Stability and Fracture Analysis of Eccentrically Loaded Brick Masonry Walls (*)

A. DI TOMMASO - E. VIOLA

Istituto di Scienza delle Costruzioni, Università di Bologna (Italy)

SOMMARIO

Scopo della presente memoria e di illustrare un procedimento numerico, basato sul metodo

delle differenze finite, per calcolare la capacità portante limite di una parete in muratura, soggetta a

presso-flessione, con comportamento geometricamente non-I ineare.

11 materiale costituente la colonna si considera reagente sia a compressione che a trazione, nelle

ipotesi di elasticità lineare e conservazione delle sezioni piane, considerando altresi la formazione di

fessure (cracks) in corrispondenza dei giunti di malta.

La distribuzione delle tensioni all'apice della fessura e conforme ai "fictitious crack model"

(F.C.M .). nel quale e preso in conto il "softening" dei material i fragili.

La trattazione consente di valutare I'influenza della snellezza della parete sul carico limite e sulla

configurazione ultima della struttura.

ABSTRACT

The purpose of this paper is to present a method of analyzing unreinforced load-bearing masonry

walls subjected to compressive axial loads and end-moments. The behaviour of the wall, made of

bricks and joints of mortar, will be represented by an eccentrically loaded beam-column model, in

the geometrically non-linear range. The material is assumed to be linearly elastic both under compres­

sion and under tensile stress. The rectangular plane section will be supposed to be plane after defor­

mation, and cracking will occur only in the mortar layers.

The distribution of stress at the crack tip is in agreement with the "fictitious crack model" [IJ, in which the softening of non-yielding materiais is considered.

This approach allows the evaluation of the influence of slenderness on the limit load and the

ultimate geometric configuration.

(*) The research support of the National (Italian) Research Council (CNR) is gratefully acknowledged.

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168

1. INTRODUCTION Although brick masonry has been largely used in the past, there is no recognized theory which

predicts the load-bearing capacity in terms of mechanical properties of the constituent materiais and

of the interface bond strength. 8rick masonry may be considered a "composite material" in which the

constituents are bricks and joint mortar. The behaviour of a load-bearing wall may be represented bya

"beam-column" strut. A crack in the interface brick morta r reduces the flexural rigidity of the beam­

-column al1d its load-carrying capacity. It will be assumed, in this paper, that cracks occur in the inter­

face between bricks and mortar, i.e. in the weakest link. Although the toughness of the interface has

been considered by severa I Authors [2, 3J, the concept of the "fictitiollS crack model" (F.C.M.). as

introdllced by Hillerborg et aI. [1] and recently deve10ped by L4>Iand [4], VIIi!! be used in order to ana­Iyse the partially cracked sections.

In order to predict the behaviour of masol1ry under eccentric load, a procedure of sllccessive corrections to the equilib rium and compatibility equat ions was forrnulated. A computer prograrnme

based on the finite difference tec:h"ique gives, for assigned values of t he load and eccentrici ty , the

deflections in a number of sections along the strut, taking into account the geometric non ·!inearity.

The pror.edure is repeated until two sets of deflections converge. Material failure occurs when the

internai force cannot be increased in order to balance the applied forces. In section 2 the fictitious

crack model and the cracking of brittle materiais are discussed. In section 3 the basic relationships

and the analysis of cracked sections are illustrated . In section 4 the particular case of a two-hinged

column with a constant initial load eccentricity is studied . In section 5 an outline of the $olu 1:ion

procedure is reported. Lastly, in sect ion 6, al1 illustrative example is presented.

2 CRACKING IN NON-YIELDING MATERIALS ~------_._ ------ -- --

Generally tlle masonry wall behaves in one of t hree ways depicted in Fig. 1 depending on the

axial eccentric force and 011 the height of the wall [6]. In case (a) the elastic sciut ion occurs; in case

(b) cracking takes place in the tension fibres, whereas in case (c) croJsh ing occllrs in the extreme fibres

subjected to compressive stress. In Ref. [7] thlee different types of sect: ol1 cr isis were co,,~ide red ;

frar.ture crisis, crushing crisis and tract iop crisis.

In order to evaluate the distribution of stress over a cracked section, the fracture zone which

develops in front of the crack tip should also be taken into consideration. It is well known that when

a cracked speciment of linearly elastic mater ial is subjected to load, the stress field near the crack tip

has a singularity in the order Of 1/0, \I\I he ~e r measures distance f rem the tip of tlle crack (se e selid

line in Fig . 2) .

In the case of real materiais, the stress distribution in front of a crack tip is affected by a fractu re

a)

Fig. 1 . Modes 01" masonry behaviour [6] .

r

~11

r x -~~ J y - ==------ -

linear elastic material

y lelding material

non-yielding material

T

Fig. 2 - Probable ay stress dlstri butlon in front of a crack [8] .

lone [8] . In the case of y ield ing material , plastic flow w lll take place at the crack tip and the probable

stress distribution of this material is represented by a dashed line in Fig. 2. In the case of non-yielding

materiais, e.g . mortar , a micro-craci<ed zone develops in front of the crack. This reduces the concentra­

tion and the distribution of stress as shown by the dotted line in Fig. 2. FI Otn the strain measurements

taken for concrete and similar materiais [8J , it has been proved tha! the fracture zone is never small

when compared with normal specimen dimensions. The deformation properties of the fracture zone

may be obtained from a stable. direct tenslle testo

3. BASIC RELATlONSHIPS ANO ANAL '51S OF SECTION -- -----

3.1 Basic relat ionship~

It is assumed that a solid wal! , made of biicl{S and mortar , ha, a stress .. strain relationship as shown

in Fig 3 .

a t (;KI}SHING ',-- ----7]

./ ! I I

---- - -,----.J<----------- .... L -----, €o €

f = aJao = €t/€o

f < O

À = a/ao

Fig. 3 - Assumed stress-strain re lationship for the brick-morta r system.

General!y the bricks and mortar have diffe:-ent propert ies. Consequently, the assumed stress-strain

re lationship is to be taken as the unit average of the brick and mortar system.

Maximum tensile and compresslve stress are denoted as aj

and aú

respective ly, and f = aI/ao' The

dimensionless parameter À = a/ao = €/€o can take any value w ithin the interval f .;;;; À .;;;; 1 It is sup­

pose that the a - € relationship represents the properties of the brick-mortar system only outside the

f racture zone.

3.2 Analysis of section

The load-carrying capacity oi the eross section of the masonry wal!, under eccentrrc axial loading,

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may be determined if the stress distribution over the cross section at the point Df failure is known. It is

assumed that plane sections remain plane after deformation. The st ress distribution of a typical

cracked cross section, is shown in Fig. 4.

A

CRACK

f x

B

Fig. 4 - Cracked section of masonry under eccentric vertical load.

ft may be observed that when tensile stress reaches its maximum value, a cross sectional area is not

capable of carrying more load, and micro-cracks will develop inside a small volume of material or f rac­ture zone. Then the stress on the material inside this fracture zone decreases elastically when strain is increased. In Fig. 4, A represents the length of the crack, R is the width of the fracture zone, E is the

load eccentricity (defined with respect to the zero stress point). and K is the distance of N from the

free edge.

The total force on the section, per unit of thickness, and the moment force around the ~eutral

axis, can be expressed as : (see Fig. 4)

aL R + D --a--=N 2 I 2

3 3

where D = ai L/ao.With the followingnon-dimensional quantities:

D L R ai a M N d= - Q= - r = - f= - X= - m= -- n=

B ' B ' B ' ao ' ao' B2 ao

' Bao

the above equations of equilibri um assume the following aspect:

XQ - f (r + i: Q) = 2 n

XQ2 _ f3Q2 _ ~ (~+ ~) = m

3 3X2 2 3 X

(1 )

(2)

(3)

(4)

(5)

where m = n (Q - k), in which k = K/B ; the constants f, r depend on the material, whereas n, k depend

on the loading. By combining equations [4] and [5]. that are non-linear in the X, Q unknowns, in order

to eliminate parameter Q, the following is obtained:

g(X) = C4X4 + C X3 + C X2 + C X + C = O 3 2 1 o (6)

where:

Co =- 2f 3 (2n + rf)2- r2f5 + 3f4 r(2n + rf) + 6nkf4

C1

= 6 n f2 (2n + rf)

C2 = f2[2fr2- 3r(2n + rf) - 12nk)

C3

= (2n + rf) (2rf - 2n)

C4 = 6nk - r2f

(7)

Equation (6) was solved for À using a computer. In the particular case of zero tensile stress (f = O). the

solution is À = 2n/3k. This value of À has been used in order to define the end-points of the interval, which contains one zero of the real valued function g(À) .

4. ANAL YSIS OF STRUT

The procedure used in this paper is based on the finite difference method. The sim pie case of the

two-hinged column with a constant initial eccentricity, was considered . The strut (Fig . 5) is divided

into 20 parts of equal length "h". In an arbitrary section, the deflection is YI' The curvature, XI' at this section may be approximately calculated as follows:

d 2 y X =- - =-

i dz2 (8)

Since the curvature will be different at stations 1 - 10, the following system of equations was ob­

tained:

[A]{y} = {b} (9)

where: I I I i j I : j 1

2 : - 1: : : : : 1 : : ---. --t-- - - -~--- - -t --- --T -----r-----r------t -----t- ----t------l i 2 :-1 : : l : : : : -- - --1-----+ - ----r --- -,.- - ---r-- -- -1---- -~- - -- -i-----t-----

:-11

2 :-1: : ~ : : I -----t-----t-----~----+--- -{----l - - ---t -- -t--- -t ----

I : - 1 : 2 t - 1 I I I J I - - --~-- ---f-----I--- ---1----1- ---..,- - - -1- - - - -+ -- - - i - --.

: 1 ! -1 1 2 :-1 l : : 1 - - -- -f------l- - ---t-- - - i------t ---- -I- --- -I-----~----~ ----

~ I : :-1:2:-1: : : -- -- -+-----t-- --+- --- -!----- -t--- - t---- -t-- - --1----:- ~---

: : : : :-1:2:-1: : --- --+ --- - -+-----1-----1--- - -t- - - - t -----f - ---,-- - -"1- ---.

: ! : : : :-1:2:-1: -----:--- - -t- --- i---- +----f-- ---4--- - ~ - - - -t- - - -~- ---

I I I I I I '-I' 2 , - I I I I I I I I I I -----+ ----~ -- ---r------t---- - ~-- -- i - --- -t------t--- --r- --

: : : : : I : : - 2: 2 I , I , I , I I I

[A) =

_ ~L

_XL _~L Y4

{y}= _~L

_.'I_L Y7

Ya yg

Y10

_~_1 _ _

--~.?_-

-~~--X4

{b}= _~.L h2

_~L (10)

X7

Xa

Xg

X10

I

~ I

h-E:·:J:=-~r]~rJ,]-FI1 - -_ _ __ a Y10

-- - - -

Fig. 5 - Oivision of strut into 20 segments.

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172

Once the displacements {y} have been determined, it is possible to obtain the deflection profile of

the strut. It will be noted that on deflecticn. geometr ica l non-!inearity results since load eccent ric ity

changes at every stat ion; t his, is known as the "second order effect" . In order to take the geornp.trical

non-l inear ity into account, a process of sur r:p's,ive correstic!1s is r3qu ired.

5. NUMERICAL,SOLUTIO[i

The procedure used isschematical ly illustrated in Fig. 6. In order to find out if a state of equil ib­

rium is possible fo~ a givel1 value of the ax ial load, "n", and a given eccentricity, "k", an ite'ative pro-

( ENO )

COMMENTS

Se" nctati(1'1 (APPENOIX)

COMpute À, Q satisfing the em.lilibr iuM equatiops (4) . (5).

Compute curvatUlp.3 ai earh station .

Convergenc'e t'=lst .

New -values or eccentricity.

Strair. test and load increments .

11* d irrensionless CI !tical load

{X} vector of cUlvatures

[y } vector o f deflect,otls

{€l vect.orofst:ai ns

{ld eccer: t r icity vector

{a } crack lel1gth vector

{d} vector of traction part of sect ion

{Q ~ v-:-c!or of compl e" IVe pa rt of secil""

-cess can be started by assuming that curvatures XI ,at ali positions, are the same, as determined from

the fo llowing equation :

(11)

where i = I, 2, ... , 10. Then, equation (6) is solved and deflections {y } at the ten stations are obtain­

ed according to equation (9). 50, in the arbitrary section, the eccentricity kl

is modified :

(12)

The equation (6) is solved with eccentricity kj in each section . This leads to new curvatures {X} from

equation (11) and new deflections {y} from equation (9) . Once convergence has been achieved, the

max imum strain of the middle section is compared with Eo. Then load N is increased, in order to

obtain a new state of equilibrium, until ElO = Eo' which is the limit value of N for the value of eccen­

tri city considered . Using this procedure, numerical results for a set of strut geometries, are obtained.

Figure 7 shows the dimensionless ultimate load n* = N/Bao versus slenderness ratio H/B, for the

load eccentr icity k = K/ B = 0.2. Two different types of brickwork, of different strengths were con­

si dered.

I I K

N n*=-- k K/B 0.2

Bao N

~ Eo = 0.002

-0-0--0- Eo = 0.004 0.3

= 0.050 H

r = 0.050

0.2 B

I" -I

0.1

10 20 30 H/B

Fig . 7 - 51enderness ratio effect on the dimensionless ultimate load.

173

B • ! N~ B=l

a r=0.05 ~, =~ ~ 0:;~1\ 00

Q CRACK "I dl- oi

N

CRITICAL LOAD n* = N/Boa = 0.1

co ! k = 0.062 À = 0.997 Q = 0.209 o

11 d = 0.011 a = 0.729 I

! k = 0.072 À = 0.919 Q = 0.220

d = 0.012 a = 0.716

i k = 0.089 À = 0.732 Q = 0.277

d = 0.018 a = 0.653

!k=0.118 À = 0.555 Q = 0.367

d = 0.033 a = 0.549

!k=0.155 À = 0.420 Q = 0.488

d = 0.058 a = 0.404

Fig. 8 - Limiting values of eccentrically loaded strut.

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6. ILLUSTRATIVE EXAMPLE

Let us consider the simple supported wall of unit width B = 1 and eight H = 10B shown in Fig . 8.

The li ne of application of the externai load N is fi xed at k = 0.2 f rom the f ree edge. ln arde r to find the

dimensionless criticai load n* = N/ao B and the ultimate geo met r ic shape of the column, Eo = 0.002

is assumed for the brick-mortar system . The computer programme developed gives n* = 0.1 and also

the numerical results presented in Fig. 8., which reproduce the crack configu rat ion and the distribu­tion of stress in each cracked section .

RE FERENCES

[1 ] A Hillerborg, M. Modéer, P.E. Peterson : Analysis of crack formation and growth in concrete by means of fracture mechanics and finite elements. Cem. and Concrete Resea rch . 6 (1976).

pgs . 773 - 782.

[2] A Piva, E. V iola: Biaxial load effects on a crack between diss imilar media. Engng. Fract ure

Mechanics, 13 (1980) pgs. 143 - 174.

[3 ] B. Hillemeier, H.K. Hilsdorf: Fracture mechanics studies on concrete compounds. Cem . and

Concrete Research, 7 (1977) pgs. 523 - 536.

[4 ] K.E. L,pland : Continuous damage model for load response estimation of concrete . Cem. and

Concrete Research . 10 (1980). pgs . 395 - 402.

[5 ] E. Viola, A. Piva: Plane strain inter-facial fracture analysis of a bimaterial incompressible body.

Engng. Fracture Mechanics, 15 (1981). pgs. 131 - 142.

[6 ] F.C. Siat Moy, L. Tall: Load-bearing masonry walls analyzed as beam-columns. labse Proceedings

P - 25 (1979), pgs. 1 - 16.

[7] A Carpinteri , A. Di Tommaso, E. Viola : Sulla capacità portante limite di pareti lapidee lesionate.

Atti Congresso Nazionale AIMETA, Palermo (1980). pgs. 93 - 104.

[8] P.E. Peterson, P.J. Gustavsson: A model for calculation of crack growth in concrete-like ma­

teriais. Numerical methods in f racture mechanics, edited by D.R.J. Owen and AR. Luxmoore

(1980) pgs. 707 - 719.

[9] R.G. Drysdale, S.E.A. Sallam, E. Karaluk : Design of masonry walls and columns for combined

axial load and bending momento Proc. 1st. Canadian Masonry Symposium (1976) pgs. 394 - 408.

[10] C.T. Grimm: Strength and related properties of brick masonry.Journ. of Struct. Div. ASCE, ST1

(1975) pgs. 217 - 232.

APPENDIX I

The following symbols are used in this paper:

K distance of N from free edge N eccentric applied load

B total depth of cross section

k K/B

A length of cracked part of section

a AlB R dimension of the process zone

R/B L depth of the compressive cross section

Q L/B

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176

n

m y

X E

e X O

d

H

n*

tensi le and compressive strength

ai/ao N/ao B dimensionless axial load

N/B2 ao dimensionless bending moment

lateral deflection

curvature load eccentricity from zero stress ax is

E/B distance from neutra I axis

t raction part of section

D/B total height of the wall

dimensionless ult imate load

current and maximum strain

current stress

E/Eo = a/ao column matrix

square matrix

inverse of a matrix