DESIGN RECOMMENDATIONS FOR MASONRY WALLSUNDER CONCENTRATED VERTICAL LOADS

C. A. Syrrnakezis”, P. G. Asteris**

Abstract

In this paper, the strength of masonry walls subjected to concentrated vertical loads isinvestigated. An orthotropic finite element model is used to simulate the wall behaviour. Non-linear deformation characteristics of masonfy material as well as its anisotropic (orthotropic)behaviour are taken into consideration. A new anisotropic yieldlfailure surface of masonry wallunder biaxial stress, in a cubic tensor polynomial form, is proposed. A parametric study iscarried out using several parameters: the loaded area length to the total length ratio, the loadlocation in relation to the end of the wall, and wall geometry. Using the results of theparametric investigation a new design rule is proposed.

Introduction

When a wall is subjected to a concentrated load (e.g. from a structural member), thecompressive resistance of the masonry immediately under the load is usually greater than thenormal compressive strength of the wall. The phenomenon occurs due to the apparent localstrengthening of the material beneath the load, restraining its lateral expansion (Poisson’seffect). The magnitude of this strength enhancement depends upon the loaded area ratio, thewall geometry and the location of the load along the wall, with the degree of enhancement

● Assoc. Professor, National Technical University of Athens, Institute of Structural Analysisand Aseismic Research, Zografou Campus, GR-1 5773, Athens, Greece.e-mail: isaarsyr@central. ntua.gr

““Research Assistant, National Technical University of Athens, Institute of Structural Analysisand Aseismic Research, Zografou Campus, GR-1 5773, Athens, Greece.e-mail: pasteris@central .ntua.gr

reducing as the load approaches the end of the wall. Although some form of stressenhancement factor is given in most masonfy design codes, its nature and magnitude variesconsiderably from country to country.

One of the difficulties in predicting the behaviour of masonry walls subjected to concentratedloads is the lack of realistic stress analysis of the masonry, since the region beneath the loadis in a state of biaxial stress with high stress gradients. The in-plane deformation and failure ofmasonry is influenced by the properties of its components, the blocks and the mortar. Theinfluence of the mortar joints is particularly significant as these joints act as weak planes. Theresults of previous studies of concentrated loading on more isotropic homogeneous material(such as concrete) are thus not directly applicable. To model the behaviour of masonry wallssubjected to concentrated loads, a more representative material model is required.

This paper describes an analytical investigation of the behaviour of masonry walls subjectedto concentrated loads, using a finite element program for non-linear anisotropic analysis. Atotal of 62 walls are analyzed, with the effect of loaded area ratio p, the load location inrelation to the end of the wall d/a, and the aspect ratio of the wall a (as defined in Figure 1)being investigated. From the results, design rule incorporating all of the above variables isproposedanalytkal

and compared to existing design rules from various masonry codes. For theinvestigation a Finite Element Model is used (Fig. 1b).

a= ah

p= a’la

If- Axo.1MPa

(a) (b)

Fig.1. Concentrated load on a wall: a) geometry and terminology, b) finiteelement mesh

Previous research

On account of the lack of comprehensive research in this area, existing design rules are semi-empirical and vary considerably from country to country. Experimental investigations havebeen hampered by the large number of variables which must be considered. Analytkalstudies have usually assumed isotropic elastic behaviour, thus ignoring the effects of materialnon-linearity. Existing rules do not take account of the aspect ratio of the wall, and most ofthem do not considered the location of the load along the wall.

IZxkting design rules for predicting the capacity of walls subjected to concentrated loadsreflect the empirical nature of the provisions, Increased stresses are typically allowedimmediately beneath the concentrated load, usually by the use of a stress enhancementfactor which is a function of the loaded area ratio. The most characteristic design rules for thestrength enhancement factor y are the following:

EC6 (1988):

y = 1.0+0.1 ~ <1,50a’

Chinese code GBJ-3-73 (Dai-Xin, 1985):

r1‘Y= –-1

P

Dai-Xin (1985):

for concentric load: Y = 1+ 0.708/ ;-1s3”00

for eccentric load:/

1y=l+ow ––1s2.00

P

Ali & Page (1988):

~ [0.80+(l-/3)d/fi](2+l/fl)’/2>Im=

1+ o.4oa-.

1.801+ o.2oa

Malek & Hendry (1988):

for central load:

for eccentric load:

for end load:

y = 0.701 /? -0”42

y = 0806 B 433

y = 0R36 p -0”2M

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Method of analysis

For the analytical investigation two requirements have to be met

● a finite element program for non-linear analysis which takes into account the distinctanisotropic nature of masonry and

. a regfJ/ar anisotropic yield surface. According to Koiter (1953) a yield surface is calledregular if it is described by a continuous function.

Finite Eiement Progmm

A finite element non-linear analysis program in two dimensions has been developed inFORTRAN. Emphasis to the anisotropic (orthotropic) nature of masonry modeling is given.Four noded isoparametric finite elements have been used, with a finer mesh immediatelybelow the loaded area where stress gradients are high. Atypical finite element mesh is shownin Figure 1(b). In the finite element program, the loads are applied incrementally. Thereference load amplitude is assumed to equal 0.1 Mpa and the load factor increment is equalto 0.50. The masonry wall is considered totally elastoplastic.

Yield surface

Brickwork is a structure which exhibits distinct directional properties, due mainly to the motirjoints acting as planes of weakness. To define yield under biaxial stress, a three-dimensionalsurface in terms of the two normal stresses and shear stress (or alternatively the two principalstresses and their orientation to the bed joints) is required. Recently a method to define ageneral anisotropic failure surface of masonry under biaxial stress, using a cubic tensorpolynomial has been proposed [Syrmakezis & Asteris 1998]. The method has beenimplemented in a specific program developed in FORTRAN.

CJy [MPa]

Fig.2. Masonty yield surface in terms of normal stresses

Using this program, the failure surface is determined for a real case of a masonry materialwhich has been studied experimentally by Page (1981). According to Syrmakezis & Asteris(1999), the yielding surface (Fig. 2) for the masonry can be described by the equation:

f[ CJ, CJ,’C)= 2.270X+ 9$7ay + 057#x + 132< + 6Q5T2- 0300xcJy+x Y

+ 0.00958tixcry + 0.00313%=< + 0243398CrxT2+ 0.4689cy2 = 1

Parametric investigation

(9)

A total of 62 masonry walls subjected to concentrated load were analyzed using the finiteelement model. As concentrated load, a distributed load on a very small area a’ (a’/a=O.O5 orO.10) is considered. The influence of the loaded area ratio & the load location d/a, and thewall aspect ratio were investigated. A summary of the investigation results are presented inTables 1 and 2. The walls were restrained horizontally and vertically at the base. The otherthree edges were assumed free.

Cellf#ld CO~CtW?ffdd /08~

A total of 32 centrally loaded masonry walls were analysed, with aspect ratios ranging from0.50 to 2.00 and loaded area ratios from 0.05 to 1.00.

Figure 3 depicts the variation of the strength enhancement factor y with the factor ~, forvarious values of the parameter ct. The strength enhancement factor y is critically dependenton the loaded area ratio $ Specifically for low values of j.1factor (@O.30), the increment of thestrength enhancement factor is excessively high. Much higher is the increment in the case oflow values of the factor a (a=O.50).

The influence of the wall aspect ratio a on the strength enhancement factor y for variousvalues of loaded area ratio p is presented on Figure 4. The reduction of the wall aspect ratio uvalue leads to an approximately linear increment of the wall strength. More significant is theincrement for low values of ~ factor. For ~=0.05 the wall’s strength is reduced up to 60Y0, withthe factor a increasing from 0.50 to 2.00.

Eccentric concenfratecf load

A total of 30 loaded masonry walls were analyzed, with aspect ratios 0.50, 1.00 and 1.50. Theloaded area ratio has been investigated for values 0.05 and 0,10 (characteristic values forconcentrated loads). Values of load eccentricity ratio d/a from 0.00 to 0.50 have been takeninto consideration.

On Figures 5 and 6, the variation of the strength enhancement factor Y related to theconcentrated load eccentricity d/a for two values of the ratio a (0.50 and 1.00) is presented.According to these figures it can be stated that the increment of the load eccentricity(reduction of ratio d/a) leads to reduction of the wall strength. For load eccentricity valuesbetween 0.20 to 0.50 the strength reduction is almost zero. This reduction becomes moresignificant for values of the ratio d/a between 0.00 to 0.20. The strength enhancement factorycan get values lower than 1 (Fig. 6), in accordance with existing experimental results (Arora,1988).

Table 1. Summary of parametric study for thecase of c%ntral load

Caaa

Length

(m)

1.50

3.00

4.50

6.00

+aigh

t

(m)

3.00

3.00

3.00

3.00

&pect

ratto

a

0.50

1.00

1.50

2.00

Loadad

area

ratii p

1.000.600.600.400.200.150.100.051.000.600.600.400.200!150.100.05

2:0.600.400.200.150.100.05

::0.600.400.200.150.100.05

Table 2. Summary of parametriceccentric load

I Length HeightI

AspactI

LoadedI

dacase (m) (m) t-wo area

a ratiop

1.50 3.00 0.50

0.100.050.100.050.100.050.100.050.100.05

0.0000.0000.1000.1000.2000.2000.3000.3000.4500.475

I I I I I 0.10 I O.000

3.00 3.00

study for the case of

1.000.050.100.050.100.050.100.05

0.0000.1000.1000.2000.2000.3000.300

I I I I 0.10 I 0.4500.05 0.475

GI 0.10 0.000

0.0000.1000.1000.2000.2000.3000.300

0.10 0.4500.05 0.475

Fig. 3.

>

Fig. 4.

3.00

2.50

2.00

1.50

1,00

0.50

0,00c

. . . . . . . .. . .. .. . . . . . . .. . . . . . . . .. . . .. . .

-. ----; -. . . . . . . ..--- . . . . . . . . . . . ...!

. . . .. . . . . . . . . .. . . .. . .. . . . . .. . .. . . . . . . . . . ....1

~ Uniaxial strength’ ~ I

. . .. ... .... .... ... ... .. .. .... .. . . .. ---- .. -;..- ------ .--. r.... . .. .-

1) 0.20 0.40 0.60 0.80 1,00

Loaded area ratio ~

Variation of strength enhancement factory with loaded arearatio P for difFerent aspect ratios a

3.00

2.50

2.00

1.50

1.00

0.50

0.00c

-----

~ \-..... J_.....__.J_............!.-.................-----

------- . .. . . . . . . . .. .. . . . .. . . . . . . . .

‘- ::’~~0.60 i

~~ Unia)(ial &@h ~ ~

.-.......... ....~.-.. . .. ....------}----..-....-....;. ..--...-.........

,

I I I Io 0.50 1.00 1.50 2.00

Aspect ratio a

Variation of strength enhancement factory with aspect ratioa for different loaded area ratios f!

,/’/

/

/’ /,/’ ,/

,/

,,.,,

‘l-1

T------”--

&_..._.----- -—_

----- - -

! p=o.05 i ~. . .. —--”’.+. —-—- ;

—’+-.-—-——— ____4.. _._.. -—--__— — ---

. .. . . . . . . . . . . . . .. . . . . .

—.—.————..

Jniaxial strength

+

——–%

a=O.50

10

Load location along the wall d/a

Fig. 5. Strength enhancement factor of eccentric loading

. . . . . .

,-‘+——

_J_.—-—-’=,... - —._+_.._.-—- ;

.--,/-~ :---k--------.---.-;-- ----------- }----- ------.,-,,

‘“ ~~ Uniaxial strength ~ ~,,,’

. . . .

--

-!

—-—-~ —-&- ~.. 0 ;~o ., {> 5’)

Load location along the wall d/a

‘----’ra=l.00

Fig. 6. Strength enhancement factor of eccentric loading

Proposed design recommendation

A general design rule for the strength enhancement factor should allow for the influence of theloaded area ratio ~, the load location along the wall d/a, and the aspect ratio of the wall ct.Using the results of the present parametric study, the following relationship providing a lowerbound to the results of the parametric study for the strength enhancement factor y can besuggested:

(lo)

where:

(p= 1.00 for the case of 0.20< ~ <0.50a

()2

(p=l.oo-12 020-: for the case of 0.00< ~ <0.20a

On Figures 7 and 8, the graphical representation of Equation 10 (Curve C) is compared to thevariation of they-values provided by Dai-Xin (Cutve A), Ali & Page (Curve B) and EC6 (Cun#eD).

Using Equation 10, the curves of Figures 9 and 10 can be derived. Figure 9 show thevariation of the strength enhancement factory with the loaded area ratio P, for various valuesof the wall aspect ratio a (8.00,4.00, 2.00, 1.00, 0.50) and central loading. Similar results foreccentric loading are shown on Figure 10.

Conclusion

An analytical parametric study of masonry walls behaviour subjected to concentrated loadshas been presented. An iterative (incremental) finite element model has been used tosimulate the wall behaviour. Through this model, the non-linear deformation of masonry canbe described.

In the parametric study, the influence of the loaded area ratio, the location of the load and thewall geometry to the strength enhancement factor Y, have been considered. All variables havebeen shown to influence signifmntly the degree of enhancement of bearing strengthappearing in the area beneath the load.

Using the results of the parametric study a new design relationship, incorporating all abovethree variables, has been proposed. This new relationship, compared to the existing designrecommendations (which do not always consider all variables), has been proved moreconservative and reliable.

Fig. 7.

Fig. 8.

3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00c

. . . . . . . .. . . . . . . .. . . . . . . . . . . . .. . .. . . . . ... . .. . . . . . . . . . .. . . . . . . .. .

T-L---------L----“”\\........;.~......................... .. . . . . . . . . . . ... . . . . . . . . . . . .

~ Uniaxial strength ~ ~.. .. .. .. ... . .. .. .... . .. . .... .. .... .. . ... .... . .. .---------

I 1 1 I0.20 0.40 0.60 0.80 1.(

3.50

3.00

2.50

2.00

Typical(a=l .00,

.50

.00

LJ.50

0.00(

o

Loaded area ratio ~

design provisions for concentrated loadscentral load).

. . ... . . . .. . .. . . . . . .. .. . . . .. . . .. . ... ... . .. .. . . . .. .. . .. . . . . . . . .. .

------ .. . . . . . -------- . . . . .. .. . . .. . . .. . . . . . .. . . . . . . ....1

...........

. .. . . .. .. . . ..L....-..---.................!..--.------.--L ----------- --~

I I I 1 Io 0.10 0,20 0.30 0.40 0.50

Load location along the wall d/a

Typical design provisions for concentrated loads(a=2.00, ~=0.05, eccentric load).

Fig. 9.

3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00c

3.50

>

Fig. 10.

3.00

2.50

2.00

1.50

1.00

0.50

t ,I I

.. L-------------!--............L ------------- . . . . . . . . . . . . .

---------- . . . L . . . . . . . . . . . . . . . . . . . . . . . . . .. L . . . . . . . . . . .

_.. ---L-------------i---------..

............ L---....................................-

I Uniaxial,strength ~ ~------ . .. . . . . . .. .. . .. . . . ----------- . ... .. .. .. .. . . . . . .-------- . .----

1 I I I

) 0.20 0.40 0.60 0.80

Loaded area ratio ~’o

Proposed design recommendation for concentric load

.- .-. . . . . . . . . . . . . . . . . .

; ct=o.50 !

. . . . . . . . .. .._.L-.. . . . . . . . . . .. . . .. . . .. .. . ..-. .. L. .. . . . . . . . . . . I

---- . . . .. . ... . . . . .. . . . . . . . . . . . .. .. . .. .. . . . . . .. L-------------- ,

! C%=l.00~L-------------L-----------.-L-------------,

~ ~=1.50 ;

! a=2000 ! (. . .. .. . . . .. .. . . . . .. . .. .. . .. .. . .. .. . .. . . .. . . .. . . I

/

~p

\;

~ Uniaxial strength ‘ ~. .. . . .. .. . . . ..-, ---------- . .. L-------

i m,

&&05 -~

0.00 I1 I 1 [ I

0.00 0.10 0.20 0.30 0.40 0.50

Load location along the wall d/a

Proposed design recommendation for eocentric load

References

Alli 1988: AlIi, S., A. W. Page, “Concentrated loads on solid masonry walls-a parametric studyand design recommendations”, Proc. Instn Civ. Enars, Part 2, 1988,85, June, pp. 271-289.

Arora 1988: Arora S. K., “Performance of Masonry Walls under Concentrated Load”,Proceedings of the British Masonrv Society, No. 2, April 1988, pp. 50-55.

Dai-Xin 1985: Dai-Xin T., “Testing and analysis of the bearing strength of brick masonry”,Proceedings of the 7th International Brick Mason IV Conferece, Melbourne, Australia, 17-20February 1985, Vol. 1, pp. 747-755.

Eurocode N06 (1988). Common unified rules for masonry structures, EEC, Re~ort EUR 9888EN.

Koiter 1953: Koiter W. T., “Stress-strain relations, uniqueness and variational theorems forelastic-plastic materials with a singular yield surface”, Quarterlv of amlied mathematics, 1953,vol. xl, pp. 350-354.

Malek 1988: Malek M. H., A. W. Hendry, ‘Compressive Strength of Brickwork Masonry UnderConcentrated Loading”, Proceedings of the ~, No. 2, April 1988, pp. 56-60.

Page 1981: Page A. W., “The biaxial compressive strength of brick mason~, Proc. Instn Civ.,~, part 2, 1981, Vol. 71, Sept., pp. 893-906.

Syrmakezis 1999: Syrmakezis, C. A., P. G. Asteris, “Masonry failure sufface under biaxialstress”, Technika Chronika, Scientific Journal of the Technical Chamber of Greece, Vol. 19,1999, (in greek).

a

p= a’/a :

l’:

a ..

a’

d:

A:

Notation

aspect ratio of the wall

loaded area ratio (ratio of loaded area to total area)

strength enhancement factor

length of wall

length of loading plate

distance of concentrated load from the nearest corner of the wall

load factor