MEEM 4405 Formula SheetMEEM 4405 Formula Sheet Base points ξinWeights wi0.0 One point formula w1=2.0 Two point formula w1=w2=1.0Uo12---σxxεxxσyyεyyσzzεzzτxyγxyτyzγyzτzxγzx+++++[]12---σ˜[]Tε˜[]==iθ() jθ()sinsin θd0π∫0 ij≠π 2⁄ ij=⎩⎨⎧= iθ()cos jθ()cos θd0π∫0 ij≠π 2⁄ ij=⎩⎨⎧=Ua12---EAxddu⎝⎠⎛⎞2= Ut12---GJxddφ⎝⎠⎛⎞2= Ub12---EIzzx22ddv⎝⎠⎜⎟⎛⎞2= Ua12---N2EA-------= Ut12---T2GJ-------= Ub12---Mz2EIzz----------=KjkEAxddfj⎝⎠⎛⎞xddfk⎝⎠⎛⎞xd0L∫= Rjpxx()fjx() xd0L∫Fqfjxq()q 1=m∑+= UAWA2--------ΩA()–12---CjRjj 1=n∑== =KjkGJxddfj⎝⎠⎛⎞xddfk⎝⎠⎛⎞xd0L∫= Rjtx()fjx() xd0L∫Tqfjxq()q 1=m∑+=KjkEIzz()x22ddfj⎝⎠⎜⎟⎛⎞x22ddfk⎝⎠⎜⎟⎛⎞xd0L∫= Rjpyx()fjx() xd0L∫Fqfjxq()q 1=m1∑Mqxddfjxq()q 1=m1∑++=Ke()[]EAL-------11–1–1= Re(){}poL2---------11⎩⎭⎨⎬⎧⎫F1e()F2e()⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫+=Ke()[]EA3L-------78–18–168–18–7= R{}poL6---------141⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫F1e()0F3e()⎩⎭⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎧⎫+=KGe()[]T[]TKe()[]T[]= RG1e(){} T[]TRe(){}= δΩe()δuGe(){}TKGe()[]uGe(){}RG1e(){}–()= vx() f1x()v1e()f2x()θ1e()f3x()v2e()f4x()θ2e()+++=f1x() 13xx1–L-------------⎝⎠⎛⎞2–2xx1–L-------------⎝⎠⎛⎞3+=f3x() 3xx1–L-------------⎝⎠⎛⎞22xx1–L-------------⎝⎠⎛⎞3–=f2x() Lxx1–L-------------⎝⎠⎛⎞2xx1–L-------------⎝⎠⎛⎞2–xx1–L-------------⎝⎠⎛⎞3+=f4x() Lxx1–L-------------⎝⎠⎛⎞–2xx1–L-------------⎝⎠⎛⎞3+=Ke()[]2EIL3---------63L 6–3L3L 2L23– LL26–3– L 63– L3LL23– L 2L2=Re(){}p0L12---------6L6L–⎩⎭⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎧⎫F1e()M1e()F2e()M2e()⎩⎭⎪⎪⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎪⎪⎧⎫+=Ke()[]ke()Le()---------11–1–1= Re(){}qxe()Le()2------------------11⎩⎭⎨⎬⎧⎫Q1e()Q2e()⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫+= Ke()[]ke()3Le()------------78–18–168–18–7= Re(){}qxe()Le()6------------------141⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫Q1e()0Q3e()⎩⎭⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎧⎫+=FT1FT2⎩⎭⎨⎬⎧⎫EAαΔT1–EAαΔT2⎩⎭⎨⎬⎧⎫= pxTEAαk-----------qx=Ke()[]EAL-------11–1–1= Re(){}poL2---------11⎩⎭⎨⎬⎧⎫pToL2------------11⎩⎭⎨⎬⎧⎫F1e()F2e()⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫FT1e()FT2e()⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫+++= Ke()[]EA3L-------78–18–168–18–7= R{}poL6---------141⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫pToL6------------141⎩⎭⎪⎪⎨⎬⎪⎪⎧⎫F1e()0F3e()⎩⎭⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎧⎫FT1e()0FT3e()⎩⎭⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎧⎫+++=13⁄()±0.0 Three point formula w1=(8/9)w2=w3=(5/9)Base points ξinWeights wi0.6±Structural AnalysisAxial (Rods) Torsion (Shafts) Symmetric Bending (Beams) Unsymmetric BendingDisplace-ments Strains StressesInternal Forces & Moments Homoge-nous Cross-sec-tion CompositeCross-sec-tion uxyz,,()ux()=φ xyz,,()φx()=uxyz,,()yxddv–=vvx()= w 0=uxyz,,()yxddv–= zxddw–vvx()= wwx()=εxxxddu=γxθρxddφ=εxxyx22ddv–=εxxyx22ddv– zx22ddw–=σxxEεxxExddu==τxθGγxθGρxddφ==σxxEεxxEyx22ddv–== τxy0 σxx«≠σxxEyx22ddv– Ezx22ddw–= τxy0 σxx«≠τxz0 σxx«≠N σxxAdA∫=T ρτxθAdA∫=N σxxAdA∫0==MzyσxxAdA∫–= VyτxyAdA∫=N σxxAdA∫0==MzyσxxAdA∫–= MyzσxxAdA∫–=VyτxyAdA∫= VzτxzAdA∫=σxxNA----=τxθTρJ-------=σxxMzyIzz----------⎝⎠⎛⎞–=q τxstVyQzIzz-------------⎝⎠⎛⎞–==σxxIyyMzIyzMy–IyyIzzIyz2–----------------------------------⎝⎠⎜⎟⎛⎞y–IzzMyIyzMz–IyyIzzIyz2–---------------------------------⎝⎠⎜⎟⎛⎞z–=q τxstIyyQzIyzQy–IyyIzzIyz2–------------------------------------⎝⎠⎜⎟⎜⎟⎛⎞– VyIzzQyIyzQz–IyyIzzIyz2–------------------------------------⎝⎠⎜⎟⎜⎟⎛⎞– Vz==xddu NEA-------=u2u1–N x2x1–()EA--------------------------=xddφ TGJ-------=φ2φ1–T x2x1–()GJ-------------------------=x22ddvMzEIzz----------=vMzEI-------xd∫dx C1xC2++∫=x22ddv 1E---IyyMzIyzMy–IyyIzzIyz2–--------------------------------------⎝⎠⎜⎟⎜⎟⎛⎞=x22ddw 1E---IzzMyIyzMz–IyyIzzIyz2–--------------------------------------⎝⎠⎜⎟⎜⎟⎛⎞=σxx()iNEiEjAjj 1=n∑--------------------= τxθ()iGiρTGjJjj 1=n∑-------------------------=σxx()iEiyMzEjIzz()jj1=n∑----------------------------–=q τxstQcompVyEjIzz()jj 1=n∑---------------------------------⎩⎭⎪⎪⎪⎪⎪⎪⎨⎬⎪⎪⎪⎪⎪⎪⎧⎫–==u2u1–N x2x1–()EjAj∑--------------------------= φ2φ1–T x2x1–()GjJj∑[]-------------------------=vMzEjIzz()j∑-------------------------xd∫dx C1xC2++∫=xddNpxx()–=xddTtx()–=xddVypyx()–=xddMzV–y=xddVypyx()–=xddMzV–y=xddVzpzx()–=xddMyV–z=Axial (Rods) Torsion (Shafts) Symmetric Bending (Beams) Unsymmetric
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