Download presentation

Presentation is loading. Please wait.

Published byDevin Norris Modified over 4 years ago

1
**INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES**

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 EQUATIONS FOR THREE-DIMENSIONAL (3D) SOLIDS**

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

10
**EQUATIONS FOR 3D SOLIDS Stress and strain Constitutive equations**

Dynamic and static equilibrium equations

11
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

14
**Constitutive equations**

s = c e or

15
**Constitutive equations**

For isotropic materials , ,

16
**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

20
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**

25
**Dynamic and static equilibrium equations**

In matrix form Note: or For static case

26
**EQUATIONS FOR TRUSS MEMBERS**

27
**Constitutive equations**

Hooke’s 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 Euler–Bernoulli theory

30
**Stress and strain sxx = E exx Assumption of thin beam**

Sections remain normal Slope of the deflection curve where sxx = E exx

31
**Constitutive equations**

sxx = E exx Moments and shear forces Consider isolated beam cell of length dx

32
**Moments and shear forces**

The stress and moment

33
**Moments and shear forces**

Since Therefore, Where (Second moment of area about z axis – dependent on shape and dimensions of cross-section)

34
**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, ,

39
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 z x y fz h xy xx xz yx yy yz O

42
**Moments and shear forces**

Moments and shear forces on a plate cell dx x dy z x y O dx dy Qy My Myx Qy+dQy Myx+dMyx My+dMy Qx Mx Mxy Qx+dQx Mxy+dMxy Mx+dMx

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

Similar presentations

OK

Bellwork Do the following problem on a ½ sheet of paper and turn in.

Bellwork Do the following problem on a ½ sheet of paper and turn in.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on eia report in malaysia Ppt on low level language of computer Ppt on forward contract foreign Ppt on different types of pollution Ppt on rc coupled amplifier definition Ppt on personal computer museum Ppt on save water save life free download Probability for kids ppt on batteries Ppt on information and communication technology in education Ppt on different types of computer softwares programs