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F inite 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 Statics and dynamics Elasticity and plasticity Isotropy and anisotropy Boundary conditions
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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 • 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 structuralmechanics
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 and static equilibrium equations • In matrix form Note: or • For static case
Constitutive equations • Hooke’s law in 1D s = Ee 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 Assumption of thin beam Sections remain normal Slope of the deflection curve where sxx= Eexx
Constitutive equations sxx = Eexx 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
z y h O fz yz yy yx xx xz xy x Moments and shear forces • Stresses on isolated plate cell
z Qx Mx Mxy Qy y O Qy+dQy Myx My My+dMy Myx+dMyx Qx+dQx dx Mxy+dMxy Mx+dMx x dy Moments and shear forces • Moments and shear forces on a plate cell dxx dy
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 (Static) where
Mindlin plate , e = -z Lq Therefore, in-plane strains where ,
Mindlin plate Transverse shear strains Transverse shear stress