37.-Safety Considerations for a Limit-State Design of Brick Masonry

by G. MACCHI

Col/ege of Architecture, Universily of Venice

ABSTRACT The design of brick masonry based on lhe limit-state method requires a definition of 'characteristic strength'· based on the statistical properties of the material and a choice of suitable factors of safety .,hich also take into accounf lhe slrue/ural behariaur.

Considérations de Sécurité paur une Construction à I' Elat Limite de Maçonnerie en Briques

Betrachtung der statischen Sicherheit von Ziegelmauerwerk

Der auf der Metlzode des Grellz, zustandes basierende EntlVurf von ZiegelmauerlVerk benotigt eine Fest­legllng clzarakteristisclzer Festigkeits· werte anhalld statistischer Material· eigenschaften und eine Reihe brallclzbarer Siclzerlzeitsfaktorell, die in die Berechmmg des VerhaltelJs der K011struktion eingehen. Ein Verg/eich der auf der ganzen Erde venvendeten Vorschriften zeigt augelJscheinlich grosse Unterschiede der Sicher­Izeitsfaktorell . Mit der Adsiclzt, vernüllftige Kriteriell festzlllegell für eine internatiolJale Empfehlung, lVie sie in almlicller Weise für andere Ballkonstruktiollell bereits vorliegt, lVurde eine Ana/yse der bestehenden Unterschiede vorgenommell.

Comparison of codes of practice in use ali over lhe world shows ,hal fllere are \Vide differences in the factor of safety. An analysis of SUelI differences is underlaken in arder to define reason­ab/e cri/eriafor Qn interna/lonal recom­mendation similar to Ihose already available for olher building struciures.

Une tous/ruetiulZ en ma~onllerie en briques basée sur la méthode de I'é/al limite nécessite une définitiolJ des valeurs caractéristiques de la résislance basée sur les propriétés statistiques du matériau el Wl choix de facteurs de sécurité cOllvenables teIJaIJt également cornpte du comportemellf de construction. Une comparaisoll des codes de pratique ell usage dans le monde enfier mollfre des di./férences apparemment importantes dans le facteur de séclIrité. Une ana/yse de tel/es différellces est elltreprise afill de définir des criteres raisonnables pour une recommandatioll illter~ nationale semblable à celles dom on dispose déjà pOlir d'autres types de constructions.

I. AN lNTERNATIONAL CODE OF PRACTICE The increasing use of probabilistic principies in the design of structures and the widely recognized capacity of this approach to express the modem safety philosophy in a better way also poses the problem of their use in the structural design of brick masonry.

Following the publication by the European Concrete Committee of various 'recommendations' for inter­national codes of practice (reinforced concrete, pre­stressed concrete, large panel buildings), which are strongly influencing the national codes, a similar docu· ment for masonry seems desirable in order that engineers do not have to apply dilferenl safety principies to dif· ferenl structural elements which often co·exist in the same building.

The work necessary to draft an 'international recom­mendation' is carried out by Committee W23A (Bearing Walls) ofthe International Council for Building Research (CIB). This work often encounters serious difficulties, for instance in respect of the not generally accepted concept of masonry as a basic material, and of the nol yet sufficiently analysed structural behaviour of a build· ing consisting of bearing walls and reinforced concrete floors.

The main difficulties arise when a comparison is made of the safety coefficients adopted by the existing codes. Such a comparison is necessary because the present codes result from wide experience and take implicit accounl of many uncertainty factors which cannot be statistically

• In lhe philosophy of semi-probalistic codes the 'characteristic strength' is statisticaJJy defined by fix.ing the probability that the actual strength would be effectively less lhan the value selected.

229

defined. (The aim of the new code is to reach a more uniform safety by means of a more consistent combino ation of various uncertainty factors, and not necessarily to reduce the mean safety factor of the structures).

The results of such an analysis of presenl codes are surprising. In 1953, THOMAS' proposed modifications to CP I I 1 (most of the modifications were incorporated in the 1964 edition2) showing that the mean load factor would be about 8·4 for a wide range of slrength, sIender· ness and eccentricity values, and never less than 4; in the same paper he stated that "the load factor of 6, when tesl panels are buill to determine lhe strength of wall construction, is reasonable". In this case, CP 111 gives load factors ranging from 5 to 12 for dilferent values ofstrength and slendemess. In many other Western countries the permissible stresses are actually based on similar load factors. The Swiss Code SIA 113 of 1965' is based explicitJy on a safety factor of 5, reduced to 4 when sample panels are tested. However, in practice the basic compressive stresses are not very different, in similar conditions, from the values of CP 1 I I.

If the USSR code (SNIP-II B·2·62) is considered, basic stresses are on average 70 % higher than in Westem countries; these values for a limit-state design, cannol be immediately compared with those of codes based on permissible stresses; in fact a partial coefficient of safety is applied to the load (1'1 to the dead load and 1·4 lo the live load), so thal the practical increase in permissible stresses is reduced to about 40 %. This dilference is not negligible as confirmed by the codes of other Eastem countries (e.g. Poland), which adopt a safety factor of 3. The common opinion in Western countries is that "the

230 Safety Considerations for a Limit-State Design of Brick Masonry overall load factor is unnecessarily high for mos! prac­tical examples1" because ofthe tendency to adopt limiting safe assumptions for each factor influencing the strength ofthe structure.

There seems to be some convergence of the professional opinions on the safety of bearing walls, but the dif­ferences between the safety factors are still so wide when compared with those on other types ofbuilding structures, that a rational probabilistic analysis seems to be the only way to reach a consistent definition of safety rules.

2. THE PROBABILISTIC APPROACH Let us consider the simple case of a waIl subjected to an eccentric axialload. If the unit strength of masonry and the load (or load e!fect) S can be considered as random quantities, statistically defined and without any cor­relation, the problem of safety is theoretically reduced4

lo ensuring that the probability of failure during the expected life of the structure is reduced to a generally accepted value,S.6 e.g. a value between lO- s and 10- 6.

A 'function of indestructibility' has been defined as the di!ference between the load-bearing capacity and the load:

R=.pAa- S. · (I)

where A = area of the cross-section and .p=reduction factor for eccentricity of loading

and slenderness.

If the distribution of probabilities of R, PR' is known, the problem is reduced to calculating the integral proba­bility V so Ihat the following inequality is verified.

R<;;O · (2) that is :

V= f;R dR · (3) _ ro

This can be done easily if the quantltles included in eqn. (I) are uncorrelated and defined by a normal fre­quency distribution with known mean values rpm, Am. Gm, Sm and known coefficients af variation ú", . SA' 8u,

3s· Figures I and 2 can be obtained if the world-wide

factor of safety y (ratio between the mean values) is introduced

y t/lmAmum

Sm · (4)

They show the values of the safety factor y which are necessary in order to obtain the required leveI of safety ; the continuous lines refer to the probability of failure V = 10- 5 and the dotted lines to V = 10- 6 in function of the coeflicient of variation 3, of unit strength. The single curves represent various leveis af uncertainty af the coeflicient for eccentricity and slenderness (3. =0,0'05, 0,10,0'15).

For the sake of simplicity the variability of lhe cross­section has been neglected (3 A = O) and only two cases of load variability have been considered (Figure I for 3s=0.05, and Figure 2 for 3s = 0·15).

Referring to Figures 1 and 2 it can be stated that, for structures made of materiaIs having a smaU strength dispersion (i.e. those which have a small coeflicient of variation 3, ) and no e!fect .p of eccentricity and slender-

7 } I I I j Y

I. , . ti~ I ~:'tzt~'"· I / lI! ' / >- 'DI/. I .., :;,L, G . o'. , " ". '1 I . <:c: ~ 'õ<Q , ~.. / /0,0

3';

-'"" ..... z.; "

. ,'

~1 1 1-1 1.5 ---- i 1 I I I I

0.05 0.10 0,15 Q.20 Ó~

FIGURE l- Relationship between the safety factor y and the coefficient of variation oa of the strength of the material (for a

coefficient ofvariation 3s= 0'05 ofthe Ioads) .

ness (e.g. a steeI bar in tension), the necessary safety factor y can be low (between I· 5 and 2 '0) and is not very dependent on variations of ao and the safety criterion (probability of failure). This expIains why there is almost general agreement when a safety factor has to be defined for steel bars in tension or for under-reinforced concrete beams in bending.

However, when the material has a high 3, the curves rise rapidly and the value of y is highly dependent on the value of 3, and on the chosen probability of failure. This explains why, in this case, some differences in experimental results and in safety cri teria can lead to very di!ferent safety factors applying in di!ferent countries.

If the reduction coeflicient .p for eccentricity and slenderness cannol be considered as deterministical1y defined the safety factor is strongly influenced by its coeflicient of variation 3 • . This can be clearly seen by the differences in the various curves in Figures 1 and 2. A coefficient 3.=0'15 can require a safety factor twice as large as the one required by 3.=0'10, all the other conditions remaining constant. This means that a rational definition of suitable safety factors (and possibIy of their reduction) can be achieved only by a 'satisfac­tory theory' able to define clearly the eccentricities in the waU and its behaviour in the case of high slenderness ratios.

The variability 3s of lhe load appears to have a smaU e!fect for high values of 3, and 3. ; this can be checked by comparing Figure I (3 s = 0,05) and Figure 2 (3s =0·15).

l i I l i 16 ;'

e" ,/ li ' ~ti I ~ '. /'/1 j// ~ . Y , ~ 4S /' b~ //1

35

~ ,o

J:~0í~ "

2.5~~~ - ~

, I I 0.05 0,1 0 0.15 020

Ó~

FIGURE 2- Relalionship between the safety factor y and the coef­ficient of varialion Ôo of the strength of lhe material (for a coef­

ficient of variation 0s=0·15 of the loads).

G. Macchi 231 [n practice this effect is also very small for the dead loads prevailing generalIy in the structures under consideratian.

3. STATlSTlCAL DATA

It has been shown how a rational definition of the safety factor depends mainly on the information on the statisti­cal data concerned with both the basic strength of masonry and the effects of eccentricity and slenderness.

As the test of a large number of storey-height walls is a long and expensive job, a general survey and statistical re­examinatian af tests carried out in different countries would bc dcsirable.

Tests on over 500 piers and walls have been made in the UK at the Building Research Station. THOMAS1

provided the histogram (Figure 3) of 151 results obtained

20 __ o

t---" " .. ,I

- J U IUI o _I r n

~

I L

í

I u

• , •

ri 'TI n o , " LOAD rACTOA

FIGURE 3- Histogram of 151 masonry walls teslcd at lhe Building Research Station. 1

in several series af tests covering a wide range af slender­ness and eccentricity; the histogram shows the load factors obtained in the tests in comparison with the loads allowed by CP 111. The coefficient of variation is 0·22 This dispersion shows the possible degree of error of a satisfactory code, which includes the effect of .p.

lt can be concluded that the safety leveis put as a basis for Figures I and 2 cannat be reached-even with safety factors larger than 6- at the actual state of strength prediction.

Tests carried out in the USSR on 747 small walls are reported by SEMENTSOV.7 The frequencies of deviations af experimental results from Onistchik's formula are plotted in Figure 4. The coefficient of variation is 0,09; it rises to 0·147 if a nominal strength of mortar (defined only by the mix proportions) is introduced in the formula instead af the actual tested strength.

These results do not necessarily contradict the above-

I I /t 1/ l~

/ 1 \ -,

/ I \ ............... I

• 0

lO

20

" -l' -2' -, o .,

FIGURE 4-Histogram of 747 small masonry walls testcd in the USSR.'

mentioned British results: they would almost agree if a coefficient of variation R. of between 0·10 and 0·15 could be attributed to the reduction coefficient for eccentricity and slenderness, which does not apply in the case of the USSR tests. This effectively agrees \Vith an evaluation done by SEMENTSOV. 6

An idea af lhe uncertainty existing in the evaluation af coefficient '" is given in Figures 5 and 6, where lhe curves

'í' � .o,.----;,.-----,----,---,--

••

" " 25 3D SLENOERNESS h,,6

FIGURE 5- Reduclion cocfficient for slenderness (comparison of dilferem codes).

obtained from the codes af practice af some European countries are compared.

In Figure 5 only the effect of slenderness is considered; in Figure 6. the combined effect of slcnderness and ec­ccntricity for m = 1 is shown . *

The wide range af values existing in some codes shows lhe expected wide variatioo of .p in relation to the quality of the masonry. A modern theory of unreinforced walls8 can define various curves for tf in relation to suitable parameters; but only a large series of tests can give a numerical measure af the possible scatter around lhe theoretical values.

'f 1.0 8~O T 08

o., ~I-

o .•

o.,

OL------,~-----,~0----~I~'----~2~0~~~2~'----~)·O SL(HOERNESS Iy'd

FIGURE 6- Reduction coefficient for slenderness and eccentricity (comparison of different codes) .

Recent tests on 288 walls are reported by WEST and others;9 a statistically based comparison of these tests with the most recem theories would be very useful. lt is worth noting lhat only tlUs type of analysis, ap­plied to a group of tests with a sufficiently \Vide range of

• In wall design it is usual to define the slenderness as the ratio h/eI af the height " to the Ihickness d. For lhe eccentricity e the parameter m=6e/elis used.

232 Safety Considerations for a Limit-State Design of Brick Masonry masonry qualities, can provide results which are im­mediately useful. Instead , small coefficienlS af variation are sometimes obtained 00 single sets Df tests; values ranging from 0·05 to 0·13 have been reported in one paper lO and values from 0·07 to 0·14 elsewhere." These values can be significant only if they are obtained on a variety af materiais having different origins but in agreement with identical specifications.

The same remarks apply to the statistical data on brick strength; coefficients af variation ranging from 0·06 to 0·25 are reported in the UK;9 values of 0·18, 0'15, and 0·12 (in a site control) can be found else­where ;ll - J] values from 0·05 to 0·17 have been obtained in the USA;IO values 01' 0·15 ror ordinary bricks and 0·24 for horizootally perforated bricks are the results of lhe tests carried oul in Belgium ;14 a mean value af 0·12, and ao exceptional value of 0'20, for a specific type of brick, have been obtained in France on horizontally perforated bricks.

The value af the coefficient af variation could decrease with increasing strengths, as it does in the case of con­crete strength, for which the standard deviation is more significant than lhe coeffident af variatian. This fact should be checked .

4. CONCLUSIONS

The present paper l11ust be considered as only a first attempt to deal with the problem ofa consistent definition of safety factors to be introduced in a limit·state code of practice. Nevertheless, the application af some elemen­tary relationships of the theory of probabilities and the analysis of some experimental data leads to provisional conclusions which can be used as a basis for flltllre work:

I. Estimating the strength 01' a masonry wall is sub­ject to considerable uncertainty, and this natllrally leads to the adoption of very large safety factors.

2. A consistenl redllction af the safety factor can only be achieved by an accurate estimation oflhe masonrystrength in relation to the qualities of the component materiais and of ali the other parameters inftuencing it ; this requires

the definition of a large number of masonry types. as is actually done by some recent codes; at the same time, a 'precise' theory has to be adopted for the definition of lhe effects of slenderness and eccentricity in the over­ali behaviour of the building, using accurate deformation parameters (which are related to the masonry type).

3. The probabilistic approach shows that a consider­able reduction of the safety facto r can be achieved in the case of 'calculated brickwork', if a sufficiently wide statistical control of the quality of specific brick­work allows the definition of its characleristic strength', and an apprapriate structural analysis is done.

REFERENCES 1. THOMAS, F. G., The Strength of Brickwork. Sfruet. ElIgr.

Feb., 1953, p. 35. 2. BR1TlSH STANDARDS INSTITUTION, Structural Recommendations

for Load-Bearing Walls. C.P.III : 1964. 3. S. 1. A. (SWITZERLAND), Technical Standard 113: 1965. 4. RJANITSYN, A. R., Ca1culation of FaiJure and 'Plasticity'

in Buildings. (Russian edition: 1954). Paris, Eyrolles, 1959.

5. PUGSLEY, A., Thc Safely of Structures. London, Arnold, 1966.

6. SEMENTSOV, S. A., Some Problems of Safety of Large Panelled Struclures. Kiev, c.l.B. (Commiltee W23A), 1967.

7. SEMENTSOV, S. A., Modern Methods of Calculating Masonry Slructures in the USSR. Moscow, c.T.B., 1958.

8. LEWICKI, B., Building Dwclling Houses wilh Large-size Prefabricatcd Unils. Paris, Eyrolles, 1965.

9. WEST, H. W. H., HODGKINSON, H. R. and DAVENPORT, S. T . E., The Performance of Walls Built of Wirecut Bricks With and Without Perforations. TraI/S. Brit. Ceram. Soe. 67, (10) , 421 , 1968.

lO. STRUCTURAL CLAY PRODUCTS RESEARCH FOUNDATION, Com­pressive, Transversc and Racking Strcngth Tests of 4-in. Brick Walls. SCPF Research Report No. 9, August, 1965.

11. HODGKINSON H. R. and others, The Design of a Wall-Testing Machine and Comparative Tests. Proc. Brit. Ceram. Soe. (I t), t. 1968.

12. STEDHAM, M. E. c., Quality Control for Load-bcaring Brick­work: Wall Tests. Proc. Bril. Cera",. Soe. (lI), 83, 1968.

13. JENKI NS, R. A. S. and HASELTINE, B. A., Recent Experiençcs with Calculated Load-bearing Rrickwork. Prac. Brir. Ceram. Soe. (11),163,1968.

14. DE GRAVE, A and MOTTEU, H., Rescarch on Load-bearing Masonry in Belgium. C.l.H. In!. Symp. on Bearing Walls, Warsaw, 1969.

15. CENTRE SCIENTlflQUE ET TECHNIQUE nu BÂTlMENT, The Resis­tance of Brick Masonry Under Lateral and Vertical Loads . (Preliminary Rcport). Paris, C.S.T.B., 1969.