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LOAD TRANSFER IN MULTIPLE LEAF MASONRY WALLS

L. Binda, A. Fontana, L. Anti

Dept. of Structural Engineering, Politecnico ~i Milano - Italy

ABSTRACT

Multiple leaf masonry walls are frequently found in historic buildings overall Europe. They consist usually of two- or three-Ieaf walls made of different materiaIs (e.g. stone-rubble-stone, stone-rubble-brick, stone­brick, etc. I. The load distribution among the leaves or wythes is strongly dependent on the mechanical, technologi cal and geometrical characteristics of each leaf. Two different limit si tuations can be detected as follows: aI transversal stiff elements transfer the load to the wythes proportionally to their axial stiffness ; bl the wythes are only connected by collar joints or by the mortar of a rubble fill ing and the load distribution is mainly dependent on the bond strength between the materiaIs. In the former case the load carrying capacity of the wall only depends on the compression strength of the materiaIs; in the latter case it relies on a parameter, the bond strength, which is difficult to be defined. In fact it is dependent on cohesion and friction properti es of the contact surface betwee n the leaves. This work has the aim to set up simple analytical models useful to study the stress distribution in multiple leaf masonries according to their technological characteristics. The efficacy of these models is based on the possibility of giving an easy and quick evaluation of the safety of the masonry walls. A comparison is made between the results obtained from the application of the studied models and from a F.E. modelo

1. INTRODUCTION Objectives When dealing with repair and/or retrofitting of masonry structures, the actual carrying capacity of the walls has to be defined prior to the intervention. Therefore the state of stress of the masonry under in-service loads must be known. This information can easily be obtained for single leaf walls when simplified hypothesis of homogeneity can be assumed. Different techniques have been used in the past for the construction of loadbearing walls in Italy, as in alI the other European countries. For the most part of the historic masonry buildings multiple leaf walls were used. Figg. 1, 2, 3 represent three different examples. Fig. 1 shows the section of a wall, made with bricks and a sort of conglomerate of lime mortar and stones; belonging to a church in Siena, Italy (XII-XVI centl. Fig . 2 and 3 show two different rubble walls, with externaI leaves made of stone, belonging to the Priory of the Blessed Virgin Mary and St. Cuthbert, Bolton-in-Wharfedale, England (XII-XVI centl. As it has been described above multiple leaf walls can totally differ one from the other; therefore it would be very difficult to synthetize this variability into few schemes representative of lhe whole . Nevertheless there

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Fig . 1 Three-leaf masonry wall of a church in Siena, Italy (XII-XVI centl

Fig . 2 Three-leaf masonry wall of Priory of the Blessed Virgin Mary and St. Cuthbert, Bolton-in­Wharfedale, England (XII-XVI centl

Fig. 3 Three - leaf masonry wall of Priory of the Blessed Virgin Mary and St. Cuthbert, Bolton-in­Wharfedale, England (XII-XVI centl

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exists the common necessity to define how loads are transferred from one wythe to the others; this in order to know the stress distribution along the wall section and subsequently to detect the safety of the wall.

Background Some research has been carried out on the above mentioned problem with reference to recently built composite walls. In these cases the geometry of the section of the wall is defined so as the construction technique; the properties of the material are also known. Studies 'have been recently developed on composite walls consisting of a wythe of clay brick and another of concrete block bounded together by a grout collar joint. In general only the block wythe is loaded and the load transfer from the block wythe to the brick wythe occurs through the collar joint . These studies have shown that the distribution of vertical stresses is not too sensitive to the size of the exact width selected for the interface elements (collar joints) so long as there value is kept relatively small [1) (2). Moreover a good fitting was found of the experimental measurements of strains by the results obtained through FE linear analyses . Finally it was observed that the results of numerical analyses do not differ substantially in the following two cases : (i) the collar joint is made of a material defined "weak" (Le. G modulus is of uncertain definition), (ii) the collar joint gives a high bond strength between the two externaI wythes (G modulus is the one of an isotropic material). The above results turn out to represent a good background also for the knowledge of the distribution of vertical loads through the different wythes of multiple leaf walls of historie buildings . In the case of an existing building the characteristics of the interface material between two wythes are unknown in terms of dimensions and mechanical properties . This aspect shows also a similarity with the following typical situation in geomechanics: the uncertainty on geometrical and mechanical characteristics of the interface between two layers of stratified rocks [3) [4). On the basis also of the results obtained by the above mentioned works a preliminary step of a wider programme is described in this paper. The aim of the research is to define the stress distribution in composite masonries of historie buildings; this problem has emerged in these last years (5) (6) and has not received so far the due attention by researchers. The possible applications of this study are the following: (i) implementation of analytical models leading to the calculation of the load-carrying capacity of multiple leaf masonry walls subjected to heavy dead load, (ii) preparation of updated recommendations for strengthening techniques of multiple leaf masonry walls.

2. DESCRIPTION OF THE PROBLEH

It is well known that the stress distribution of vertical loads among the different wythes of a multiple leaf masonry depends on the morphological and mechanical properties of the wythes. But this distribution is in most cases practically unknown. Nevertheless the range of the occurrences can be bounded within two limits, on which an investigation is needed: a) the wythes are horizontally connected by elements of high stiffness (such as stone blocks crossing the whole section of the wall) which can transfer the load from one wythe to the other (Fig. 4). There flexural behaviour allows them to carry out the task without significant dimensional variations. The load transfer can be analytically determined through simple models by taking into account only the vertical deformation of the wythes.

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In other words the load carrying capacity of the masonry only depends on the compressive strength of the weakest material. This structural models do not take into account the type and nature of the connections at the interfaces. b) the connection between the wythes is realized through vertical mortar joints. The distribution of the vertical forces is exclusively assigned to the bond between the joints and the leaves of the wall. Nevertheless the geometryand the mechanical properties of the joint of historic walls are practically unknown. Two typical cases are presented in Fig 3 and Fig 5. In the first case the dowel action of the projection of the largest stones, provides to the mechanism of shear transfer. In the second case the transfer can be realized by a system of diagonal members. Horizontal tensioned members are then necessary for the equilibrium of the system; these last elements can only be introduced successively by a repair intervention. Nevertheless it is difficult to detect in practice the structural behaviour of the system; infact the geometry and the mechanical characteristics of the joint beco me important input data. Their determination is left to an experimental research which will be carried out in the future and will particularly concern the relationship T-r. It seems reasonable that a constant thickness has to be assumed for the equivalent joint. Meanwhile the constitutive matrix of the joint material should be defined. In other words it will be assumed that the interface between two wythes is a weakness plane and can be represented by a thin joint with convenient characteristics.

3. NUHERICAL ANALYSIS

A recent numerical and experimental research on new composite masonry walls [1] has shown that the interface can be considered a perfectly bonded joint rather than modeled as a weak joint when the components are not cracked. Hence when cracking does not occur it is not necessary to use interface elements to obtain a better accuracy from the finite element analysis. AIso the linear mechanical behaviour assumed for wythes seems to be comparable with experimental results. In fact it is well known that the mechanical response of brick masonry walls to vertical loads is linear up to approximately the 70+80% of the compressive strength. This also in the case of decayed masonries subsequently repaired by injection techniques [6]. For linear elastic materiaIs it has been shown that the numerical analysis is not sensitive, "in practice", to the thickness of the joint, which has to be assumed among the values usually adopted [1], [2].

analysis was in the joint

assumed between

Following alI the above mentioned suggestions, the numerical carried out with the assumption of a linear elastic material considered perfectly bonded, that is no weak interface was the wythes of the wall . The type and dimensions of the model adopted for the numerical analysis are reported in Fig. 6. First of alI the mechanical effects of the dead load among the wythes was analysed. The wall is a three leaf wall; the externaI leaves were assumed as made of the same material 1. The following ratios were assumed: r

1/r

2 = 1.5 between the bulk densities ; E

1/E

2 = 3 between the

elastic moduli. A value of 0.1 for the Poisson coefficient u was chosen for alI the wythes. As it is shown in Fig. 7 the transfer of normal vertical stresses takes place from material 2 to material 1· negligible shear and normal horizontal stresses were detected . When the thicknesses of the bearing wythes are not large and their values are comparable, a simple rough explanation of the behaviour can be obtained as follows. Normal stresses can be defined as: ~1= r

1x + ~~1; ~2= r

2 x - ~~2

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k1=E111

k 2= E212 k3= E3 13 a =(11)1 2)/2 b=(12+13)/2

Fig. 4 A simp l e model of a compo­site masonry when the interface is assumed "normally" bounded

~,....

E <J co .. _ o.

'" '" '" . w - wlm õ w õi cô w m <D ~ .. ... o w m <D '" '" ())

"' ! I .., 1

! .. I 1

'" I m I 1 ~ . . - . .- .- -- . '-' . ~ . ~ ' -,---, - _ . _- - " r------= " -'" ())

:l5 1 1

w o

~ I w '" w m tO W õi W '" '" '" w w

w m tO w m .., m tO '" '" ()) - .. ... o w

,.,';',l;,=~ "1'.'; "A "" ",,,j,j ,,1'.

12 eni 20em 12em

-0.0

1.0

Ô Q.. 2.0 2 ~

3.0 * b

4.0

5.0

ties

compressed ==~ .. ~~ __ ~~ .. ~~~~elem_e_nt_s __

.~ J - • : -

Fig. 5 Shear stresses transferred by diagonal compressed members

'EI! ....,

o

~ , ~ ~ ~ ~ lJii.

~

"111.

~ ~

" , ~ , "

20 40 60 ao 100 120 140 160 180

x (em)

~1 _ 12 _ .!.., , G) ® G)

'--.....

-18 1-

rn=EEJ el. 1 - 33 _el. 3 4- 66 ~el. 67- 99 ~eI.IOO- 1 32 G:EEf) el. 133- 165

200""""" el. 1 66- 198

Fig. 6 Mesh used to deve­lop the numerical analysis

Fig. 7 Vertical stresses due to dead load

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where "I' "2 are the bulk densities of the materiaIs and â~I' â~2 the stress

variations due to stress transfer.

The respect of equilibrium requires: t

2J 1 â~1 dy -o

t +t J 1 2 âer

2 dy

t 1

= O (1)

When âer1

and âer2

is not depending on y as it happens in the case of pure

dead load, then i t can be assumed that 2t1

âer1

= t2

â~2 '

The respect of continuity in the vertical direction (ver is negligible in y

comparison to er ) for the externaI and internaI wythe is x E IE

"212-"1 following relationship: âer

2 ElE

1 2 + t 12t

2 1

. x

expressed by the

For the chosen values "1/"2' EI/E2' v, the following results are obtained

for er and er: er = (0,018 + O,004)10-3 ·x er = (0,012 - O,0047)10-3 ·x 1 2 1 . 2

MPa. The values obtained from these equations agree with those determined by F.E.M. represented in Fig . 7. Several load conditions were tested. For the externaI wythes an elastic modulus E

1 = 1500 MPa was assumed; for the internaI wythe the following

three different cases were taken into account E2 = 1000; 500; 250 MPa. In

Figs. 8, 9 the most significant results are reported in the absence of a stiff horizontal distributor and with eccentric loads. It is easily observed that the ratio er Ip is rather low, while the ratio T/p

y

can reach even values near to 1 at the top of the wall and becomes very low after a rapid decrease. The same conclusions are reached in the case of more than 2 wythes, and are confirmed by the results obtained when 3 different materiaIs and 5 symmetrical wythes are considered.

4. SIMPLE MATHEMATICAL HODELS

Two simple models are proposed which can represent the phenomenon of concentrated load transfer in two typical limit situation: presence or absence of horizontal stiff elements below the application point of the load (the intensity of the load is referred to the length unit). a) Presence of stiff horizontal elements (such as blocks of stones or courses of bricks built-in the masonry) . The average value of the normal st ress in the wythes can be easily calculated if the model shown in Fig. 4 is adopted; the stiffness of every spring represents that of the corresponding wythe. Let us consider, as an example, a three leaf masonry and a concentrated load P applied at a distance e from the centre of the masonry; within the limit of the elastic response of the material, the stress distribution is realized in the following proportions:

F k ea + k (a+b) (e+b) F k ata-e) + k b(b+e) 1 k 2 3 2 k 1 3

P 1 k [k a 2 + k b2 + k (a+b)2] p 2 [k a 2 b2 2 ' k + k + k (a+b) ]

1 2 3 3 1 2 3 3

F k (a+b) (a-e) - k eb (3)

3 k 1 2 P 3 k [k a 2 k b2 + k (a+b)2] +

1 2 3 3

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According to eqs. (3), when the two externaI wythes are made of the same material, if t

1 = t

3= 12 cm, t

2 = 20 cm, e = 4 cm, E

1/E

2 = 6, a = b = 16 cm,

the vaIues of the average stress in the three leaf can be calculated as: ~1 = 0,455 MPa, ~2 = 0,060 MPa, ~3 = 0,263 MPa.

If these values are compared to the correspondent values obtained from the F.E. model, a 3% difference is found . If more accurate results are desired, the maximum and minimum value of the normal stress within each wythe have to be calculated. In the case of the above described example and for the most loaded wythe these values are given

2below:

~ = ~ + Pe/(t + t) ; ~ = ~ - Pe/(t + t )2 . (4) 1max 1 1 2 1mln 1 1 2

The value for the remaining wythes can be anaIogously obtained. AIso in this case the comparison with the values caIculated by applying the F.E. analysis (see sec o 2) shows a good agreement. When a stiff horizontal element is present connecting the wythes and for the material of the wythes a linear ~-c law can be assummed, then the proposed simple model can be applied with good results . The conclusion is drawn also in accordance with the results obtained in [7]. Three main goals are achieved: (i) the possibility of implementing programs of simple use on home-computers; (ii) the reduction of computer costs in comparison with a corresponding F. E. analysis; (iii) the availability of a fast mean to judge the efficacy of repair and strengthening techniques, provided that a linear elastic behaviour be assumed for the strengthe ned masonry.

b) Absence of the stiff horizontal elements. A limit condition of weak interface between the wythes is assumed in order to study the stress distribution of concentrated loads. This criterion represents an extension to masonry walls of the one proposed in [8]. A system of selfequilibrated normal stresses ~ (x) is introduced along the height e of the masonry (Fig .

y

10). These stresses, representing a sort of normal bond, ensure the contact between the wythes; hence the hypothesis of friction transferring shear stresses along e is reliable, provided that the system of ~ be adequate to

y

the system of T. If the invariancy of ~ and the linear behaviour of the y

materiaIs are assumed, with reference to Fig . lO, it is possible to obtained the transferred internaI force N(x) at a distance x from the top and the associated i~ternal moment M(x) = Nd if the condition of minimum elastic energy is imposed along e. The energy U is expressed by :

12d2] + 2(P_N)2 + ~ [~]2 ]

t 2 E t G dx dx . 2 2

(5)

1

Then the following differential equation has to be satisfied:

o? = _G _ _ [1 + 12d2

+ 2E1 tl] E1 t 1s t 2 E t

N" - 2 N 2P G ex. =- Ets'

2 2

where (6)

1 2 2

under the boundary conditions N(O) = O; (dN/dx)e = O; being G and s

respectively the shear modulus and the thickness of the weak joint. Since d=s+t

1/2 (see Fig. 10), then the ratio N/P representing the internaI

force N transferred to the externaI wythes can be obtained as follows:

-0.5

Li

""-

0 .0

l- 0.5

Q

""-

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~ -~

IV

~ k.. ~

b BE!88El el. 1 - 33

Q

'----<-Q

'----b

1.0 -- el. 34 - 66 ~ C~ el. 67 - 99

~el.l00-132 BE!88El el. 133- 165 ~ e1.166-198 - CJ,

15 ......... 7.y O 20 40 60 80 '00 '20 '40 '60 '80 200

X (em)

- 1.0

- 0.5 1\

0.0 ~ ~

~ 0.5 1f

14 ~ ,Aâ

I~ 1. 0

, .5 o 20 40 60 80 '00 '20 140 , 60 '80 200

X (em)

p=~ ~---'.ill.l.j.l.~i-+-_ t 2

t, t2 t,

BE!88El el. 1 - 33 ~ el. 34- 66 C~ el. 67- 99 ~el.l00-132 GeeEl0 el. 1 33- 1 65 ~ e1.166-198 - CJ, .......... T~y

e

Fig . 8 sional stresses

Non-dimen-vaIues of

when the Ioad is eccentric

Fig. 9 Non-dimen­sional vaIues of stresses when the Ioad are appIied on the externaI wythes

Fig. 10 Mechanism of shear transfer in the case of "weak" inter­face

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N P

---------------[l+tghae senh(ae I) - cosh(ae I)] + 2(1+3 s/t + 3 sZ/tz)E t /E t

(7)

1 1 Z Z 1 1

where the function between brackets is called "transfer function". The multiplying coefficient of the transfer function does not depend on G; also since s«\ the value of the coefficient is not inf.luenced by the values of

s. The difficulty of a reliable experimental determination of G can be partially overcome by taking into account that a nominal value G/E

j(i=l,2)

can be proposed. A nominal value deduced by the in situ observation of existing walls can be assumed for s, which is usually unknown. In fact s is generally very small compared to ti and tz; the possible difference between the assumed value and

the real one has a minor influence on the value of distance along which the shear stress T is distributed . The following comments can be made on the parameters of Eq. (7): (i) N/P ~ 1 if E

1 ~ 00; (i i) for a e 2:: 4, tghae '" 1 and senhae '" coshae, then the

bracketed transfer function becomes 1 and when x=e, then the value of the ratio N/P depends only on the coefficient of the transfer function: the greater is a the smaller is e; (iii) if the internaI wythe is strengthened by injection of grout, the length e along which stresses due to transfer forces develope will be reduced. In a generic section along the distance e the between the joint and the wythe are ~ N(t +6d)/tZ and ~ = 2(P-N)/t , provided

lx 1 1 Zx Z

stresses at the expressed by: that an adequate

interface T = dN/dx; value of

~ be assigned. The state of stress is determined and the leveI of safety y

can be assessed. The results reached allow to draw for the historic multiple leaf masonries the same conclusions as in [1] for the composite masonry: when a linear elastic response can be assumed for the material, then the ratio N/P can be considered independent on the mechanical properties of the weak interface between wythes, if an adequate system of self-equilibrated stresses ~ is

y

real ized.

5. CONCLUSIONS

The following final remarks can be derived on the transfer of vertical loads among the wythes of a multiple leaf masonry (when a linear elastic behaviour of the material is assumed): a) the proposed simple mathematical models can satisfactorily describe the

state of stress in the wythes of the walls and allow for the evaluation of the load carrying capacity of the walls;

b) the presence of stiff transversal elements can successfully distribute the vertical loads along the wythes of the section; in this case shear stresses along the interfaces of the wythes can be disregarded;

c) in the absence of stiff horizontal elements, when a perfect bond can be assumed between the wythes, the value of the height along which shear stresses are distributed does not exceed the total thickness of the wall; this behaviour reminds the principIe of De S. Venant for homogeneous solids;

d) when a weak bond has to be considered between wythes, the distance e can be previously calculated; in this case a strengthening intervention is

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necessary in order to assure the distribution of T along the contact surfaces;

e) a systematic in situ experimental investigation is suggested on multiple leaf historic masonries for a better knowledge of weak interfaces; this in order to determine nominal values of G as a function of the leaves properties and also to assume a nominal relationship for T-~.

REFERENCES

(1) Anand, S.C.; Rahman, M.A., "Interface behavior in concrete block mortar joints. A comparison of analytical and experimental results" 50 North Amer. Masonry Cenf., Univ. of Illinois, Urbana, 3-6 June, 1990.

(2) Rahman, M.A.; Anand, S . C., "Some problems in estimatin§ peak shear stresses in collar-joints of composite masenry walls", 5 North Amer. Masonry Conf., Univ. of Illinois , Urbana, 3-6 June, 1990.

t 3) Desai, C.S. et. alii, "Thin-Iayer element for interfaces and joints", Int . Jour. of Numerical and Analytical Methods in Geomechanics, Vol.8, 1984 .

4) Goodman R. E., et aI ii, "A model for the mechanics of pointed rocks" , ASCE Journ., 5M3, May 1968.

5) Benzoni, G. M.; Fontana, A., "Load carrying capacity of masenry piers wi th epenings", 7

0 I BMaC , Melbourne, 1985 .

6) Binda, L.; Fontana, A., "Mechanical behaviour of brick masonries derived from units and mortar characteristics", Proc. 8

0 Int.

Brick/Bloc Masonry Cenf. Dublin, 1988 . (7) Miltiadou; A.E., "Contribution à l'étuede des coulis hydrauliques pour

la réparation et le renforcement des structures et des monuments historiques en maçonnerie", Thêse Doctorat de l'Ecele Nationale des Ponts et Chaussées, Paris 2.10.1990.

(8) Goland, M.; Reissner, E., "The stresses in cemented joints", Jour. of Appl . Mech., n.66, 1944 .