Second Order Analysis

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1 Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to be perfect, and there were no lateral deflecton untl the load reached P cr. At P cr, the orgnal confguraton of the frame becomes unstable, and wth a slght perturbaton the deflectons start ncreasng wthout bound. However, f the system s not perfect, or t s subjected to lateral loads along wth gravty loads, then the deflectons wll start ncreasng as soon as the loads are appled. However, for an elastc frame, the maxmum load capacty wll stll be lmted to that correspondng to P cr Second Order Analyss To trace ths curve, a complete load-deflecton analyss of the frame s necessary. A second order analyss wll generate ths load-deflecton curve In a second order analyss procedure, secondary effects as the P- and P- effects, can be ncorporated drectly. As a result, the need for the B and B factors s elmnated. In a second-order analyss, the equlbrum equatons are formulated wth respect to the deformed geometry, whch s not known n advance and constantly changng, we need to use an teratve technque to obtan solutons. In numercal mplementaton, the ncremental load approach s popular. The load s dvded nto small ncrements and appled to the structure sequentally.

2 Repeat untl convergence, where Q j approx. 0 Second Order Analyss Dscretze frame nto beam-column elements For each beam-column element, formulate stffness matrx k = k o + k g Assemble structure stffness matrx K usng k and d array Begn Iteraton for load step R = K D D = [K ] - R Where n the load step, R = ncremental load vector K = structure secant stffness matrx D = ncremental dsplacement vector D = D + D Form the structure nternal force vector R usng all element r compute element end forces r usng T F and P, M a, M b compute P, a, b from e, a, b usng element local k e Compute e, a, b from d usng T d Extract d from D Second Order Analyss Form the structure nternal force vector R usng all element r Form the structure external force vector R + = R + R Form the structure nternal force vector R usng all element r compute element end forces r usng T F and P, M a, M b Evaluate unbalanced force vector Q = R + - R usng the current value of axal force P Update element k Assemble element k to form updated structure stffness matrx K compute P, a, b from e, a, b usng element local k e Compute e, a, b from d usng T d Extract d from D D =[K ] - Q D =D + ( D k)

3 Second Order Analyss After convergence, D + = D n = D + D k Assumed another load ncrement, and go back to begnng AISC (00) Specfcatons Chapter C Desgn for Stablty 3

4 C. General Stablty Requrements. General Requrements Stablty shall be provded for the structure as a whole and for each of ts elements. The effects of all of the followng on the stablty of the structure and ts elements shall be consdered: () Flexural, shear, and axal member deformatons, and all other deformatons that contrbute to the dsplacements of the structure. () second order effects (both P- and P-δ effects). (3) geometrc mperfectons (4) stffness reductons due to nelastcty (5) uncertanty n stffness and strength All load-dependent effects shall be calculated at a level of loadng correspondng to the LRFD load combnatons C. General Stablty Requrements Any ratonal methods of desgn for stablty that consder all of the lsted effects s permtted; ths ncludes the methods dentfed n Sectons C. and C. C. Drect Analyss Method of Desgn Consst of calculaton of the requred strengths n accordance wth Secton C, and calculaton of avalable strengths n accordance wth C3 s permtted for all structures C. Alternatve Methods of Desgn The effectve length method and the frst-order analyss n Appendx 7 are permtted as alternatves to the drect analyss method for structures that satsfy the constrants specfed n that appendx. 4

5 C. Calculaton of Requred Strengths For the drect analyss method, the requred strengths of the components of the structure shall be determned from Analyss conformng to Secton C.. Intal mperfectons accordng to Secton C. and stffness accordng to Secton C.3. C. General Analyss Requrements () Account for member deformatons (flexural, axal, shear) and all other component and connecton deformatons that contrbute. Account for stffness and stffness reductons (accordng to C.3) that contrbute () Second-order analyss that consders both the P-Δ and P-δeffects. Use of approxmate second-order analyss provded n App. 8 s permtted as an alternatve to a rgorous second-order analyss (3) The analyss shall consder all gravty and other loads (4) For desgn by LRFD, the second order analyss shall be carred out under LRFD load combnatons 5

6 C. General Analyss Requrements () cont Second-order analyss requrements A P-Δ only second-order analyss (one that neglects the effects of P-δresponse on the structure) s permtted when: (a) structure supports gravty load usng vertcal members (b) rato of maxmum second-order drft to maxmum frstorder drft s.7 (c) no more than /3 of the gravty load s supported by columns part of the MRFs n the drecton of translaton. However, t s necessary n all cases to consder the P-δ effects n the evaluaton of ndvdual members subject to flexure and compresson. Ths can be satsfed by usng the B factor n App. 8. C Consderaton of Intal Imperfectons The effects of ntal mperfectons on the stablty of the structure shall be taken nto account by : (a) drect modelng of mperfectons n the analyss as specfed n C.a (b) applcaton of notonal loads as specfed n C.b These are mperfectons n the locaton of ponts of ntersecton of members. Out-of-plumbness of columns. Intal out-of-straghtness of ndvdual members s not addressed n ths secton. It s covered n the desgn of columns (compresson members). 6

7 C.a Drect Modelng of Imperfectons In all cases, t s permssble to account for the effect of ntal mperfectons by ncludng them drectly n the analyss. The structure shall be analyzed wth ponts of ntersecton of members dsplaced from ther nomnal locaton. The magntude of the ntal dsplacement shall be the maxmum amount consdered n the desgn. The pattern of ntal dsplacements shall be such that t provokes the destablzng effect User Note: Intal dsplacements smlar n confguraton to both dsplacements due to loadng and antcpated bucklng modes should be consdered n the modelng of mperfectons. The magntude of the ntal dsplacements should be based on permssble constructon tolerances (AISC CoSP), or on actual mperfectons f known C.a Drect Modelng of Imperfectons If the structure support gravty loads usng vertcal columns, and the rato of maxmum second-order drft to frst-order drft s.7, then t s permssble to nclude mperfectons only n analyss of gravty-only load combnatons and not n analyss for combnatons that nclude appled lateral loads. 7

8 C.b Use of Notonal Loads For structures that support gravty loads usng vertcal columns, t s permssble to use notonal loads to represent the effects of ntal mperfectons n accordance wth ths secton. Nomnal geometry used. () Notonal loads shall be appled as lateral loads at all levels. These loads shall be addtve to the other loads appled n all load combnatons. The magntude of the notonal loads shall be: N = 0.00 a Y N = notonal load appled at level (kps) Y = gravty load appled at level from load combos () The notonal load N shall be dstrbuted over that level n the same manner as the gravty load at that level, and appled n the drecton provdng greatest destablzaton C.b Use of Notonal Loads (3) The notonal load coeffcent of 0.00 s based on the nomnal ntal out-of-story plumbness rato of /500. If other value s justfed, t s permtted to adjust the notonal load coeffcent proportonally. (4) If the rato of the maxmum second-order drft to maxmum frst-order drft s less than or equal to.7 n all stores, t s permtted to apply the notonal lateral load n gravty only load combnatons and not n combnatons usng other lateral loads 8

9 C.3 Adjustments to Stffness The analyss of structures to determne requred strengths shall used reduced stffness as follows: A factor of 0.80 shall be appled to all stffness that contrbuted to the structure stablty. Ths factor can be appled to all stffness n the structure. An addtonal factor, τ b shall be appled to the flexural stffness of all members whose flexural stffness are contrbutng to the stablty When ap r / P y 0.5, t b =.0 When ap r / P y > 0.5, t b = 4(aP / P ) r Y -ap r / P Y (3) Instead of usng τ b <.0, t s permssble to use τ b =.0 and an addtonal notonal load of 0.00αY, whch are to be added n the drecton that causes more destablzaton and also for gravty load combos. 9

Finite Element Modelling of truss/cable structures

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