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THE LOAD-CARRYINGI3~~HAVIOUR OF TJ:lE BED,JOINTS JNMASONRY W!\Lt.ING SUBJECTED TO ARBITRARY LOADING

8th I:\TER "-: :\ TIO \'_\L BRIO: '.L\SO:\RY CO'\'FEREl\'CE DtBLI'\' 1988

DIPL.-ING. \!.\TTHlAS PFUFER Technical Uni\-ersit~- Darmstadt

6100 Darmstadt. Pet e rsenstraf',e 15, Fed e ral Re publi c of Ge rma ny

ABSTRACT .. " . . , - " " " . . . " ....

-

The bed joints within a masonry wall are subjected to various internaI forces. When the wall is loaded in its own plane (sheet-effect), normal vertical and horizontal stresses and shear stresses are produced; on loading perpendicular to the wall plane e. g. wind or soi! pressure or even with buckling (plate-ef­fect) bending moments in both directions, twisting moments and shear forces are produced, dependent on the support conditions. Vertical bending moments are here only possible in conjunction with simultaneous application of vertical compressive forces, if no tensile strength of the mortar is given consideration. If the load eccentricity attains a certain value, the joint cracks open so that only a small part of the cross section remains to transfer the loading. The horizontal moments as well as the shear forces resulting from the sheet and the pIa te effect are transferred by shearing stresses via this compressed cross-sectional area. Thus cracking and failure criteria for those internai forces which produce shcaring stresses in the horizontal joint are dependent on the remaining internaI forces. This report investigates the interaction of ali possible internaI forces in the horizontal joint. A theory is developed which describes the complicated stress condition taking into account the pla ­sticization of the mortar. lt is seen that when considering this plasticization, much larger forces and moments can be transferred via the opened joints than in accordance with the theory of elasticity.

Kx K y

Ex Ey W P vx Vy

d h e c

Kxy 2H GXY

Ü

NOMENCLATURE

stiffness va lues of the orthotropic plate moduli of elasticity and shear bending surface and loading of the orthotropic plate transverse strain coefficients wall depth, height, length and overlap of the blocks eccentricity of the vertical load compressed residual cross section with opened joint

m. mv m.v qx qv

fiiX If/y qP ax av lxv nx ny txv ííY ijS Mx Mv Nv Qp Qs

aR II 12

MPL Qsp Qpp

Wr ( Yxv re XD yD

/3R 1.

Il K" f A

1. INTRODjJC.'I'ION

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internai plate forces per unit length referenced internaI plate forces in -plane stresses of the sheet in-pla ne int ernaI forces per unit length r e fe renced in-plane internaI for ces internaI forces in the bed joint boundary stress in the bed joint wi th eccentric loading shearing s tresses in the bed joint plastic momen t plastic shear forces (sheet and plate) torsionaI resistanc e moment strain, slipping and curvature position of rotationaI centre in the bed joint compressive strength of the walling co hes ion coe ffi cient of friction referenced cohesion reduct io n . fa ctor

~lasonry walls act simultaneously as sheets a nd plates when subjected t o loading in the wa11 plane as we 11 as perp endi cularI:-·. \Vi th the pIa te - bearing effect a uni- or biaxial load transfer occurs de pendent on the s upp ort ing conditions, whereby bending moments in both directions a nd twi s ting momen ts can be produced. Even with slender ';ralls 'g hi ch are only subjected to hi gh \'ertical i.e. axial loadings, bending and twisting morn e nt s are produce d by th e preliminary deformations which are una\'o ida ble in building construction. At first the masonry ;;:a11 can be cal cu lated with the aid of the plate or she et theory, if one inserts t he specific qualiti e s of the ',\'a ll jointing into the plate stiffness values K" K" and 2H. which are used in the differential eq ua tion for the orth otropic pl a te , thus :

= p (1)

Appropria t e derÍ\'ations for these values and for th eir changes dependent on the state of loading are gi\'e n as examples in [11. This is aIs o in principIe app licab le for the s hee t effec t . As one can see, for instance in [21. in detail however the "normal " proce dures for the dc termin a tion of st ress es from the sheet effec t or of ben din g stl'esses f rom th e pla t e effec t is no longe r usa bl e . Here more exa ct in vestigations are nee de d regarding the interac tion of th e indi\'id ua l \.v alling blocks and mortar joints. In this report the behavio ur of th e bed joi nt s su bj ec t ed t o arbitrary intern ai forc es is examined and a th eo ry for th eir fa ilure derived, \\' hereby the plasti ­cization of the morr ar is considered.

2. HOMOÇENEOUS SHEETS AND PLATE$

The loading of a homogeneous \r all within an d perpen dicul ar t o i ts O\\'D plane leads to s tresses as we ll as bending and t wis ting rnoments as shown in figur e l. These are give n wi th th e charact e r ist ic moduli of elastici ty E", E,. th e trans\'erse st rain coefficients h, I ', and the stiffness \'alues of the plat e 1\"

L

1386

Ky and Kxy in accordance with the sheet a nd plate theory, namely with the following equations :

êhe~t

(]z = Ex " (h + Vy"ty) / (l-vxVy) (]y = Ey " (ty + Vx" €x) / (1- Vx Vy)

txy= GXY " YXY

(2)

o, f t f f t t t

o,

~k~~~~J

plªtg:

mz Kz (ih + Vy "i!y)

(i!y + Vx "i!x) my Ky

mXY= K"y" i!XY

""'" ~~~*-,XY

$ m, f t , ~~-----tJ&

444$44$ Figllr~) stresses in th e shee t and inte rnai forces of the plate

(3)

The strain E of th e sheet is attain ed from AIRY 's stress function , the CUl'va­ture re of the plate from the bending surface w. These can be cal culated , for ex ample , by nume ri cal me th ods (differential calculus, finite e lement method) or , in special cases, by closed analytical so lutions, but this is hO 'w'eve l' not a subjec t matte r of this reporto

3. MA~ONRY WALL AS PLATE AND~H~!';T 3.1. GENI':R.AI,.

If one conceives the masonry 'N all as ::tn orthotropic plate OI' sheet, the n ODe can in th e first in s tanc e ca lculate the internai forces OI' th e sheet strcsses as for a horn oge neo lls con s tructi on e leme nt in a cco rdance with the above equations. Then a "trans lation " must be ma de of these int ernai force s onto the walling bond and on the for ces acting with .in the joints. The \' e rtical joints should hereby not be used for anr force transfer (butt joints as "\'er­ti ca l slits "), as, du e to shrinkage of the mortal' , t he re is oft e n no r eliabl e tensional conne ction 01' waUs are even constructed \v ith out mortal' fil!. The she et s tress es are compil ed as internai forces. namely:

ny (]y"d nx (]x "d t XY tx y" d (4)

3.2 . MATERIAL B!';HAVIOUR OF THE MORTA R

For the subs equent investigations the deformati on behaviour of the mortal' is ne8 ded. As numerous differcnt materiaIs are used in masonry construction one ca nnot give a gene rally \'alid stl'ess-strain law for compressi ve force . For th is

..

1387

reason a simple linear relationship is used as shown in figure 2, even though morta r on its own would more likely show a curvilinear slope, as for concrete, but decisive is however the behaviour of the blocks , for which the linear­elas ti c law is applicable. AIso , this relationship is ' on the safe side for all cases. Tensile stresses a re not considered in the bed joints. When subjected to shear stresses, mortar behaves as shown in figure 3. At first there is an approximately elastic zone until the maximum shear stress max T is reached. At this point the stress drops to the pure friction component. From here no further load increase is possible, the stress remains constant and is inde­pendent of displacement travel.

<Il <Il <l> ;:; <Il ,

,"/ Y

~~ V , /' ./

s train

Figure~ , : stress-strain la '-.- for compressio n

3.3 ., INTE~NAJ." , fQRC~§JNTHE BEI)JQIN'r

r '\ \

I I

displacemen t

f i gllfe3 st l' ess - displacement law for shear

The internaI for ces fr om the plate calculation and the stresses from the sheet ca lculation give the for ces for the bed joint as shown in figure 4 . From the equili brium conditions and considering the results fr om the "shear-failure ­theo ry " [3 J one obtains:

My my ·Ü + mXY ·h Mx mx ·h + mxy · ü Qp qy ·ü + q" ·h Ny ny ·ü ± t"y ·h Qs nx ·h ± t XY ·Ü

Qs

~ (5)

figure 4 : intern a i for ces in the bed joint

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ª.4,EL.i\STIÇ .. ~.H.~ABING STRESSES IH .'l'I:IEIlE:P.JQIN'l'

From these internal forces one can determine the stresses in the bed joint in accordance with t he theol'Y of elasticity. Vertical compressive stresses result fr om the bending

e<~ . i ~ \, I fi"

flg\ll'e . 5 : vertical compressi ve stres­ses

mome nt :\1;- and the normal force Ny . Due to the lin ear o - E-rela ­ti onship these are distributed in a triangular manner for opened joint OI' trapezoidaUy for full J' compressed joint , depe ndent on the eccentricity of the ve rti ca l force. The determination of these stresses is simple and needs no further explanation. (see Figure 5)

The determination of the sheal'ing st res s es is more difficult: for a beam with rectangular cross section, a paraboiic shearing stress dirtribution is produc ed by a shear force. For the plate effect (shear force Qp) this distribution could be used as a basis, but this is not applicable for the sheet effect (shear force Qs) . Exac ter investigation, whereby a waU was analysed as a "slit sheet" subjected to horizontal tension using an electronic calculation program [4J show that the shearing stress de ve lopment is dependent on the relation­ship of rigidity of blocks and mortar. Similar results are met with in [5J. With very "soft" bl oc ks compared to the mortal', shearing stress peaks occur at the joint ends, with a r ev ersed rigidity relati onship the stresses are constan t. (Figure 6)

~/(p/~// ~ : ///4// //i

1 = co nsto

1111111111111111111111111111 111

s tress pea ks

~ /T?7~ ~

7/~/;r~ ~

"rigid block, soft joint" "soft block , rigid joint"

fig\1rg6: shearing stresses in the bed joint from Qs

The stress peaks at the joint ends are only valid with material which has an unlimited elastic behaviour and in fact do not occur in this magnitude in practice. They are rapidly reduced by local plasticization. It is therefor felt justifiable , in the interest of simplicity, to always calculate with constant shear stresses.

1389

The tOl'siona l mornent Mx in the bed joint is Pl'Od11Ccd fr om the h') :'i zo ntal bcnding moment mx and the twisting momcnt m,;,. f or t his cond itio n. th e shcaring stresscs can be cal.culated ac co l'din g to St. \ 'c nant 'N ith the use of the follo \\'ing formulas:

\/'r Ih

I1 I1

Mx / Ih I1 • Mx / \/'r with

C2 'ü'(0,33 - 0,21(c/ü) + 0,083(c/Ü)2) ü2 'c'(0,33 - 0,21(ü/c) + 0,083(ü/C)2)

0.743 + 1.297 e(-1.619.C/ü)

0.743 + 1.297 e(-1.6IQ'ülcl

c;S;ü c>ü

c;S;ü ü;S;c

The sheari ng stress es from ',I, are pl'esente d in figur e, i.

The ultimate shear stress for the elastic conditioll is sho '.v n in figure 3 and is dependent on the simu l-ta neo llsly acting vertical compl'essi\'e srre s s. With an open cd bed joint (figure 5) there is no such r; om pressivc stres s on ane sid e , 50 that the cohesion represents the ultimate s heal' stres s .

12

and

(6)

1 !

Figure I sh ear s tress cs fl' orn 1,1.

3.5, CRAÇj(INGCRITERIAFOR THE BED JOINT ("FLOW CONDITION")

Due to the cons id eration of the plastici zation prope rt y of the mort ar, one must distinguish between crackin g and failure criteria. The at tain ing ~) f th e "elastic state limit" is defined as cl'acking cri teria and is the equi\'a le nt to the "flow condition" in steel construction, whereas the failure criteria rcpre ­sents the absolute maximum \'al ue for the load carrying capaci ty. Wi th arbi ­trary combination of the internaI forces, the elast ic state limit can be pl'e­sented as a three-dimensional diagram, if two of the five internaI forces are assumed as constant parametel's. It is here advantageous to introducc refe ­renced internaI forces:

~ my + mu' h/ü qy + qX'h/ü

d2 '/JR qp

d· W /JR

mx m" + m"y'ü/h n" + tu • ü/h

d2 • W /JR qs

d'W/JR

ny ny + t"y 'h/ü d· /JR (7)

Figure 9 shows ex amples for the ultima te elastic internai forc es. Here a further dimensionless value was introduced, namely the "refere nc ed cohes ion" K' = To / Jl.{3R . The lo we r leveI of the diag ram is formed by the axis tines for mx and J'iY, i.e, the internaI forces that oceur in a uniaxial \' ertical

spanned masonry wal!. The ext ernaI boundary re­presents th e failure cri teria of the j oint under compres ­sion, which is giv en by the ultimate compressi\'e strength /3R of the wal!. (Figure 8) . The vertic a l axis gi ves the co inc ide ntal transferable horizontal mome nt m, under simultan eously conside ration of the shear forces qs and C}p. On e can see that with elas ti c cal culated stresses no large \'al ues are possible when considering e> d! 6 (opene d joint).

m,

qp O qs = O

I~

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o ~~ ·-"i<.-------r------~------...,

lI)

o

o o ,.........--

0 ,00 0.05

, ', ~ /. \.} .

e/

e ",ó, i?'

0,10 0 .16 1fh

Figure 8: ultima t e load carrying capacity 'A'ith unia xial bending

m)'

ü/h =: 0.4 d/h =: 1.0 K" =: 0 .1

Jif,

qp 0.5

iiY

M

qs O

Figure9 : elastic state limit with biaxial bending

3.6. PLASTIC Sli)';ARING$'I'R)';Sl')ES INTIi)';~)';D JOIN:T

As one can see in figure 3, the shearing stress is reduced after exceeding the ultImate elastic shear stress to apure friction component. then remains con ­stant ana IS independent 01' the displacement trave!. If, for example, the ulti ­mate elas ti c moment m, is reached for a particular internaI force combination, then the mortal' begins to plasticize. The transitional state between elastic and ful! y plasticized bed joint is rather difficult to determine by calculation, For the "ful!y plasticized" condition it is assumed that both blocks grind against one another like two rigid forms, 50 that circular shaped shear streS S

!ines are formed as shown in figure 10.

The shearing stresses thus produced can not be calcu­lated with the theory of elasticity, they are linearly dependent on the simul­taneouslj" acting vertical compressive stresses. In or­der to determine the maxi­mum plastic "torsional mo­ment" MPL in the bed joint wi thou t taking a shear force into consideration, the posi­tion of the centre of rotation must be found first.

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1 ,/ '- \ / --I- '- \

/ ( /1-" "\ \ \ \ \ \ \....... 1 J \\'- _/ // \'-'- --- // " '- - --- /

ü l 1

aR

l,"i.glJJ.:gJ.9: plastic shearing stresses in the bed joint

This point is found with the aid of the equilibrium conditions, where with exclusive moment effect the forces in both directions must be zero , in accor­dance with the following equations (for opened bed joint) :

(c-yo) ii/2

o J J ./x2 + y2 I

y - dx-dy

-yo o (8)

The solution beeing looked for can be found by iterative means (even for a non-opened joint), by subdividing the joint area into small elements and numerically calculating the integral, supposing various values for Yo. yo/ c is with opened bed joint between 0.68 for ü/ c = 3 and 0.71 for ü / c = 0 .1. .Ao.

good approximation is given in [1 J with 1/.[2 = 0.707 . The accompanying ulti­mate plastic moment for opened bed joint is obtained from the equation:

(c-yo) ii/2

2 . J J -yo o

/

/'

/ ..

~ ~

... ;;:p

~ ~ N , o

0,0 0,5 1,0 1,5

ü/ c (resp . ü/ d)

+ y2 I

#

V -.. ...

2,0

- dx-dy

(9)

This can aIs o be determined by numerical means for arbitrary com-pressive stress distributions. In Figure 11 oné can see as an example the ultimate plastic moment as a func­tion of the joint dimensions for a constant distribu­tion of the com­pressiv e stresses.

fJgldfE!JJ: ultimate plastic moment with approximations (see section 4)

1392

The ultimate plastic shear forces Qsp and Qpp can be determined very simply as they are given from the normal force ~y in the joint multiplied by the coefficient of friction , thus:

Qsp Qpp

p-ny ·ü ll·ny ·ü

for for

Qpp Qsp

o O (10)

This dependence must also be valid for every arbitrary inclined shear force , so that the resulting plastic shear force follows like this:

QpL I Qs p 2 + Qpp2 ll·ny ·ü

:> 00 l

ox Xo

~í xf ~ l l ü ~ 'I

flgllfe J .. ;:! .. :. numerlcal evaluation 01" the ultimate internaI forces

'-' ;:.,

(11)

To determine the ultimate plastic moment with simul­taneously acting arbitrarr shear force combinations it is also assumed that the shearing stress lines run circularly shaped around a specific centre of rotation . As the posi tion of this poin t cannot be evaluated directly , one must proceed in a rever­se manner. The joint is once more subdivided into numer­ous small areas as shown in figure 12.

Presuming the position for the centre of rotation and the distribution of the compressive stresses (and thus the size of the shearing stresses), the accom­panying resulting shear forces are obtained by the addition of the force components in each direction, the plastic moment br addition of these compo­nents multiplied by their relevant lever arms. The reference point for the lever arms is given by the point of intersection of the shear forces if these act singly, for example the 1/3rd point of the compressed area in case of an opened joint. Using this approach, all combinations can be attained, the transition to pure shear force is given by the siting of the rotation centre in infinity. If one relates every single internaI force to its relevant maximum value excluding all others, and thus normalizing them to values between zero and one, the ultimate pIas ti c moment is given as a function of one of the shear forces as shown in figure 13, in which many individual values are entered from various presumed stress distributions (e.g. including parabolic shaped distribu­tion). It becomes apparent that the values lie approximately on a circular line. As the shear forces and the torsional moment can be positive as well as negative, the limiting surface of the normalized plastic internaI forces can be given approximately by a sphere (figure 14), in other words:

<Xl

-J o .. ::E >'. <D ~ . E o

...... -J .. <t ~ o

"'. o

o o

~~ ~

1--'--- ~

!- ._ .

1-

, 0.0 0.2 0 .4

~ plr fie

= [ ~ ~ I'B-J i>.

i?!I

9. ~ 'I '-

o.e 0.8

QPl/ maxQPL

figllfElJ:3 : ultimate plastic moment

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The vector of the normalized ultimate plastic internaI forces in the bed joint aIways has the value one.

MPL / maxMPL

1

J Qsp! maxQsp

Qpp / maxQpp

figllI.",J4: normalized uItimate plastic internaI forces (Uplastic sphere U)

As already described in detail previously, in contrast to the elastic behaviour no direct determination of stresses is possible from the internaI forces within the joint in the plasticized state. Even if this were possible for the indivi­dual internaI forces, a superposition of the stresses for arbitrary internaI force combinations could not be carried out. For this reason it is desirable to find an approximate solution to evaluate the stresses due to arbitrary force combinations directly in order to avoid the lengthy iterations necessary to t"ind the centre of rotation and the numerical integration of the stresses. As here alI possible distributions of the vertical compressive stresses have to be considered, Le. with both compressed and also opened bed joint, it is purposeful to show the plastic moment firstly for the fully compressed joint with constant stress (as already presented in figo 11) dependent on the joint dimensions ü and d as an approximation, and the dependence on the size of the vertical normal force and on the vertical bending moment by a correction termo Various equations are possible for this approximation :

approximation by straight lines:

IIrL o 0,15-(1,5 + 1,0-ü/d) ü/d ~ 1 '" ü-d2 -Il-PR 0,15-(1,0 + 1,5-ü/d) ü/d ~ 1 (12)

or approximation by parabola:

0,25 + (0,1 + 0,033'ü/d)'ü/d (13)

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Both possibilities are presented in figure 11 as a broken or a dotted line. It is seen that the deviation from the "exact" solution is minima!. and is less than 5 % in the interesting area. The equations are valid for vertical compressive stresses a = !3R. As the shearing stress is directly proportional to the compressive stress, the dependence of the plastic moment on the size of the vertical normal force Ny can be given quite easily. On the other hand it is quite difficult to exactly evaluate the influence of the vertical bending moment My, Le. the eccentricity of Ny by one equation for fully compressed and opened joint. Inserting the above mentioned referenced values, a suitable approximation is given as follows:

ii& = ü (_

!fPL o - - - Dv-h

iüY 0,35 + ü/d (14)

Figure 15 shows the ultimate plastic moment for the condition Qp = qs = O as a function of the normal force ifY and the vertical bending moment my . This three-dimensional diagram is comparable to the diagram for the elastic state limit (figure 9) and is presented similarly scaled. It is seen that, especially with opened bed joint, much larger moments can be transferred than in accordance with an elastic calculation.

mx

my ..

ü/h = 0.4 d/h = 1.0

mx

figllr~J§: plastic moment and deviation of the approximation

n;:

my

In order to finally determine the plastic momen t, the dependence on the si­multaneously acting shear forces (resp. the resulting shear force) must be taken in to consideration, which is given by the "plastic sphere"; in other words, a reduction factor has to be determined which indicates how much of the shear bearing capacity of the joint is already been "used up" by shear forces . Considering the equations (5), (7), (10) and (11), the resulting existing shear force is given by:

Qe" d-w/h- I i1.;2-h2 + iji>2-ü2 (15)

1395

and the ultimate plastic shear force:

(16)

The quotient Qex/ QPL must be less than 1 and supplies, with the relationship from figure 13, the reduction factor for the plastic moment:

I 1 - Qex/Qpl (17)

The equations in this presentation describe the stress conditions in the bed joint with arbitrary loading of masonry walling , and give the ultimate values for arbitrary internaI force combinations, 1.vhereby the flow capacÜy of the mortar is taken in to full account. The formulas should assist in providing realistic solptions to various problems in masonry walling construction e.g. the buckling behaviour of walls with biaxial load transfer [6J or to determine the load carrying properties of externaI wall leaves subjected to wind loading .

R~f~R~NÇ~S

I 1/ PFEIFER M.: Analytical investigations of masonry walls subjected to axial compres­sive forces and biaxial bending moments. Proceedings of the Fourth North American Masonry Conference, Los Angeles, 1987

[ 2) PFEIFER M.: Investigations on the stresses in masonry walls subjected to con­centrated loads . Proceedings of the 7th International Brick Masonry Conference, Melbourne, 1985

3) MüLLER H.: Untersuchungen zum Tragverhalten von querkraftbeanspruchtem Mauer­werk. Dissertation Darmstadt 1974

( 4) PFEIFER M.: PLUS2, Programmsystem zur Berechnung von Platten und Scheiben ·nach Theorie n. Ordnung. Darmstadt 1986

[ 5) SCHWING. H.: Zur wirklichkeitsnahen Berechnung von Wandscheiben aus Fertigteilen. Dissertation Darmstadt 1975

[ 6) PFEIFER M.:

-

Theoretische Untersuchungen über das Knickverhalten von beliebig gelagerten Mauerwerkswanden. (Disserta tion in preparation)