1 FINITE ELEMENT APPROXIMATION Rayleigh-Ritz method approximate solution in the entire beam –Difficult to find good approximate solution (discontinuities in derivatives) Finite element approximates solution in an element –In a small element simple functions are acceptably accurate Beam element –Divide the beam using a set of two-node elements –Elements are connected to other elements at nodes –Concentrated forces and couples can only be applied at nodes –Positive directions for forces and couples –Constant or linearly distributed load F1F1 F2F2 C2C2 C1C1 p(x) x

2 BEAM ELEMENT DEFINITION Nodal DOF of beam element –Each node has deflection v and slope  –Positive directions of DOFs –Vector of nodal DOFs Non-dimensional coordinate s Will write deflection curve v(s) in terms of s v1v1 v2v2 22 11 L x 1 s = 0 x 2 s = 1 x

3 BEAM ELEMENT INTERPOLATION Deflection interpolation –Interpolate the deflection v(s) in terms of four nodal DOFs –Use cubic polynomial: –Relation to the slope: –Apply four conditions: –Express four coefficients in terms of nodal DOFs

4 BEAM ELEMENT INTERPOLATION cont. Deflection interpolation cont. Shape functions (Check) –Hermite polynomials –Interpolation property N1N1 N3N3 N 2 /L N 4 /L

5 Quiz-like problems What are the common properties of all four Hermite interpolation functions? What is the slope of N 1 at x=L? What is the value of N 2 at x=L? What is the slope of N 2 at x=L? A race car sensors show that at t=0 it had zero speed and an acceleration of 2g, and at t=2sec it had a speed of 30m/s and an acceleration of 1g. Use Hermite interpolation to estimate its speed at t=1sec (replace x by t, L=2sec, v is speed, and replace  by the acceleration a) Answers in notes page.

6 BEAM ELEMENT INTERPOLATION cont. Properties of interpolation –Deflection is a cubic polynomial (discuss accuracy and limitation) –Interpolation is valid within an element, not outside of the element –Adjacent elements have continuous deflection and slope Approximation of curvature –Curvature is second derivative and related to strain and stress –B is linear function of s and, thus, the strain and stress –Alternative expression: –If the given problem is linearly varying curvature, the approximation is exact; if higher-order variation of curvature, then it is approximate B: strain-displacement vector

7 BENDING MOMENT AND SHEAR Approximation of bending moment and shear force –Stress is proportional to M(s); M(s) is linear; stress is linear, too –Maximum stress always occurs at the node –Bending moment and shear force are not continuous between adjacent elements Linear Constant

8 EXAMPLE – INTERPOLATION Cantilevered beam Given nodal DOFs Deflection and slope at x = 0.5L Parameter s = 0.5 at x = 0.5L Shape functions: Deflection at s = 0.5: Slope at s = 0.5: L v1v1 v2v2 22 11

9 EXAMPLE A beam finite element with length L Calculate v(s) Bending moment Bending moment caused by unit force at the tip

10 Quiz-like problem What is the tip force and couple that we will need to apply to a cantilever beam of length L and stiffness EI so that it will have a unit displacement and zero slope? Answer in the notes page