The Finite Element Method for the Analysis of Non-Linear and ...

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Special Considerations-Geometric Stiffness Beam Element In the case of a beam element with bending properties in which the deformed shape is assumed to be a cubic function due to the rotations φ i and φ j at the ends, additional moments M i and M j are developed. The force-displacement relationship is given by the following equation: ⎡ ⎢ ⎣ F i M i F j M j ⎤ ⎥ ⎦ = T 30L ⎡ ⎢ ⎣ ⎤ 36 3L −36 3L 3L 4L 2 −3L −L 2 ⎥ −36 −3L 36 −3L ⎦ 3L −L 2 −3L 4L 2 ⎡ ⎢ ⎣ v i φ i v j φ j ⎤ ⎥ ⎦ or FG = KGv The well-known elastic force deformation relationship, for a prismatic beam without shearing deformations, is ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ F i 12 6L −12 6L v i ⎢ M i ⎥ ⎣ F j ⎦ = EI ⎢ 6L 4L 2 −6L −2L 2 ⎥ ⎢ φ i ⎥ L 3 ⎣ −12 −6L 12 −6L ⎦ ⎣ v j ⎦ or FE = KEv M j −6L −2L 2 −6L 4L 2 φ j Therefore, the total forces acting on the beam element will be: F T = F E + F G = [K E + K G]v = K T v Institute of Structural Engineering Method of Finite Elements II 52
Special Considerations - Geometric Stiffness Conclusion - Geometric Stiffness In the case of constant (dead) loads where T is usually constant the calculation of the large displacement effect for cable and beam elements is performed by the addition in the FEM code of an appropriate extra “geometric” stiffness term. Institute of Structural Engineering Method of Finite Elements II 53
Page 1 and 2: The Finite Element Method for the A
Page 3 and 4: 2D Axisymmetric, Plane Strain and P
Page 13 and 14: The Beam Element *The section on th
Page 15 and 16: Beam Elements Euler Bernoulli Beam
Page 17 and 18: Beam Elements - Strong Form Equilib
Page 19 and 20: Beam Elements - Strong Form to Weak
Page 21 and 22: Beam Elements - FE Formulation Phys
Page 23 and 24: Beam Elements - Shape Functions The
Page 25 and 26: Beam Elements - FE Matrices From Fr
Page 27 and 28: Beam Elements - Example Consider a
Page 29 and 30: Beam Elements - Example Boundary fo
Page 31 and 32: Beam Elements - Example The global
Page 33 and 34: Timoshenko Beam Chapter 9: THE TL T
Page 35 and 36: Timoshenko Beam Exact integration S
Page 37 and 38: The Beam Element in Large Displacem
Page 49 and 50: Special Considerations-Geometric St
Page 51: Special Considerations-Geometric St

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