EWP IIB-7 Walling Shape and Materials Usage R5 · PDF fileEWP IIB-7 Walling Shape and Materials Usage R5.docx 02/08/2016 2 1 Introduction ... Fired-clay bricks, mortar and lean concrete

EWPIIB-7WallingShapeandMaterialsUsageR5.docx 02/08/2016

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Energy&Low-incomeTropicalHousing(ELITH)WorkingPaperEWPIIB-7

WallingShapeandMaterialsUsage-Plane,Buttressed,CrenelatedandWavyWalls

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 Introduction

2 Fourshapevariants:plain,buttressed,crenelatedandwavy

3 Wallingmaterialsandmethodsofconstruction

4 Single-storey,two-storeyandmulti-storeywalling

5 Criteriaforcomparison

6 Plainwalls

7 Buttressedwalls

8 Crenellatedwalls

9 Wavy(serpentine)walls

10 Numericalcomparisonsofthe4wallplans

11 Conclusions

12 Bibliography

Appendix:CalculationsforConstantRadiusWavyWalls


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1 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

Fig1b Planewall–Full-brickthickness,suitablybonded(m=2)

Fig1c Buttressedwall(example:n=5,m=3)

Fig1d Crenelatedwall(example:n=5,m=3)

Fig1e Serpentinewall(example:n=10,m=5)

W

D=mxW

Extension=(m-1)W

Buttressspacing=nxbricklength

Spacingofset-backs=nxL

Set-back

=(m-1)W

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

varietyof ‘bonds’ canbeused tohold thewall strongly together.WithanEnglishbond (alternate

coursesofheadersandstretcherswithoutinternalorexternalcontinuousverticaljoints)theout-of-

planestiffnessofthefullbrickwallisabout8timesthatofthehalf-brickoneandthelateralloading

resistanceis4timeshigher.

Brickwork is ancient. However bonded fired-clay bricks were not commonly used in Europe for

housewallinguntilabout1800andthecorrespondingdates inAsiaandAfricamightbe1950and

1990 respectively.Wattle and daub (poles andmud), dried soil blocks,matting on poleswere all

dominantwalling techniques in thetropicsuntil late in the20thcentury.Moreover inmanyplaces

the full-brick, bonded, fired-clay brick wall (and its ‘cavity wall’ variant) have given way tomore

complexdesignsor to theuseof larger cementitioushollowblocks. Blockwalls, that are typically

150mmthick,haveastiffnessintermediatebetweenthethoseof112mmhalf-brickand225mmfull-

brickwalling.

Other brickwalls can be described by theirmean thickness tm expressed as amultipleu of brickthicknessW. The ratio u = tm/tD will also equal the number of bricks per course divided by the

numberofbricks inadatumwallof thesame length.Thus foradouble-thicknessplanewall,U=2andm=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 achieveu< 2,much of thewall lengthmust be of single-brick thickness. However to achieve

adequateoverallstiffnessandstrength,thelocalaxesofsuchsingle-bricksectionsneedtobeoffset

fromtheoverallwallaxis.AmeasuremisthereforeusedtodescribethewalldepthDasamultiple

ofW (i.e. D =m xW). Thus (see Fig 1c) for a buttressedwall, (m-1) xW is the protrusion of the

buttresses.Foracrenelatedwall,seeFig1d,Dthedistancebetweenthe‘front’and‘rear’faces;forawavywall(Fig1e)itisthemaximumdistancebetweenthosefaces.Forallthreewallshapes,the

walldepthisD=mxWandthemeanthickness isuxW.As increasingmresults ingreater lateral

stiffnessandstrength,yet increasingu incursgreatercost,weare interested inwaysof increasingthe‘materialsproductivity’ratiom/u.

The energy consumed, and the GHGs emitted while producing house walling, depend on the

quantity ofmaterial used. This in turn, for a givenwall area, is roughly proportional to themean

thicknesstmofthewall.InthisWorkingPaperwelookonlyattheimpactofwallshapeonthatmean

thickness. It is however of little value to compare the mean thickness of walls of very different

performances.Thereforeweshouldeitherspecifyaparticularperformanceandcomparethemean

thicknessesof rivaldesignsthatachieve it,orweshouldgeneratesomesortofproductivity figure

whose numerator is a performance measure and whose denominator is wall cost or mean wall

thickness.


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Tokeepcomparisons simple,wewill restrictourselves towallsof a specifiedheight– thedatum,

buttressed,crenelatedandsinuouswallsareassumedtobethesameheight.

As housewalling andboundarywalling are typically 225mm thick,wemight 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 arehighly affectedby shapeor thickness, namely structural strength and stiffness.We choosenot to

considerheretheeffectofusingdifferentwallingmaterials.

Theresistanceandstiffnessofevenplanewallingiscomplextocalculate,forexampleaccordingto

Eurocode6. Fornon-uniformwalling, suchasbuttressed, crenelatedand 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.

Crenelatedwalls (Fig1d) carry thebuttressing idea furtherwithmostof thewall beingdisplaced,

alternately, to one side or the other of the wall’s axis. This substantially increases stiffness and

resistancetooutofplaneforceswithoutmuch increasingthemeanwall thicknessandhencewall

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) ismany

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 tomore complex failuremodes.

However for some application (garden walls), and some architects, the very sinuosity can be

consideredavirtue.Zig-zagwallingcanbeconsideredaspecialcaseofserpentinewalling.

3 Wallingmaterialsandmethodsofconstruction

Wallingbeingaveryancienttechnology,comesinamyriadofforms.Thetwomost-commonforms

ofwallingassemblyaremonolithicandblock. In the former therelevantmaterial isassembledon


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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 asmortar or an arrangementof interlocks. Becauseeachblock/brick is

small,before it isbroughttosite itcanbesubjectedtosubstantialpressuresortemperatures ina

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, ithassincesupersededbyothertechniques it isuncommoninthetropicsandsowill

notbeconsideredfurtherinthisWorkingPaper.

Wallscanbenon-homogenous,forexamplehavingtwoleavesmadeofdifferentmaterialsorhaving

surfacerenders/plasters.Eventhoughtheyareusuallythinsuchrenderscancontributesignificantly

towallperformance.AgaininthisWorkingPapersuchrefinementswillbeignored.

Themainpurposesofwallingare(i)asaclimate-excludingcurtainor(ii)asacombinationofcurtain

andstructuralsupport.Curtainwalling(i)onlyhastocarryitsownratherlightweightandwithstand

localforcesincludingimpacts;moststructuralloadsinsuchbuildingsarecarriedbyameshofbeams

andcolumnsbetweenwhichthewalling isstretched.Curtainwalling isusuallyofbrick/blockorof

factory-madepanel,orofglass.Bycontrast,‘structural’or‘load-bearing’walling(ii)carriesnotonly

itsownweight(includingtheweightofthewallingofhigherfloors)butalsothetransferredweight

ofsuspendedfloorsandroofing. Italsoresiststhehorizontal forcesdueto itsconnectiontothose

more horizontal elements. A further horizontal force on load-bearing walls is that due to wind-

loading,seismicaccelerationsandimpacts.

4 Single-storey,two-storeyandmulti-storeywalling

Veryhighbuildingscompriseastructuralframeandacoveringofcurtainwalling,hangingfromthat

frame.Historically load-bearingwalling (withno frame)wasuseduptoaboutsix 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 load-

bearingbrickwallshasbeenadvanced[Eurocode61996,Hendry1997]andthereisaslightrevival

of interest inthem.Forthepurposesofthispaperwewillconsideronly1,2and4storeyhousing

withload-bearingwalls.Single-storeyrepresentsthegreatbulkofruralhousinginthetropics.Two-

storeyconstructionisnowcommonintownsandisalsoobservableinstiltforminruralSEAsia.Four

storeyhousingisusuallyintheformofapartments,oneormoretoeachfloor.ELITHWorkingPaper

IIB-3comparesthepropertiesofhousingofthesethreeheights.


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5 Criteriaforcomparison

Aswellascost(foragivenmaterial,thisisstronglycorrelatedwithmeanthicknesstm),weareinterestedforhouse-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,footprintperunitlengthofwallequalsatleastthemeanwallthicknesstm.Forcertainpurposeswemightwanttoincreasethismeasurebya

factorofabout2(forboundarywalls)and1.5forexternalhousewalls)toaccountforthatpartof

thewall’sfoundationwidthnotunderneathafloor.Inthecaseofbuttressed,crenelatedandserpentinewalls,thereisanadditionalland-taketowhichitisdifficulttoassignacost.

Inthecaseofawavyboundarywall,thewallcreatesbaysthatarealternatelyonthe‘inside’andthe

‘outside’of thewall. Theareasof thesebaysmightbeplantedand sonotbe ‘lost’ to thegarden

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.

VerticalStrength (iii) Crushing isnot thecommonmodeofwall failure: thevertical load-carryingcapacitydependson(perunitlengthofwall)theEulerbuckingresistanceF = π2 E I / Heff

2 ,whichforanygiveneccentricityandconstraints isproportionaltothematerialstiffness(E)andthe(perunitlength)secondmomentofareaI.TheeffectiveheightHeffforawallofheightHcantileveredfromits

baseandlaterallyconstrainedatitstop,equals0.71H,orifnotsolaterallyconstrainedequals2H.SoincomparingwallshapesweneedtocomparetheirrespectivevaluesofI(e.g.Ip,Ib,Ic,Is,).Thesevalues are tabulated below for a range ofwall 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-carryingcapacity (iv), forexample themaximumacceptablewindpressureacting

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,wallweight, causes a compressive pre-stress of σcomp = ρgH (whereH is thewallheight above the level chosen for analysis, g is gravity, ρ is wall density).H can be convenientlychosenastotalwallheight,sothattensileseparationisassumedtooccuratthebottomofthewall.

If thebrickwork is assumed tohaveno tensile strength, thenultimate failurewill occurwhen 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 momentM (per unit length) at the chosen level

dividedby themodulusZ per unit length (Z = I/ymax). So the sustainablemoment isMg = σcompZHowever if thewallmaterialdoes have some tensile strength σtens then themoment that canbe

resisted is increasedby Mt=σtensZ,givingMfailure= (σtens+σcomp)Z. Ineithercase, thecomparative

abilitytoresistlateralforcesofdifferentwallshapesisproportionaltotheirrespectivevaluesofper

unitmodulusZ(namelyZp,Zb,Zc,Zs,)

Conventionalwallanalysisoftenrepresentstheeffectofgravitybyaverticalpressurepg,sothatatanychosenheighthwithinthewall,themomentthatcanbecarriedcanbewritten

M=pgZ where pg=ρg(H–h)forawallwith100%brick-to-brickcontact.

Ifhowever thebrick-to-brickcontactareaAc is less than thebrick’splanareaA, 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 andother design parameters. It is therefore too extensive a topic to be covered by this

Paper.Howeverwenotethatitisgenerallylateralaccelerationthatdamageswallingduringtremors

and theeffectof suchaccelerations is similar towind loading.Thus seismicperformancedepends

upontheabilityofthewalltoresistlateralforces(justarguedtodependuponZ)andonthewall’smass,sincehorizontalseismicaccelerationsareconvertedintolateralforcesviathismass.Mass,for

agivenmaterialandwallheight,willbeproportionaltomeanwallthicknesstm.ThusresistancetoseismicfailureisproportionaltoZ/tm.

Lateralstiffness(vi)is,likeEulerload,proportionaltoEIor(forconstantE)justtotheperunit2ndmomentofareI.Formortaredwalls,lateralstiffnessisusuallyadequatetopreventanymalfunction

e.g. crackingof internalplaster.However forunmortaredwalls (of interlockingblocks), 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 havebeendeveloped to give at least 25%overlapwithin

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 2ndmoment I), and its lateral strength by

factorx4.

Verticalloadingisduetoself-weightandtheweightofsupportedupperfloorsandroof.

Lateralloadingarisesfrom:

windforces,whosepressurecentroidishalfwayupthewallandwhosemomentisgreatestatthe

wallbase,havingvalueMw=pressure.H2/2perunitlength;horizontalseismicaccelerationforces,whosecentreofaction ishalfwayupthewallandwhose

momentabove thewallbasehas thevalueMe=acceleration.ρgtmH2/2perunit lengthofwall,wheretmisthemeanwallthickness(againhingingisequallyprobableatallheights);

horizontallineforcestransmittedfromceilings(spanfloors)androof.Hinging/overturningdueto

anyof these forces isequallyprobableatallheights,as therestoringmomentduetowall

weightatanycourseisproportionaltothewallheightabovethatcourse.

7 Buttressedwalls

Abuttressedwall isshowninFig1c:theparticularexampledependsuponthebuttressdepthm-1(expressedas amultipleofbrickwidthW) and thebuttress spacingn (expressedas amultipleof

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

notaveragedalong thewall, as itdependsupon themaximumdistance (ymax)on the tensionside

between the surface of thewall and thewall’s neutral axis. Since I riseswith the third power ofthickness, awallwithdeepbuttresseshasamuch larger stiffness thanonewith smallbuttresses.

Andofcoursemakingbuttressesmorefrequentalsomakesthewallstiffer.

Let I0 be theunit 2ndmomentofarea (unitsarem3)ofawallwithbuttresses spaced infinitely far

apart.Let Ibbe theunit2ndmomentofareaofabrickwallwithonebuttress (asinglebrickwide)

every n bricks of its length. Letm-1 be the depth of the buttress,measured in brick thicknesses

perpendicular to thewall’s face.Thus foraplanewall,m=0andmeanthickness= to.Whenm is

increasedornisdecreased,thestiffnessmeasureIb/I0increasesstrongly,thestrengthmeasureZ/Z0


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increasesmoderatelyandthecostmeasuretb/t0 increasesweakly.Inthetables2to5thestiffnessandstrengthmeasuresrealsonormalisedtothenumberofbricksused.

Table2 Propertiesofbuttressedwalling

Walldepth/W=m 2 3 4 5

Spacing n in bricklengths

5 10 5 10 5 10 5 10

Stiffness & Euler

ratioIc/I03.2 2.2 10.3 6.3 24.4 14.8 47.1 28.8

StrengthratioZ/Z0 1.2 0.8 2.5 1.4 4.4 2.4 7.0 3.8

CostratioUb=tb/t0 1.2 1.1 1.4 1.2 1.6 1.3 1.8 1.4

Stiffness ratio per

brick(Ib/I0)/Ub

2.7 2.0 7.4 5.3 15.3 11.4 26.2 20.6

Strength ratio per

brick (also seismic

ratio)(Zc/Z0)/Ub

1.0 0.7 1.8 1.2 2.8 1.9 3.9 2.7

Sofor theexampleshadedabove,namelyaddinga3-brickbuttressevery10bricks,will increase Iand 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-storeyhousing.

Howeverwallanalysisnormally takesaccountof thepositionofnotbuttressesbut ‘returns’ (well-

bondedperpendicularsupportwalls)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.

Abuttresshas tobondwith the restof thewall. The simplestbond (assuminga single-brickmain

wall)istohaveadoubleheaderonalternatecourses,ordiagonallyspacedsingleheadersonevery

course.Thisisnotastrongjointintension(wherewall-to-buttressforcesaretransmittedbybrick-

to-mortar-to-brick shear). If ¾ bricks are available, other bonds including those with double-

thicknesswallsarepossible.

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 foundationsmust be substantial, rigid and if possible extend beyond their ends.One

methodofextending thesupport leverage (foranygivenbuttressvolume) is toemployslopingor

triangularbuttresses.Howeverasbuttressesareusuallyshort,theyhavepoorresistancetoracking

causedbyslidingatcourse joints.Suchsliding/shearing is reducedbyhavingmuchweightonthe

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

theaxisof its frontrowtotheaxisof itsbackrow(seeFig1c)andexpress itasamultiplem-1ofbrickthickness;D=mxWisthereforethedepthofthewall.

Table3 Propertiesofcrenellatedwalling

Walldepthm 2 3 4 5

Spacing n of set

backs(bricklengths)5 10 5 10 5 10 5 10

Stiffness & Euler

ratioIc/I04.8 4.4 15.8 14.4 35.2 31.6 64.0 56.6

StrengthratioZc/Z0 2.4 2.0 5.3 4.8 8.8 7.9 12.8 11.3

CostratioUc=tc/t0 1.2 1.1 1.4 1.2 1.6 1.3 1.8 1.4

Stiffness ratio per

brick(Ic/I0)/Uc

4.0 4.0 11.3 12.0 22.0 24.3 35.6 40.4

Strength ratio per

brick (also seismic

ratio)(Zc/Z0)/Uc

2.0 2.0 3.8 4.0 5.5 6.1 7.1 8.1

Again takingm = 4 i.e. (a setback-depth of 3W) andn = 10 (i.e. a setback-spacing of 10L) as ourexample,wenowhaveavery large increase in stiffnessandaquite large increase in strength for

onlyamodest (30%) increase inbrickuse. Moreover thesymmetricalnatureofacrenelatedwall

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.


11

9 Wavy(serpentine)walls

Serpentinewalls 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’s localaxisandthewall’saxisdoesnotusuallyexceed300,sotheiralong-the-wall length

andhencebrick-countiscommonlyonlyafewpercentmorethantheirlengthalongthewall’saxis.

Ifnormalstraightbricksareused, it isdesirablethattheanglebetweensuccessivebricksdoesnot

exceedabout90,otherwise thewallwill appear tooknobblywithbricksoverhanging thosebelow

thembyover2mm.Asinusoidalwallplanisprobablymostvisuallysatisfying.Howeveraserpentine

wallconstructedofalternatearcseachofconstantradiusRwillresultinalowermaximumcurvature

than a sinusoidal wall of the same depth and wavelength. Using such constant-curvature arcs is

normalpractice.

TheAmericanBrick Institute’stechnicalnotes(1982&1999)discusstheproportionsofserpentine

wallsthathaveshownverylong-termdurability.Thesearegenerallyverydeep,withdepthsnotless

thanhalfofwallheights.

Fig3 Serpentineboundarywallinseismiczone(FortPortal,Uganda2015)100mm-thick

Analysisoftheperunit2ndmomentofareaIandthemodulusZofserpentinewallsisdifficult,unless

werestrictourselvestocyclelengths(‘wavelengths’)correspondingtointegernumbersofbricksper

halfwave.

As for the otherwall types,we definem as the number of brickwidths thatmake up thewall’s

overalldepthDandnthenumberofbrickstomakeuptwoofthewall’sarcs(i.e.acompletecycle).


12

Table4 Propertiesofwavywalling

Recommended

Radiusofarc/t 13 40 10 20 6 15

Arc-angle/degrees 44 26 74 52 120 74

Maxangletowallaxis 22 13 37 26 60 37

Walldepth/W=m 3 5 7

Bricks/cycle=n 10 18 12 18 12 18

Wavelength/bricklength 10 18 11.2 17.5 10 16.7

Stiffness & Euler ratio

=Is/I07.1 7.5 25.8 26.2 57.6 57.3

StrengthratioZs/Z0 2.4 2.5 5.2 5.3 8.2 8.2

Costratious=ts/t0 1.03 1.01 1.07 1.03 1.21 1.07

Stiffness ratio per brick

=(Is/I0)/us6.9 7.4 24.1 25.4 47.6 53.6

Strength ratio per brick

(Zs/Z0)/us2.4 2.5 4.9 5.1 6.8 7.7

Angle between

successivebricks

8.80

2.90

12.30* 5.7

020.0

0*

8.30

Depth/wavelength 0.15 0.09 0.21 0.14 0.35 0.22

*unacceptablylarge

Astheanglebetweensuccessivebricksinacourseshouldnotexceedabout9owhichrequiresthat

brick-lengthsperhalfwave(n)wouldnormallybeabout9,givingtypicallya4mwavelength.

Figure4 Anglebetweensuccessivebricks

AssumingameanhorizontalspacingbetweenbricksofCmean=10mm,andaminimumspacingCmin

of2mm,requiresCmax=18mm.HenceforbricksofthicknessW=100mm,themaximummortar

wedgeangleshouldnotexceed(Cmax-Cmin)/W=0.16rad=90.Thisgivesanoutwards‘protrusion’ofabout 8mm between the joint between two bricks and the brick in the course below. Such

protrusionisratherlarge,givingthewallaknobblyappearance.

Normal shear friction between coursesmay not be sufficient to hold thewall together, so some

horizontalreinforcementinamortarcoursenearthetopofthewallisdesirable.

Experimentswithasection,similar to thedimensions ‘recommended’ inTable4,of thewavywall

shown inFig3, indicateda lateral stiffnessabout10 times thatofanadjacentstraightwallof the

Cmax

Angleαbetweensuccessivebricks

shouldnotexceed0.16c=9

0

Cmin

Protrusion=Lα/4


13

sameheight and thickness. This is less than the factorof around57 shown in the theory. Further

experimentswillbeperformed.

The brick count of a serpentinewall is low – often only a few percentmore than a single-wythe

planewall.Indeedthemainattractionofthewavywallshapeisthatitallowsalargeincreaseinits

effectivedepth (m) tobemadewithout significantly increasing itsuseofmaterials. So serpentine

wallsareusuallybuiltdeeperthancrenelatedwalls;e.g.forthecasem=7,shownhighlitinTable3

above,thedistancefromfrontfacetobackfacewouldbe70cm(ifbricksare10cmwide).If,fora

boundarywall,thewholeserpentinemustlieinsidetheplotboundary,thenthereissignificantextraland-take,about50%ofwhichiseffectivelydonatedtotheneighbouringplot.Ifhoweverthewall’s

axis lies along the plot boundary, then both this and the neighbouring plot experience alternateintrusionandretreat(bays).

Exampleofasimplehouseplanwith

serpentineexteriorwallsand

straightinternalwalls.

10 Numericalcomparisonsofthe4wallplans

Wecannowcompare(Table5)the4shapevariants–plane,buttressed,crenelatedandserpentine.

Table5 Comparisonofwallshapes(m=D/W=3shownshaded).

Wallshape Plain Buttressed

every5bricks

(n=5)

Crenulated

every5bricks

(n=5)

Wavy

wavelength

(n=18)

Wall-depthm 1 2 3 2 3 5 2 3 5 3 5 7

Costratious=ts/t0 1 2 3

1.2 1.4

1.8 1.2 1.42

1.8 1.01

1.03 1.07

Stiffnessratio(I/I0) 1 8

27 3.2 10.3+

47.1 4.8+

15.8+

64.0 7.5+

26.2 57.3

Strengthratio(Z/Z0) 1 4* 9

1.2 2.5*

7.0* 2.4* 5.3* 12.8 2.5*

5.3* 8.2

(Is/I0)/us 1 4 9

2.7 7.4

26.2 4.0 11.3 35.6 7.4

25.4 53.6

(Zs/Z0)/us 1 2 3

1.0 1.8

3.9 2.0 3.8

7.1 2.5

5.1 7.7

Notes:Stiffness-ratio/cost-ratio=(Zs/Z0)/usisalsotheseismicaccelerationtoleranceratio

Wavywallwavelengthislong(n=18),elsethebrick-to-brickanglewouldbetoohighForplainwallsm=2iscommonlyused–asshownbyboldproperties

Tosimplifythiscomparisonletusfirst lookatthecasem=3, i.e.wherethewalldepthD,fronttoback,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).


14

If however we had taken amore 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

tooverturning forces equal to thatof a double-thickness (m = 2) planewall, forwhichZ/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, asmarked

+inTable5):

abuttressedwall,withm=3,n=5,costing30%lessthantheplanewall;

acrenelatedwall,withm=2.5,n=5,costing35%less;

awavywall,m=3,n=18,costing50%less.

Ingeneral therefore,buttressing isnotaveryefficientuseofbrick,whereasbothcrenulationand

employingwavywallingoffersubstantialsavings.Themainpricetopayforthissavingisthepossible

inconvenienceofhavingsteppedorcurvingsurfaces inaroomor losinga littleplotareanexttoa

boundarywall.

10 Conclusions

Crenelatedandwavywallsuseupto50%lessbricksthanplanewallstoachievethesameresistance

tobuckling,samestiffnessandsameresistancetolateralforces.Buttressedwallsareintermediate

intermsofbrickeconomy.Indeedthesedesignscanbeusedwithasinglebrickthickness(e.g.100

mm) to get satisfactory performance in both housing and boundarywalls; by contrastwith plane

wallingadoublethickness(e.g.200mm)isnormallyrequired..

Bothcrenelatedandwavydesignsare suitable for increasing strengthandstiffnessbymaking the

walldeeperyetwithoutusingmanymorebricks.Theshapeofthesematerially-efficientwallplans

mayconflictwiththerectangularnormofmostmodernbuildings.Conversely theyofferscopefor

architecturaldistinctiveness.

Thesimplifiedanalysisusedinthispaperassumesthatduringloading,thewallsholdtogetherand

actasawhole.Rackingduetoslippagebetweencourseshasthereforenotbeenconsidered.These

assumptionsmayproveunfoundedinthecaseofverydeepcrenelatedorserpentinewalls–e.g.m

valuesexceedingsay5.

11 Bibliography

HendryAW,SinhaBP,DaviesSR,1997,DesignofMasonryStructures,3rdEd,E&FNSpon

BSEN1996-3:Simplifiedcalculationmethodsforunreinforcedmasonrystructures


15

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


16

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)) where λ = R/t 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 e = (1 - 3(sin(θ)*cos(θ)/θ) + 2cos^2(θ)

As a close

approximation μ = μ' = 1+1.59*(m-1)^2

Relative stiffness per brick used is μ / σ = 1+6λ^2*e

I / Io relative relative relative

wall radius half stiffness brick stiffness Wall relative Wave depth/ thickness R/t arc angle per meter usage per brick depth strength length wavelth

t R λ θ θ e μ σ μ / σ D m μ / m W D / W μ'/μ

m m rads degree m d / t

0.1 0.6 6 1.05 60 0.2620 57.6 1.21 47.6 0.70 7.0 8.19 2.08 0.34 0.979

0.1 1 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 0.1 1 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

0.1 1 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

0.1 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

0.1 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 1.002 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 1/6 circs 1.000 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 0.998


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