EWPIIB-7WallingShapeandMaterialsUsageR5.docx
02/08/2016
Energy&Low-incomeTropicalHousing(ELITH)WorkingPaperEWPIIB-7
WallingShapeandMaterialsUsage-Plane,Buttressed,Crenelated
andWavyWalls
ByT.H.Thomas,WarwickUniversity,2016
ABSTRACT
Astraightlongboundarywallshouldgenerallyhaveaslendernessratiooflessthan14toattain
sufficientlateralstrengthandstiffness.Thusa2.4mwallwouldneedtobe17cmthick,i.e.fullbrick
(20cm)ratherthanhalfbrick(10cm).Athinner,cheaper,half-brickwallcouldbeusedbutrequires
buttressesplacedtypically3mapart.Therearehowevertwoprovenalternativestobuttressing,
namelyemployingawavyoracrenelatedwallplan,thatuselessmaterialbutmayincursuch
penaltiesascreatingroomsthatarehardtofurnishorboundariesthatdonotmatchplotownership.
Thepaperidentifies4structuraland3non-structuralperformancecriteriaforwallsandagainst
thesecomparesthe4wall-planshapeslistedinthetitle.Takinga20cmfull-brickwallasdatum,the
relativestiffnessperbrickusedfora40cm-deepbuttressedwallisaboutx3andtherelativelateral
strengthisaboutx1.For40cm-deepcrenelatedwalls,thecorrespondingratiosarex6andx3.For
40cm-deepwavywallsthecorrespondingratiosarex13andx4.Thusthinwavyandcrenelatedwall
plansoffersignificantlybetterperformance‘perbrick’underlateralforcesthandothinbuttressed
orthickstraightwalls.
CONTENTS
1
2
7
Introduction
Fourshapevariants:plain,buttressed,crenelatedandwavy
Wallingmaterialsandmethodsofconstruction
Single-storey,two-storeyandmulti-storeywalling
Criteriaforcomparison
Plainwalls
Buttressedwalls
8
9
Crenellatedwalls
Wavy(serpentine)walls
3
4
5
6
10
Numericalcomparisonsofthe4wallplans
11
Conclusions
12
Bibliography
Appendix:CalculationsforConstantRadiusWavyWalls
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Introduction
ThisELITHWorkingPaperisoneofaset(WPIIB-4)examiningtheenergyembodiedinwallingand
howitmightbereduced.WPIIB-4-1(Wallingstiffnessandstrengthintropicalhousing)developeda
simplisticanalysisforplanewalling.Thispaper,WPIIB-4-3,usesthattomakeacomparisonofthree
other walling plans. For purposes of comparing these different plans (shapes) we can take as a
datumastretcher-bonded‘half-brick’wallofthicknesstD=100mm(Fig1abelow).
Fired-clay bricks, mortar and lean concrete blocks all have a high (per litre) energy content and
carbonfootprint,sothereshouldbeastrongincentivetousingbothonlysparingly.Oneinteresting
optionistobuildmainlyinhalf-brickthicknessbuttoawallplanthatsomehowgivesthestiffnessof
afull-brickwall.Thepotentialsavingofmaterial,firingenergyandcarbonfootprintisupto50%.
Fig1a Planewall–Half-brickthickness
W
Fig1b Planewall–Full-brickthickness,suitablybonded(m=2)
D=mxW
Fig1c Buttressedwall(example:n=5,m=3)
Extension
=(m-1)W
Buttressspacing=nxbricklength
Fig1d Crenelatedwall(example:n=5,m=3)
Set-back
=(m-1)W
Spacingofset-backs=nxL
Fig1e Serpentinewall(example:n=10,m=5)
RadiusR
amplitude
=(m-1)W
Brickaxis
Wall’sNeutral
axis
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Howeverforbothload-bearinghousewallsandforboundarywalls,afull-brick(Fig1b)ratherthan
half-brick wall is usually employed where, provided the brick proportions are properly chosen, a
variety of ‘bonds’ can be used to hold the wall strongly together. With an English bond (alternate
coursesofheadersandstretcherswithoutinternalorexternalcontinuousverticaljoints)theout-ofplanestiffnessofthefullbrickwallisabout8timesthatofthehalf-brickoneandthelateralloading
resistanceis4timeshigher.
Brickwork is ancient. However bonded fired-clay bricks were not commonly used in Europe for
house walling until about 1800 and the corresponding dates in Asia and Africa might be 1950 and
1990 respectively. Wattle and daub (poles and mud), dried soil blocks, matting on poles were all
dominant walling techniques in the tropics until late in the 20th century. Moreover in many places
the full-brick, bonded, fired-clay brick wall (and its ‘cavity wall’ variant) have given way to more
complex designs or to the use of larger cementitious hollow blocks. Block walls, that are typically
150mmthick,haveastiffnessintermediatebetweenthethoseof112mmhalf-brickand225mmfullbrickwalling.
Other brick walls can be described by their mean thickness tm expressed as a multiple u of brick
thickness W. The ratio u = tm/tD will also equal the number of bricks per course divided by the
number of bricks in a datum wall of the same length. Thus for a double-thickness plane wall, U=2
andm=2.
For external house walls and for boundary walls, such double-depth brick (i.e. 200mm deep) is
commonly used to achieve adequate stiffness and strength. Neglecting renders or plasters, this
requiresabout420kgofbrick+mortarpersquaremeterofwalling.Byusingasingle-brickthickness
(i.e.100mm),themasspersquaremeterofbrickwallingcouldbereducedtoabout230kg.Orwith
some more complex wall-shape for which 1<U<2, we might achieve a lesser, but still valuable,
weightsavingrelativetothedouble-depthnorm.
To achieve u < 2, much of the wall length must be of single-brick thickness. However to achieve
adequateoverallstiffnessandstrength,thelocalaxesofsuchsingle-bricksectionsneedtobeoffset
fromtheoverallwallaxis.AmeasuremisthereforeusedtodescribethewalldepthDasamultiple
of W (i.e. D = m x W). Thus (see Fig 1c) for a buttressed wall, (m-1) x W is the protrusion of the
buttresses.Foracrenelatedwall,seeFig1d,Dthedistancebetweenthe‘front’and‘rear’faces;for
awavywall(Fig1e)itisthemaximumdistancebetweenthosefaces.Forallthreewallshapes,the
walldepthisD=m xWandthemeanthicknessisu x W.Asincreasingmresultsingreaterlateral
stiffnessandstrength,yetincreasinguincursgreatercost,weareinterestedinwaysofincreasing
the‘materialsproductivity’ratiom/u.
The energy consumed, and the GHGs emitted while producing house walling, depend on the
quantity of material used. This in turn, for a given wall area, is roughly proportional to the mean
thicknesstmofthewall.InthisWorkingPaperwelookonlyattheimpactofwallshapeonthatmean
thickness. It is however of little value to compare the mean thickness of walls of very different
performances.Thereforeweshouldeitherspecifyaparticularperformanceandcomparethemean
thicknessesofrivaldesignsthatachieveit,orweshouldgeneratesomesortofproductivityfigure
whose numerator is a performance measure and whose denominator is wall cost or mean wall
thickness.
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To keep comparisons simple, we will restrict ourselves to walls of a specified height – the datum,
buttressed,crenelatedandsinuouswallsareassumedtobethesameheight.
As house walling and boundary walling are typically 225mm thick, we might choose to adjust the
various performance criteria to “that achieved by a 225mm-thick plane wall”. This will simplify
visualisationofwallproperties,astb,p,tc,pandts,pbecometherespectivemeanwallthicknessesof
buttressed,crenelatedandsinuouswallstogivethesameperformanceasa200mmplanewall.
Inatropicalsetting,someaspectsofperformance(forexamplewindandnoiseexclusion)arelittle
affected by wall design. In this paper we therefore concentrate on performance factors that are
highly affected by shape or thickness, namely structural strength and stiffness. We choose not to
considerheretheeffectofusingdifferentwallingmaterials.
Theresistanceandstiffnessofevenplanewallingiscomplextocalculate,forexampleaccordingto
Eurocode 6. For non-uniform walling, such as buttressed, crenelated and sinuous, analysis is even
morecomplex.Thereforethispaperemployshighlysimplifiedanalysis,whereverpossiblelinkedto
footprintareaA,secondmomentofareaIandbending-sectionmodulusZ.
2
Fourshapevariants:plain,buttressed,crenelatedandwavy
Plane walling (Fig 1a and 1b) is of uniform thickness (mean thickness = maximum thickness). To
achieveadequateresistancetoout-of-planeforces,thisthicknessmaybequitelargeentailingmuch
material.Planewallingiseasytoerectandinhousingproducesconvenientroomshapes.
Buttressedwalling(Fig1c)hasuneventhickness.Itseffectivethicknessisincreasedbytheaddition
ofperiodicbuttressesinawaythat–foragivenstiffnessorlateralstrength–useslessmaterialthan
a plane wall. However incorporating well-bonded buttresses requires some skill, may result in
inconvenientroomshapesandmayconsumemorelandareathanaplainwall.Acellularfloorplan
(many tiny rooms) is the ultimate form of buttressing but is rarely acceptable to a house’s
occupants.
Crenelated walls (Fig 1d) carry the buttressing idea further with most of the wall being displaced,
alternately, to one side or the other of the wall’s axis. This substantially increases stiffness and
resistancetooutofplaneforceswithoutmuchincreasingthemeanwallthicknessandhencewall
cost.Againskillisrequiredtoachieveproperbondingandroomshapeissomewhatcompromised.
Plane,buttressedandcrenelatedwallingisnormallystraight,wavyserpentinewalling,onceknown
as‘crinkle-cranklewalling’(Fig1e)isnot.Awavywalluseshardlymorematerialthanastraightone
of the same thickness but (provided it holds together and does not ‘rack’ when loaded) is many
times stiffer and more force-resisting. The penalty is the inconvenient shape (in a world where
almost all fittings and furniture are rectangular) and any liability to more complex failure modes.
However for some application (garden walls), and some architects, the very sinuosity can be
consideredavirtue.Zig-zagwallingcanbeconsideredaspecialcaseofserpentinewalling.
3
Wallingmaterialsandmethodsofconstruction
Wallingbeingaveryancienttechnology,comesinamyriadofforms.Thetwomost-commonforms
of walling assembly are monolithic and block. In the former the relevant material is assembled on
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siteasacontinuum,eitherusingshutteringorusingaformersuchasaverticalmesh.Thereissome
scopeforlightcompressionofthewallingmaterialbyapplicationofsloworimpactforces,butany
such compression is limited by the bursting strength of the shuttering. By contrast with block or
brickorstonewalling,theindividualunitsareassembledwithoutshuttering,usuallybyusingeither
a jointing compound such as mortar or an arrangement of interlocks. Because each block/brick is
small,beforeitisbroughttositeitcanbesubjectedtosubstantialpressuresortemperaturesina
press or furnace, thereby cheaply improving its mechanical properties (strength, hardness,
accuracy).Howeverblockwallingmayhavealmostnotensilestrengthatthejointsbetweenblocks.
Its structural analysis commonly entails assuming a complete absence of tensile bonding unless
reinforcingorposttensioningisemployed.
Voidswithinawallweshallregardas‘materialproperties’affectingmaterialdensities,strengthsetc.
For the purposes of comparing wall shapes, we will assume all rival shapes employ the same
materials.Aproblemmayapplywith‘cavity’walls,inthatcavitiescaneasilybecombinedwithplane
walling but not with buttressing, tessellation or serpentine form. However although cavity
construction was widely used in temperate countries in the 20th century to restrict damp
penetration,ithassincesupersededbyothertechniquesitisuncommoninthetropicsandsowill
notbeconsideredfurtherinthisWorkingPaper.
Wallscanbenon-homogenous,forexamplehavingtwoleavesmadeofdifferentmaterialsorhaving
surfacerenders/plasters.Eventhoughtheyareusuallythinsuchrenderscancontributesignificantly
towallperformance.AgaininthisWorkingPapersuchrefinementswillbeignored.
Themainpurposesofwallingare(i)asaclimate-excludingcurtainor(ii)asacombinationofcurtain
andstructuralsupport.Curtainwalling(i)onlyhastocarryitsownratherlightweightandwithstand
localforcesincludingimpacts;moststructuralloadsinsuchbuildingsarecarriedbyameshofbeams
andcolumnsbetweenwhichthewallingisstretched.Curtainwallingisusuallyofbrick/blockorof
factory-madepanel,orofglass.Bycontrast,‘structural’or‘load-bearing’walling(ii)carriesnotonly
itsownweight(includingtheweightofthewallingofhigherfloors)butalsothetransferredweight
ofsuspendedfloorsandroofing.Italsoresiststhehorizontalforcesduetoitsconnectiontothose
more horizontal elements. A further horizontal force on load-bearing walls is that due to windloading,seismicaccelerationsandimpacts.
4 Single-storey,two-storeyandmulti-storeywalling
Veryhighbuildingscompriseastructuralframeandacoveringofcurtainwalling,hangingfromthat
frame. Historically load-bearing walling (with no frame) was used up to about six stories, however
the exact analysis of such high walls was not possible and they had to be built with large but
indeterminatesafetyfactors.Thisincreasedtheircostandprovokedtheearly20thcenturychangeto
framed construction for all high-rise buildings. More recently the analysis of multi-storey loadbearingbrickwallshasbeenadvanced[Eurocode61996,Hendry1997]andthereisaslightrevival
ofinterestinthem.Forthepurposesofthispaperwewillconsideronly1,2and4storeyhousing
withload-bearingwalls.Single-storeyrepresentsthegreatbulkofruralhousinginthetropics.TwostoreyconstructionisnowcommonintownsandisalsoobservableinstiltforminruralSEAsia.Four
storeyhousingisusuallyintheformofapartments,oneormoretoeachfloor.ELITHWorkingPaper
IIB-3comparesthepropertiesofhousingofthesethreeheights.
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5 Criteriaforcomparison
Aswellascost(foragivenmaterial,thisisstronglycorrelatedwithmeanthicknesstm),weare
interestedforhouse-wallingin
Table1
Criteriaforassessingwallperformance
Criterion
(i) Land-area(footprint)occupiedbythewallandtheexternalpartofitsfoundations
(ii) Theconveniencetothehouse’soccupants,forexampleofresultantroomshapes
(iii) Verticalload-carryingcapacity
(iv) Horizontalload-carryingcapacity,includingfailuremodes
(v) Seismiccapacity,naturalfrequencyanddamping
(vi) Lateralstiffness
Othercriterialikeappearance,thermalinsulation,durability,cleanlinessetcaremoreaffectedby
choiceofwallmaterialthanbychoiceofwallshape,sotheylieoutsidethescopeofthisWorking
Paper.
Footprint&Convenience(i&ii):Thefirstcriterionabove,footprintperunitlengthofwallequalsat
leastthemeanwallthicknesstm.Forcertainpurposeswemightwanttoincreasethismeasurebya
factorofabout2(forboundarywalls)and1.5forexternalhousewalls)toaccountforthatpartof
thewall’sfoundationwidthnotunderneathafloor.Inthecaseofbuttressed,crenelatedand
serpentinewalls,thereisanadditionalland-taketowhichitisdifficulttoassignacost.
Inthecaseofawavyboundarywall,thewallcreatesbaysthatarealternatelyonthe‘inside’andthe
‘outside’ of the wall. The areas of these bays might be planted and so not be ‘lost’ to the garden
area.Oronlytheinsidebaysareplantedandtheoutsidebaysareregardedassacrificedland.There
isasimilarissuewiththespacebetweenbuttresses/piersonboundarywalls.Thatbetweeninwards
facingbuttressesisusableinvariouswaysbutthatbetweenoutwardsfacingbuttressesmighthave
toberegardedaslostbecausebuttressesmustbekeptwithintheplotboundary.
Forhousewalls,theinteriorbays(serpentine)orrecesses(buttressedorcrenelated)inroomswill
certainlybeusedbutmaybeseenasinconvenientlyshapedandthereforeoflowervaluepersquare
meter.
Discussingthe‘inconveniencepenalty’ofdifferentwallshapesiscontinuedinsections6to9below.
Thecriteria(iii)to(vi)inTable1arestructuralandallareusuallymorerelevanttoload-bearingwalls
thantocurtainwalls.
Vertical Strength (iii) Crushing is not the common mode of wall failure: the vertical load-carrying
capacitydependson(perunitlengthofwall)theEulerbuckingresistanceF = π2 E I / Heff2 ,whichfor
anygiveneccentricityandconstraintsisproportionaltothematerialstiffness(E)andthe(perunit
length)secondmomentofareaI.TheeffectiveheightHeffforawallofheightHcantileveredfromits
baseandlaterallyconstrainedatitstop,equals0.71H,orifnotsolaterallyconstrainedequals2H.
SoincomparingwallshapesweneedtocomparetheirrespectivevaluesofI(e.g.Ip,Ib,Ic,Is,).These
values are tabulated below for a range of wall configurations. The Euler formula above does not
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apply to a wall buckling under its self-weight, which is however rarely the dominant failure
mechanism.
Horizontal load-carrying capacity (iv), for example the maximum acceptable wind pressure acting
on the whole wall face or top-edge lateral force (per unit of length) to overturn the wall, is
determined by the onset of tensile cracking in the wall-face to which the load is applied. Load
capacity depends on compensation by wall weight and on resistance due to the vertical tension
strengthofwall’smaterial.
The first factor, wall weight, causes a compressive pre-stress of σcomp = ρgH (where H is the wall
height above the level chosen for analysis, g is gravity, ρ is wall density). H can be conveniently
chosenastotalwallheight,sothattensileseparationisassumedtooccuratthebottomofthewall.
If the brickwork is assumed to have no tensile strength, then ultimate failure will occur when the
tensile stress in a wall face due to out-of-plane forces equals the pre-stress due to gravity. The
maximum tensile stress will be the overturning moment M (per unit length) at the chosen level
divided by the modulus Z per unit length (Z = I/ymax). So the sustainable moment is Mg = σcomp Z
However if the wall material does have some tensile strength σtens then the moment that can be
resisted is increased by Mt = σtens Z, giving Mfailure = (σtens+σcomp) Z. In either case, the comparative
abilitytoresistlateralforcesofdifferentwallshapesisproportionaltotheirrespectivevaluesofper
unitmodulusZ(namelyZp,Zb,Zc,Zs,)
Conventionalwallanalysisoftenrepresentstheeffectofgravitybyaverticalpressurepg,sothatat
anychosenheighthwithinthewall,themomentthatcanbecarriedcanbewritten
M=pgZ
where pg=ρg(H–h)forawallwith100%brick-to-brickcontact.
If however the brick-to-brick contact area Ac is less than the brick’s plan area A, then the ‘gravity’
equivalentpressureishigher: pg=(A/Ac)ρg(H–h).
orwherethewallalsohassometensilestrengthσtensand/orissubjecttopost-tensionofσpost-tens
M=(pg+σtens+σpost-tens)Z
whereonlypgvarieswithpositionhupthewall.]
Earthquake resistance (v) is a complex phenomenon that depends on wall stiffness, internal
damping and other design parameters. It is therefore too extensive a topic to be covered by this
Paper.Howeverwenotethatitisgenerallylateralaccelerationthatdamageswallingduringtremors
and the effect of such accelerations is similar to wind loading. Thus seismic performance depends
upontheabilityofthewalltoresistlateralforces(justarguedtodependuponZ)andonthewall’s
mass,sincehorizontalseismicaccelerationsareconvertedintolateralforcesviathismass.Mass,for
agivenmaterialandwallheight,willbeproportionaltomeanwallthicknesstm.Thusresistanceto
seismicfailureisproportionaltoZ/tm.
Lateralstiffness(vi)is,likeEulerload,proportionaltoEIor(forconstantE)justtotheperunit2nd
momentofareI.Formortaredwalls,lateralstiffnessisusuallyadequatetopreventanymalfunction
e.g. cracking of internal plaster. However for unmortared walls (of interlocking blocks), stiffness is
much lower. This may result in walls having too low a natural frequency and hence resonance
problemsduring‘quakes.
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6 Plainwalls
Despite the bricks in a plane wall normally having no interlocks (although sometimes ‘frogs’ are
indentedtotheirtopsurfaces),themortarhassufficientshearbondingtothebrickstoholdthewall
together. This adhesion is enhanced by overlap between bricks on successive courses and the
avoidanceofcontinuousperplines.Forsinglethicknesswallsastretcherbondgives50%overlaps.
For double thickness a variety of bonds have been developed to give at least 25% overlap within
eachleafandtotiethetwoleavestogether.Single-thickness(e.g.100mm)planebrickwallsarenot
normally used for external house walling or for boundary walling. By contrast single-thickness is
normalforblockwalls(typically150mmthick)andmortaradhesionmaybeassistedbyinterlocking.
Doublingthethicknessofawallraisesitscrushingstrengthbyafactorx2,itsEulercollapseloadand
lateral stiffness by factor x8 (both being dependent on 2nd moment I), and its lateral strength by
factorx4.
Verticalloadingisduetoself-weightandtheweightofsupportedupperfloorsandroof.
Lateralloadingarisesfrom:
windforces,whosepressurecentroidishalfwayupthewallandwhosemomentisgreatestatthe
wallbase,havingvalueMw=pressure.H2/2perunitlength;
horizontalseismicaccelerationforces,whosecentreofactionishalfwayupthewallandwhose
moment above the wall base has the value Me = acceleration.ρgtm H2/2 per unit length of
wall,wheretmisthemeanwallthickness(againhingingisequallyprobableatallheights);
horizontallineforcestransmittedfromceilings(spanfloors)androof.Hinging/overturningdueto
anyoftheseforcesisequallyprobableatallheights,astherestoringmomentduetowall
weightatanycourseisproportionaltothewallheightabovethatcourse.
7 Buttressedwalls
AbuttressedwallisshowninFig1c:theparticularexampledependsuponthebuttressdepthm-1
(expressed as a multiple of brick width W) and the buttress spacing n (expressed as a multiple of
brick length L). Buttresses are also called piers, except that piers, like columns, can be inserted
symmetricallyinthewalllineandnotjusttooneside.‘Pier+Panel’isafurtherspecialcasewhere
onlythe(reinforced)pierhasfullfoundations.
Providedawallbehavesasalaterallyrigidcontinuum,thenthemeasuresofgreatestinteresttothe
structuraldesignerareits(mean)unitsecondmomentofareaIanditsmodulusZ.By‘unit’wemean
perunitlengthofwallingandby‘mean’thatthisisaveragedalongthelengthofthewall.Modulusis
not averaged along the wall, as it depends upon the maximum distance (ymax) on the tension side
between the surface of the wall and the wall’s neutral axis. Since I rises with the third power of
thickness, a wall with deep buttresses has a much larger stiffness than one with small buttresses.
Andofcoursemakingbuttressesmorefrequentalsomakesthewallstiffer.
Let I0 be the unit 2nd moment of area (units are m3) of a wall with buttresses spaced infinitely far
apart. Let Ib be the unit 2nd moment of area of a brick wall with one buttress (a single brick wide)
every n bricks of its length. Let m-1 be the depth of the buttress, measured in brick thicknesses
perpendicular to the wall’s face. Thus for a plane wall, m = 0 and mean thickness = to. When m is
increasedornisdecreased,thestiffnessmeasureIb/I0increasesstrongly,thestrengthmeasureZ/Z0
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increasesmoderatelyandthecostmeasuretb/t0increasesweakly.Inthetables2to5thestiffness
andstrengthmeasuresrealsonormalisedtothenumberofbricksused.
Table2
Propertiesofbuttressedwalling
Walldepth/W=m
2
Spacing n in brick
5
10
lengths
Stiffness & Euler 3.2
2.2
ratioIc/I0
StrengthratioZ/Z0
1.2
0.8
CostratioUb=tb/t0
Stiffness ratio per
brick(Ib/I0)/Ub
Strength ratio per
brick (also seismic
ratio)(Zc/Z0)/Ub
3
4
5
5
10
5
10
5
10
10.3
6.3
24.4
14.8
47.1
28.8
2.5
1.4
4.4
2.4
7.0
3.8
1.2
2.7
1.1
2.0
1.4
7.4
1.2
5.3
1.6
15.3
1.3
11.4
1.8
26.2
1.4
20.6
1.0
0.7
1.8
1.2
2.8
1.9
3.9
2.7
Sofortheexampleshadedabove,namelyaddinga3-brickbuttressevery10bricks,willincreaseI
and hence stiffness about 15 fold, Z and hence lateral strength (assuming force applied on the
buttressed side) about 2.4 fold, yet the number of bricks needed increases by only 30%. This
buttressspacing(10bricklengths)isapproximatelyequaltothelikelywallheightHforsingle-storey
housing.
Howeverwallanalysisnormallytakesaccountofthepositionofnotbuttressesbut‘returns’(wellbondedperpendicularsupportwalls)assumedtobeinfinitelystrongandrigid.Yetwithsuchreturns
spacedclosely(spacingequalsapprox.thewallheight),thereturnsincreasethelateralpressureload
thewall’sweightcanresistonlybyafactorofabout3[Hendry,Fig7.6].Sinceabuttresshasrather
low stiffness, it is structurally worth less than a return. Therefore even a very deep buttress is
unlikelytoincreaselateralstrengthmorethan3-fold.
A buttress has to bond with the rest of the wall. The simplest bond (assuming a single-brick main
wall)istohaveadoubleheaderonalternatecourses,ordiagonallyspacedsingleheadersonevery
course.Thisisnotastrongjointintension(wherewall-to-buttressforcesaretransmittedbybrickto-mortar-to-brick shear). If ¾ bricks are available, other bonds including those with doublethicknesswallsarepossible.
Boundarywallsoftenfailbyexcessiveleaning,duetoveryslowsubsidenceundertheirfoundations,
rather than collapse under unusually high loading. (The bell tower of Pisa Cathedral is a famous
example, where leaning developed over eight centuries.) If buttresses are to resist this form of
failure their foundations must be substantial, rigid and if possible extend beyond their ends. One
method of extending the support leverage (for any given buttress volume) is to employ sloping or
triangularbuttresses.Howeverasbuttressesareusuallyshort,theyhavepoorresistancetoracking
causedbyslidingatcoursejoints.Suchsliding/shearingisreducedbyhavingmuchweightonthe
joints–whichpointstowardsusingfull-heightbuttresses.
Buttressesarethoughtto‘intrude’andarethereforeusuallyplacedontheoutsideofahousewall.
Placed internally they slightly reduce the flexibility of rooms, especially if their depth exceeds 0.3
meters(3bricks).Placedexternally,andwithfoundations,theydirectlyoccupyanextralandareaof
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approximatelytwicetheirplanarea.Howeverifthehousewallisalsoontheplotboundarythenit
mayhavetobemovedinwardsbythebuttressdepth,therebylosingconsiderablymorelandarea.
8 Crenellatedwalls
Crenulation,seeFig1d,isameansofincreasingawall’seffectivethicknesswithoutusingmuchextra
material.Thecontributiontothewall’sI,byanyindividualbrick,riseswithapproximatelythesquare
ofthedisplacementofthatbrick’saxisfromthewall’saxis.Soacrenelatedwallwithlargesetbacks
hasamuchlargerstiffnessthanonewithasmallsetback.Wedefine‘set-back’asthedistancefrom
theaxisofitsfrontrowtotheaxisofitsbackrow(seeFig1c)andexpressitasamultiple m-1of
brickthickness;D=mxWisthereforethedepthofthewall.
Table3
Propertiesofcrenellatedwalling
Walldepthm
2
Spacing n of set
5
10
backs(bricklengths)
Stiffness & Euler 4.8
4.4
ratioIc/I0
StrengthratioZc/Z0
2.4
2.0
CostratioUc=tc/t0
Stiffness ratio per
brick(Ic/I0)/Uc
Strength ratio per
brick (also seismic
ratio)(Zc/Z0)/Uc
3
4
5
5
10
5
10
5
10
15.8
14.4
35.2
31.6
64.0
56.6
5.3
4.8
8.8
7.9
12.8
11.3
1.2
4.0
1.1
4.0
1.4
11.3
1.2
12.0
1.6
22.0
1.3
24.3
1.8
35.6
1.4
40.4
2.0
2.0
3.8
4.0
5.5
6.1
7.1
8.1
Again taking m = 4 i.e. (a setback-depth of 3W) and n = 10 (i.e. a setback-spacing of 10L) as our
example, we now have a very large increase in stiffness and a quite large increase in strength for
only a modest (30%) increase in brick use. Moreover the symmetrical nature of a crenelated wall
meanstheforwardandreversepropertiesarethesameaseachother.
Unlikewithbuttresses,thereisnoneedforaparticularlystifffoundationunderanypartofthewall.
Howevertheinconvenienceofthewallshape–itsimpactonroomusageandfurnishing-isgreater
thanforbuttressing.Indeedincountriesthatusedtohavecoalfireplacesineveryroom,theirdisuse
has led to the expensive demolition of ‘chimney breasts’, suggesting that residents found their
presence too intrusive. If the crenulations are ‘inwards’ from the external wall-line of a typical
house,andthereisonecrenulationperroom,theneachroomareaisreducedbytypically3%.For
boundarywallingthereislittlearealoss,sincethewall’sdirectfootprintislessthanthatofastraight
wall.Howevertheindividualrectangularbaysonboththeinsideandoutsidemayhavelittleutility
andcometobecountedaspartofthefootprint.
Constructionofastretcher-bondcrenelatedwallisstraightforward,usingclassiccornerbonds.The
possibilityofrackingishoweversignificant.Theconnectionbetweenthetwoleavesofacrenelated
wallcanalsobeareinforcedbrickpier,forexamplea300mmx300mmpierconnectingtwo100mm
wythes.
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9 Wavy(serpentine)walls
Serpentine walls are uncommon, although a number (under the name ‘crinkle-crankle wall’) that
were built over 200 years ago still survive in England, the Netherlands and USA. They achieve
effectivedepthviatheircurvature,notviatheadditionofextrabricks.Howevertheanglebetween
thebrick’slocalaxisandthewall’saxisdoesnotusuallyexceed300, sotheiralong-the-walllength
andhencebrick-countiscommonlyonlyafewpercentmorethantheirlengthalongthewall’saxis.
Ifnormalstraightbricksareused,itisdesirablethattheanglebetweensuccessivebricksdoesnot
exceed about 90, otherwise the wall will appear too knobbly with bricks overhanging those below
thembyover2mm.Asinusoidalwallplanisprobablymostvisuallysatisfying.Howeveraserpentine
wallconstructedofalternatearcseachofconstantradiusRwillresultinalowermaximumcurvature
than a sinusoidal wall of the same depth and wavelength. Using such constant-curvature arcs is
normalpractice.
TheAmericanBrickInstitute’stechnicalnotes(1982&1999)discusstheproportionsofserpentine
wallsthathaveshownverylong-termdurability.Thesearegenerallyverydeep,withdepthsnotless
thanhalfofwallheights.
Fig3 Serpentineboundarywallinseismiczone(FortPortal,Uganda2015)100mm-thick
Analysisoftheperunit2ndmomentofareaIandthemodulusZofserpentinewallsisdifficult,unless
werestrictourselvestocyclelengths(‘wavelengths’)correspondingtointegernumbersofbricksper
halfwave.
As for the other wall types, we define m as the number of brick widths that make up the wall’s
overalldepthDandnthenumberofbrickstomakeuptwoofthewall’sarcs(i.e.acompletecycle).
11
EWPIIB-7WallingShapeandMaterialsUsageR5.docx
Table4
02/08/2016
Propertiesofwavywalling
Radiusofarc/t
13
Arc-angle/degrees
44
Maxangletowallaxis
22
Walldepth/W=m
3
Bricks/cycle=n
10
Wavelength/bricklength
10
Stiffness & Euler ratio
7.1
=Is/I0
StrengthratioZs/Z0
2.4
Costratious=ts/t0
1.03
Stiffness ratio per brick6.9
=(Is/I0)/us
Strength ratio per brick 2.4
(Zs/Z0)/us
Angle
between 8.80
successivebricks
Depth/wavelength
0.15
40
26
13
10
74
37
20
52
26
6
120
60
18
18
7.5
12
11.2
25.8
18
17.5
26.2
12
10
57.6
Recommended
15
74
37
7
18
16.7
57.3
2.5
5.2
5.3
8.2
8.2
1.01
7.4
1.07
24.1
1.03
25.4
1.21
47.6
1.07
53.6
2.5
4.9
5.1
6.8
7.7
2.90
12.30*
5.70
8.30
0.09
0.21
0.14
20.00*
0.35
5
0.22
*unacceptablylarge
Astheanglebetweensuccessivebricksinacourseshouldnotexceedabout9owhichrequiresthat
brick-lengthsperhalfwave(n)wouldnormallybeabout9,givingtypicallya4mwavelength.
Figure4
Anglebetweensuccessivebricks
Cmax
Protrusion=Lα/4
Angleαbetweensuccessivebricks
shouldnotexceed0.16c=90
Cmin
AssumingameanhorizontalspacingbetweenbricksofCmean =10mm,andaminimumspacingCmin
of2mm,requiresCmax=18mm.HenceforbricksofthicknessW=100mm,themaximummortar
wedgeangleshouldnotexceed(Cmax-Cmin)/W=0.16rad=90.Thisgivesanoutwards‘protrusion’of
about 8mm between the joint between two bricks and the brick in the course below. Such
protrusionisratherlarge,givingthewallaknobblyappearance.
Normal shear friction between courses may not be sufficient to hold the wall together, so some
horizontalreinforcementinamortarcoursenearthetopofthewallisdesirable.
Experimentswithasection,similartothedimensions‘recommended’inTable4,ofthewavywall
shown in Fig 3, indicated a lateral stiffness about 10 times that of an adjacent straight wall of the
12
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same height and thickness. This is less than the factor of around 57 shown in the theory. Further
experimentswillbeperformed.
The brick count of a serpentine wall is low – often only a few percent more than a single-wythe
planewall.Indeedthemainattractionofthewavywallshapeisthatitallowsalargeincreaseinits
effective depth (m) to be made without significantly increasing its use of materials. So serpentine
wallsareusuallybuiltdeeperthancrenelatedwalls;e.g.forthecasem=7,shownhighlitinTable3
above,thedistancefromfrontfacetobackfacewouldbe70cm(ifbricksare10cmwide).If,fora
boundarywall,thewholeserpentinemustlieinsidetheplotboundary,thenthereissignificantextra
land-take,about50%ofwhichiseffectivelydonatedtotheneighbouringplot.Ifhoweverthewall’s
axis lies along the plot boundary, then both this and the neighbouring plot experience alternate
intrusionandretreat(bays).
Exampleofasimplehouseplanwith
serpentineexteriorwallsand
straightinternalwalls.
10
Numericalcomparisonsofthe4wallplans
Wecannowcompare(Table5)the4shapevariants–plane,buttressed,crenelatedandserpentine.
Table5
Comparisonofwallshapes(m=D/W=3shownshaded).
Wallshape
Wall-depthm
Costratious=ts/t0
Stiffnessratio(I/I0)
Strengthratio(Z/Z0)
(Is/I0)/us
(Zs/Z0)/us
Plain
1
Buttressed
every5bricks
(n=5)
2
3
1
2
1
1
1
1
8
4*
4
2
3
27
9
9
3
2
1.2
3.2
3
1.4
5
10.3
1.2
2.5*
2.7
7.4
1.8
1.0
1.8
+
47.1
7.0*
26.2
3.9
Crenulated
every5bricks
(n=5)
2
1.2
4.8
+
2.4*
4.0
2.0
3
1.4
5
2
15.8
+
5.3*
11.3
3.8
1.8
64.0
Wavy
wavelength
(n=18)
3
5
7
1.03
1.07
+
26.2
57.3
1.01
7.5
12.8
2.5*
5.3*
8.2
35.6
7.4
25.4
53.6
2.5
5.1
7.7
7.1
Notes: Stiffness-ratio/cost-ratio=(Zs/Z0)/usisalsotheseismicaccelerationtoleranceratio
Wavywallwavelengthislong(n=18),elsethebrick-to-brickanglewouldbetoohigh
Forplainwallsm=2iscommonlyused–asshownbyboldproperties
Tosimplifythiscomparisonletusfirstlookatthecasem=3,i.e.wherethewalldepthD,frontto
back,is3timesabrickthickness.Inthetable,thedatacolumnsforthiscasearehigh-lit.
Intermsofcost,foragivenlengthofwallandwalldepthm=3,theorderis:
wavy(best),crenelated,buttressed,plane(muchtheworst).
Intermsofperformanceasmeasuredby2ndmomentIandsectionmodulusK,theorderis:
plane/crenelated(good),buttressed/wavy(poor).
If we combine performance and cost by expressing performance per brick used, we get:
crenelated(best),plane,wavy,buttressed(worst).
13
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If however we had taken a more typical 2-brick (m=2) plain wall as our datum, then the
‘performanceperbrick’testwouldhaverankedtheoptions: crenelated(best),wavy,plane,buttressed(worst)
Adifferentandmorerealisticapproachistofirstdefineaperformanceandthenassessthematerials
costofmeetingit,usingeachofthefourwallshapes.Thuschoosing(forconvenience)aresistance
to overturning forces equal to that of a double-thickness (m = 2) plane wall, for which Z/Z0 = 4,
where Z0 is the modulus of our datum single-wythe plain wall. We get, approximately, by
interpolation(ofcandidatesmarked*inTable5),theoptionsof:
abuttressedwall,withm=4,n=5,costing20%lessthantheplanewall
acrenelatedwall,withm=3,n=5,costing30%less
awavywall,withm=4,n=18,costingnearly50%less
Applyingthesameapproachtogettingastiffness(andalsoquakeresistanceandEulerbucklingload)
equal to that of the double (m=2) plane wall, we could use (candidates with I/I0 = approx 8, as
marked+inTable5):
abuttressedwall,withm=3,n=5,costing30%lessthantheplanewall;
acrenelatedwall,withm=2.5,n=5,costing35%less;
awavywall,m=3,n=18,costing50%less.
In general therefore, buttressing is not a very efficient use of brick, whereas both crenulation and
employingwavywallingoffersubstantialsavings.Themainpricetopayforthissavingisthepossible
inconvenienceofhavingsteppedorcurvingsurfacesinaroomorlosingalittleplotareanexttoa
boundarywall.
10 Conclusions
Crenelatedandwavywallsuseupto50%lessbricksthanplanewallstoachievethesameresistance
tobuckling,samestiffnessandsameresistancetolateralforces.Buttressedwallsareintermediate
intermsofbrickeconomy.Indeedthesedesignscanbeusedwithasinglebrickthickness(e.g.100
mm) to get satisfactory performance in both housing and boundary walls; by contrast with plane
wallingadoublethickness(e.g.200mm)isnormallyrequired..
Both crenelated and wavy designs are suitable for increasing strength and stiffness by making the
walldeeperyetwithoutusingmanymorebricks.Theshapeofthesematerially-efficientwallplans
mayconflictwiththerectangularnormofmostmodernbuildings.Converselytheyofferscopefor
architecturaldistinctiveness.
Thesimplifiedanalysisusedinthispaperassumesthatduringloading,thewallsholdtogetherand
actasawhole.Rackingduetoslippagebetweencourseshasthereforenotbeenconsidered.These
assumptionsmayproveunfoundedinthecaseofverydeepcrenelatedorserpentinewalls–e.g.m
valuesexceedingsay5.
11
Bibliography
HendryAW,SinhaBP,DaviesSR,1997,DesignofMasonryStructures,3rdEd,E&FNSpon
BSEN1996-3:Simplifiedcalculationmethodsforunreinforcedmasonrystructures
14
EWPIIB-7WallingShapeandMaterialsUsageR5.docx
02/08/2016
http://en.wikipedia.org/wiki/Crinkle_crankle_wall,2014
BIA,1982,Structuraldesignofserpentinewalls–non-loadbearing,TN33,Renton,VA,USA
BIA,1999,BrickinLandscapeArchitecture-GardenWalls,TN29A,Renton,VA,USA
Photosofwavywalling:
http://www.geograph.org.uk/gallery/crinkle_crankle_or_serpentine_walls_15374,acc’dDec2016
15
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Appendix
Calculations for Constant Radius Wavy Walls
Assume wall is a sequence of 'facing' arcs each of radius R, arc angle 2θ, and thickness t, where
R = λt
2nd moment of wavy wall is μ times that of a straight wall of thickness t and of the same 'axial' length; .
I = µ Io
However as the curved wall is slightly longer (for the same axial length), we have a brick usage multiplier of
σ = θ/sin(θ)
Depth, front-face to back-face of wall , D = t + 2*R*(1-cos(θ)); datum wall depth = t; relative depth is m = D/t
m = 1 + 2*λ*(1-cos(θ))
Relative stiffness μ = (θ/sin(θ))*{1+6λ^2(1 - 3(sin(θ)*cos(θ)/θ) + 2cos^2(θ)}
μ = σ*(1+6λ^2*e)
where λ = R/t
where e = (1 - 3(sin(θ)*cos(θ)/θ) + 2cos^2(θ)
As a close
approximation
Relative stiffness per brick
used is
μ = μ' = 1+1.59*(m-1)^2
μ / σ = 1+6λ^2*e
I / Io
wall
radius
thickness
R/t
t
R
m
m
0.1
0.6
0.1
1
0.1
1
0.1
1
0.1
0.1
λ
relative
relative
relative
half
stiffness
brick
stiffness
Wall
relative
Wave
depth/
arc angle
per meter
usage
per brick
depth
strength
length
wavelth
μ
σ
μ/σ
μ/m
W
D/W
θ
θ
rads
degree
e
6
1.05
60
0.2620
57.6
1.21
47.6
0.70
7.0
8.19
2.08
0.34
10
1.571
90
1.0004
601.2
1.57
382.7
2.10
21.0
28.62
4.00
0.53
semicircs
0.943
10
0.785
45
0.0900
55.0
1.11
49.5
0.69
6.9
8.02
2.83
0.24
1/4 circles
0.991
10
0.523
30
0.0189
12.4
1.05
11.8
0.37
3.7
3.36
2.00
0.18
1/6 circs
1.000
5
50
0.1
6
0.0000
1.4
1.00
1.4
0.15
1.5
0.93
2.00
0.08
1.002
5
50
0.2
11
0.0004
7.4
1.01
7.3
0.30
3.0
2.46
3.97
0.08
1.005
0.1
5
50
0.284
16
0.0017
26.6
1.01
26.3
0.50
5.0
5.32
5.60
0.09
1.004
0.1
5
50
0.3
17
0.0021
32.8
1.02
32.4
0.55
5.5
6.01
5.91
0.09
1.004
0.1
5
50
0.4
23
0.0066
100.3
1.03
97.7
0.89
8.9
11.28
7.79
0.11
1.002
0.1
5
50
0.6
34
0.0323
484.9
1.06
456.3
1.85
18.5
26.26
11.29
0.16
0.998
0.1
2.5
25
0.1
6
0.0000
1.1
1.00
1.1
0.12
1.2
0.88
1.00
0.13
1.001
0.1
2.5
25
0.2
11
0.0004
2.6
1.01
2.6
0.20
2.0
1.30
1.99
0.10
1.003
0.1
2.5
25
0.3
17
0.0021
9.0
1.02
8.8
0.32
3.2
2.77
2.96
0.11
1.004
0.1
2.5
25
0.4
23
0.0066
25.8
1.03
25.1
0.49
4.9
5.22
3.89
0.13
0.1
2.5
25
0.523
30
0.0189
72.0
1.05
68.8
0.77
7.7
9.37
4.99
0.15
0.1
2.5
25
0.6
34
0.0323
122.0
1.06
114.8
0.97
9.7
12.53
5.65
0.17
16
D
m
m
d/t
μ'/μ
0.979
1.002
1/6 circs
1.000
0.998
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17