ENGINEERS FILE NOTE NO .3 .
eISfB I
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372 I (21) I
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Published by The Brick Development Association
THE CALCULATION OF ECCENTRICITIES IN LOAD BEARING WALLS
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by Professor A.W. Hendry BSe phD OSe CEng FICE FIStruetE FRSE
B55628: Part 1 states that the eccentricity of loads on walls should preferably be calculated but does not indicate how this may be done. This note, based on recent research
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work , addresses this problem. It discusses the background work and gives a method of calculating the eccentricity on load bearing walls between reinforced concrete floors.
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INTRODUCTION Oneofthe basic problems in the design of load bearing brickwork walls is that of allowing forthe effects ofslenderness and eccentricity ofloading. In BS 5628 Part 1 designersare advised to determine eccentricity ofloading by calculation but, recognising the difficulty ofdoing this at the time ofpublication, the Code gives an empirical ruleforassessing eccentricity. Recently, researchwork has been completed which permits calculation of eccentridties on a rational basis and it is the purpose ofthis Note to summarise the problem and a suitable method of calculating eccentricity for the case ofload bearing brickwork wallscompressed between reinforced concretefloor slabs.
END FIXITY FROM FLOOR SLABS Where a wall is compressed between reinforced concretefloor slabs, two interrelated aspects ofthe problem can be discerned.Thefirst is illustratedin Fig 1 where the strengthening effect offloor slabs on a wall is suggested. Thus, when a wall is tending to fail in buckling, the ends
tend to rotate;with floor slabs present, some degree ofrestraint is introduced by flexure ofthe floor slabs. Thegreater the relative stiffnessofthe floor slabs, the greater the restraint and the nearer the situation approaches that ofa fixed ended column.
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WALL END ECCENTRICITY Thesecond aspect ofthe problemrelates to the effect ofthe floor slabs on the eccentridty ofloading at the wall ends. Loading on a floor slab,as in Fig 2,will apply a momentto the top ofa wall the magnitude ofwhich will depend on the rigidity ofthe wall/floorslab joint. Ifthe load from a section ofwall above that under consideration is eccentric, it will introduce a bending moment into the slab. Thetotal load appliedto the top ofthe wall, (PcJ will be the sum ofthe load from above (Pul plus the slab reaction (Ps) and will be applied at an eccentricity eL such that Pu ' e, + PL' eL= M s The moments developed, and thus the eccentricities ofloadingat the wall ends, depend on a number offactors, including the relative stiffnessofthe wall and slab and the jointfixity.
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P,
• y';::> (0) Unrestrained end rotation
~ Restraining moment
(b) End rotation restrained by lloorslabs
Rg 1: Res1raining effectof floorslobs on wallbuckling.
Ag 2: Condifions at wall/floor slobjoint.
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10
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08
~
I
~ 0.6
~
rr;
c
Q 0.4 Ie - 1.ON/mm 2
I, - O.5N/mm2
02
0 "'---_ _- - _ - _ -_ _1.0
40
50
70
6.0
_
'0 0
90
8.0
Rg 3: Joint fixity V ~a b/wa ll stillness ralJa n
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30
20
Ie - Q.25Nfmm 2
_ _-
J O I N T FIXITY
CALCULATION OF ECCENTRICITY
Aconsiderable amount ofexperimental research has been devoted to the study of wall/floorslab interaction and in particular to the evaluationof jointfixity. Theresults are summarised in Fig 3 which shows the joint fixity, definedas the ratio ofthe actual moment transmitted by a joint to that whichwould exist ifthe joint was fully rigid, plotted on a base ofslab/ wall stiffnessratio. Thejointrigidity also depends on the compression on the joint from loadingabove and three curves are shown for joint compressivestresses between O.25N/mm2 and lON/mm 2•
Various methods have been developed for the calculation ofeccentricityofloadingat a joint. Avery easilyapplied partialframe analysisis show in Fig 4 for external walls from which eccentricities resulting from slab loads may be calculated, provided that the rigid frame moments are reduced to allow forthe joint fixity as defined in the preceedingparagraph. Formulae are also available forinternal walls but in this case eccentricities are generally negligible. Thejoint fixity factor may be obtained from Fig 4 but, as an approximation, the following formula may be used: F=
-
1
1
0.44ex + II where ex is the slab/wall stiffness ratioas defined in Fig 4.
r-
- -
w/m
,I
g/m
h/2
I
I
A
(EI),
h
I A
I
(EI),
h
(Ell .
I
8
(EI).
I
g/m
B
I
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I
-
h
-
'-
I·
·1 Superimposed load
MA -
1 cr.
+2
2' - N - ' wl2
n -~ . (EI), l (EI).
I·
·1
Deed Load
M _ _1_ . , A 8 (0 +3) ·l N - 4 (0
+ 2)' -
(0
+ 1)'
Rg 4: CalC1Jlatlan or EccentricityIn Exterior MasonryWalls
SLENDERNESS RATIO Theslenderness ratio ofa wall is defined as the ratioofthe effective height to effective thickness. For solidwalls the effective thickness is the same as the actual thickness whilstfor a cavitywall it is conventionally taken as two thirds of the combinedthickness ofthe leaves. The effective height theoretically depends on the eccentridties ofthe loading at the top and bottom ofthe wall.Thus, referring to Fig 5, if the eccentridties are ofthe same sign, (orzero at one end), the wall will deflectin singlecurvature (Fig 5 (all. If the eccentridties are ofopposite sign, the wall will deflectin double curvature (Fig 5 (bl). Inthe first case the effective height is equal to the actual height ofthe wall whilstin the second case the effective height is less than the actual height. To simplify calculations, the effective height ofa wall compressed between floor slabs is usuallyassumed to be three quarters of the actual height. This is in reasonable agreement with experimental and theoreticalresults. Also to simplify calculations, it is reasonable to assume that the eccentricity . ofloadingabove a wall/floorslab joint is zero.
DESIGN ST RENGTH OF A WALL Having determined the slenderness ratio and eccentrid ty ofloading, the design strength per unit length ofa wall is then calculatedby multiplying the design strength ofthe material (fkl)'ml by the thickness ofthe wall (t) and by a capadty reduction factor (p) whichis a function of the slenderness ratioand eccentridty. Many theories have been produced for determining capadty reduction factors and these give rather disparate values. Fortunately, in the majority of practical cases the eccentrid ties will be rather small and the reduction factors by allmethods willbe near to unity where the stress in the brickwork is significant.
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REDUCTION FACTORS By way ofillustration, capacity reduction factors forthe variousstorey height sections ofthe outer wall ofa building Fig 61have been calculatedby the method described in this Note in conjunctionwith B55628 Part 1. Two wall thicknesses have
(0)
Single curvature
Fig 5: EIIedive heights in sing le and In double curwture
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been considered, namelya 280 mm cavity wall and a 215 mm solidwall. Incalculating the rigidity ofthe joint between a floor slab and a cavitywall consisting oftwo 103 mm thickleaves, experimental results have shown that such a wall will behave as a single leafwall when the jointcompression exceeds 0.3N/mm2• This is the case in all but one ofthe jointsin this example. Theresults are summarised inTable 1 from which it will be seen that eccentricities are generally quite small. The reduction in load bearing capacity resultingfrom slenderness and eccentricity does not exceed 14 per cent below the third level from the top ofthe building. Aspecimencalculation is given in an Appendix to this Note to show the procedureto be adopted in calculating eccentricities in design.
Slab:Span 35 0< 4.5 m ftlickness 150 or 1'10 mm
LeveI
(
0
J 1
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2r
(
l,...-280 mm Cavity Wall or 215 mm Solid Wall
2
(
3
(
-
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Ie
Slab Loading D.L 4kN/m' S.L 2kN/m'
L..
(b)
Double curvature
7 Ag 6: Dimensions and loa ding tor outer wall aI hypalhehca l bYilding
CONCLUSION In855628Part 1, paragraph 31 states that the eccentridty ofloading in walls should preferably be calculated. As described in this Note, practical means are now available for doingthis. As compared with the conventional rulegiven in the Code, the resulting eccentridtieswill usually be higherwhen calculated by a method giving valuesconsistentwith experimental results. In most cases, however, the capadty reduction factor for slenderness and eccentricity will not be greatlydifferent as between the two methods wherebrickwork strength is critical. This is because the Code reduction factors do not decrease until the eccentridty ratiois greater than 0.05and in many cases the correctly calculated valuewill not exceed this figure. Notwithstanding this conclusion, it is clearly more satisfactory to adopt a rational method ofcalculation in preference to a rule ofthumb, the limitations ofwhich are unknown.
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Full scale structural lesl on wall/floor slab loin! behavIOur.
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APPENDIX
ECCENTRICITIES AND
CALCULATION OF
CAPACITY REDUCTION
ECCENTRICITY AT A
FACTORS FOR THE OUTER WALL
WALL/FLOOR SLAB JOINT
OF A BUILDING Level beiow Roof Slob
The load conditions at the fi" t floor ievel at the outer wall in a seven storey bric kwo rk building are as shown in Rg 7 The floor slo b baa" o n th e inner ieat of a 280 mm cavity wa ll. Tests have been shown that in suc h a case the jo int fixity sho uld be calculated with reference to the loaded leaf only
1 2 3 4 5 6 7
Referring to the partial fra me fo rmula shown in Fig 4 and ta king the ratio of elastic moduli for concrete and brickwork as 2:
cx - ~ . L
Floar Slob: Span 3.5 m: thic kness l SO mm 280 mm Cavity Wall " 215mm Solid Wallt
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0.13 0.075 0.057 0046 0.038 0.034 0.025
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0.17 0.16 014 0.11 0.094 0 079 0.060
0.78 0.86 0.88 089 0.89 0.89 0.89
0.73 0.75 0.79 0.86 0.89 0.92 0.96
(EQ, _2.85 . 2 . ( 100 ) ' _ 176 (EQw 3.0 102.5 . Level beiow Roof Slo b
N-4(cx + 2)'-(cx + 1)"-4(1.76+2)'-(1.76+ 1)'-48.93 Moment due to superimposed load : 1 cx + 2 - . - - . wL' 2 N
1 2 3 4 5 6 7
_ ]. . ( 1.76 + 2) X32 x3' 2 48.93 . 4 . -0.37 kNm Moment d ue
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to dead load:
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Floor Slob: Span 4.5 rn; thickness 170mm 280 mm Cavity WoII" 215 mm Soid Wollt
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0.22 0088 0.069 0.055 0.047 0.041 0.031
S1endemess ratio : 14
8(cx + 3) g L
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0.62 0.84 0.87 0.88 0.89 089 0.89
0.20 0.1 4 0.12 0.094 0.078 0.067 0.060 t
0.66 0.80 0.84 0.89 0.93 0.95 0.96
Slenderness rolio 9
Eccenlridly ratio - eft 1 x 9 07x 3' 8(1.76 + 3) .
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"" Copocify reduction foetor (BS 5628 Port 1. )
- 0.72 kNm
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Total rigid frame moment - 0.37 + 0.72 - 1.09 kNm
load from 'tore~ O.9G + t6Q, - 82.35kN(m
Joint fiXity foetor 1 (0.44 cx + 1.1)
-0.53 1 0.44 x 1.76 + 1.1
m
Joint moment - 0.53 x 1.09 - 0.58 kNm 0.58 X 10' e82.35 - 704 mm
Aoor soan: 3.0 m Slob load : lAG,
+ t6 Q"
- 12.3tkN(m e 704 - - 102.5 - 0.069
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t 4G, - 9.07kN(m t6Q, - 3.24kN(m
Fig 7: loading at wall/floor slab join!
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The Assodation wou ld be inte res ted to hear. from Engine ers or Architects, of pro jects which they consider worthy of inclusion in The BDA Engineers' File Note Series . All initial submissions should contain reference to t he particular area of the design wh ich. it is considered. would be of interest to the design profession as a whole. All enquiries should be addressed to The Technical Editor. J Morton BSc PhD CEng MICE MlnstM
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The views expressed in this File Note are those of the Author's. Readers are expressly advised that they do not necessarily reflect the views of the Association. To demonstrate the initia l engineering decis ions taken, scheme calculations have been included in the Note . They are NOT intended to be full and detailed calculations and they should NOT be read as such.
THE ENGINEERS FILE NOTES SERIES BINDER Aspecially designed binder has been produced to hold the File Note Series and is available from the Promotional Manager, the Brick Development Association, Woodside House . Winkfield. Windsor, Berkshire SL4 2DX. Please enclose £2.50 remittance per binder to cover post and packing. Readers are expressly advISed that . whilst the contents of this publICation are behevt'd to be accurate. CO!TKt and ccenpiete. no relianceshould be placl!duponits ecete nts as beIngappbcable to any particular d rcumstances Any advke. opinionor information containt'd is publish«fonly on the footingthat the Brick Dl!velopment Assodahon. Its servants Of agents and all contributors to this pubbcaoon shall be under no IutMlity whatsoever in respect of its contents
DeSigned and Produad for the Bnck ~Iopment Associauon. YIOodstde House . wlnkfJek1. Windsor . Brrkshlre Sl4 20X, Tel~ : Winkheld RO¥iI' (0]44) 885651 by frank Walter C The Blick Dl!vl!lopmrnt Assodauon.
~tin
Lnmted
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