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Mat Chapter 4
  INTRODUCTION TO LIMIT STATES INTRODUCTION TO LIMIT STATES 1.0INTRODUCTION A Civil Engineering Designer has to ensure that the structures and facilities he designsare (i fit for their !ur!ose (ii safe and (iii econo ical and dura#le$ Thus safet% is oneof the !ara ount res!onsi#ilities of the designer$ &o'ever it is difficult to assess at thedesign stage ho' safe a !ro!osed design 'ill actuall% #e ) consistent 'ith econo %$There is in fact a great deal of uncertaint% a#out the an% factors 'hich influence #othsafet% and econo %$ *irstl% there is a natural varia#ilit% in the aterial strengths andsecondl% it is i !ossi#le to !redict the loading 'hich a structure (e$g$ a #uilding a% #esu#+ected to on a future occasion$ Thus uncertainties affecting the safet% of a structureare due to ã uncertaint% a#out loading ã uncertaint% a#out aterial strength and ã uncertaint% a#out structural di ensions and #ehaviour$These uncertainties together a,e it i !ossi#le for a designer to guarantee that astructure 'ill #e a#solutel% safe$ All that the designer could ensure is that the ris, of failure is e-tre el% s all des!ite the uncertainties$An illustration of the statistical eaning of safet% is given in *ig$ .$ Let us consider astructural co !onent (sa% a #ea  designed to carr% a given no inal load$ /ending o ents (/$M$ !roduced #% characteristic loads are first co !uted$ These are to #eco !ared 'ith the characteristic resistance or strength (R$M$ of the #ea $ /ut thecharacteristic resistance (R$M$ itself is not a fi-ed 0uantit% due to variations in aterialstrengths that ight occur #et'een no inall% sa e ele ents$ The actual resistance of these ele ents can #e e-!ected to var% as a conse0uence$ The statistical distri#ution of these e #er strengths (or resistances 'ill #e as s,etched in ( a $ Si ilarl% the variation in the a-i u loads and therefore load effects (such as #ending o ent 'hich different structural ele ents (all no inall% the sa e ight encounter intheir service life 'ould have a distri#ution sho'n in ( b $ The uncertainty here is bothdue to variability of the loads applied to the structure, and also due to the variability of the load distribution through the structure.  Thus if a !articularl% 'ea, structuralco !onent is su#+ected to a heav% load 'hich e-ceeds the strength of the structuralco !onent clearl% failure could occur$Unfortunatel% it is not !ractica#le to define the !ro#a#ilit% distri#utions of loads andstrengths as it 'ill involve hundreds of tests on sa !les of co !onents$ Nor al designcalculations are ade using a single value for each load and for each aterial !ro!ert%and a,ing a!!ro!riate safet% factor into the design calculations$ The value used ister ed as 1 Characteristic Strength or Resistance 2 or 1 Characteristic Load  2$3 Co!%right reserved Version II4 - 1   4  INTRODUCTION TO LIMIT STATES Characteristic resistance of a material (such as Concrete or Steel) is defined as that value of resistance below which not more than a prescribed percentage of test resultsmay be epected to fall.  (*or e-a !le the characteristic %ield stress of steel is usuall%defined as that value of %ield stress #elo' 'hich not ore than 5%  of the test values a% #e e-!ected to fall$ In other 'ords this strength is e-!ected to #e e-ceeded #% 95%  of the cases$Si ilarl% the characteristic load is that value of the load, which has an accepted  probability of not being eceeded during the life span of the structure.  Characteristicload is therefore that load 'hich 'ill not #e e-ceeded 95%  of the ti e$ 2.0STANDARDISATION Most structural designs are #ased on e-!erience$ Standardisation of all designs isunli,el% 'ithin the foreseea#le future hence design rules #ased on e-!erience #eco euseful$ If a si ilar design has #een #uilt successfull% else'here there is no reasons 'h%a designer a% not consider it !rudent to follo' as!ects of design that have !rovedsuccessful and ado!t standardised design rules$ As the conse0uences of #ad design can #e catastro!hic the societ% e-!ects designers to e-!lain their design decisions$ It istherefore advantageous to use ethods of design that have !roved safe in the !ast$Standardised design ethods can hel! in co !aring alternative designs 'hile ini isingthe ris, of the chea!est design #eing less safe than the others$Most 4overn ents atte !t to ensure structural safet% through regulations and la's$Designers then atte !t to achieve a-i u econo % 'ithin the range of designs that Version II4 - 2 Curve (a) Risk of  failure  F  r  e  q  u  e  n  c  y Variation of resistance (R.M.) between nominally ientical materials !oa use in calculation tren#t$ (or resistance) use in calculationsCurve (b)Variation of maimum life time loa effects (&.M.) !ig. Statistical #eaning of Safety  !oa effects ivie by Resistance moment   INTRODUCTION TO LIMIT STATES the regulations allo'$ *re0uentl% the !rofessions are allo'ed to regulate the selves5 inthese a cases the Regulations or Codes of $ractices  are evolved #% consultation andconsensus 'ithin the !rofession$ 3.0 ALLOWABLE STRESS DESIGN ASD! 6ith the develo! ent of linear elastic theories in the '9 t$  centur% the stress7strain #ehaviour of ne' aterials li,e 'rought iron 8 ild steel could #e accuratel%re!resented$ These theories ena#led indeter inate structures to #e anal%sed and thedistri#ution of #ending and shear stresses to #e co !uted correctl%$ The first attain entof %ield stress of steel 'as generall% ta,en to #e the onset of failure$ The li itations dueto non7linearit% and #uc,ling 'ere neglected$The #asic for of calculations too, the for of verif%ing that the stresses caused #% thecharacteristic loads ust #e less than an 1 allowable stress 2 'hich 'as a fraction of the%ield stress$ Thus the allo'a#le stress a% #e defined in ter s of a 1  factor of safety 9'hich re!resented a argin for overload and other un,no'n factors 'hich could #etolerated #% the structure$ The allo'a#le stress is thus directl% related to %ield stress #%the follo'ing e-!ression:In general each e #er in a structure is chec,ed for a nu #er of different co #inationsof loading$ The value of factor of safet% in ost cases is ta,en to #e around .$;<$ Man%loads var% 'ith ti e and these should #e allo'ed for$ It is unnecessaril% severe toconsider the effects of all loads acting si ultaneousl% 'ith their full design value 'hile aintaining the sa e factor of safet% or safet% factor$ Using the sa e factor of safet% or safet% factor 'hen loads act in co #ination 'ould result in unecono ic designs$A t%!ical e-a !le of a set of load co #inations is given #elo' 'hich accounts for thefact that the dead load live load and 'ind load are all unli,el% to act on the structuresi ultaneousl% at their a-i u values:(Stress due to dead load = live load > allo'a#le stress(Stress due to dead load = 'ind load > allo'a#le stress(Stress due to dead load = live load = 'ind > .$?? ti es allo'a#le stress$In !ractice there are severe li itations to this a!!roach$ These are the conse0uences of  aterial non7linearit% non7linear #ehaviour of ele ents in the !ost7#uc,led state and thea#ilit% of the steel co !onents to tolerate high theoretical elastic stresses #% %ieldinglocall% and redistri#uting the loads$ Moreover the elastic theor% does not readil% allo'for redistri#ution of loads fro one e #er to another in a staticall% indeter inatestructures$ Version II4 - 3  safetyof   Factor  stress(iel  stress )llowable  =  INTRODUCTION TO LIMIT STATES 4.0 LIMIT STATE DESIGN An i !roved design !hiloso!h% to a,e allo'ances for the shortco ings in the  %llowable Stress &esign #  'as develo!ed in the late '9*+ @s and has #een e-tensivel%incor!orated in design standards and codes for ulated in all the develo!ed countries$Although there are an% variations #et'een !ractices ado!ted in different countries the #asic conce!t is #roadl% si ilar$ The !ro#a#ilit% of o!erating conditions not reachingfailure conditions for s the #asis of 1  Limit States &esign 2 ado!ted in all countries$1Li it States9 are the various conditions in 'hich a structure 'ould #e considered tohave failed to fulfil the !ur!ose for 'hich it 'as #uilt$ In general t'o li it states areconsidered at the design stage and these are listed in Ta#le .$ Table ' Limit States U$%i&'%e Li&i% S%'%eSer(i)e'*i$i%+ Li&i% S%'%e  Strength (%ield #uc,lingSta#ilit% against overturning and s'a%*racture due to fatigue/rittle *ractureDeflectioni#ration*atigue chec,s (including re!ara#le da age due to fatigueCorrosion ,-ltimate !imit tates are t$ose catastro/$ic states0 w$ic$ require a lar#er reliability inorer to reuce t$e /robability of its occurrence to a very low level. , erviceability !imit tate1 refers to t$e limits on acce/table /erformance of t$e structure.  Not all these li its can #e covered #% structural calculations$ *or e-a !le corrosion iscovered #% s!ecif%ing for s of !rotection (li,e !ainting and #rittle fracture is covered #% aterial s!ecifications 'hich ensure that steel is sufficientl% ductile$ ,.0 ARTIAL SAET/ ACTOR  The a+or innovation in the ne' codes is the introduction of the !artial safet% factor for at$ A t%!ical for at is descri#ed #elo':In general calculations ta,e the for of verif%ing that    2   ≤    R B Version II4 - 4
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