Finite Element Method

1
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 2

• INTRODUCTION TO MECHANICS
• FOR SOLIDS AND STRUCTURES

2
CONTENTS

• INTRODUCTION
• Statics and dynamics
• Elasticity and plasticity
• Isotropy and anisotropy
• Boundary conditions
• Different structural components
• EQUATIONS FOR THREE-DIMENSIONAL (3D) SOLIDS
• EQUATIONS FOR TWO-DIMENSIONAL (2D) SOLIDS
• EQUATIONS FOR TRUSS MEMBERS
• EQUATIONS FOR BEAMS
• EQUATIONS FOR PLATES

3
INTRODUCTION

• Solids and structures are stressed when they are
  subjected to loads or forces.
• The stresses are, in general, not uniform as the
  forces usually vary with coordinates.
• The stresses lead to strains, which can be
  observed as a deformation or displacement.
• Solid mechanics and structural mechanics

4
Statics and dynamics

• Forces can be static and/or dynamic.
• Statics deals with the mechanics of solids and
  structures subject to static loads.
• Dynamics deals with the mechanics of solids and
  structures subject to dynamic loads.
• As statics is a special case of dynamics, the
  equations for statics can be derived by simply
  dropping out the dynamic terms in the dynamic
  equations.

5
Elasticity and plasticity

• Elastic the deformation in the solids disappears
  fully if it is unloaded.
• Plastic the deformation in the solids cannot be
  fully recovered when it is unloaded.
• Elasticity deals with solids and structures of
  elastic materials.
• Plasticity deals with solids and structures of
  plastic materials.

6
Isotropy and anisotropy

• Anisotropic the material property varies with
  direction.
• Composite materials anisotropic, many material
  constants.
• Isotropic material property is not direction
  dependent, two independent material constants.

7
Boundary conditions

• Displacement (essential) boundary conditions
• Force (natural) boundary conditions

8
Different structural components

• Truss and beam structures

9
Different structural components

• Plate and shell structures

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EQUATIONS FOR 3D SOLIDS

• Stress and strain
• Constitutive equations
• Dynamic and static equilibrium equations

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Stress and strain

• Stresses at a point in a 3D solid

12
Stress and strain

• Strains

13
Stress and strain

• Strains in matrix form

where
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Constitutive equations

• s c e

or
15
Constitutive equations

• For isotropic materials

,
,
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Dynamic equilibrium equations

• Consider stresses on an infinitely small block

17
Dynamic equilibrium equations

• Equilibrium of forces in x direction including
  the inertia forces

Note
18
Dynamic equilibrium equations

• Hence, equilibrium equation in x direction

• Equilibrium equations in y and z directions

19
Dynamic and static equilibrium equations

• In matrix form

Note
or

• For static case

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EQUATIONS FOR 2D SOLIDS
Plane stress
Plane strain
21
Stress and strain
(3D)
22
Stress and strain

• Strains in matrix form

where
,
23
Constitutive equations

• s c e

(For plane stress)
(For plane strain)
24
Dynamic equilibrium equations
(3D)
25
Dynamic and static equilibrium equations

• In matrix form

Note
or

• For static case

26
EQUATIONS FOR TRUSS MEMBERS
27
Constitutive equations

• Hookes law in 1D
• s E e

Dynamic and static equilibrium equations
(Static)
28
EQUATIONS FOR BEAMS

• Stress and strain
• Constitutive equations
• Moments and shear forces
• Dynamic and static equilibrium equations

29
Stress and strain

• EulerBernoulli theory

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Stress and strain
Assumption of thin beam
Sections remain normal
Slope of the deflection curve
where
sxx E exx
?
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Constitutive equations

• sxx E exx

Moments and shear forces

• Consider isolated beam cell of length dx

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Moments and shear forces

• The stress and moment

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Moments and shear forces
Since
Therefore,
Where
(Second moment of area about z axis dependent
on shape and dimensions of cross-section)
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Dynamic and static equilibrium equations
Forces in the x direction
Moments about point A
?
?
35
Dynamic and static equilibrium equations
Therefore,
?
(Static)
36
EQUATIONS FOR PLATES

• Stress and strain
• Constitutive equations
• Moments and shear forces
• Dynamic and static equilibrium equations
• Mindlin plate

37
Stress and strain

• Thin plate theory or Classical Plate Theory (CPT)

38
Stress and strain
Assumes that exz 0, eyz 0
,
Therefore,
,
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Stress and strain

• Strains in matrix form

e -z Lw
where
40
Constitutive equations

• s c e
• where c has the same form for the plane stress
  case of 2D solids

41
Moments and shear forces

• Stresses on isolated plate cell

42
Moments and shear forces

• Moments and shear forces on a plate cell dx x dy

43
Moments and shear forces
s c e

• s - c z Lw

?
Like beams,
Note that
,
44
Moments and shear forces
Therefore, equilibrium of forces in z direction
or
Moments about A-A
45
Dynamic and static equilibrium equations
46
Dynamic and static equilibrium equations
?
(Static)
where
47
Mindlin plate
48
Mindlin plate
,
e -z Lq
Therefore, in-plane strains
where
,
49
Mindlin plate
Transverse shear strains
Transverse shear stress