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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

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**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

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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

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**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.

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**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.

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**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.

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

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**Different structural components**

Truss and beam structures

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**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:

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Stress and strain Strains

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Stress and strain Strains in matrix form where

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**Constitutive equations**

s = c e or

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**Constitutive equations**

For isotropic materials , ,

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**Dynamic equilibrium equations**

Consider stresses on an infinitely small block

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**Dynamic equilibrium equations**

Equilibrium of forces in x direction including the inertia forces Note:

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**Dynamic equilibrium equations**

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

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**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

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Stress and strain (3D)

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Stress and strain Strains in matrix form where ,

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**Constitutive equations**

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

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**Dynamic equilibrium equations**

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**Dynamic and static equilibrium equations**

In matrix form Note: or For static case

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**EQUATIONS FOR TRUSS MEMBERS**

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**Constitutive equations**

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

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**EQUATIONS FOR BEAMS Stress and strain Constitutive equations**

Moments and shear forces Dynamic and static equilibrium equations

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Stress and strain Euler–Bernoulli theory

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**Stress and strain sxx = E exx 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

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**Dynamic and static equilibrium equations**

Therefore, (Static)

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**EQUATIONS FOR PLATES Stress and strain Constitutive equations**

Moments and shear forces Dynamic and static equilibrium equations Mindlin plate

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

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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

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**Constitutive equations**

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

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**Moments and shear forces**

Stresses on isolated plate cell z x y fz h xy xx xz yx yy yz O

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**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

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**Moments and shear forces**

s = c e s = - c z Lw Like beams, Note that ,

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**Moments and shear forces**

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

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**Dynamic and static equilibrium equations**

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**Dynamic and static equilibrium equations**

(Static) where

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Mindlin plate

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

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Mindlin plate Transverse shear strains Transverse shear stress

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