Presentation on theme: "Finite Element Method CHAPTER 5: FEM FOR BEAMS"— Presentation transcript:
1Finite Element Method CHAPTER 5: FEM FOR BEAMS for readers of all backgroundsG. R. Liu and S. S. QuekCHAPTER 5:FEM FOR BEAMS
2CONTENTS INTRODUCTION FEM EQUATIONS Shape functions construction Strain matrixElement matricesRemarksEXAMPLE
3INTRODUCTION The element developed is often known as a beam element. A beam element is a straight bar of an arbitrary cross-section.Beams are subjected to transverse forces and moments.Deform only in the directions perpendicular to its axis of the beam.
4INTRODUCTIONIn beam structures, the beams are joined together by welding (not by pins or hinges).Uniform cross-section is assumed.FE matrices for beams with varying cross-sectional area can also be developed without difficulty.
5FEM EQUATIONS Shape functions construction Strain matrix Element matrices
6Shape functions construction Consider a beam elementNatural coordinate system:
7Shape functions construction Assume thatIn matrix form:or
8Shape functions construction To obtain constant coefficients – four conditionsAt x= -a or x = -1At x= a or x = 1
15RemarksTheoretically, coordinate transformation can also be used to transform the beam element matrices from the local coordinate system to the global coordinate system.The transformation is necessary only if there is more than one beam element in the beam structure, and of which there are at least two beam elements of different orientations.A beam structure with at least two beam elements of different orientations is termed a frame or framework.
16EXAMPLEConsider the cantilever beam as shown in the figure. The beam is fixed at one end and it has a uniform cross-sectional area as shown. The beam undergoes static deflection by a downward load of P=1000N applied at the free end. The dimensions and properties of the beam are shown in the figure.P=1000 N0.5 m0.06m0.1 mE=69 GPa=0.33