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MANE 4240 & CIVL 4240 Introduction to Finite Elements Introduction to 3D Elasticity Prof. Suvranu De

# MANE 4240 & CIVL 4240 Introduction to Finite Elements Introduction to 3D Elasticity Prof. Suvranu De

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### Text of MANE 4240 & CIVL 4240 Introduction to Finite Elements Introduction to 3D Elasticity Prof. Suvranu De

• Slide 1
• MANE 4240 & CIVL 4240 Introduction to Finite Elements Introduction to 3D Elasticity Prof. Suvranu De
• Slide 2
• Reading assignment: Appendix C+ 6.1+ 9.1 + Lecture notes Summary: 3D elasticity problem Governing differential equation + boundary conditions Strain-displacement relationship Stress-strain relationship Special cases 2D (plane stress, plane strain) Axisymmetric body with axisymmetric loading Principle of minimum potential energy
• Slide 3
• 1D Elasticity (axially loaded bar) x y x=0 x=L A(x) = cross section at x b(x) = body force distribution (force per unit length) E(x) = Youngs modulus u(x) = displacement of the bar at x x 1. Strong formulation: Equilibrium equation + boundary conditions Boundary conditions Equilibrium equation F
• Slide 4
• 3. Stress-strain (constitutive) relation : E: Elastic (Youngs) modulus of bar 2. Strain-displacement relationship:
• Slide 5
• Problem definition 3D Elasticity x y z Surface (S) Volume (V) u v w x V: Volume of body S: Total surface of the body The deformation at point x =[x,y,z] T is given by the 3 components of its displacement NOTE: u= u(x,y,z), i.e., each displacement component is a function of position
• Slide 6
• 3D Elasticity: EXTERNAL FORCES ACTING ON THE BODY Two basic types of external forces act on a body 1.Body force (force per unit volume) e.g., weight, inertia, etc 2.Surface traction (force per unit surface area) e.g., friction
• Slide 7
• BODY FORCE x y z Surface (S) Volume (V) u v w x X a dV X b dV X c dV Volume element dV Body force: distributed force per unit volume (e.g., weight, inertia, etc) NOTE: If the body is accelerating, then the inertia force may be considered as part of X
• Slide 8
• x y z STST Volume (V) u v w x X a dV X b dV X c dV Volume element dV pypy pzpz pxpx Traction: Distributed force per unit surface area SURFACE TRACTION
• Slide 9
• 3D Elasticity: INTERNAL FORCES If I take out a chunk of material from the body, I will see that, due to the external forces applied to it, there are reaction forces (e.g., due to the loads applied to a truss structure, internal forces develop in each truss member). For the cube in the figure, the internal reaction forces per unit area(red arrows), on each surface, may be decomposed into three orthogonal components. x y z Volume (V) u v w x Volume element dV xx yy zz yz yx xy xz zy zx
• Slide 10
• 3D Elasticity x y z xx yy zz yz yx xy xz zy zx x, y and z are normal stresses. The rest 6 are the shear stresses Convention xy is the stress on the face perpendicular to the x-axis and points in the +ve y direction Total of 9 stress components of which only 6 are independent since The stress vector is therefore
• Slide 11
• Strains: 6 independent strain components Consider the equilibrium of a differential volume element to obtain the 3 equilibrium equations of elasticity
• Slide 12
• Compactly; where EQUILIBRIUM EQUATIONS (1)
• Slide 13
• 3D elasticity problem is completely defined once we understand the following three concepts Strong formulation (governing differential equation + boundary conditions) Strain-displacement relationship Stress-strain relationship
• Slide 14
• 1. Strong formulation of the 3D elasticity problem: Given the externally applied loads (on S T and in V) and the specified displacements (on S u ) we want to solve for the resultant displacements, strains and stresses required to maintain equilibrium of the body. x y z SuSu STST Volume (V) u v w x X a dV X b dV X c dV Volume element dV pypy pzpz pxpx
• Slide 15
• Equilibrium equations (1) Boundary conditions 1. Displacement boundary conditions: Displacements are specified on portion S u of the boundary 2. Traction (force) boundary conditions: Tractions are specified on portion S T of the boundary Now, how do I express this mathematically?
• Slide 16
• x y z SuSu STST Volume (V) u v w x X a dV X b dV X c dV Volume element dV pypy pzpz pxpx Traction: Distributed force per unit area
• Slide 17
• n nxnx nyny nznz STST TSTS pypy pxpx pzpz If the unit outward normal to S T : Then
• Slide 18
• nxnx nyny STST In 2D dy dx ds x y n TSTS pypy pxpx dy dx ds xx yy xy Consider the equilibrium of the wedge in x-direction Similarly
• Slide 19
• 3D elasticity problem is completely defined once we understand the following three concepts Strong formulation (governing differential equation + boundary conditions) Strain-displacement relationship Stress-strain relationship
• Slide 20
• 2. Strain-displacement relationships:
• Slide 21
• Compactly; (2)
• Slide 22
• x y A B C A B C v u dy dx In 2D
• Slide 23
• 3D elasticity problem is completely defined once we understand the following three concepts Strong formulation (governing differential equation + boundary conditions) Strain-displacement relationship Stress-strain relationship
• Slide 24
• 3. Stress-Strain relationship: Linear elastic material (Hookes Law) (3) Linear elastic isotropic material
• Slide 25
• Special cases: 1. 1D elastic bar (only 1 component of the stress (stress) is nonzero. All other stress (strain) components are zero) Recall the (1) equilibrium, (2) strain-displacement and (3) stress- strain laws 2. 2D elastic problems: 2 situations PLANE STRESS PLANE STRAIN 3. 3D elastic problem: special case-axisymmetric body with axisymmetric loading (we will skip this)
• Slide 26
• PLANE STRESS: Only the in-plane stress components are nonzero x y Area element dA Nonzero stress components h D Assumptions: 1. h ##### Prof. Suvranu Dedes/Trusses.pdf1 MANE 4240 & CIVL 4240 Introduction to Finite Elements Prof. Suvranu De Development of Truss Equations Reading assignment: Chapter 3: Sections 3.1-3.9
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