. The aim is to provide test results with well defined boundary and loading conditions. Furthermore, tests to determine the fundamental properties of the masonry components are also being preformed in order to define an accurate material description. It is expected that the experi­mentai data will further verify the accuracy of the lower bound limit model as well as provide a valuable bank of data for future modelling.

Ultimately, it is hoped that this research will improve masonry design procedures and overcome the shortcomings of the widely adopted empirical design methods used at present.

7. CONClUSION

A general finite element formulation of the lower bound theorem for analysis of the in-plane strength of unreinforced masonry has been presented . By assu­ming that failure only occurs on two orthogonal planes, the masonry composi­te has been treated as an anisotropic, inhomogenous material. The importance of the lower bound limit theorem as a design tool lies in its ability to provide a safe estimate of the collapse load. By definition, the lower bound approach will always underestimate the true collapse load and hence lead to a conservative designo The numerical example developed provides results consistent with both the previously presented lower bound formulation (where anisotropic inhomo­genous material properties were assumed) and the experimental data. This in­dicates that the new technique produces an accurate estimate of the actual co­IIapse load of unreinforced masonry structures under in-plane loading . Finally the ultimate application of the method to design procedures for unreinforced masonry shear walls is discussed, including a brief outline of the future direc­tion of the project.

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8. REFERENCES

Ganz H.R. and Thurliman B. 1985. Plastic Strength of Masonry Shear Walls, Proceeding of the 7" International Brick and Block Masonry Conference, 77·20 February, 7 985, Melbourne, Australia, 837-846.

Lourenco P.B. 1996. Computational strategies for masonry structures. PhD Dissertation, Civil En­gineering Department, Delft University of Technology.

Page A.W. 1989. A parametric study of the behaviour of masonry shear walls. Proceedings of the 5" Canadian Masonry Symposium, 5-7 June, 7989, Vancouver, Canada, 341-352.

Raijmakers T.I.M. and Vermeltfoot A.Th ., 1992. Deformationa contro lled tests in masonry shear walls. Report B-92-7 756, TNO-Bouw, Delft, The Netherlands.

SIA V1 77/2, Design of Masonry Walls, 1989. Swiss Standard, Swiss Society of Engineers and Ar­ch itects.

Sloan S.W. 1988. Lower bound limit analysis using finite elements and linear programming. In­ternational Journal for Analytical Methods in Geomechanics, 12:61 -77.

Sutcliffe D.l., Yu H.S. and Page A.W. 1999a. Lower bound limit analysis of unreinforced masonry shear walls. Proceedings of the 7'h International Symposium on Numerical Models in Geomechanics, 7 -3 September 7999, Vienna, Austria. 639-644.

Sutcliffe D.l., Yu H.S. and Page A.w. 1999b. Computational mechanics - Limit analysis of un­re inforced masonry shear walls. Proceedings of the 76'h Australasian Conference on the Mechanics of Structures and Materiais, 8-7 O December 7999, Sydney, Australia. 29-34.

Vermeltfoot A.Th. and Raijmakers T.I.M., 1993. Deformation controlled tests in masonry shear walls, Part 2. Report TUE/BKO/93 .08, Eindhoven University of Technology, Eindhoved, The Nether­lands.

Yu H.S. and Sloan S.W. 1994. Bearing capacity of jointed rock. Computer Methods in Advanced Geomechanics, Siriwardane and Zaman (eds), Rotterdam. 2403-2408.