II th INTERNA TIONAL BRlCKJBLOCK MASONR Y CONFERENCE TONGJI UNIVERSITY, SHANGHAl, CHINA, 14 - 16 OCTOBER 1997

LOADBEARING MASONRY WALLS SIMULTANEOUSLY SUBJECTED TO VERTICAL AND LATERAL LOADS

I 2 Arne Cajdert and Olof Sjõstrand

1. ABSTRACT

This paper presents a design method for loadbearing masonry walls subjected to simultaneous vertical and lateral loads, developed as an extension to Eurocode 6 [1]. The method pay regard to the following possible failure cases: o overall stability of the wall o ultimate stresses in field sections and support sections o plate-buckling based on yield-line analogy according to [2], {3]. Eurocode 6 does not provide for control of ultimate stresses in field and for plate­buckling.

2. INTRODUCTION

Traditional calculation methods for masonry generalIy deal either with loadbearing walIs with predominant vertical loads, or with lateralIy loaded non-Ioadbearing walls. In fact, most masonry walIs are subjected to both vertical and horizontalloads. This paper is an attempt to overbridge the gap between the current models and thus give consistent design methods for various load combinations.

Keywords: Masonry walls, Design methods, Eurocode 6, Stability, Plate-buckling

ISenior Lecturer, PhD, University ofOrebro, S-701 82 Orebro, Sweden. Fax +4619303463 . 2Senior Design Engineer, MSc, AB Jacobson & Widmark, Hebsackersgatan 24, S-254 37 Helsingborg, Sweden. Fax +46 42133191.

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ControI of overaIl stability is based upon Eurocode 6 with regard to slenderness, field eccentricity due to end moments, horizontal load, initial imperfections and creep. Ca1culation of field eccentricity due to end moments is shown.

UItimate stresses Field section capacity will be controlled for vertical loads and field moment due to horizontal loads, end moments and eccentricity due to initial imperfections and creep.

Support section capacity will be controlled at low slenderness in accordance to Eurocode 6.

Plate-buckling At low vertical loads plate-buckling, i.e. plate-action with due regard to second-order effects, may be determining.

3. NOTATIONS

Notations of constants and variables explained in the following paragraphs deviate from those in Eurocode 6 in order to meet the whole concept of this paper.

4. DESIGN LOAD ACTION

Loadbearing rnasonry walls are normally subjected to a combination of design vertical loads N d and design lateralload p caused by wind or earth pressure.

Nd = Gd+Qd+Sd+Wd (I) Nd = N I' N2 and Nrn at upper and lower horizontal support and at rnid-height,

see figure ] a and b Gd design dead load Qd = design imposed load Sd = design snow load W d = design wind load

The verticalload N d may have the eccentricity at the horizontal supports, e l and e2, thus causing eccentricity end moments M] = N] e] and M2 = N") e2. At mid-height the verticalload N may have the eccentricity e thus causing the field moment

m m M = N e . The lateral load p causes the field moment M = N,n e and at the m m m p p supports the reaction moments M ip and M 2p respectively.

The initial deflection eo and the deflection due to creep ek causes the field moment Nm (eo + e,).

The total moment at mid-height with regard to second-order effects will be: Mtot =. Nm edim/c = Nm (em + ep + eo + ek)/c . . (2) accordmg tofigure 3a and b (c = second order dIvIsor, diagram I).

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NO/Rr(J

o ____ ._0 _ _ _____ O- _ _ __ o

. 0.1

02

06

08

aaL---------~~~--------~~~--------~~-­~ (1000-ICk \G~

1e4 [\)

Diagram 1. Second- order divisor c

Accumulaled verlical load Nd

I M2res ; M2p N2+N 1 +G watl

Figure la. Design /IIodel for loadhearillg /IIoso/lry lI'alls silll/lllwzco/lsly slIh;ectcd lo vertical (J/ullolcrallollds.

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The lateral load is assumed to give end moments in horizontal supports except for the upper support in the top storey wall, where M 1 normally is zero, because the top joint with the roof support cannot absorb more end goment than the eccentricity offers. The

moments Mp' M 1p and M2p may be determined from plate diagrams.

wç e 2~2------- Momenl 01 lop eccenlrlClly

M12 = Nl2" e12

12nd FLOORI

Nm2 = N12 +0.sG wa1l 2 -WSUCiJon

(il lhe vertical load IS lavourablel

M11 = N11 ' ell

11s1 FLOOR 1

Nm1 = Nn'Nll ·O.sGwa1l1 'Wpressure +S

(il lhe verllCal load is unlavourableJ

Figure 1 b. Design verticalloads. E,v calculalillg momelll capucities for cases where verticalloads are favourable. on/y permanelll /oatls lVi/h redllced partial coefficielll \ViII he /aken in/o COllsideratioll.

Wino suclion may bc determining in f1cld and wind prcssurc may bc detem1ining at lhe upper support in walls below the uppermost storey, figure 2a and b.

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N, • Nd ' Go ,00 'Sd -Wo (kN/m'l lar mosl unlavOU"able 10aO comblflaloon

WINQ SUCTlON

D' Wd

(kN/m'J

D

I N2 .N,-G watl

e2 • O in exlernal wall

Loads by WlllO suclion

a) Wind suction

@

Mamenl 01 100 eccenlricily

o

Momenl 01 wind sue I,on D

o

Mm-Mp

}-'...-",,-I-~Mlot =Nm" elol ~ mv

Resulllflg momenl 01 [orresDonO,ng eccenlflc,ly

10D eccenlflClly . w,nO suchon. e, eo on,l,al defleCl,on. creeo anO secanO -order aefleCl'CIIl

N, • Nd • Go ,00 -So -Wd (kN/m'J for mosl unlavourable load combonal,on I M, res·M, -M" I

~~-t P-1 IMv !M, i

WINO

~ PRESSuRE EARTH PRESSURE

~ D •

via el De I G",ail @ {kN/m 7 }

,

U I 1 N? =N1·G w .::1!

t.>7. ; O In êx ler il21 wall

loads o, wlnd pí'essu:-e or earlh ccesscre

:-:c:nenl o: :::::9 eccen:nClly

b) Wil1d pressure or earlh preSSllre

Momenl 01 D wlfld pressure ar ear lh pressure

, I N2 '---' i f"'12 rf?~ =M7o= M1P

Resullrng mamen! 01 10p eccen lr,c,lv <1no v .. ',nd press • .Jfe·

i l t~ 2 ;---. . M2 re~ ~ r.,\,

Resul Itng l7Iomenl Dl :OD f?CCenlnoly. \"LIlU pres~ ... wr: "!" Iral deltecl,on. creeo Jnd secoí1d-orOer (ÍF?:I.~:ilor

Figure 2. Momellts il1 loadbearillg l11aSO/1/)' wall below the /op s/orey ,,,.(dl. simul/alleously subjecled to vertical and lateralloarls. 11/ v =- vertical moment capacity (horizontal vector)

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5. ECCENTRICITIES

elres~

~

e<lm' OOSI

h-+---v"'-7"--l fiai ' E{)tII/C

f()ITI ' em -e ú "ek -Po

el( ; Hem .EO I

t f2re.}.e:?p&e\l

(((enlrcd~s

E'ôm • em ·eO 'ek -e o ek • Hem. E'o I

e':)l ~ fiam I (

fI( • He",. E'I)- -eOt ' oy earlh oreSSlI"E

a) Wind suction b) Wind pressure or earth pressure

Figure 3 Eccentricities in wall below top storey wall. simultaneollsly sllbjected to vertical and lateralloads

End eccentricities e I' e2 top and bottom eccentricities of verticalload e lp' e2p top and bottom eccentricities oflateralload

elres' e2res resulting top and bottom eccentricities

Mid-height eccentricities etot total eccentricity at mid-height including second-order deflection

etot edim/c (3) c second-order deflection divisor, diagram 1 edim design eccentricity at mid-height exclusive second-order def1ection,

determined by the condition: em + eo + ep + ek 2: 0,05 t (4) eccentricity at mid-height caused by eccentric end moments M! and M2 MmlNm (5) moment at mid-height determined by diagram 2 fram the end moments M I and M2 by various support conditions

Mm K(M! + M2) M I N I e! = top eccentrici ty moment M 2 N2 e2 = bottom eccentricity moment For walls only spanning vertically: em = 0,5 (M! + M2) Co initial deflection at mid-height

(6) (7a) (7b) (8)

ep = calcu!ated eccentricity at mid-height caused by lateral load Illolllent

el? MplNm (9) Mp vertical field moment of lateral load (horizontal vector)

2 ~ ph coefficicnt dClcmlined by diagralll 3a ar 3b

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(10)

ek = increase ofmid-height eccentricities em' eo and earth pressure ep

due to creep _ _ _ _ 1/2

ek = 0,002 <Dk(hef\f)[t(em + eo + ep)] (11) <Dk = creep factor As wind load does not cause any creep, e by wind load will be zero in eq (11).

p Sign rules

AlI eccentricities are positive when they give tension in outward surfaces: e + by wind suction p - by wind pressure or earth presure

em + by normal fIoor eccentricity in top ofwall

For wind exposed walls the determining load case normally will be wind suction in combination with eccentric vertical load (ep and em both have + sign).

eo most unfavourable sign will be used, i.e. the sign giving the greatest value ofed·

1m

If e p has another direction than em and is greater, eo wilI have the saroe sign as ep'

~ + if em + eo + ep > O - if e + e + e < O m o p

where ep only is taken into consideration for earth pressure.

I

-'-----.-----1----+---- ---------1.-I --"-' I -

o,.----~~~~--f_~----i-----+-----T-----~----t__ ! Hm : I( ( M,'M 1 j !

O]

elll'lC ';f!,-e}) L -\l-\-+,,------:!---i------!----L--u~---i-

I I , 1 ' ,

o'I-----i------i---\-'r-'ri-'- ' , L--.j __ j_ I j! i

o' ~---_--ii-_-_-_-_ ----i-----\ __ '_,\; '_""'_-_-._t-!~--------.;-:---------- i-- - _ ~--r-, ; ~____ I ~ i n/I

0<>'> O

I - I­

O'. ~---=-:;:: __ -.l_ '

I·} l~ 7C

Diagram 2_ Coefficielll K for lI1id-heighl momen/ callsed by endmomen/s M 1 alld M2-

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H '(I • D}.t.1P" MlI"PU,,~f H '(I " P""P't"

}l 1'12\1" t1 11

ulA U< A U6A UlB u'B U6B

Q P Q p tOll/ TAl ((o: ((lH((NT I p .. , -------------~

tOll/lJ<! AI. [IH (11: F((Nf I=-~ ~ ~v(RT(Al !M (II:FFClNI

O" "i." V~ ! L / ~ 0 481 V i

/L ~'" I ~~a"" I 4' "4«'f-« 1

O"

a~Al

V[RICAI. 'UO tD'( (lI:ff((NI I .", -oOS --=~

~ ~ i 00<;

~ ;:::--- ~ v~ ~>(JlI/[NIA f(lO /1)"{N J ( OCfF((NJ

I VL:::'11', I

---t6l!/CI<IAl 'El9 tD'(NI (lI:ff ((NI

I r-------l "lA O~A a" t-=--=': -~ I

I

O 10 I, 10 2, IIh o

lO I I, 10

a) Fixed edge at upper support b) Free edge at upper support

Diagram 3. CoejJicients p and a for moments in field and at supports by lateral loads ..

6. CONTROL OF VERTICAL LOAD CAPACITY­WITH REGARD TO MEMBER STABILITY

eom 1I

10r----.-----r----,---~----~--_,

09 · 00<;

08 " 01

07 - O~

06 -01

<I> OS - 01';

0< - OJ ' 0))

01

02 ·0(.0

01 · o's

o o 10 15 20 7S lO

~ (lOCO Ir')" 'ti. (lo;

Diagram 4. Reductioll facLOr t1J lVit/z regard to slellderness 11 ejt ef and relative field eccellfricily edu/I.

The control ofmember s tability is based upon Eurocode G witll regard la slcndemess helter· rclative field ccccntricity cdim/t due to : end ccccntricities, initial imperfecti0!~S .

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'''' ..1-

: I

I ! i I

I .r~1:

!

i , :

: :

: 1. ;In

lateralload and creep. Second-order effects are considered in diagram 4.

The control of member stability is carried out as a control of the verticalload capacity with regard to buckling according to

(12)

Nd = most unfavourable verticalload in actualload combination according to paragraph 4.

~d = vertical load capacity . <D = reduction factor considering slenderness hef\r relative eccentricity edim/t

and second-order deflection The slenderness ratio is modified with regard to another elasticity modulus Ek than 1000 fck.

b = length of masonry wall, normally 1 m t = thickness of loadbearing wall fcd = design compression strength ofmasonry in vertical direction

fcd = fck(Ym·Yn) hef = effective buckling height of masonry wall tef = effective thick?e~ fI~masonry waIl tef = [(tI + (E21E1) t2 ] (13) tI = thickness ofloadbearing waIl t2 = thickness of laterally supporting wall, alternatively the wall with the least

stiffness, when both leaves are loadbearing, e.g. internaI double leafwalls. edim according to equation (4).

The verticalload capacity will be very low and even zero, if edim/t approaches 0,5 or exceeds this value. This will formaIly be the case when N

d is relatively small. The

design method then is not applicable. The lateralload capacity should instead be controlled by the plate action model according to paragraph 8.

7. CONTROL OFVERTICAL LOAD CAPACITY-WITH REGARD TO ULTIMATE STRESSES IN END SECTIONS

The ultimate compressive stresses wiIl be controlled in end sections by the condition due to compressive strength valid for both triangular and rectangular stress block, figure 4:

Nd :::: R"d = b t 0,75 fcd (1 - 2e/t) (14)

The control may practically be carried out as a control ofthe moment capacity at the top and bottom supports ofthe wall according to figure 2a ar b:

mv = ml for uncracked section and m2 for cracked section at end sections

The moment capacity m2 for cracked section according to figure 4:

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(15) (16)

( 17)

and the mopent capacity ml for uncracked section according to figure 5:

m I = ~d t /6 + Nd tJ6 (18)

where Nd = N 1 or N2 respectively.

The resulting eccentricities, see figure 3a and b. will be el res = el + el p e2res = e2 + e2p

(I9a) (I9b)

where e2 nonnally is zero in exterior walls.

eIp = MIpIN1 e2p = M2pIN2

(20a) (20b)

M 1 P alld M2p may be determined fIom diagram 3a ar b. In top storey walls, where M1p normally is zero, diagram 3b is applicable.

I t

t 2 Nd --e=--2 3 f ed

t 2 Nd t e = ---- e l - - e =

2 3 feri 2

t 2 Nd 3'(--e) -2 fed

t - 2e =

i '1 ! l j 1 j i

Figllre 4. Vertical load capacily R"d and momelll capaci/y m 2 for cracked seclion due /0 CO!1lpressive s/rcnglh for Iri/ongular and rec/ongular s /ress block respectively.

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0.75 f ed

0.75 f ed

Figure 5. Momell! capacity m 1 for uncracked section.

8. CONTROL OF LATERAL LOAD CAPACITY­WITH REGARD TO PLA TE BUCKLING

The vertical load capacity may formally be very low and even zero according to paragraph 6, when the design vertical load is smalL In such a case the lateral load capacity should be checked by controlling plate buckling with a plate action mode!.

The lateral load capacity by simultaneous vertical and lateral loads may be calculated according to yield line analogy, taking into account the varying moment capacities for lateralloads p in horizontal and vertical directions mh and mvp respectively, see [3]. In [3J these capacities are denoted mh and mv. p

The vertical moment capacity mvp (horizontal moment vector) will be detelmined for mid-height sections with regard to cOITesponding eccentricities and second-order deflection:

mvp = m2ec = lateral load moment capacity for cracked mid-height section

mvp = m 1ec = lateralload moment capacity for uncracked mid-height section

1t is easy to understand that plate buckling is considered, as the second-order deflection divisor eis dependent on the slendemess ratio, see diagram 1.

According to figure 20 and 2b lhe total desi gn vertical momcnt Mtnt

should not cxceed the moment capaci ti es 1112 OI' 111 1 respectivcly as shown below.

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8.1 Vertical moment capacity for lateralload by cracked section, m2ec

(21)

The design lateralload moment M should not exceed the moment capacity m2ec with regard to field eccentricities lue to support eccentricities em' initial deflection eo' creep deflection ek and second-order deflection:

M < m2 =cm2-N (e +e +ek) = - lJ- ec mm o

m2 according to equation (17).

8.2 Vertical moment capacity for lateralload by uncracked section, m1ec

(23)

Correspondingly the condition for Mp at uncracked section will be

(24)

ml according to equation (18).

9. REFERENCES

[1]

[2]

[3]

Eurocode 6 (EC 6), ENV 1996-1-1 Design ofmasonry structures­Part 1-1 General rules for buildings - Rules for reinforced and unreinforced masonry.

Cajdert, Ame: LateralIy loaded masonry walIs. Doctoral thesis, Chalmers University, Division ofConcrete Structures, Gothenburg 1980.

Sjostrand. OloI Generalized Design Method of Laterally Loaded Masonry WalIs. Proceedings ) oth Intemational BrickIBlock Masonry Conference, Calgary ) 994.

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