11th INTERNA T10NAL BRICKlBLOCK MASONRY CONFERENCE

TONGJI UNIVERSITY, SHANGHAI, CHINA, 14 - 16 OCTOBER 1997

NuMERICAL MODELLING OF UNREINFORCED MASONR Y W ALLS SUBJECT TO LATERAL !MP ACT

Matthew Gilbertl, Tom Molyneaux2 and Brian Hobbs3

1. ABSTRACT

This paper describes the development of a three-dimensional non-linear finite element model suitable for predicting the response of freestanding unreinforced masonry walIs to vehicIe-like impact loadings. As performance of unreinforced walIs subject to lateral loadings wiIl often be largely govemed by the mechanical behaviour of the joints, a three­dimensional interface formulation capable of modeIling joint fracture has been developed, and is described in the paper. Crack propagation between units is modelIed according to a cohesive crack type law. The performance of stretcher bonded concrete blockwork walIs in particular is studied in the paper. It has been found that reasonable agreement between experimental test results and the numerical predictions can be obtained.

2. INTRODUCTION

The overaIl aim of the work described has been to develop an understanding of the response of plain brickwork and blockwork waIls to localized impact loading. One of the primary areas of application is in assessing the behaviour of masonry parapets 011 bridges and roadside retaining walIs when subjected to vehicIe impact. For example, of the order of 50 percent of the bridges in the UK are of unreinforced masonry construction and yet there is currentIy no method of analysing the impact performance of their parapets.

The stimulus for the numerical work described in this paper arose folIowing initial use of a

Keywords: Masonry; Walls; Impact; Finite Elements

I Lecturer, Department of Civil and Sttuetural Engineering, University of Sheffield, Mappin Street, Sheffield SI 310, UK

2Lecturer, Department of Civil Engineering, University of Liverpool, PO Box 147, Brownlow Street, Liverpool L69 3BX, UK

3Professor and Head, Division of Civil Engineering and Building, The University of Teesside, Middlesborough, TSl 3BA, UK

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general purpose dynamic finite element package (DYNA3D) to predict the performance of masonry walls subject to actual vehicle impacts l Discrete modelling of the constituent masonry units in walls allowed remarkably good qualitative predictions of failure modes to be obtained. However it was found that rather high values for the bond strength were required in order to c10sely replicate the experimentally observed failure modes. Thus a decision to attempt to improve the joint failure criterion was taken. Three changes to the interface model described previously have therefore been implemented:

'ti) Provision of a normal pressure dependent relationship for the shear strength (Mohr Coulomb)

(i i) Provision of a post-peak softening branch following initial fracture (iii) Inclusion of a rate effect model

It should be noted that the explicit solution strategy (using the central difference time integration scheme) employed removes many of the numerical difficulties associated with the modelling of crack propagation, and is particularly suitable when tackling dynamic problems involving impacts.

Associated with the analytical work, an extensive programme of tests on full scale walls has been carried out recently2 The results from the laboratory tests have been used to develop and verify the analytical methods described in this paper. Small scale material tests have also been undertaken to provide the necessary data for the modelling work.

3. FINITE ELEMENT MODELLING

3. 1 Overview

The performance of masonry structures is to a large extent governed by the mechanical behaviour of the joints between the constituent units. Several different analytical approaches have been developed for modelling joint fracture in masonry over the years. Notably, smeared crack models3 and discrete crack models4

•5 have proved popular. The

work described here has adopted a discrete mo dei in which individual masonry units are modelled separately, and which incorporates a cohesive type cracking mode!.

The use of cohesive crack models, first developed by Hillerbourg6 in the 1970's enabled objective results to be obtained when using the finite ~lement method to model crack propagation. Using a cohesive crack model implies that disconnection of a linkage connecting adjacent blocks will occur as a gradual processo Strength based failure criteria may be used to signal crack initiation, but complete failure will only occur when ali the fracture energy deemed to be present in the joint has been dissipated.

The justification for use of a discrete mo dei may be put forward as follows : When cracks in masonry walls are examined it will normally be found that either ali the cracking is confined to the mortar joints, or that mortai joint cracking is accompanied by cracks running only through the centres of some of the masonry units. An exception to this occurs when weak masonry units are used in conjunction with relatively strong mortar. In this case cracks appear to be unaffected by the unit bonding pattern, and the wall may be deemed to have approximately isotropic material properties. In this latter case the use of a smeared

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approach may seem attractive. However, in the former cases the use of a discrete method of analysis is more attractive because potential failure zones are known in advance - i.e. the mortar bed and perpendicular joints, together with the mid points of the masonry units . The discrete model described here allows the growth of individual cracks in the joints to be tracked. The current model does not however allow for spalling ofmasonry.

In many cases it is possible to carry out reasonably realistic analyses of masonry structures using two-dimensional analyses (for example when modelling the , response of masonry shear walls). However, in other cases a more general three-dimensional model, such as the one described here, will be required (for example when modelling masonry wall panels subjected to lateral loading). In this study individual masonry units have been modelled using linear-elastic solid elements.

3.2 Development ofModel

3.2.1 Interface Formulation

A penalty stiffuess contact/interface formulation as described by Hallquist 7 is adopted as· the starting point for the work described in this paper. Using the formulation, the surfaces of adjacent blocks are defined with either 'slave' or 'master' properties. Prior to fracture, individual slave surface nodes are tied to the master surfaces of an adjacent block. This is achieved via the penalty stiffuess method which involves applying a restoring force to any slave node which becomes displaced trom its initial position on the master surface. Prior to fracture the restoring force is proportional to the separation of the slave node and master surface. The formulation then allows adjacent blocks to debond and then to slide with friction .

3.2.2 Joint Fracture

When the interface force at a tied node reaches a predefined failure surface, the failure surface may be designed to shrink gradually to model the dissipation of tracture energy, using a cohesive crack type model. The proposed model has similarities to that proposed by Rots4

, except that in this model the specialised contact algorithm referred to in 3.2.1 is used, rather than explicit interface elements. Additionally the model is fully three dimensional. The important features ofthe model are described below.

(a) An exponential softening relationship is applied to tied nodes failing in shear (mode I) or tension (mode 11). The mode-I and mo de-lI displacements at which the residual interface stress has reached 0.1 percent ofthe maximum stress (in a quasi-static analysis) is specified for each analysis (uj and uJ/f respectively). The values of these displacements can readily be ca1culated trom the mode-I and mode-II fracture energies and the ultimate tensile and shear bond strengths bfthe joint. i.e:

1 - log.(O.OOI)G; UI = t

UI (1)

(2)

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where UI is the tensile strength of the joint at fracture, c is the cohesion. Values used by

Rots4 for a medium strength mortar were ej =O.OlN/mm eJl! = O.05N/mm. Bocca5

recorded experimentally values for ejfor brick masonry units ofapproximately O.05N/mm. Softening parameters ,I and ,lI are introduced:

{IOg, (O 0014}

1<1 = e "f (3)

{lag,(O DOI4} 1<11 = e "f (4)

Where u{ , u: are the 'plastic' deformations normal (tensile) and tangential to the interface

respectively. Parameters ,I and ,lI are then used to calculate a global softening parameter I<

used to scale down the failure surface. Thus I< may be calculated from :

(5)

(b) A Coulomb friction type relationship has been adopted to model the shear strength in the presence of normal compression; a cutoff is used in the presence of normal tension. 1. e. the failure criterion (also shown diagramatically on Fig. 1) :

1 <5, f = (U:nal)2 +( {< trial - U~al tan~})2

I<U! I<C (6)

where {} are McCauley brackets, and where the trial normal stress at the interface is defined by Uma/, and where:

and where <tria/ is the trial shear stress at the interface, c is the cohesion, and tant,6 is the (static) coefficient of friction . The relationship allows a smooth transition between principally cohesive and purely frictional shearing resistance - which occurs once a given joint has fully 'softened'.

An initial trial stress state is calculated using a penalty formulation . Thus the interface force at a slave node is calculated from:

F=Du (7)

Where D=diag(k,k,k), u=(ul,u2,u3)T, the elastic displacement between a given sI ave node and associated master segment. Thus subsequent trial interface forces (e.g. at timestep n+ 1) can be obtained from :

F;;; = Fn + Dt:.u (8)

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rn rn ~ -rn "­cu Cl)

.r:. Cf)

Soltening

Fig. 1 Proposed failure envelope

jD w Initial failure surface, u=O

disPIaCemeUn: " I SOltenin,.....- ...

,/ ,/

.....-,;' Residual failure

,/ ,/ surface, u>llt ,/

Normal stress

and where the interface stitlhess, k, can be calculated from:

(9)

where Pfis a penalty factor (taken as 0.1), A is the area ofthe master segment, K is the bulk modulus of the master segment, and Ve is the element volume. The interface stitlhess is designed to be sufficiently high so as to concentrate deformations in the attached bodies, rather than in the interface itself

The trial shear and normal stresses can readily be calculated from the interface :orces. If this stress state is found to lie outside the failure surface, the stresses are scaled down to the failure surface as follows:

t a:nal a =--f (10)

(11 )

(12)

'Plastic' deformations (which lead to the progressive shrinkage of the failure surface) are calculated at each subsequent timestep. The interface model softens isotropically, and exhibits an elastic unloadinglreloading response (penalty interface stitlhess = k).

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3.2.3 Handling Interface Definitions

Defining interfaces becomes a non trivial matter when dealing with up to several hundred separate definitions. Automatic search/numbering schemes are available, but at present, because of the regular layout of units in masonry walls a manual scheme is adopted. Figure 2 shows an exploded elevation of a small section of stretcher bonded wall. Typical layouts of the separate contact surfaces are shown (separate surfaces are allocated different numbers - numbering scheme shown arbitrary).

201 801 201 802

III X

1I1

101 II 11102 II 11

51 Y

101 102 701

(a) No unit fracture permitled (b) Unit fracture permitted

Fig. 2. Typical definition of contact surfaces - possible numbering scheme shown

The layout shown in Fig. 2 (b) is similar to that used by Schwer and Lindberg9 in a rock mechanics problem. They found, using purely mctional interfaces, that this layout would allow some overlap at the comers (e.g. comer overlap of block X and half block Y [after fracture] as there is no contact surface separating them), leading to results which were similar to those found using the discrete element method in which the comers of the blocks are rounded to prevent 'grid lock'.

A consequence ofusing a specialised (non symmetric) contact formulation is that care must be exercised in defining the assignrnent of master and slave properties to the interfaces in the model, and in the subdivision of blocks into e\ements. For exliltlple, in Fig. 2(a), suppose the lower block is subdivided lengthways into two elements. Suppose that the top surface of this block is defined as a slave surface, so that the top surface nodes are to be 'fixed ' to the upper blocks. However, using the current formulation the top surface nodes at midlength of the lower block may in fact be arbitrarily fixed to either of the two top blocks - and the block left unconnected to these nodes will be able to fracture away from the other two blocks at an erroneously low stress leveI. In this case the problem can simply be avoided by providing an odd number of elements across the length of the lower blocks, ensuring initial connectivity of nodes is predictable.

3.2.4 Mesh Size Sensitivity

From numerical modelling of triplet specimens8 indications are that, when the softening branch is inc1uded, the proposed model is not especially sensitive to mesh size, even when rather coarse meshes are used.

3.2.5 Utilizing Symmetry

Invariably fracture pattems found in practice in masonry will not be symmetric. Forcing the numerical model to form symmetric pattems, by specifying planes of symmetry, will be

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likely to lead to the dissipation of fracture energy wruch would not occur in practice. When symmetry is not utilized, numerical truncations and rounding errors - analogous to physical imperfections - will often lead to one crack opening in preference to another symmetrically positioned crack.

4. VALIDATlON AGAINSTLABORATORYRESULTS

4.1 Description ofConcrete Blockwork Walls

The performance of the 3 laboratory test walls constructed using ' strong' (37.4N/mm2)

concrete block will be examined (designated as wall C5, C6, and C7) The units were solid and the walls were constructed using a designation (iii) mortar to BS5628 ptl. In the case of these walls the number of physical units was relatively small. As through-block fracture was thought to be unlikely to occur the interfaces in the numerical model were prescribed to coincide only with the mortar bed and perpend joints (although in the case of one of the walls tested this proved not to be the case in practice). Additionally, as all 3 walls were essentially physically identical, and were subjected to nominally identical loading regimes, some indication as to the variability of the walls could be taken into account when comparing theoretical and experimental behaviours. Each wall was subject to a concentrated out-of-plane impact loading at mid length. The impulse applied to each wall was 2. 77kNs. Pulse lengths were of approximately 36mS duration. Small differences between the walls were: in the case of wall C6 the base was coated with mould oil prior to laying blocks; the bonding pattem ofwall C7 was slightly different to that used for walls C5 and C6. It is evident from the Fig. 3 that the failure modes of the 3 walls were broadly similar, although by no means identical. Note the unit fracture in the case ofwall C6.

WallC5

WallC7

Fig. 3. Experimentally recorded crack pattems (showing only 'principal' front face tensile & shear cracks)

4.2 Description ofF.E. Models

Physical blocks were modelled by 3x3x2 (LxBxH) 8-noded solid elements. Computationally cheap (but rather low accuracy) single point integration was used with viscous hourglassing control.

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Material density and coefficients of friction were taken as the measured values. For other parameters estimated values were used . Note that the Youngs modulus 1P0issons ratio are smeared properties, relating to the composite masonry material. Refer to Tables I and 2.

Parameter Strong Concrete Steel impactor Blockwork abutments

Youl1g's modulus (kN/mm' ) 20 30 ri-.B!d Poissons ratio 0.3 0.3 rigid

Density (kRm3) 2270 2370 7800

Table I . Properties of solid e1ements

Para meter unit-unit mortar-base steel-wall Abutment-base interface interface interface interface

Tan1 0.64 0.85 0.3 0.47

O"~ (N/mm' ) 0.714 0.143 O O

G~ (N/nun) 0.05 0.01 O O

T«(N/mm') 1.0 0.2 O O

GJ ~/mm) 0.01 0.005 O O

Table 2. Properties of interfaces

The load applied to the back face of a rigid striker in the wall models was: 0-+ I OOkN-+ 130kN-+OkN, over 0-+ 12mS-+24mS-+36mS (impulse 2.76kNs).

Figure 4 shows the predicted deformed shapes of the walls with and without the inc1usion ofthe softening branch at 0.2s (corresponding approximately to maximum displacement -subsequently the walls rock back under the influence of gravity) . It is evident that the inc1usion of the branch reduces the apparent brittleness of the joints, leading to large panel formation, rather than the more localized failure mode observed without softening. Note the discontinuous contours of out-of-plane displacement shown in Fig. 4. A1though not readily visible in the Figure it should be noted that partial or localized fracture occurred in many other joints (this phenomenon was identified by plotting all interface nodes where failure had occurred, after the completion of an analysis). Additional hairline cracks (not shown on Fig. 3) were also observed in the case of the experimental walls.

Figure 5 shows the experimental and predicted displacement time histories !!for the walls (out-of-plane displacement, recorded at rnid-height ofwall, 250mm from centre ofimpact) . Examination of the Figure reveals that although there is broad agreement between the experimental and predicted responses, it appears that initial fracture may be occurring prematurely in the numerical model. Implementation of a rate effect model may overcome this problem, and may be justifiable on physical grounds (rather than merely on the grounds of numerical expediency) as significant dynarnic enhancement has been recorded when carrying out dynamic triplet tests in the laboratory At the present time these experimental results can only be replicated numerically when a rate-effect model is implemented.

It has been found that the influence of the se1ected magnitude of fracture energy is not dramatic. For example, using values one order of magnitude lower, the failure mode is similar (large panel formation) , the only difference being that the distance between fracture

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lines is reduced slightly. Altematively usmg values one order of magnitude higher the distance between fracture lines increases.

Wall model - softening branch included

Wall model - no softening branch included

KEY: 10mm 2 30mm 3 50mm 4 70mm 5 90mm 6 110mm

FigA. Predicted wall failure modes (time=O.2s)

120 .-----------,'---:-N7"o-s-o....,ft:-e-ni:-n-g"";""b-ra-n-c"""h

Ê 100

§. 80 c Q)

E 60 Q) to co

40 C. (J)

o 20

O O 0.05 0.1 0.15

Time (s)

Softening branch

Experimental

0.2 0.25 0.3

Fig. 5. Predicted and experimentally recorded displacement vs time responses

5. CONCLUSIONS

The discrete modelling approach for masonry walls subject to out-of-plane impact appears to be capable of providing reasonable estimates of wall performance. In respect of the developments described in this paper, preliminary findings are that the addition of a

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softening branch allows closer estimates of the response of the test walls to be obtained than when a fully brittle model is adopted.

Initial fracture (occurring prior to the softening phase) sometimes appeared to occur prematurely in the current numerical model. This may indicate the need to incorporate a reliable rate effect model into the interface algorithm. Development work is therefore being directed towards this area at the present time.

6. ACKNOWLEDGEMENTS

The support of EPSRC, under grant reference GRlJl0587 is acknowledged Also acknowledged is the support of Ove Arup and Partners, who provided OASYS-LS­DYNA3D for use in the project. Thanks are also due to the Lawrence Livermore National Laboratory who provided a version of LLNL-DYNA3D for development purposes during the project.

7. REFERENCES

l. Molyneaux, T.C.K. , Gilbert, M., and Hobbs, B.,"Modelling vehicle impacts on freestanding masonry walls", 3rd Computer methods in Structural Masonry conference, Lisbon, 1995.

2. Hobbs, B., et al., "Simulation ofvehicle impacts on masonry parapets", Subrnitted to the 11th Int. Brick Block Masonry conference, Shanghai, 1997.

3. Pande, G.N., Middleton, l , Lee, J.S. , and Kralj, B. , "Numerical simulation of cracking and collapse ofmasonry panels subject to lateralloading", Proc. 10th Int. Brick Block Masonry conference, Calgary, pp107-115, 1994.

4. Rots, J.G. , and Lourenco, P. B., "Fracture Simulaticins of Masonry Using Non­Linear Interface Elements", Proceedings 6th North American Masonry Conference, Drexel University, Philadelphia, 1993.

5. Bocca, P., Carpinteri, A, and Valente, S., "Fracture mechanics of brick masonry: size effects and snap-back analysis", Materials 'Uld Structures, Vol. 22, pp364-373 , 1989.

6. Hillerbourg, A. , "Analysis of crack formation and crack growth in concrete by means offracture mechanics and finite elements", Cement and concrete research, 6, Pergamon, 1976.

7. Hallquist, lO. , Gooddreau, G.L. and Benson, D.J., "Sliding interfaces with contact-impact in large-scale lagrangian computations'~ , Computer meths.in App. Mechanics and Engineering, Vol. 51, ppl07-137, 1985.

8. Molyneaux, T.c.K. , Gilbert, M., "Modelling masonry joints under impact loading", Oasys DYNA3D Users Conference, London, 1997.

9. Schwer, L.E. and Lindberg, H.E., "A finite element approach for calculating tunnel response injointed rock", Int. J. Numerical & Analytical Methods in Geomechanics, Vol. 16, pp529-540, 1992.

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