354066632x by Unknown

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A Computational Procedure for Instationary 375 V~ V, l Ax3 Ayl AY3 Fig.4. Constant straintriangle. From (3) the relationship betweenthe side strainsand N-directionalstrains can then be established {e} = [D] {E} = [D][F]-1 {e} = [Tn~]{e}. (18) The N x 3 matrix [Tn~] resolvesthe side strainsinto the N-directions.Note that the elements of this matrix are only functions ofthe element shape at time t = 0, the undeformed configuration.It must be determined only once at the beginning of each calculation. Ifa set of virtualnodal displacements, {Sa}T = {5U1,5Vl, 5U2, (~V2,5U3, 5V3} is applied tothe finiteelement, the virtualside strainsare Ax~ 5e~ = OL---~ax~ + 5Ay~ = xs(~Ui - ~U~) + y~(~ - ~). where xs and y~ are defined by setting Axs Ay 8 x~ = OL ] and y~- 0L 2 Thus, the virtualstrainscan be expressed in matrix form as -Yl 5U2 {~e}= ~e2 = x2-y2 0 0 -x2 ~V2 5e3 X3 Y3 x3 --Y3 0 (~U3 5Vz (19) (20) (21) = [*B] (22)

376 Perry Bartelt andMarc Christen The superscripttisplaced on the [/)]matrix to emphasize thatitisa function of the deformed element configuration.The virtualN-directionalstrainsare subsequently relatedto the virtualfiniteelement displacements according to {&} = [Tn~] {~e} = [T~] [tB] {~a} = [tB] {Sa}. (23) Rewriting the internal virtualwork by using the N-directionalstressand strainmeasures and then substituting theabove epressionrelatingthe virtual element displacments to the virtual strainsallows the definitionof the finite element internalnodal forces {tpi}at time t: W~= /A{S}T {SE}dA= /A{a}T {5~}dA = {Sa}T f [tB]T{ta}dA = {Sa}T{tpi}. (24) JA Moreover, {%} = ['B] 'A. (25) The tangent-element-stiffnessmatrix [/(T]isfound by differentiatingthe internalforces (see Crisfield[20]),i.e. d{tpi}=(d[tB]T{ta}+[tB]Tdlta})tA=[KTld{a}. (26) In the numerical calculationswe neglectthe geometric stiffnesscontribution, that is, d[tB] T{ta} ~0. (27) The increment in stressisfound from the N-directionalstressstrainrelationship(7), [da e &3 7e'~ Z ~ dTe (2S)d{(y}=d(�~ee-t-~"/e)=~eee�Ct+-~e]dee+--dcn n=l Assuming that the lastterm of the above equation issmall, N d7 ~ e /~E d-~ de'~ ~ 0, (29) the diagonal elements Cn of an NxN materialmatrix [CT] can be defined by f dc~ e d~ e'~

A Computational Procedure forInstationary 377 The N-directionalincremental stress-strainrelationshipisthen d{ta} = [tCT]dee= [tCT] [tB]d{a}. (31) This equation can be substitutedinto (26) to find d{tpi} = [[tB]T [tCT] [tB]tA]d{a}. (32) The element stiffnessmatrix isthen by definition = ['B] (33) The approximations (27) and (29) are not significant forthe accuracy of the numerical calculationssince the stiffnessmatrix is used only to approximate the internalstressstate.As willbe shown in the next section,the equilibrium between the internaland externalforces is always strictlyenforced. However, the approximations could influencethe number of computational iterationsrequiredto find equilibrium.In the numerical calculationsa Poisson's ratiofor snow of u = 0.25 is assumed, thus, /~ = 0 and (29) is exactly zero. The increment in load produced by a creep deformation is d{tpc} = [tB]T [tCT] d{e'} tA. (34) These forcesare applied to thefiniteelement mesh in order toproduce creep deformations corresponding to thetime dependent viscous strainrates.In the calculationprocedure we use an explicitscheme to find d {ev}: d{~ v} = {~} At, (35) where At isthe time step increment of the time integrationscheme. Finally,a self-weightload, {tpe},always acts on the system. This

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