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