Microstructural model for dry block masonry walls with in-plane loading

J. Mech. Phys. Solids, Vol. 42. No. 7, pp. 1159 1175, 1994

Copyright c' 1994 Elsevier Science Ltd Printedin GreatBritain.All rightsreserved

Pergamon

0022-5096194

$7.00+0.00

0022~5096(94)E0019-Z

MICROSTRUCTURAL MASONRY

WALLS GIOVANNI

Istituto

MODEL

di Scienza delle Costruzioni,

WITH

ALPA

IN-PLANE

BLOCK

LOADING

and ILARIA MONETTO

Universita’

(Recked

FOR DRY

di Genova,

Via Montellegro I, Genova,

Italy

22 June 1993)

ABSTRACT for dry block masonry walls loaded by in-plane forces. A microstructural approach is followed to account for the displacement jumps taking place at small flaws inside the blocks as well as at block edges. Displacement jumps are treated, for a sufficiently large sample of masonry, as strain contributions by means of a proper homogenization procedure. Blocks are modelled as elastic solids containing small frictional cracks distributed at random. The strain contributions due to displacement jumps occurring at block edges are shown to be both contained and uncontained. After discussing the former, the limiting stress conditions for the mechanisms of uncontained strains become active and the corresponding flow laws are analysed. Under the restriction that strains are not so large that the block assembly is substantially changed, the derived constitutive equations can be employed in continuum analyses of masonry walls. CONSTITUTIVE EQUATIONS are obtained

1.

INTRODUCTION

MOST OF THE CONTRIBUTIONS

in modelling the mechanical behaviour of masonry start from a phenomenological point of view. The main feature is seen in the weak, or quite absent, tensile strength. Accordingly, masonry bodies are often analysed as continuous media having constitutive equations based on the main assumption of no-tension material. An outline of problems, proposed models and computational methods concerning equilibrium, deformation and limit analysis of masonry structures may be drawn from the following papers: HEYMAN (1966), LIVESLEY (1978), DI PASQUALE (1982), GIAQUINTA and GIUSTI (1985), COMO and GRIMALDI (1985), PANZECA and POLIZZOTTO ( 1988) and STEIGMANN (199 1). Methods employing phenomenological stress-strain relationships allow the gathering of important features of the mechanical behaviour of masonry without the need to specify shape, size and mechanical properties of the elementary parts of its microstructure. This applies properly to masonry solids made of irregular stones, with or without a mortar or concrete filling, characterized by a random microstructure. Nevertheless, masonries constituted by an assembly of regular blocks may exhibit a behaviour strongly affected by the shape and size of the constituting elementary parts. In such cases, the need arises for a microstructural approach involving both the block geometry and assembly in a proper description of the masonry mechanics, although 1159

G. ALPA and I.

1160

MONETTO

it will lead to constitutive equations requiring more complex computational procedures for solving structural problems. In this paper, constitutive equations concerning the elastic and inelastic deformation, as well as the limiting states of strength, are derived for dry block masonry walls loaded by in-plane forces based on microstructural considerations. Masonry walls are modelled as a perfectly regular assemblage of rectangular, full thickness blocks with frictional forces acting on the joint surfaces. The block material, in turn. is modelled as an elastic solid with plane stable frictional microcracks (block crushing is not considered). Displacement jumps which may occur at cracks and joints are treated as inelastic strain contributions accounted for by a proper homogenization procedure. According to this model, the constitutive equations have: (i) an elastic range bounded by the stress states activating displacement jumps at microcracks and joints ; (ii) an inelastic range characterized by contained plastic strains due to displacement jumps at microcracks and joints; (iii) a third range, the main object of the paper. characterized by an uncontained plastic flow related to well defined mechanisms depending on the block dimensions and assembly. As occurs in the theory of elastic perfectly plastic materials, the plastic flow, uncontained (undetermined) in the constitutive equations, may be contained (determined) in the boundary value problem for a given body. Of course, the third range attains its validity limit when the strains are so large that the corresponding displacement jumps change substantially the masonry web. Under the restriction of small deformations, the derived constitutive equations may be employed for continuum incremental and limit analyses. Some comparisons with theoretical and experimental results available in the literature show that the proposed model is able to explain some typical features of the mechanical behaviour of masonry walls depending on their microstructure (specifically on the block side ratio).

2.

MICROSTRUCTURAL

THEORY

OF MASONRY

WALLS

In this section the problem of determining constitutive equations for dry masonry walls is shown to find its rightful place in the microstructural theory of materials. specifically within the model of a perfectly homogeneous elastic matrix containing flaws. In this context, there is a condition in order that the mechanical behaviour of a body can be analysed through the concept of continuum with a local constitutive equation : the body can be divided in parts, small enough with respect to the body size, each one having an autonomous behaviour expressible through a relation between a stress tensor and a strain tensor (intended as overall entities for each part). In our analysis, there are two kinds of constitutive equations : the first kind concerns small portions 5’ of a body 98 containing a number of flaws much smaller (microcracks) than the size of the body portion ; the second kind concerns small portions of body containing parts of larger flaws [Fig. l(a)]. Constitutive equations of the first kind account for the contribution to strains due to the microcracks contained in Y‘ and must be obtained, according to the requirement of local character (independent

Model for dry block masonry

1161

b)

a) FE.

walls

I. (a) Small portions jumps

Y of a body d containing at microcracks and flaws;

microcracks and parts of large flaws (c) portion A%. of a masonry wall.

; (b) displacement

of shape and size of Y), considering an infinite volume of solid weakened by a proper crack distribution. Constitutive equations of the second kind account for the strain contribution of the parts of flaws in V and must be based on the local frictional contact conditions at any point of the flaw surfaces. The above considerations refer to the general concept of material but have a great relevance for block masonries because : (i) except for particular situations, masonry cannot be considered as a material with a local constitutive equation for structural elements having minimum size a little larger than the block size ; (ii) the acting forces and the boundary conditions for a masonry structure must be compatible with a stress-strain field varying smoothly enough so that it can be considered homogeneous for masonry portions including a number of blocks; (iii) families of discontinuity surfaces corresponding to the joint array cross the whole masonry structure and they can be included in a continuum analysis only if local contact conditions can be stated. Then, aiming to model masonry walls as elastic plates containing frictional flaws, let us summarize assumptions and mathematics of this model starting from the general case of a tridimensional body [for further details of the general treatment see MURA (1991)]. Denoting by r the position vector [Fig. 1(a)], under the assumption of small deformation, the (microscopic) field variables describing point by point the stress and strain state in the elastic matrix are : the displacement vector u(r), the Cauchy stress tensor T(r) and the infinitesimal strain tensor E(r). To these variables, some macroscopic field variables can be associated describing the overall behaviour of the portion Y‘ previously defined, namely: a stress tensor E(r) and a strain tensor T(r) well defined by the following functionals of the tractions t and displacements u at points of the boundary surface 5” of Y [for the mathematical formalism see GURTIN (1972)] :

s s

z=’

v

r@tdA,

(1)

‘F

sym (u @ n) dA,

.v

where V is the volume of V and n the outside normal unit vector on Y. By applying the mean stress theorem, it can be shown that (1) defines X as the mean of any stress field satisfying both equilibrium (in the absence of body forces)

1162

G. Awn

and I. MONETTO

and the boundary conditions involving t; by virtue of the mean strain theorem, (2) defines r as the mean of any strain field associated with a continuous displacement field satisfying the boundary conditions involving u. With some implicit approximations related to boundary effects and to the presence of body forces, the stress and strain state in a body can be analysed through the macroscopic field variables C(r) and T(r), providing that proper stress-strain relationships are stated accounting for elastic properties, shape and array of flaws as well as contact-friction conditions. Considering that at flaw surfaces contact forces are equal and opposite and displacement jumps can occur, by an application of the mean stress and mean strain theorems to ‘/; (bounded by .‘/ and by the internal contours corresponding to Aaw faces) there follows :

x = T. r=R+iisyrn

(i:,AuOn.dA)+t~sym(S,huOn,ilA).

(3) (4)

I’ where T and E are, respectively, the means of T and E in the elastic matrix; %, indicates one of the opposite faces (reference face of unit normal n,) of the ith microcrack having a fixed location point of position vector r, in V; 9, is the portion of theith large flaw falling in 3‘. having n, as normal unit vector of the reference face ; Au is the displacement jump ; N and A4 are the number of microcracks and large flaw parts falling in $? [Fig. 1(b)]. Denoting by I6 the compliance fourth order tensor of the elastic matrix, from (3) and (4) the following relation is obtained :

which puts forward that there are three strain contributions: the elastic one by the matrix; the contribution due to microcracks; the contribution due to the large flaw parts falling in the considered volume. If the array of microcracks and large flaws is specified and the contact-friction conditions are considered, it is possible (in principle) to express Au as a function of the assigned tensor X, making so explicit the stressstrain relationship (5). With reference to Fig. 1(c), where a portion A %’ of masonry wall is shown, specializing the relation (5) leads to : .c

r=

iM!:+Zsym(Au~@ni)+~sym(Aii,Oe,)+~sym(Ae,@e,), I,

(6)

where N is the number of cracks per unit volume; Au,* = SC6AudA is the total displacement jump at the ith (plane) microcrack; Aii, and Ati,: are the mean displacement jumps at joints having unit normal vector n K e, (staggered joints) and n z e,. (aligned joints), respectively. The strain contribution due to microcracks in rock-like materials, such as stone or concrete blocks in masonry, has received a great deal of attention; the available

Model for dryblockmasonry walls

1163

results in the literature will be summarized in the following section. The strain contribution due to displacement jumps at joints will be analysed in Sections 4 and 5.

3.

MICROCRACK STRAIN CONTRIBUTION

The idea of explaining some features of the mechanical behaviour of rock-like materials considering them as elastic solids with frictional microcracks was originated by WALSH (1965). Among various subsequent contributions [see, for example, HORII and NEMAT-NASSER(1983), KACHANOV(1987, 1992)] we summarize here a mechanical model which applies for arbitrary triaxial stress paths (ALPA and TAFANELLI,1984; ALPA and GAMBAROTTA,1988). Stable plane microcracks are assumed randomly distributed (in shape, size, location and orientation) within an isotropic linear elastic matrix; crack interaction is disregarded. The problem is to determine Au,* of (6) as a function of the applied stress IL To this purpose, let us write the second term of the right side of (6) as follows : r* =

sym (Au*(n) 0 n) dQ,

(7)

sR

where Q is the half-unit sphere representing all the orientations and Au*(n) dQ is the total displacement jump due to the cracks having unit normal vector n inside the infinitesimal solid angle dQ [Fig. 2(a)]. By virtue of the matrix linearity, the displacement jump can be expressed by : Au*(n) = R(n)CRT(n)(Cn-f*(n)),

(8)

where R(n) is the rotation of a reference orthogonal system having e,* = n [Fig. 2(a)] ; C is a compliance tensor relating, in such a defined reference, the resolved stress acting on the plane of unit normal vector n to the strain contribution (per unit solid angle) due to opening and sliding of cracks laying on such a plane (C is not a function of n because of the matrix isotropy and of the random distribution of microcracks) ; f*(n) represents the contact-frictional reactions at crack surfaces, the actual values of which must be obtained by integrating proper incremental equations over the whole loading path. To make explicit the governing equation describing the evolution off*, it is cone, t

FIG.2. (a) Unitsphere of orientation

; (b) i* in ( IOd)for 6

(c) i* in (10d) for b < 0.

G. ALPA

1164

venient to rewrite the equation from sliding effects : r: = ah

alz,n,dR+k i (2

and I. MONETTO

(7), with the expression

(8) of Au*, separating

~~--~*~+(~1,t,+t,n,)dR s <1

(i,,j=

s,_v.I),

opening

(9)

where x = 0 for r~ < 0 and x = 1 for CJ> 0; /I and k are two constants, depending on the matrix properties as well as on the crack density, expressing opening and sliding, respectively ; CTand t are the normal and tangential resolved stresses acting on planes of unit normal vector n (which are related to the applied stress C through: 0 = (En) * n and z = IEn - an) ; r* is the frictional force ; t,, t, are the components of the unit vector of (z-z*). Equation (9) may be derived from (7) or directly stated as a modified version of the plastic slip idea by BATDORF and BUDIANSKY (1954). The constant k is related to h because both are consequences of the same cause: the presence of cracks. For penny-shaped cracks, it can be deduced that k/l? = 2/(2-v), where v is the Poisson ratio which can be assumed to approximately coincide with that of the matrix. The evolution of z* can be drawn from the Coulomb frictional conditions which need to be written in an incremental form involving the stress increments ci and f : f* = 0

f* = -W+

for

(T > 0

or

CJ= 0,

for

0 = 0,

ci < 0,

or

g < 0.

it*1 < -/UJ

or

0 < 0,

IT*1 = -/_Kr.

for

g = 0,

d- < 0.

for

0 < 0,

IT*/ = -pa.

(IOa)

ci 3 0,

I%/ < -,&

2.E

2*

< --6,

ItI 3 -@,

t*f:r

(lob)

(1Oc)

3 -,M,

(1Od)

where 11is the microcrack friction coefficient [the mechanical meaning of the relation (10d) is shown in Figs 2(b)-(c)]. So. r* is shown to play the role of an internal variable memorizing the loading history. For proportional loading, it is possible to integrate in closed form the above relaticns (10) and obtain finite incremental constitutive equations to employ in a step by step analysis (ALPA and GAMBAROTTA. 1988). For static and dynamic finite element analyses, the derived stress-strain relationships must be inverted, requiring a proper iterative numerical procedure.

Model for dry block masonry

1165

walls

cl

b)

FIG. 3. (a) Hysteresis loop obtained through a closed form solution for uniaxial compression (E: Young’s modulus of the matrix) ; (b) evolution of the elastic (no sliding) domain in a deviatoric plane for pure shear superimposed to a hydrostatic compression ; (c) free shear vibrations of a homogeneously in-plane stressed sample for different values of the lateral stress (ALPA and TAFANFLLI. 1984 ; ALPA and GAMBAROTTA. 1988).

In the case of simple stress state, the integration over 0 in (9) can be performed analytically, so obtaining a closed form solution (uniaxial or pure shear stress superimposed to a hydrostatic compression). For general states of stress it needs a numerical integration joined to a proper discretization of $2. If h, k and p are stated, as well as the elastic properties of the matrix, it is possible to employ the above relations in a fully computational procedure. Alternatively, the microstructural stress-strain relationships can be employed, joined to proper experimental tests, in characterizing the material behaviour. Figure 3 shows closed form and computational solutions derived from this model.

4.

JOINT

CONTRIBUTION

TO INELASTIC

CONTAINED

STRAINS

Contained displacement jumps may take place also at the staggered joints parallel to the ?: axis [Fig. 1(c)l. As a first approximation, these joints can be considered as a doubly periodic array of full thickness cracks in an elastic plate [Fig. 4(a)]. Accordingly, specializing the relation (6) leads to

a) FIG. 4. (a) Doubly

periodic

b) array of flaws simulating the staggered joints; due to microcracks and to aligned joints.

(b) additional

displacements

G. ALPA and I. MONETTO

II66

(the index where pC ‘\ and +lC 1x1are, respectively, the extension and the shear deformation c means “contained”) : Aa,, and AU,,. are the mean displacement jump components in the x and J’ directions. respectively. Disregarding any elastic interaction, the expressions of AU,,, and AtiT, as functions of the applied stress X (of components CJ~,rrV,r,,) are given by the solution of a single crack of length b in an elastic plate of Young’s modulus E:

(12) where 6 = 0 for 0, -C 0 and 6 = 1 for C, > 0; F is the friction stituting in (I 1) there follows :

force at joints.

Sub-

(13) The evolution following :

of F is governed

by (10) which,

in the actual

p’=o

for

C, > 0

P = i,,.

for

0, = 0,

6, < 0,

or

0, < 0,

IFI < -P/,(T,

or

0-, < 0.

IFI = -,/&o,,

P=

or

f7, = 0,

case, reduce

d-, 2 0,

to the

(14a)

Ii,,.1 < -,u~c?,

t,, < --I*,,&,,

(14b)

-,Ll,,c!, (l4c)

where ph is the friction coefficient. By integrating over the loading path. any actual value of F can be obtained and the relations (I 3) give the actual value of the strains. The above relations can be considered as a first approximation because three circumstances have been neglected : (i) the elastic flaw interaction; (ii) the influence of microcracks on the mechanical properties of the plate simulating the masonry wall ; (iii) the presence of the aligned joints which allows additional contained displacement jumps [Fig. 4(b)]. The shortcoming of point (i), even if of little relevance, could be eliminated by computing the displacement jumps for the given doubly periodic array of flaws. To account approximately for the presence of microcracks. an effective Young’s modulus E* may be introduced replacing E in (13). The additional displacement jumps may be approximately accounted for by introducing in (12) a magnification factor nr (n? > 1) which could be obtained experimentally or based on theoretical considerations. Each of the three points is worth performing a deeper specific analysis which exceeds the aim of this paper.

Model for dry block masonry walls 5.

MECHANISMS OF UNCONTAINED

1167

FLOW

As the main purpose of this paper, the possible mechanisms of uncontained flow and the related rules are analysed in this section. Figure 5 shows that four fundamental mechanisms of rigid relative displacement between two parts of a portion AW of a masonry wall are compatible with the block assembly : (1) pure opening; (II) pure sliding ; (III) opening at staggered joints and sliding along the aligned ones; (IV) opening at aligned joints and sliding along the staggered ones. Other mechanisms will be combinations of the fundamental ones. The conditions of (local) possible equilibrium, with respect to the relative movements associated with such mechanisms, are obtained by comparing, at a separating path crossing AW‘, the resulting force given by a homogeneous applied stress state with the resulting force given by the frictional reactions. Let CJ_ or, z,,. be the plane stress state components and p<,, ph the friction coefficients of the aligned and staggered joints, respectively, for the mechanisms (I) and (II) one obtains immediately : mech. (I) :

g, < 0,

(15)

mech. (II) :

Jz,,.\ +~~cr,, d 0.

(16)

More complex are the mechanisms (III) and (IV), the typical picture of which is shown in Fig. 5, also because several different paths are possible. However, with

mechanism

x

(I)

mechanism

1

(II)

Jgeneric separating path t

+x

mechanism

(III)

separat with returns

‘t LX

1

mechanism

FIG. 5. Fundamental mechanisms of uncontained flow.

Tw

( path (IV)

-

G. ALPA and I. MONETTO

I168

reference to the paths shown in the figure, it is easy to argue that paths which return can be excluded since the corresponding resulting force due to the frictional reactions is greater than that corresponding to the same path after eliminating any return. Furthermore, the equilibrium equations for an arbitrary path forming a mean angle c( with the ?;-axis are written as : mech. (III):

o,A~.+]z,,.]A~ltanr

,< -~,~,A.):tancc-~1,I~,,lA~,,

]r,,.]A_)‘+o,.AJ,tana

< --~h~,A~~~b)~,,,IA?‘tanr

mech. (IV) : -! ]-r,,(A~+o,A~tan~

< 0

(17)

for

0,+/r,,]tanr

d 0

for

o,+)z,,.]tancx

3 0,

(18)

from which it is easy to verify that the limiting conditions correspond to a minimum value CL*of M defined by an equal step path running along the block edges (Fig. 5). Therefore, the conditions of possible equilibrium are : mech. (III):

o,+

(19)

(20) for

cr,+$r,,l

2 0.

In the above relations z,, appears with its absolute value. For z,, > 0 the possible mechanisms correspond to Fig. 5; for z,, < 0 the symmetric ones can occur. In the case of the mechanism (III), for z,,. = 0 two symmetrical mechanisms characterized by x* and - LY*are possible as well as a separating path in the y-direction (vertical in current cases) running along the block edges. In the space of the stress components, each limiting condition of (local) possible equilibrium corresponds to two planes symmetrically placed with respect to the plane z,,. = 0. The result is a piecewise linear limiting surface [Fig. 6(a)]. Figure 7 shows the typical cross section (lying in a plane g, = constant) of the limiting surface for varying N. h and putting ,U = ,ui, = ph. The plotted graphs show that, for fixed u and h and varying ,LLthere is a modification in the typical cross section of the limiting surface. The enclosed admissible stress state domain becomes more and more narrow as p tends to vanish. There is, furthermore, a strong dependence of the domain shape on the ratio a,lh. It is worth while to note that, for some particular combinations of geometrical and frictional strength parameters (u/b < 2, p = pC,= ,u,, = l), the pitches corresponding to the mechanisms (III) and (IV) are placed one upon the other; this means that, at the same limiting stress state, two distinct mechanisms or some of their combinations can become active. This is an interesting feature of the dry masonry behaviour, deriving from the geometrical properties of its constituent elements, displayed by the microstructural approach here followed.

Model for dry block masonry

walls

1169

C ffY

I

0)

\ \ \ \ \ \ \ \ \

El

-Ly=o

\

b) FIG. 6. Piecewise linear limiting surface of local possible equilibrium for a/h = 0.5 and p = 0.5 : (a) in the space of the stress state components CT,,CT,,t,, ; (b) on a plane CT,= constant; (c) on the plane T,, = 0.

Once the stresses have attained the limiting conditions, uncontained plastic flow can take place according to the mechanisms of Fig. 5. Of course, local flow, uncontained for free boundary conditions at the contour of the considered small portion of “material” A%/, may be contained by the surrounding portions of masonry wall in which different stress state may occur depending on the boundary conditions at the structure contour. This is a current situation in the theory of elastic-perfectly plastic solids. Turning to (6) and considering the specified mechanisms shown in Fig. 5, one obtains the following expressions of the uncontained strain rate components :

(21) where the index u means “uncontained”; 6; = 1 if the suffix i (i = I, II, III, IV) corresponds to the mechanism activated by the applied stress state, otherwise 6, = 0 ; i is a scalar multiplier related to the strain amount. The physical meaning of 3. is easily argued from Fig. 5: it represents the mean value of the sliding or opening displacement jumps at joints. By superimposing, as usual, the strain rate space to the space of stresses, it can be seen that the strain rate vector is not normal to the corresponding pitch of the limiting stress surface (Fig. 7 shows the projection of the strain rate vector on a plane g, = constant). This situation is current in frictional materials for which, as is well known, the flow rules are of a non-associated kind (for pu and ,u,, vanishing or not

1170

G. ALPA and I. MONETTO

FJC;.7. Typical cross section in a plane u, = constant

of the limiting

surf&e

for varying

rr:h and il.

involved, the strain rate vector is normal to the corresponding limiting stress surface). Corners appearing in the limiting domains (Fig. 7) correspond to stress states for which two mechanisms are possible as well as each of their combinations (current situation in the plasticity theory too). As a result of the present analysis, as observed in the above, a more problematic indeterminacy may occur: two mechanisms and the related strain rate vectors are possible corresponding to the same limiting stress state [as an example, for a/h = 0.5, ,u = 1 both the mechanisms (III) and (IV) may become active]. These indeterminacies are, probably, reasons of the known difficulty in interpreting the block masonry behaviour in real structures and in experimental tests. In employing the above stress-

Model for dry blockmasonry walls

1171

strain relationships for a continuum analysis, the underlined circumstances must be carefully considered since they may lead to a plurality of solutions. The obtained results are based on deterministic considerations. Of course, each strength or flow parameter is a random variable characterized by some degree of statistical dispersion. This is a current situation in strength of material theories, where a probabilistic approach involving both mean values and coefficient of variations of the microstructural parameters leads to some rounding of the limiting state domains depending on the dispersion (ALPA, 1984; ALPA and GAMBAROTTA,1990). A certain imperfection in the alignment of courses is, instead, a peculiar characteristic of block masonry. This is the reason for the currently observed smaller uniaxial compressive strength in the course direction than that in the direction normal to them [see the experiments on brick masonry by PAGE (1983)]. In the context of the present analysis, where block crushing has not been considered, the deterministic point of view has led to equal infinite resistance against uniaxial compressive gr and G,.. The introduction of some measure of imperfection in the alignment of courses will lead to a finite value of the compressive strength in their direction. As a first approximation, this kind of imperfection may be simulated as a small rotation c of the given stresses with respect to the reference axes x and _Vassumed to establish the course direction. This way of considering the imperfection in the alignment of courses has a significant consequence only on the limiting condition (16) corresponding to the mechanism (II) of pure sliding (Fig. S), which actually becomes :

/Tyl.lf/w,,-Eeo,

< 0.

(22)

On a plane (r, = constant in the space of stresses, the above relation is plotted as a couple of symmetrical straight lines [dashed lines in Fig. 6(b)] intersecting each other at the point T,, = 0, o, = gJ P,/E. So, the considered imperfection leads to a finite value of the compressive strength in the direction of courses. A more detailed probabilistic analysis exceeds the aim of the paper. Nevertheless, the above considerations may be useful in interpreting experimental results which will fall near the smoothly rounded domain qualitatively plotted in Fig. 6(b).

6.

COMPARISONSAND CONCLUSIONS

The microstructural approach here followed shows that: (i) dry block masonry walls with in-plane loading behave as an anisotropic material without tensile strength in the direction normal to the courses and endowed with a tensile strength in the course direction depending on friction and state of stress; (ii) four mechanisms of uncontained flow, involving the block geometry, characterize the limiting strength states and situations may occur where a multiplicity of mechanisms correspond to a single state of stress; (iii) uncontained sliding flow can occur both pure shear and combined with elongation ; (iv) the flow rules at limiting strength states are of a nonassociated kind; (v) under the assumption of small flow, the uncontained flow rules derived from the microstructural approach can be employed in a continuum analysis. Comparisons with theoretical and experimental results will be useful to clarify the above.

G. ALPA and

1172

(a) Comparisons

I. MONETTO

with theoretical results

Referring to Fig. 8(a), let us consider a wall with a rectangular opening of width 1. A classical problem is to determine the separating line from the active and inactive zone, the so called “natural free boundary” which would take place in the absence of any architrave. The components of the reaction R related to the opening presence (at the prolongation of the opening upper side) due to the loads p and y are: V = qLj2 ; H = Stan $ (0 < $ < p,,), where L is an ideal width which may be approximately evaluated by an elastic analysis of the vertical compression Do near the opening sides [Fig. 8(a)]. From a qualitative point of view, we can define an ideal elliptical thrust line of equation : p,t” + qs’ + qL tan $y -q/s

= 0.

(23)

As a current approximation. the natural free boundary can be considered running close to an ellipse touching the opening corners a little lower than that corresponding to (23). A more refined analysis leads to an elliptical or hyperbolic free boundary depending on (I, q and c (VILLAGGIO, 1981). Such approximate evaluations of the natural free boundary line are based on Heymann’s assumptions of no-tension material and no-sliding (or negligible shear stress z,,.). From the point of view of the microstructural approach, it needs to be verified if an admissible stress state occurs along the evaluated free boundary. Considering that only a normal stress g, in the tangential direction is imposed from the boundary conditions. the following effective limitations are derived from the limiting states of (16), (19), (20) : mech. (II) :

tan I44 G P<,,

(24)

mech. (III) :

tan /VI < min (u,, a/26),

(25)

mech. (IV) :

tan Icpl < &.

(26)

b) FIG. 8. The

problem of the natural

free boundary at the upper side of a rectangular masonry wall.

opemng in a dry block

Model for dry block masonry

1173

walls

where ‘p defines the direction of cr, with respect to the y-axis. The limitation corresponding to the mechanism (III) is, of course, the only one to be considered. The above limitations lead to a couple of possible situations : (i) if tan ]$I > min (pL,,a/26) the free boundary defined as a thrust line by (23) is not admissible and the effective one is formed by two symmetrical straight lines characterized by tan (cp] = min (pL,,a/2b) [Fig. 8(b)] ; (ii) if tan ]$I < min (,u~, 426) the free boundary line is formed by two symmetrical curved tracts, starting from the opening corners, corresponding to (23) with tan (cp] < min (pL,,a/2b) and by two symmetrical straight tracts with tan ]q] = min (pa, 42b). Curiously, in this latter case the resulting free boundary line is like a pointed arch.

(b) Comparisons

with experiments

on dry masonry models

Carefully manufactured models are suitable to judge the approach followed here as well as the validity of the obtained issues. In the following, some comparisons are shown with experimental results from tests on panels made by regular blocks of “peperino” with a friction coefficient p = p<, = ~0, = 0.6 (see BAGGIOet al., 1991). Eight tests have been performed with various panel and block sizes : H/B = 0.48, 1, 1.5, 2; a/b = 2,4. As Fig. 9 shows, the limiting state of equilibrium has been obtained by rotating the support plane until critical angles ‘p* [the table in Fig. 9(a) contains ‘p* for all tested panels] have been reached. In Fig. 9(b) the typical picture of the sliding paths is shown for one of such panels (H/B = 1.5, a/b = 4). The sliding picture allows one to argue that the mechanism (III) becomes active. This agrees with the limiting conditions stated in Section 5 by (15), (16), (19) and (20), as shown in Fig. 9(c) where the limiting domains are plotted for a/b = 2. 4. In

b) FIG. 9. Dry block panels of peperino

c) at a limiting

state of equilibrium.

1174

G. ALP..+and I. MONETTO

fact, (i) the situation of tests is such as Q, 2 0 and z,,./cT, z tan ‘p* ; (ii) the limiting surfaces intersect the z,,/Ig, 1axis at 0.38 and 0.46, respectively for a/h = 2 and a/h = 4, corresponding to the mechanism (III) of opening at staggered joints and sliding along the aligned ones. This explains the following: (i) all the observed critical angles (p* [Fig. 9(a)] are less than arctan ,D (31 ), whereas for the simple sliding mechanism (II) it would be (p* = arctan ,D: (ii) the critical angle increases as the ratio u/h increases [Fig. 9(a)] in agreement with the limiting condition of the mechanism (III) [Fig. 9(c)] : (iii) there is a triangular zone of panel in which uncontained flow does not take place, in contrast with that which would occur if the simple sliding mechanism (II) became active. From a quantitative point of view, there is a fairly good agreement between theoretical and observed critical angles (p* : for u,ih = 2, 0.38 (theor.) against a mean experimental value tan (p* z 0.35 : for a/b = 4, 0.46 (theor.) against tan ‘p* z 0.48.

REFEREN(.ES

ALPA, G. (1984) On a statistical approach to brittle rupture for multiaxial states of stress. Enyng Frcrcture Md. 19, 88 I-90 I. ALPA. G. and GAMBAROTTA. L. (1988) Theoretical evaluation of the frictional damping in rocks. Proc. VI ht. Cmf. Num. Meth. Geomd7ar~ics (ed. G. SWOBO~A), Vol. I, pp. 391 396. Balkema, Rotterdam. ALPA, G. and GAMBAROTTA.L. (1990) Probabilistic failure criterion for cohesionless frictionals materials. J. McL./I. P/I~~.s.Solids 38, 491-503. ALPA, G. and TAFANELLI, A. (1984) Equazioni costitutive per solidi elastici microfratturati. Prw. VII Conywsso Nazionak AIMETA. Trieste, pp. 165%175. BAGGIO, C., MASIANI. R. and TRWALLWI. P. (1991) Modelli discreti per lo studio dclla muratura a blocchi. Proc,. V C~I~W~FIONrrrioncrk rii Ingc~,ynerirrSi,smiw. Palermo. Vol. 2. pp. 1205~1217. BATDORF,S. B. and BUDIAUSKY,B. (1954) Polyaxial stress-strain relations of a strain hardening metal. J. Appl. Md7. 21, 323 ~326. COMO. M. and GRIMALDI. A. (19X5) An unilateral model for the limit analysis of masonry walls. Proc. 2nd mectir7,y of1 ~~iihcvd Prohkrm in Structurd Ad.v.si.c. Ravello. 1983.CISM No. 288. pp. 25-38. Springer, Berlin. Dr PASQUALE, S. (1982) Questioni di meccanica dei solidi non reagenti a trazione, Prm. VI Conyw.sso Ncionak AIMET,4. Genova. Sez. II, pp. 25 I-263. GIA~UISTA, M. and GIUSTI, E. (19X5) Researches on the equilibrium of masonry structures. Arch Rution. Mccii. Anat. 88, 359 392. GURTIN. M. (1972) Linear theory of elasticity. Mec,lzanic~s q/’ Solids /I (ed. C. TRUESDELL, Encyclopedia of Physics). Vol. Via/2, pp. 1 295. Springer, Berlin. HEYMAN. J. (1966) The stone skeleton. Int. J. Soliu’?,Struct. 2, 249 279. How. H. and NEMAT-NASSER, S. (1983) Overall moduli of solids with microcracks: loadinduced anisotropy. J. M&7. Ph~s. Solids 31, 155 171. KAWANOV. M. (1987) Elastic solids with many cracks: a simple method of analysis. ht. J. Solids Sttwt. 23, 23-43. KACHANOV. M. (1992) Effective elastic properties of- cracked solids: critical review of some basic concepts. Appl. Mdl. Rw. 45, 304 335. LIVESLEY,R. K. (1978) Limit analysis of structures formed from rigid blocks. ht. J. Nuns. Methods EF7gF7g 12, 1853 187 1. MURA. T. ( 199 I ) Micromwhu~7ic~.s of’Dc~f>cts il7 Solids. Kluwer Academic, Dordrecht.

Model for dry block masonry walls

1175

PAGE, A. W. (1983) The strength of brick masonry under biaxial tension