26.- Shear Design of Brick Lintels

by C. ZELGER

Materilllprüfimgsanr für das BauWf!sen der Techll. Unil'ersitiit Müncnell , W. Germany

A.BSTRA.CT

For brick finteis where lhe brickwork carried does lIor differ from normal brickll'ork asfar as bedding and bond­ing are concerned (plaill brick limeis), a simple Iheory /tas been developed for Ilte calculatiol1 of lhe transverse stress lha' CGIl be taken up. This takes accawlf of lhe Ir;clioll in file bedjoints due to lhe verlical pressure from load. The Iransverse stress tllal call be laken up incl'eases wirll decrease ill shear slenderness ralio .

The tlzeoretical trearmem is based 011 fesl resulls obtained aI lhe Buildillg Materiais Testing Laboratories Df the Technical UniverJity, Munich. A brief repor! is given 011 lhe fests. Final/y it is SIIOWII , lha! tire formula for lhe shear desigl1 recelltly il11rodllced in Germany withil1 the framework of the design codes, which reslIlts [rom a sfight variatioll on the above theory, qllalitively corresponds to the load­bearing capacity recorded Ofl numerous brick finteIs ullder predominalll/y trallsverse loading. Theory and prac­tice also tally lVell quantitatively, ir suitable constal11s for friction and ad­hesiol1 are substituted in the destgn formu la.

NOTATION

b = width of cross-section h=static eH'ective depth

Calcu/ du Cisaillement de Linteaux en Briques

Pour les Iil11eaux ell briques ou la maçonnerie supportée ne difJere pas de la maçonnerie Ilormale ell ce qui conceme I'encastrement et I'adhérellce (Iinteaux en briques plans) . une /héorie simple a é/é élaborée pour le calcul du cisaillement transl'ersal admissible. Celle-ci rient compre du frol1ement dans les joints d'assise dú à la pressioll verticale exercée par la charge. Le cisaillemeJil qui peul êrre assumé augmente al/ec la diminu-1ion de I'élallcemem de cisaillemem. Le traitement théorique esl [ondé sur des résulta/s d'essa; obrenus au Laboratoire d' Essai des matériaux de construction de I'Université Tecllllique de Munich. Vn brefrapporl des essais eSl [oumi. On montre finalemeJil que la formule récemment imroduite en Allemagne pour le calcul du cisaillement, dans le cadre de directives de calcul déril'ées, avec de légêres modijicatio1Js, de la théorie sus-mentionnée correspond qualitatil'e­ment, à la charge por/ante enregistré 5ur de nombreux Iinteaux en briques sous une charge à prédominance de cisaillement. La théorie el la pratique s'accordent également bien du point de vue quantitarif à condition d'intro­duire, dans les formules de calcul, des constantes convenables pour le frotte­ment et J'adhérence.

Berechnung der Biegespannungen in Ziegelstürzen

Für Ziegelstürze, deren Mauerwerk sich beziiglich Steinanordl1l1ng und Billdung nichl VOII norma/em Mal/er­werk unterscheidet (flache Ziegel­sliirze) , wurde eine einfaclze Theorie entwickelt , um jene Durchbiegespan­mmgen zu errechnen, die der Sturz aufne/unen kann. Sie beriicksichtigt die der vertikalen Be/astung entgegen­wirkende Reibung in den Fugen. Die aulnehmbaren Spannungen Ivachsen mit fallelldem S chlankheits l'erhiiltnis der S cheerfliiche. Die theoretische Behandlung des Problems basiert auf Versuchsresultaten, die im Bau­material-Priiflabor der Teclmisclzen UnÍl'ersitiit München gewonnen wurdell. Ein kurzer Bericht dieser Versuche lVird gegeben. SchliejJlich wird geztÚgt, da'p die Forme/ zur Bereclmung ohne Schaden aulne/un­barer Scheerspanntmgen, die kürzlich in Del/tsch/and im Rahmen der Entwurfs-Richtlinien, die mit einer geringen Abweichung aus der l'orgellannten Theorie hervorgingen, eingeführt wurde, qua/itativ der an zahlreichen Ziegelstürzen ermittelten BelastbarkeÍl auf Biegung entspricht. Theorie und Praxis stimmen ebenfalls gut überein , wenll richtige Werte fiir die Reibung und Adhiisioll in die FormeI eingesetzt werden.

to slip between bodies adhering to one another. here also termed cohesion

,.=coefficient of friction ' I = safety factor V=(as subscript) breaking condition

z= lever arm of the internai forces at lhe vertex of the funicular curve

k:=z/h a = shearing distance À=a/h = shear slenderness ratio Q = transverse force M = bending moment H = horizontal thrust

max = before M and Q means 'maximum' in relation to lhe place of occurrence. Thus, for exampIe. max Q is lhe transverse force on the bearing surface, independent of the loading

per=(as subscript) permissible under normal tions

condi- '/)\>\C .:, ..

V = vertical force F = eH'ective area of adhesion 1. INTRODUCTION ~ , = correction term for lhe description of the During the past 10 years brick lintels have been more ".,.~.

eH'ective area of adhesion widely manufactured and used in West Germany. They T= shear stress have the advantage of being made of the same material

11. = strength value for the description of the resistance as the surrounding masonry, so that defects due to 161

162 Shear Design of 8rick Lintels differenlial shrinkage and thermal expansio n are avoided . T he prefabricated tensile strap spans openi ngs up to clear widths of aboul I !TI (3·3 ft) withoul the aid of assembly supports, and with supports beyond this width. The shuttering involved in lhe making af concrete lintels and lhe waiting time necessary before they are sufficiemly hard are eliminated . Disadvantages are lhe low resistance of the joints to shear stresses and the praclical impos­sibility af accommodating shear reinforcement in lhe brickwork structure.

The fo llowing ideas a re restricted to so-call ed plain brick Iintels. In this type lhe reinforcement is laid in a prefabricated tensile strap, made of cha nnel-shaped brieks whieh are laid in a row and fi lled with concrete. The slaek or pretensioned re inforeement is bedded into this concrete. 00 lhe building site brickwork is erected above th. lensile slrap in the usual manner. The brieks thus have lhe same type af bedding as in conventional walls. A part o f the brickwork above lhe tensile strap forms lhe compression zo ne of the p lai n briek lintel. The bending compressive force affects lhe bricks in a direction parallel to lhe bed joint. Ali shear forces must be earried by lhe mortar joints (see Figure I).

Brickwork Tokirq Port \O Cornbrrwd

~ :,fr: : ~ ~ Pr!l'fobrrCOle<1 Tmsill! StfOp Wlth

Rll inforumcmt

FIGURE I- Plain brick lintel.

Cooc rvte

In loading tests plain briek lintels a lmost always break as a result af sli p in the joints. It therefore seems very urgent and necessary lhat a method af measuring shear should be developed, which wi ll take into aeco unt lhe peculiarities of the briek Iintels and will open up the widest possible scope for their use, taking every adva ntage possible but also paying due atlention to normal safety requirements.

2_ THEORY OF SHEAR FAILURE OF PLAIN BRlCK LINTELS

If ODe assumes lhat brickwork cannOl absorb shea r stresses in any direetian, lhen the lintel must bear lhe load like an areh with a tie. The tie eonsists of the pre­fab ricated tensile slrap with lhe reinforcemem , the arch is formed by the brickwork above it.

Such supporl fa ils if

(a) lhe compressive force destroys the areh ;

(b) the tensile foree destroys the p refabricated tens ile strap;

(c) the ho rizontal shear destroys the bonding between arch and tie at lhe springing.

Possibi li ties (a) and (b) can be excluded by an ap­propriate bending calculation, as is always dane for reinforced brickwork. The fo llowi ng considerations serve to ensure aga in sl the most common case af failure , (e). If z is the díslance of the line of thrust of the areh [rom lhe tie rod at lhe paint af lhe greatest bending mament max M, lhen according to Figure 2 lhe hori­zontal thrust

h

max M H = -­

z

_ Q + ", ar Q.

"Lm

no

· (i)

FII .. "Ul-u. 2- Idealized play of forces in a plain brick UnteI. Only lhe area between a bearing surface and lhe cross·section with the

maximum moment of resislance is given .

The vertical reaction of lhe support is

V= mox Q · (2)

Each hori zontal joint, which is intersected by the line of lhrusl must transmit shearing force H and normal force V. Therefore the following desígns a re not o nly valid for the d irect area of support. The more levei trend of the line of thrust at the point of intersect ion with higher posi tioned bed joinls must nOl lead lo the assumption that with a constant horizontal force H the vertical force is less. J n each sect ion it is the sarne as V beca use of the balance. The so-called extended Cou lomb's law of friclion tells us lhe value of lhe shearing force H , for a given normal force V, which overcornes the resistance to shear by presupposing alo ng with the frietion a n ad­ditional eohesion between lhe jointed bodies (see Figure 3):

(3)

v / 1 /

H 1 _,;lO =--= -= -=-=-~ ~ - ,;" F H

I )- - --- -

V / .... ';':...

I v FIGU RE 3- Explanation of Cou lomb's extended law of rriction.

By substiluting eqns . (I) and (2) in eqn. (3) o ne gelS:

maxM -- ="..mox Q+ fJ, . F

z · (4)

lt is convenient to e liminate the bending moment I/IOX M from eqn . (4) by int roducing the term 'shear dislance' which is orten used in reinforced concrete bui lding.

max M G= --- -

IIIGX Q · (5)

lhe shear slenderness ratio

a max M " = - = ----;:-;

h /IIOX Q.h · (6)

C. Zelger 163

and lhe ralio

· (7)

The shear distance a is explained by Figure 2. If the load is taken to be concentrated ai one poinl, which is ai a dislance a from lhe supporl, then this point load causes the same maximum moment, I11GX M, and the same maximum transverse force, max Q, as the actual load. The line of applicalion of the concenlraled load and the tangents to lhe funicular curve at the supporl and at the apex intersect in a point.

It is stil! necessary to discuss the question: What value should be put on the area of adhesion F in eqn. (4)? Tests on !inlel beams show thal Ihere is a tendency for a stepped crack (see Figure 4) or several stepped cracks to

FIGURE 4--Fracture pattern of a plain brick lintel with a 1-23 shear slenderness.

form, and their projection on plan exceeds the shear distance a by a certain length which will be called t. In each case the overhang of the beam beyond the Iheoretical support point, and with uniform loading probably an even greater distance, is added to the shear distance. The following is a reasonable formula

F=(a+ t).b · (8)

in which the value of t can remain open for the time being. By substituting eqn. (8) in eqn. (4) and eliminating

max M from eqns. (4) and (6) one gets

A max Q.hfz = p..max Q+ ,8,.(a+ t).b . (9)

Solving for max Q gives

ajh+ tjh max Q = ,8, . b . Z -----jl

/\ - fLZ I · (lO)

Taking eqns. (6) and (7) in to accounl one finally gets:

A+ tjh max Qu=,8,.b.z . (11)

A-p..k,

The subscripl U indicates lhe condition of failure. If one takes a practical case, eqn. (11) should be divided by a suitable safety faclor v.

max Qu = ,8' .b.z A+ tjh (12) v v À-p..kz

and, by introducing permissibIe values, one gets:

A+ tjh Qptr = 'Tper·b.z. ~, --'-=:'"

/\ - p..kz

(13)

Eqn. (13), apart from the fraction on the right-hand side, corresponds with the well-known shear-calculation formula , = Qj(b. z). The fraction can be taken as a cor­rection term to take account of slenderness ratio and frjction.

3. FRlCTION AND COHESION Tests have been carried out in Munich to determine the value of J-L and {3.r (see Figure 5). The angle between the

FIGURE 5- Compression test on masonry with an angle of about 32<) between lhe direction of force and the bed joints.

line of action of the force and the plane of lhe bedding joints was varied. The bricks had a strength of around 200 kgfjcm' (2845 lbfj in') ai right angles to the bedding joint and around 60 kgfjcm' (855 lbfj in') parallel lo it. The compressive and bending strengths of the lime­cement-mortar were around 20 and 7 kgfjcm' (285 and 100 lbfjin') respectively.

The validity of Coulumb's law of friction is confirmed. The tesl values are reproduced in the form of a ' u - Uu diagram in Figure 6. The besl slraighl line intersects lhe

FIGURE 6-Interrelation of shear and normal stresses in lhe bed joint in a state of failure.

'u-axis ai cohesion ,8, = 2·0 kgfjcm2 (28Ibfjin') and the gradient p.=0·51 is equivalent to the friction co­efficient.

164 Shear Design af Brick Lintels 4. COMPARISON OF THEORETlCALLY ANO EXPERIMENTALLY OETERMINEO

TRANSVERSE FORCES AT FAILURE

Te' ls on plain brick lintels, as shown in Figure 4, have beeo carried out in Munich .1 The onlyvalues to be varied were lhe shear-slenderness fatio and lhe load-distribution (point loading by two symmelrical single loads ar an approximalely uniform load-distribulion). Bricks and mortar were lhe same as those described in Section 3 and shown in Figure 5. The transverse forces at failure max Qu/b.z found fram the lests are plolled as points against the shear slenderness ratio in Figure 7, The

,

- ., -"--:-;O=C7'C=1 o 2 SingllZ I,.ood: .~

-+ EqnJ1tJ

~,

, L-____ -L ______ L-____ ~ ___

, 2' FIGURI: 7- lnterrclation af recordable transverse force maximulll Qu/b. z and shear slenderness according to eqn. (11); wherciJ.= O· 51 ; Br= 2-0 kgfjcm2 (281bf/in2); k::= O' 735. Comparison with tcst valucs.

curves show lhe theoretical relationship according to eqn. (i I). Theory and praclice agree well if wilh poinl loading l/h is seI equal lo I and with approximalely uniform loading l/h = 2.

5. SHEAR OESIGN IN GERMANY Vnlil recenl ly lhe manufacture and use of brick linlels in Germany still required building permission. To obtain this in each case tests were necessary. Ali lhe test results 2 available to lhe author are surnmarized in Figure 8. The scaller is exceplionally wide. Eqn. (I J) with

kgtj cmt * JP -

,

, , "

,., "

• •

•• N " N "

FIGURe 8- Survey af ali test results known to lhe aUlhor on plain brick lintel s wilh variolls Iypes af brickwork and differcnt makcs af

tensile Slraps. Comparison with eqns . (11) and (14) .

the constants found in Munich (f' = 0·51 ; /l, = 2·0 kgf/cm' (28Ibf/ in') ; l/h = 2) is shown as lhe top curve in Figure 8. It can be seen lhat the resuIts obtained in M unich correspond roughly to average conditions. Furthermore it can be seen that the qualily of the morta r apparenlly has no decisive effect, for the range af distribution for mortar strenglhs between 25 and 50 kgf/cm' (355 lo

710 Ibf/ in') does not conlrast in any way with those for mortar strengths over 50 kgf/cm2 From earlier results 3

it follows lhat lhe friclion coefficient does nOl vary to any great extent. 11 was more Dr less lhe same in tests on bonded concrete bodies with very different mortars and moreover agreed appraximately with the value of around 0·5 established in lhis paper. The property chiefly responsi ble for lhe wide scaller mUSl be adhesion. This depends on the surface configuralion of lhe brick and above ali on lhe workmanship. lts cOlltribulion to the shear resistance can be grasped from eqn. (3) as the praduct of the cahesion f3 't and the area of adhesion, F. Whether one attributes lhe uncertainty, evident [rom the wide distribution in Figure 8, to bOlh lhe values /l, and F ar lO only one of lhem, is fillally a queslion of judgement. For lhe purpose of defining a uniform value of cohesion ar perrnissible shear stress in lhe German code for the design of brick linlels , lhe simple assumption has been made lhat lhe wide scaller is allribuled only to lhe randam variatian in the area af adhesian, F. This c1arified the matter in so far as lhe workmanship affecls lhe more materially-delermined cohesion ilself less than lhe effeclive area, e.g. by more ar less complete filling of the bed joinl with mortar. The value Il h, which helps 10 determine the value of F according lo eqn. (8), and the magnitude af the permissible stress Tpt!r were so defined that lhe lower limit of varialion of lhe lransverse forces at failure found in lhe lests (brolcen line in Figure 8) is 2·1 times as great as lhe permissible transverse force (Iower curve in Figure 8). This condition leads lo the following design formula:

>' + 0'4 Qpu =Tpu.b.z "-0'4 . (14)

wilh "P"= 1·1 kgf/cm' (!5·5Ibf/ in'). [n lhe region ofvery small shear slenderness ratios of aboul >." 0·6 slip fractures are no longer possible, or at least onIy at such shear stresses that the compressive force acting in the nOlional arch destroys lhe masonry around lhe springing. To exclude such failures lhe hyperbolic curves were cul off in lhe range 0 < >. < 0·6 by parallels lo lhe>. axis.

The safety factor of 2·1 is considered necessary in Germany, unless there are other c1ear visible warning signs. Wilh shear failures in unreinforced joints there is no early warning. Many manufacturers of plain briclc lintels argue against a uniform calculalion based on eqn. (14), beca use acceptance tesls on lheir products have perhaps given results in the middle or even in the upper part of lhe range of scaller. Such an argument does no! sland up to examination. The prefabricated tensile straps are sold. The quality of the wall above, which alone de­lemines the shear slrenglh of lhe whole beam cannot be influenced ar controlled by the manufacturer. lt must be bome in mind lhat from each make of lensile strap, plain brick lintels can in practice be produced, the slrenglh of which in each individual case can just as easily lie on the upper ar lower limil of the range of scatter depending on haw much caTe is taken.

REFERENCES I. ZELGER, c., Reinforced Brick Untels. Ziegelind/lstrie (24), 749

1967. ' 2. SCHELLBACH, G. and ZELGER, c., Recommendations for the

Shear Design of Brick Lintels. Ziegelindustrie (24), 767, 1967. 3. ROSCH, H. and ZELGER, c., The lnflLlence of Joints on the

Slrenglh of Prefabricated Shells. Beton li. Stahlbetonb. (ta), 234. 1961.