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1 Finite Element Method FEM FOR BEAMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 5:

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Presentation on theme: "1 Finite Element Method FEM FOR BEAMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 5:"— Presentation transcript:

1 1 Finite Element Method FEM FOR BEAMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 5:

2 Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS INTRODUCTION FEM EQUATIONS – Shape functions construction – Strain matrix – Element matrices – Remarks EXAMPLE – Remarks

3 Finite Element Method by G. R. Liu and S. S. Quek 3 INTRODUCTION 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.

4 Finite Element Method by G. R. Liu and S. S. Quek 4 INTRODUCTION In 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.

5 Finite Element Method by G. R. Liu and S. S. Quek 5 FEM EQUATIONS Shape functions construction Strain matrix Element matrices

6 Finite Element Method by G. R. Liu and S. S. Quek 6 Shape functions construction Consider a beam element Natural coordinate system:

7 Finite Element Method by G. R. Liu and S. S. Quek 7 Shape functions construction Assume that In matrix form:or

8 Finite Element Method by G. R. Liu and S. S. Quek 8 Shape functions construction To obtain constant coefficients – four conditions At x= a or = 1

9 Finite Element Method by G. R. Liu and S. S. Quek 9 Shape functions construction or or

10 Finite Element Method by G. R. Liu and S. S. Quek 10 Shape functions construction Therefore, where in which

11 Finite Element Method by G. R. Liu and S. S. Quek 11 Strain matrix Therefore, where (Second derivative of shape functions)

12 Finite Element Method by G. R. Liu and S. S. Quek 12 Element matrices Evaluate integrals

13 Finite Element Method by G. R. Liu and S. S. Quek 13 Element matrices Evaluate integrals

14 Finite Element Method by G. R. Liu and S. S. Quek 14 Element matrices

15 Finite Element Method by G. R. Liu and S. S. Quek 15 Remarks Theoretically, 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.

16 Finite Element Method by G. R. Liu and S. S. Quek 16 EXAMPLE Consider 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 N 0.5 m 0.06 m 0.1 m E=69 GPa =0.33

17 Finite Element Method by G. R. Liu and S. S. Quek 17 EXAMPLE Step 1: Element matrices

18 Finite Element Method by G. R. Liu and S. S. Quek 18 EXAMPLE Step 1 (Contd): Step 2: Boundary conditions

19 Finite Element Method by G. R. Liu and S. S. Quek 19 EXAMPLE Step 2 (Contd): Therefore, Kd=F where d T = [ v 2 2 ],

20 Finite Element Method by G. R. Liu and S. S. Quek 20 EXAMPLE Step 3: Solving FE equation – Two simultaneous equations v 2 = x m 2 = x rad Substitute back into first two equations of Kd=F

21 Finite Element Method by G. R. Liu and S. S. Quek 21 Remarks FE solution is the same as analytical solution Analytical solution to beam is third order polynomial (same as shape functions used) Reproduction property

22 Finite Element Method by G. R. Liu and S. S. Quek 22 CASE STUDY Resonant frequencies of micro resonant transducer

23 Finite Element Method by G. R. Liu and S. S. Quek 23 CASE STUDY Number of 2- node beam elements Natural Frequency (Hz) Mode 1Mode 2Mode x x x x x x x x x x x x 10 6 Analytical Calculations x x x 10 6

24 Finite Element Method by G. R. Liu and S. S. Quek 24 CASE STUDY

25 Finite Element Method by G. R. Liu and S. S. Quek 25 CASE STUDY

26 Finite Element Method by G. R. Liu and S. S. Quek 26 CASE STUDY


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