Ultimat strngth analysis & dsign of rsidntial slabs on ractiv soil This documnt prsnts an ovrviw of thory undrlying ultimat strngth analysis and dsign of stiffnd raft and waffl raft slabs, as commonly usd in rsidntial construction. 1. Notation & units B w h L L s M u m u P s p 0 p * W W s w f y s α α s γ Δ = Stiffning rib width (m) = Width of soil-slab contact ara (m) = Soil stiffnss (Pa/m) = Width of quivalnt rctangular floor plan (m) = Isolatd slab panl width (m) = Ovrall stiffnd sction ultimat bnding momnt pr mtr (Nm/m) = Isolatd slab panl ultimat bnding momnt pr mtr (Nm/m) = Total foundation load (N) = Initial stat soil prssur = Pa ultimat soil prssur (Pa) = Edg load pr unit primtr lngth (N/m) = Total wight of suprstructur (N) = Uniformly distributd floor load (Pa) (Dad plus liv) = Surfac movmnt (mm) = Aspct ratio of quivalnt rctangular floor plan ( 1.0) = Aspct ratio of isolatd slab panl ( 1.0) = Ratio h L = Dflction (mm) 2. Equivalnt rctangular floor plan Most rsidntial floor plans consist of ovrlapping rctangls; thrfor ultimat strngth analysis of an quivalnt rctangular plat modl is a good starting point. Figur 1 shows a typical rsidntial floor plan and quivalnt rctangl. Th quivalnt rctangl is calculatd as follows: Aspct ratio: α = 19.39 192.0 = 1.325 Width: L = = 12.04m Lngth: αl = 15.94m 14.63 α Figur 1 - Typical rsidntial floor plan and quivalnt rctangl Copyright 2013 Fran Van dr Woud All rights rsrvd Pag 1 of 5
3. Thory 3.1 Limit stat mthodology Limit stat mthodology is built on idntifying th govrning failur mchanism and applying appropriat factors of safty and dsign critria to nsur adquat in-srvic prformanc. Th two major limit stats in structural prformanc ar srvicability and strngth, and in th prsnt contxt ths two limit stats ar inxtricably lind through soil-structur intractiv prformanc. Th thr slf-vidnt conditions for failur of a slab ar: Yild condition Equilibrium condition Mchanism condition Th yild lin mthod satisfis ths conditions prfctly. 3.2 Global hogging failur 3.2.1 Yild lin pattrn Figur 2 shows th fully dvlopd yild lin pattrn for global hogging mod failur of a rctangular floor slab. Th shadd ara rprsnts th soil-slab contact ara. Assuming surfac movmnt outsid th rctangular soil-slab domain is constant all around th primtr, th soil-slab contact ara is also rctangular, with dgs quidistant from th slab dgs. Hnc th cornr yild lins roughly pass through th cornrs of th soil-slab contact rctangl. Figur 2 - Yild lin pattrn & soil-slab contact ara Th yild lin pattrn for global hogging failur dvlops as follows: Th first yild lin mor than lily starts along th cntrlin paralll to th long sid of th quivalnt rctangl, whr th bnding momnt is gratst. As surfac movmnt incrass, th cntral yild lin lngthns and thn splits into two yild lins hading at about 45 dgrs angls to th cornrs. Whn th yild lins ar fully dvlopd, distortion of th slab is li th motion of a thrdimnsional mchanism consisting of four plan sgmnts hingd togthr along th yild lins and rotating rlativ to ach othr about th yild lins. 3.2.2 Ultimat prssur distribution on soil-slab intrfac In common with most intractiv soil-structur analyss it is assumd that soil is a linarly lastic matrial. It follows from prvious paragraphs that th ultimat prssur on th soil-slab intrfac varis linarly from zro at point B on th dg of th soil-slab contact rctangl to a pa at point A at th cntr of th slab, as shown in Figur 3. Copyright 2013 Fran Van dr Woud All rights rsrvd Pag 2 of 5
A B C P * 3.2.3 Ultimat dflction analysis Figur 3 - Ultimat prssur distribution on soil-slab intrfac Figur 4 shows th half-sction through th fully dvlopd yild lin pattrn. A 0 B 0 C 0 Undisturbd soil surfac A 1 B 1 C 2 h/2 L/2 C 1 y s Subsidd soil surfac Figur 4 - Half-sction through fully dvlopd yild lin pattrn Points A 0, B 0, and C 0 li in a straight lin along th undisturbd soil surfac. Points A 1 and B 1 li on th displacd soilslab intrfac in th contact zon. Point C 1 is on th displacd soil surfac at th dg of th non-contact zon. Point C 2 lis on th xtnsion of th straight lin through points A 0 and B 1. Th soil surfac profil in th non-contact zon is indtrminat. Th soil-slab intrfac prssur at A is p* and thrfor, by dfinition of soil stiffnss, p* is th amount th intrfac at A at th cntr of th slab is dprssd into th soil. Th intrfac prssur at B is zro and thrfor p* is also th rlativ dflction of th intrfac btwn A 1 and B 1. Th rlativ dflction across th full width of th slab is dtrmind by scaling along th straight lin A 1 B 1 to th outr dg of th slab: Δ = p*. Surfac movmnt is γ dtrmind by scaling along th straight lin A 0 B 1 C 2, which includs a rasonabl allowanc for th indtrminat gap btwn th undrsid of th slab and soil surfac: y s = 2p* γ 3.2.4 Equilibrium analysis Pa prssur is drivd by quating th total foundation load on th slab to th upward soil prssur rsultant on th soil-slab contact rctangl. Ultimat bnding momnt is drivd from convntional yild lin analysis. Ths drivations involv intgration of products of aras of rctangls and triangls. Th nd rsults ar: p * = 6P s γ(3α + 2γ 3)L 2 M + = 6W + [(3α +1)w γ 2 (α + γ 1)p * ]L 2 s f u 3.3 Global sagging failur Th yild lin pattrn for global sagging failur is idntical to that for hogging failur, xcpt as follows: Th yild lins ar ngativ bnding momnt, th soil-slab contact ara is th unshadd ara in Figur 2, and th pa soil-slab intrfac prssur is at th slab dgs. Th nd rsults ar: p * = 6P s γ(3α 2γ + 3)L 2 M = 6W {[3α +1+ 3γ(α +1)]w γ 2 (3+ α γ )p * }L 2 s f u Copyright 2013 Fran Van dr Woud All rights rsrvd Pag 3 of 5
3.4 Local slab panl failur Th failur mchanism for isolatd intrnal slab panls is similar to global hogging failur. Th critical slab panl is at th cntr of th floor plan, whr th soil-slab intrfac prssur is gratst. Yild lin analysis includs ngativ bnding momnt yild lins on th panl dgs in addition to positiv bnding momnt yild lins in th intrior. Th rsult is: m u + + m u = (3α s 1)p* L s 2 24(α s +1) 3.5 Local cornr and dg failurs Figur 5 shows th yild pattrns for local cornr and dg failur mchanisms. Ths typs of failurs may occur for xampl on a cut/fill sit whr th fill has not bn adquatly compactd around a cornr or along an dg, or on a sit with gilgais. Th following rsults ar obtaind from straightforward quilibrium analysis: Figur 5 - Yild lins for local cornr and dg failurs Local cornr failur: M u = W 2 + w f 2 12 Local dg failur: M u = W (1+ L ) + w f 2 2 Th problm with ths failur mchanisms is that thr is no way of rlating thm to surfac movmnt. Dsignrs wishing to xplor cornr and dg failurs might considr taing distanc in Figur 5 as th dg distanc dfind in AS-2870. 3.6 Govrning failur mchanism Li th strngth of a chain is that of its wast lin th strngth of a slab is that of its wast failur mchanism According to th uppr/lowr bound thorm in structural mchanics th govrning failur mchanism for slabs on ractiv soil is th on xhibiting th lowst ultimat surfac movmnt capacity. Convrsly, th govrning failur mchanism is th on rquiring th highst ultimat strngth to sustain th most svr combination of dsign loading and surfac movmnt. It is impossibl to prdict a-priori which failur mchanism govrns. Global hogging failur usually govrns on building sits undr normal soil moistur conditions at tim of construction. Global sagging failur may govrn on building sits that ar abnormally dry at tim of construction. Local cornr and dg failurs may govrn on building sits xhibiting non-uniform soil charactristics, such as for xampl inadquatly compactd soil on a cut-fill sit, or th prsnc of gilgais. Local slab panl failur may govrn whn stiffning ribs ar too widly spacd or whn th slab thicnss is too small, or whn it supports xcssiv local point loads. 4. Ultimat strngth prformanc Th ultimat strngth prformanc of a stiffnd slab on ractiv soil is convnintly dscribd as a function of surfac movmnt. A typical graph of ultimat bnding vrsus surfac movmnt at which th slab fails is shown in Figur 6. It Copyright 2013 Fran Van dr Woud All rights rsrvd Pag 4 of 5
is an S-curv shap, consisting of a short straight-lin part on ithr sid of th vrtical axis, followd by xponntial curvs to positiv and ngativ asymptots. Th asist way to plot this graph is to ma γ in th quations in Sction 3 th indpndnt variabl. Calculat th valus of all th othr variabls in th quations, starting with γ qual to 1.00 and dcrmnting it by small amounts. Continu until th calculatd y s valus covr th dsird rang btwn ngativ and positiv valus. This procdur producs th xponntial parts of th graph, and a straight lin drawn btwn th points γ qual to 1.00 on ithr sid of th vrtical axis complts th graph. Figur 6 - Ultimat strngth prformanc graph Th straight-lin part of this graph is in th rang of surfac movmnts: 12P s (3α +1)L 2 y s 12P s (3α 1)L 2 Th asymptotic limits ar: Subsidnc: M u + = 6W s + (3α +1)w f L2 Hav: M u = (3α 1)w f L2 5. Ultimat strngth dsign Th first phas of ultimat strngth dsign is to calculat th graph in Figur 6. Th nxt phas is to intrpolat th graph to dtrmin th dsign ultimat hogging and sagging bnding momnts to rsist failur at th dsign surfac movmnts applicabl to th building sit. Th final phas is to calculat th following slab dsign dtails: Ovrall stiffnd sction dpth, slab thicnss, minimum numbr of stiffning ribs in both dirctions, stiffning rib width, amount of top and bottom stl rinforcmnt, and othr dtails in accordanc with th gnral construction rquirmnts in AS 2870. Copyright 2013 Fran Van dr Woud All rights rsrvd Pag 5 of 5