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beams-bars

# beams-bars - Chapter 2 Bars and Beams Static analysis...

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1 Chapter 2: Bars and Beams Static analysis : forces are constant in time or change very slowly. Linear analysis: deflections are small so that material behavior is elastic. No failure, no gaps that open or close. Truss elements (bars, rods) : pinned(hinged) at connection points; resist axial forces only. Hence it has axial dofs only. Frame elements (beams) : welded (or, connected with multiple fasteners) at connection points; resist axial and transverse forces and bending moments. Has axial, transverse and rotational dofs.

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2 Bar Element: Nodal displacements and nodal forces Nodal displacements and nodal forces of a finite element are related through the stiffness matrix of the element. We’ll derive the stiffness matrix of a bar element now: In Fig. (a), the left node is displaced while u 2 =0. In Fig. (b), it is the opposite. The forces to maintain these displacements: F 11 = -F 21 =(AE/L)u 1 ; -F 12 = F 22 =(AE/L)u 2 L x u 1 F 21 F 11 A, E 1 2 (a) L x F 22 F 12 u 2 A, E 1 2 (b)
3 Bar Element: Stiffness Matrix Derivation If both u 1 and u 2 are nonzero, the nodal forces are F 1 = F 11 +F 12 =(AE/L)(u 1 -u 2 ) ; F 2 = F 21 +F 22 =(AE/L)(u 2 -u 1 ) Writing these equations in matrix form: where the coefficient matrix is called the element stiffness matrix. = 2 1 2 1 1 1 1 1 F F u u L AE = = 2 1 2 1 , with and 1 1 1 1 F F u u L AE r d r kd k

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4 General Stiffness Matrix Formulation The above is a direct method to compute the element stiffness matrix. This method is feasible for simple elements only. There is also a formal procedure which uses the following: where B: strain-displacement matrix for the element E: stress-strain matrix dV : volume element = dV T EB B k
5 Bar Element: Shape (interpolation) functions To derive B we interpolate axial displacement u of an arbitrary point on the bar between its nodal values u 1 and u 2 : L u 1 u 2 x u=N 1 u 1 +N 2 u 2 where N 1 and N 2 are called the shape functions: N 2 =x/L 1 x N 1 =(L-x)/L 1 x

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6 Bar Element: Strain matrix B Rewriting u: Nd = = 2 1 u u L x L x L u where N is the shape function matrix. Then, = = = = = L L dx d dx d dx du x 1 1 where ) ( B Bd d N Nd ε Now, since there is only one stress component in an axial bar, σ =E ε and, therefore, the stress-strain matrix is just the elastic modulus E.
7 Bar Element: Stiffness Matrix derivation Substituting B and E into the integral expression for the element stifness matrix, = = 1 1 1 1 1 1 / 1 / 1 0 L AE Adx L L E L L L k which is the same matrix as before.

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8 Bar Element: When is element exact?
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