Euler-Bernoulli element with shear/mass center offsets

From standard beam formulas and using a stiffness transformation rule, the stiffness and mass matrix of the element can be computed. This is done in the Octave script beam-matrices-calc.m in \testdata\TestElement. The resulting stiffness matrix can be found in stiffnessmatrix2.txt. 

The verification is done in the unit test TestFrameElt::TestElementStiffnessWithShearCenterOffset()

Element properties

E = 3
A = 2
G = 4
J = 1
L = 1
I_x = 3
I_y = 4
shearOffsetX = s_x = 2
shearOffsetY = s_y = 3

1 Stiffness matrix

Using symbolic multiplication, the resulting stiffness matrix becomes
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Table
StiffnessMatrix
 6 0 0 0 0 0 -6 0 0 0 0 0
 0 108 0 -324 0 54 0 -108 0 324 0 54
 0 0 144 288 -72 0 0 0 -144 -288 -72 0
 0 -324 288 1552 -144 -162 0 324 -288 -1552 -144 -162
 0 0 -72 -144 48 0 0 0 72 144 24 0
 0 54 0 -162 0 36 0 -54 0 162 0 18
 -6 0 0 0 0 0 6 0 0 0 0 0
 0 -108 0 324 0 -54 0 108 0 -324 0 -54
 0 0 -144 -288 72 0 0 0 144 288 72 0
 0 324 -288 -1552 144 162 0 -324 288 1552 144 162
 0 0 -72 -144 24 0 0 0 72 144 48 0
 0 54 0 -162 0 18 0 -54 0 162 0 36



Pass criterion: The full stiffness matrix computed in Ashes shall be equal to the values above.

2 Mass matrix

Element properties
Mass = M = 42
massOffsetX = s_x = 2
massOffsetY = s_y = 3

In the same fashion, for a lumped mass element: Analytical mass matrix is found in massmatrix2.txt.

The verification is done in TestFrameElt::TestEulerElementMassWithMassOffsetLumpedMass()


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Table
MassMatrix
 21 0 0 0 0 0 0 0 0 0 0 0
 0 21 0 -63 0 0 0 0 0 0 0 0
 0 0 21 42 0 0 0 0 0 0 0 0
 0 -63 42 273 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 21 0 0 0 0 0
 0 0 0 0 0 0 0 21 0 -63 0 0
 0 0 0 0 0 0 0 0 21 42 0 0
 0 0 0 0 0 0 0 -63 42 273 0 0
 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0



Pass criterion: The full mass matrix computed in Ashes shall be equal to the values above.