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> 

with(LinearAlgebra); restart
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v1 := x^3*a[3]+x^2*a[2]+x*a[1]+a[0]:
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r := diff(v1, x):
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v[i] := subs(x = 0, v1):
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theta[i] := subs(x = 0, r):
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v[j] := subs(x = L, v1):
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theta[j] := subs(x = L, r):
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MD11 := subs(a[0] = 1, a[1] = 0, a[2] = 0, a[3] = 0, v[i]):
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MS11 := subs(a[0] = 0, a[1] = 0, a[2] = 0, a[3] = 0, theta[i]):
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MT11 := subs(a[0] = 1, a[1] = 0, a[2] = 0, a[3] = 0, v[j]):
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MR11 := subs(a[0] = 1, a[1] = 0, a[2] = 0, a[3] = 0, theta[j]):
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TM := Matrix(4, 4, [[MD11, MD12, MD13, MD14], [MS11, MS12, MS13, MS14], [MT11, MT12, MT13, MT14], [MR11, MR12, MR13, MR14]]):
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with(MTM):
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TM1 := inv(TM):
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b := Typesetting:-delayDotProduct(TM1, c):
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c := `<,>`(v[1], theta[1], v[2], theta[2]):
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a[0] := b(1):
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a[1] := b(2):
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a[2] := b(3):
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a[3] := b(4):
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vshape := x^3*a[3]+x^2*a[2]+x*a[1]+a[0]:
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XX1 := collect(vshape, v[1]):
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XX2 := collect(XX1, theta[1]):
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XX3 := collect(XX2, v[2]):
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XX4 := collect(XX3, theta[2]):
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v[1] := 1:
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NN1 := XX4:
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v[1] := 0:
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NN2 := XX4:
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v[1] := 0:
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NN3 := XX4:
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v[1] := 0:
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NN4 := XX4:
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Mat := `<,>`([NN1, NN2, NN3, NN4]):
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Mat1 := diff(Mat, x):
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Mat2 := diff(Mat1, x):
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segma := Mat2:
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KK := EI*expand(Typesetting:-delayDotProduct(segma, transpose(segma))):
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KG := int(KK, x = 0 .. L)

KG := Matrix(4, 4, {(1, 1) = 12*EI/L^3, (1, 2) = 6*EI/L^2, (1, 3) = -12*EI/L^3, (1, 4) = 6*EI/L^2, (2, 1) = 6*EI/L^2, (2, 2) = 4*EI/L, (2, 3) = -6*EI/L^2, (2, 4) = 2*EI/L, (3, 1) = -12*EI/L^3, (3, 2) = -6*EI/L^2, (3, 3) = 12*EI/L^3, (3, 4) = -6*EI/L^2, (4, 1) = 6*EI/L^2, (4, 2) = 2*EI/L, (4, 3) = -6*EI/L^2, (4, 4) = 4*EI/L})

 (1)
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``
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NULL


 

Download stiffnesmatrice.mw

 Stiffness matrix implementation of a finite element beam based on Euler-Bernoulli theory stiffnesmatrice.mw
beam mechanics finite-element

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