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Finite Element Method by G. R. Liu and S. S. Quek 1 Finite Element Method INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES for readers of all backgrounds.

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Presentation on theme: "Finite Element Method by G. R. Liu and S. S. Quek 1 Finite Element Method INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES for readers of all backgrounds."— Presentation transcript:

1 Finite Element Method by G. R. Liu and S. S. Quek 1 Finite Element Method INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 2:

2 Finite Element Method by G. R. Liu and S. S. Quek 2 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

3 Finite Element Method by G. R. Liu and S. S. Quek 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 Finite Element Method by G. R. Liu and S. S. Quek 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 Finite Element Method by G. R. Liu and S. S. Quek 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 Finite Element Method by G. R. Liu and S. S. Quek 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 Finite Element Method by G. R. Liu and S. S. Quek 7 Boundary conditions Displacement (essential) boundary conditions Force (natural) boundary conditions

8 Finite Element Method by G. R. Liu and S. S. Quek 8 Different structural components Truss and beam structures

9 Finite Element Method by G. R. Liu and S. S. Quek 9 Different structural components Plate and shell structures

10 Finite Element Method by G. R. Liu and S. S. Quek 10 EQUATIONS FOR 3D SOLIDS Stress and strain Constitutive equations Dynamic and static equilibrium equations

11 Finite Element Method by G. R. Liu and S. S. Quek 11 Stress and strain Stresses at a point in a 3D solid:

12 Finite Element Method by G. R. Liu and S. S. Quek 12 Stress and strain Strains

13 Finite Element Method by G. R. Liu and S. S. Quek 13 Stress and strain Strains in matrix form where

14 Finite Element Method by G. R. Liu and S. S. Quek 14 Constitutive equations = c or

15 Finite Element Method by G. R. Liu and S. S. Quek 15 Constitutive equations For isotropic materials,,

16 Finite Element Method by G. R. Liu and S. S. Quek 16 Dynamic equilibrium equations Consider stresses on an infinitely small block

17 Finite Element Method by G. R. Liu and S. S. Quek 17 Dynamic equilibrium equations Equilibrium of forces in x direction including the inertia forces Note:

18 Finite Element Method by G. R. Liu and S. S. Quek 18 Dynamic equilibrium equations Hence, equilibrium equation in x direction Equilibrium equations in y and z directions

19 Finite Element Method by G. R. Liu and S. S. Quek 19 Dynamic and static equilibrium equations In matrix form or For static case Note:

20 Finite Element Method by G. R. Liu and S. S. Quek 20 EQUATIONS FOR 2D SOLIDS Plane stress Plane strain

21 Finite Element Method by G. R. Liu and S. S. Quek 21 Stress and strain (3D)

22 Finite Element Method by G. R. Liu and S. S. Quek 22 Stress and strain Strains in matrix form where,

23 Finite Element Method by G. R. Liu and S. S. Quek 23 Constitutive equations = c (For plane stress) (For plane strain)

24 Finite Element Method by G. R. Liu and S. S. Quek 24 Dynamic equilibrium equations (3D)

25 Finite Element Method by G. R. Liu and S. S. Quek 25 Dynamic and static equilibrium equations In matrix form or For static case Note:

26 Finite Element Method by G. R. Liu and S. S. Quek 26 EQUATIONS FOR TRUSS MEMBERS

27 Finite Element Method by G. R. Liu and S. S. Quek 27 Constitutive equations Hookes law in 1D = E Dynamic and static equilibrium equations (Static)

28 Finite Element Method by G. R. Liu and S. S. Quek 28 EQUATIONS FOR BEAMS Stress and strain Constitutive equations Moments and shear forces Dynamic and static equilibrium equations

29 Finite Element Method by G. R. Liu and S. S. Quek 29 Stress and strain Euler–Bernoulli theory

30 Finite Element Method by G. R. Liu and S. S. Quek 30 Stress and strain Assumption of thin beam Sections remain normal Slope of the deflection curve where xx = E xx

31 Finite Element Method by G. R. Liu and S. S. Quek 31 Constitutive equations xx = E xx Moments and shear forces Consider isolated beam cell of length dx

32 Finite Element Method by G. R. Liu and S. S. Quek 32 Moments and shear forces The stress and moment

33 Finite Element Method by G. R. Liu and S. S. Quek 33 Moments and shear forces Since Therefore, Where (Second moment of area about z axis – dependent on shape and dimensions of cross-section)

34 Finite Element Method by G. R. Liu and S. S. Quek 34 Dynamic and static equilibrium equations Forces in the x direction Moments about point A

35 Finite Element Method by G. R. Liu and S. S. Quek 35 Dynamic and static equilibrium equations Therefore, (Static)

36 Finite Element Method by G. R. Liu and S. S. Quek 36 EQUATIONS FOR PLATES Stress and strain Constitutive equations Moments and shear forces Dynamic and static equilibrium equations Mindlin plate

37 Finite Element Method by G. R. Liu and S. S. Quek 37 Stress and strain Thin plate theory or Classical Plate Theory (CPT)

38 Finite Element Method by G. R. Liu and S. S. Quek 38 Stress and strain Assumes that xz = 0, yz = 0, Therefore,,

39 Finite Element Method by G. R. Liu and S. S. Quek 39 Stress and strain Strains in matrix form = z Lw where

40 Finite Element Method by G. R. Liu and S. S. Quek 40 Constitutive equations = c where c has the same form for the plane stress case of 2D solids

41 Finite Element Method by G. R. Liu and S. S. Quek 41 Moments and shear forces Stresses on isolated plate cell z x y fzfz h xy xx xz yx yy yz O

42 Finite Element Method by G. R. Liu and S. S. Quek 42 Moments and shear forces Moments and shear forces on a plate cell dx x dy z x y O dx dy QyQy MyMy M yx Q y +dQ y M yx +dM yx M y +dM y QxQx MxMx M xy Q x +dQ x M xy +dM xy M x +dM x

43 Finite Element Method by G. R. Liu and S. S. Quek 43 Moments and shear forces = c = c z Lw Like beams, Note that,

44 Finite Element Method by G. R. Liu and S. S. Quek 44 Moments and shear forces Therefore, equilibrium of forces in z direction or Moments about A-A

45 Finite Element Method by G. R. Liu and S. S. Quek 45 Dynamic and static equilibrium equations

46 Finite Element Method by G. R. Liu and S. S. Quek 46 Dynamic and static equilibrium equations where (Static)

47 Finite Element Method by G. R. Liu and S. S. Quek 47 Mindlin plate

48 Finite Element Method by G. R. Liu and S. S. Quek 48 Mindlin plate, Therefore, in-plane strains = z L where,

49 Finite Element Method by G. R. Liu and S. S. Quek 49 Mindlin plate Transverse shear strains Transverse shear stress


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