# INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES

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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

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

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

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.

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.

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.

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

Different structural components
Truss and beam structures

Different structural components
Plate and shell structures

EQUATIONS FOR 3D SOLIDS Stress and strain Constitutive equations
Dynamic and static equilibrium equations

Stress and strain Stresses at a point in a 3D solid:

Stress and strain Strains

Stress and strain Strains in matrix form where

Constitutive equations
s = c e or

Constitutive equations
For isotropic materials , ,

Dynamic equilibrium equations
Consider stresses on an infinitely small block

Dynamic equilibrium equations
Equilibrium of forces in x direction including the inertia forces Note:

Dynamic equilibrium equations
Hence, equilibrium equation in x direction Equilibrium equations in y and z directions

Dynamic and static equilibrium equations
In matrix form Note: or For static case

EQUATIONS FOR 2D SOLIDS Plane stress Plane strain

Stress and strain (3D)

Stress and strain Strains in matrix form where ,

Constitutive equations
s = c e (For plane stress) (For plane strain)

Dynamic equilibrium equations

Dynamic and static equilibrium equations
In matrix form Note: or For static case

EQUATIONS FOR TRUSS MEMBERS

Constitutive equations
Hooke’s law in 1D s = E e Dynamic and static equilibrium equations (Static)

EQUATIONS FOR BEAMS Stress and strain Constitutive equations
Moments and shear forces Dynamic and static equilibrium equations

Stress and strain Euler–Bernoulli theory

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

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

Moments and shear forces
The stress and moment

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

Dynamic and static equilibrium equations
Forces in the x direction Moments about point A

Dynamic and static equilibrium equations
Therefore, (Static)

EQUATIONS FOR PLATES Stress and strain Constitutive equations
Moments and shear forces Dynamic and static equilibrium equations Mindlin plate

Stress and strain Thin plate theory or Classical Plate Theory (CPT)

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

Stress and strain Strains in matrix form e = -z Lw where

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

Moments and shear forces
Stresses on isolated plate cell z x y fz h xy xx xz yx yy yz O

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

Moments and shear forces
s = c e s = - c z Lw Like beams, Note that ,

Moments and shear forces
Therefore, equilibrium of forces in z direction or Moments about A-A

Dynamic and static equilibrium equations

Dynamic and static equilibrium equations
(Static) where

Mindlin plate

Mindlin plate , e = -z Lq Therefore, in-plane strains where ,

Mindlin plate Transverse shear strains Transverse shear stress