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ECIV 720 A Advanced Structural Mechanics and Analysis(2)

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ECIV 720 A Advanced Structural Mechanics and Analysis(2) Powered By Docstoc
					          ECIV 720 A
  Advanced Structural Mechanics
          and Analysis


Final Review
            Final Review

Fundamental Concepts
    Stresses and Equilibrium
    Boundary Conditions
    Strain Displacement Relations
    Stress-Strain Relations
    Interpolation
    Potential Energy and Equilibrium
    Rayleigh Ritz Method
    Galerkin’s Method
               Engineering Systems



  Lumped Parameter
                                 Continuous
      (Discrete)

• A finite number of      • Differential Equations
  state variables           Govern Response
  describe solution
• Algebraic Equations
Matrix Structural Analysis - Objectives



Use
Equations of Equilibrium
                             Basic Equations
Constitutive Equations
Compatibility Conditions

Form
[A]{x}={b}

Solve for Unknown Displacements/Forces
{x}= [A]-1{b}
            FEM Analysis - Objective

Governing
Equations of                                  System of
Mathematical                                  Algebraic
                    FEM Procedures
Model                                         Equations

Differential
Equations

 Where is it used?
 Analysis of Solids and Structures
 Soil and Rock Mechanics
 Heat Transfer
 Fluids
 Virtually in any field of engineering analysis…
                  Assumptions


• Linear Strain-Displacement Relationship

• Small Deformations

• Equilibrium Pertains to Undeformed Configuration
              FEM Process of Analysis



Reliability of Solution
depends on choice
of Mathematical
Model



Accurate
Approximations
of Solutions
Basic Relationships of Elasticity Theory
Equilibrium

                     Equilibrium
              Write Equations of
              Equilibrium

                     SFx=0
                     SFy=0
                     SFz=0
              Boundary Conditions
                           Equilibrium at Surface




Prescribed Displacements
Strain-Displacement Relations




   Assumption
   Small Deformations
             Stress-Strain Relations

        Isotropic Material:         E, n
Generalized Hooke’s Law




                              (a)

                          Equations (a) can be solved for s...
          Stress-Strain Relations – Material Matrix

s x                                             x 
s                                               
 y                                              y
s z 
                                                z 
                                                  
                                               
 yz                                            yz 
                                               
 xz                                             xz 
 xy 
                                                xy 
                                                  



    Material Matrix
                Special Cases


e.g. Two Dimensional – Plane Stress
Thin Planar Bodies subjected to in plane loading
                   Strain Energy


In the general state of stress for conservative systems



               1 T
            U   σ εdV
               2 V
Function Interpolation/Approximation


Interpolation or approximation of state
   variables is a key operation in FEM
                procedures


           Common Approach:
 Interpolation of State Variables Using
                Polynomials
Shape Functions – A General Approach
Assume a polynomial



     For example for 1-D

 u   a0  a1  a2  an
                      2          n
 Shape Functions – A General Approach
Enforce Boundary Conditions


u             
   1  a0  a1 1  a2 1
                       2
                                         u1
                                  an 1n


u             
   2  a0  a1 2  a2 2
                       2
                                        u2
                                  an 2n



                       

un           
           a0  a1 n  a2 n
                          2
                                        un
                                  an n
                                        n
 Shape Functions – A General Approach

Solve for ai

   1 1    1   a1   u1 
              n

             n      u 
   1 2     n  a2   2 
                   
                     
          
             n 
   1  n
            n  an  un 
                       
 Shape Functions – A General Approach

Substitute in
 u   a0  a1  a2  an2            n




u   N1  u1  N 2  u2    N n  un



        Shape Functions
 Shape Functions – A General Approach

Substitute in
 u   a0  a1  a2  an2            n




u   N1  u1  N 2  u2    N n  un



          Degrees of Freedom
            (nodal values)
               2-D The Pascal Triangle
 ux, y   a0  a1 x  a2 y  a3 xy an x                      n

               Pascal Triangle                            Degree
                     1                                      0
                      x        y                            1
               x2         xy         y2                     2
          x3       x 2y        xy2        y3                3
     x4     x 3y      x2y2         xy3          y4          4
x5   x 4y      x3y2        x2y3           xy4        y5     5

          …….
     Shape Functions – Important Points

The continuous displacement field is expressed
in a discrete form as a linear combination of
shape functions.
                   n
       u      
                  i 1
                         N i   ui   Or in matrix form

                                  u1 
                                  u 
                                   2
        u    N1 , N 2  N n    Nu
                                   
                                  un 
                                   
Shape Functions – A generalized concept


  • Shape Functions are not necessarily
    Lagrange Polynomials

  • Shape functions may be any functions
    that approximate the variation of state
    variables and satisfy Boundary
    Conditions
     • e.g. trigonometric, logarithmic, incomplete
       polynomials, etc
   Shape Functions – Important Points


Depending on choice of Shape Functions
the representation is exact or approximate at
intermediate points
                      n
          u       N   u
                     i 1
                            i     i



Representation at discrete nodes is exact
                      n
         u        N   u
                     i 1
                            i   i      ui
     Solution of Continuous Systems –
           Fundamental Concepts
Objective: Determine displacement u…

                     …satisfying equilibrium equations




                              Governing Equations
s=f()   =g(u)
                              2nd order PDE
      Solution of Continuous Systems –
            Fundamental Concepts

                 Exact solutions
limited to simple geometries and boundary & loading
                     conditions
             Approximate Solutions
Reduce the continuous-system mathematical model to a
                 discrete idealization


        Rayleigh Ritz Method
        Galerkin Method
Potential Energy P & Rayleigh Ritz Method


P = Strain Energy           -       Work Potential
           U                             WP

                            WP      u fdV
                                         T       Body Forces
                                     V

    Strain Energy Density            uT TdV    Surface Loads
                                     V
         U 1
                                     u Pi
      u    s                              T
         V 2                                    Point Loads
                                             i
                                     i
                                   (conservative system)
   1 T
U   σ εdV
   2 V
        Total Potential & Equilibrium

   1 T
P   σ εdV   u fdV   u TdV   u i Pi
                 T         T         T

   2 V         V         V
                                  i
   Principle of Minimum Potential Energy
For conservative systems, of all the kinematically
      admissible displacement fields, those
 corresponding to equilibrium extremize the total
  potential energy. If the extremum condition is
     minimum, the equilibrium state is stable
            P
Min/Max:         0 i=1,2… all admissible displ
            ui
   The Rayleigh-Ritz Method for Continua

        1 T
     P   σ εdV   uT fdV   uT TdV   uT Pi
                                            i
        2 V         V          V
                                         i


The displacement field appears in

  work potential WP      uT fdV   uT TdV   uT Pi
                                                  i
                          V         V
                                               i


                          1 T
 and strain energy     U   σ εdV
                          2 V
  The Rayleigh-Ritz Method for Continua

Before we evaluate P, an assumed displacement
         field needs to be constructed
               Recall Shape Functions
                             For 3-D
      For 1-D
                          u   N i  x, y , z  u i

               
         n
u x   N i x ui          v   N j  x, y , z  u j
        i 1

                         w   N k  x, y , z  u k
 The Rayleigh-Ritz Method for Continua

Interpolation introduces n discrete independent
displacements (dof) a1, a2, …, an. (u1, u2, …, un)

Thus

u= u (u1, u2, …, un)


and

P= P (u1, u2, …, un)
  The Rayleigh-Ritz Method for Continua

For Equilibrium we minimize the total potential
            P(u,v,w) = P(a1, a2, …, an)
     w.r.t each admissible displacement ai

  P
      0
  a1               Algebraic System of
  P            n Equations and n unknowns
      0
  a2
     
  P
      0
  an
  Weighted Residual Formulations

 Consider a general representation of a
   governing equation on a region V


            Lu  P
     L is a differential operator
                        d     du 
eg. For Axial element       EA   0
                        dx    dx 

             L  EA       
                d  d
                dx dx
Weighted Residual Formulations


          Lu  P
Assume approximate solution
  (Recall shape functions)
      n
u  i i
~ Nu              then
     i 1

      ~  P'
     Lu
        Weighted Residual Formulations


    Lu  P          ~  P'                  n
                   Lu                  u   N i ui
                                       ~
                                           i 1

    Exact          Approximate

                ~   P  L N u   P
                                   n
  Error =     Lu           i i 
                            i 1 
Objective:
Define ui so that weighted average of Error vanishes
        Weighted Residual Formulations
Objective:
Define ui so that weighted average of Error vanishes

 Set Error relative to a weighting function Wi=0

           ~  P dV  0
     Wi Lu
    V
                                    i  1,.., n

If we choose shape functions as weighting functions
             GALERKIN FORMULATION
                Galerkin Formulation

    Let function Wi=Ni

 N1 Lu  P dV  0
       ~
V                            Algebraic System of

  N 2 Lu  P dV  0
                         n Equations and n unknowns

V
        ~




 N n Lu  P dV  0
        ~
V
            Galerkin Formulation


Set Error relative to a weighting function Wi=0


        ~  P dV  0
    Wi Lu                         i  1,.., n
  V

                          n
In General define      N ii
                         i 1


                 ~  P dV  0
               Lu
            V
         Principle of Virtual Work




 A body is in equilibrium if the internal virtual
work equals the external virtual work for every
 kinematically admissible displacement field
        Galerkin’s Method in Elasticity
                 Virtual Work

 Virtual Total Potential Energy




Compare to Total Potential Energy
   1 T
P   σ εdV   u fdV   u TdS   u i Pi
                 T         T         T

   2 V         V         S
                                  i
             Galerkin’s Formulation

•More general method

•Operated directly on Governing Equation

•Variational Form can be applied to other
governing equations

•Preffered to Rayleigh-Ritz method especially
when function to be minimized is not available.
FEM General Procedure
             Finite Element Formulation

 Rayleigh-Ritz
   1 T
P   σ εdV   u fdV   u TdS   u i Pi
                 T         T         T

   2 V         V         S
                                  i
P
    0
ui



 Galerkin

0    σ εφdV   φ fdV   φ TdS   φ
         T               T         T          T
                                              i   Pi
     V               V         S
                                          i
                   Discretization

     Define a number of nodes and
     elements
   Interpolate real and virtual displacement
    field within each element by same type
           of interpolating functions Ni

   u   N i u i  Nue
   ~                          φ   N i φ i Nφ e
                For Each Element

x  Nxe       = B ue         s = E B ue

B
   dN
             εφ  Bφe
   dx
             Introduce into variational forms

 Rayleigh Ritz
   1 T
P   σ εdV   u fdV   u TdS   u i Pi
                 T         T         T

   2 V         V         S
                                  i
       1 T
 P   u e k eu e   uT f e   uT Te  uT P
                       e          e       e
     e 2            e          e


 Galerkin
0   σ εφdV   φ fdV   φ TdS   φ
             T                   T                   T       T
                                                             i   Pi
         V                   V               S
                                                         i

0   φ k u   φ f   φ T  φP
             T
             e   e   e
                             T
                             e e
                                         T
                                         e       e
     e                   e           e
                   Assemble…
                                              u1
u  u1 u2  un 
 T
                                              u2
                                              u3
φ  1  2   n 
 T
                                         P4
                                              u4
P  P P2  Pn 
 T
      1                                  Pi   ui

K  ke          F   f e  Te   P
                                              ui+1
                 
                                              un-1
         e                e              Pn   un
   1 T                    P
P  u Ku  u F
            T
                              0
   2                      ui
                                    0  Ku  F
0  φ Ku  φ F  φ Ku  F 
     T       T        T
Boundary Conditions – Elimination Approach

      kii kij kik kil kim    ui   Pi
          k
      kji Kjj kjk kjl kjm   u
                            ujf   P
                                  Pfj
            ff    Kfs
      kki kkj kkk kkl kkm    uk = Pk
      kli klj klk kll klm   ul    Pl
         Ksf        Kss     us    Ps
      kli klj klk kll klm   u m   P m

                              -1
Kffuf+ Kfsus=Pf       uf =   Kff (Pf +   Kfsus)
Ksfuf+ Kssus=Ps         Ksfuf+ Kssus=Ps
                       Elements


• Axial Element – Linear & Higher Order
• 2D Constant Strain Triangle
• 2D 4-node Quadrilateral
• 2D Higher Order Elements
• 3D Solid Elements
• Beam Elements with and without Shear Deformation
• Plate/Shell Elements with and without Shear
  Deformation
• Special Elements (rigid links, elastic supports, infinite
  boundaries, etc)
    1
                 4   Element Stiffness Matrix ke

2                   For Example 2-D Plane Strain
             3
         

     1 T                   Ue 
U e   ε σDdV               1 T T
     2 Ve                      q e  B DBtdA q e
                             2      le

        dV  tdA               1 T             
                             q e t  B DBdAq e
                                        T

         = B qe               2  A            
        s = D B qe                          ke
                               8x8 matrix
        Element Stiffness Matrix ke

Furthermore
dV  tdA  t det Jdd

and                    Jacobian

k e  t  B DBdA 
              T

       A
       1 1
                                      Numerical
      t   B DB det Jdd
                  T
                                      Integration
      1 1
      Jacobian of Transformation


u u x u y       u   x     y   u 
       
 x  y           
                                     x 
                                          
u u x u y
       
                     u    x
                      
                        
                      
                                J        u 
                                    y 
                                          
                                       y 
                                        
 x  y 
                 Jacobian of Transformation
v v x v y       v   x     y   v 
                      
 x  y                        x 
                                          

v v x v y
       
 x  y 
                      
                        
                      
                                J
                     v    x        v 
                                    y 
                                          
                                       y 
                                        
             Gaussian Quadrature




  Karl Friedriech Gauss discovered that by a

special placement of nodes the accuracy of the

numerical integration could be greatly increased
                       Gaussian Quadrature
    Theorem on Gaussian nodes
     Let q be a polynomial of degree n such that
           b

            qx x dx  0          k  0,1,..., n - 1
                        k

           a

      Let x1,x2,…,xn be the roots of q(x). Then
b

 f x dx   w f x   w f x   w f x     w f x 
a              i
                   i        i   1   1     2    2         n   n



    with xi’s as nodes is exact for all polynomials of
    degree 2n-1.
     Gaussian Quadrature 2-point

                             W2=1   f(2)   f()
 W1=1      f(1)




       1   1 3                2  1 3
-1                                          1

       f x dx  1 f         1 3
      1
                            1 3 1 f
      1
Gauss Points and Weights
 2-Dimensional Integration

     Gaussian Quadrature
     1 1

           f  , dd 
     1 1


                   
 1     n

 1  wi f i ,  d 
   i 1           

 w w f  , 
n      n

                j   i     i    j
j 1 i 1
           2-D Integration 2-point formula
                            
                                             w1  1
                 w1w2  1       w2 w2  1    w2  1
2  1 3

                                               
                 w1w1  1       w2 w1  1
1   1 3


               1   1 3   2  1 3
              2-D Integration 2-point formula

                            

 1   1 3

                                             
 1   1 3                                 1 1


               1   1 3   2  1 3
                                              f  , dd 
                                            1 1


w1w1 f 1 ,1   w2 w1 f 2 ,1   w1w2 f 1 ,2   w2 w2 f 2 ,2 
      Choices in Numerical Integration



• Numerical Integration cannot produce exact
  results
• Accuracy of Integration is increased by using
  more integration points.
• Accuracy of computed FE solution DOES
  NOT necessarily increase by using more
  integration points.
Modeling Issues: Element Shape

     Square : Optimum Shape
     Not always possible to use

       Rectangles:         Larger ratios
       Rule of Thumb       may be used
       Ratio of sides <2   with caution


        Angular Distortion
        Internal Angle < 180o
    Modeling Issues: Degenerate Quadrilaterals
Coincident Corner Nodes
           4                                   4


           x             3
                                               x
                   x

       x                                   x       x
               x                                   x
1                                      1

                   2                                   2   3


                                       Integration Bias

                       Less accurate
    Modeling Issues: Degenerate Quadrilaterals
Three nodes collinear
          4
                                Integration Bias
          x           3                                    3
                  x
                                           4           x
                                           x
      x                               x
              x                                x
1                                 1

                  2                                2




                      Less accurate
  Modeling Issues: Degenerate Quadrilaterals


        2 nodes




Use only as necessary to improve representation of
geometry
      Do not use in place of triangular elements
              A NoNo Situation
y
                          3   (7,9)

                                             

             (6,4)                            
                      4

                                          Parent

      1                               2
                                          J singular
    (3,2)                        (9,2)

                                                   x
            All interior angles < 180
Another NoNo Situation


                            




                            

                         x, y
                         not uniquely
                         defined
     Convergence Considerations


For monotonic convergence of solution

         Requirements

Elements (mesh) must be compatible

Elements must be complete
           Monotonic Convergence

                 FEM Solution

                          Exact Solution


             No of Elements

For monotonic convergence the elements must be
   complete and the mesh must be compatible
       Mixed Order Elements

Consider the following Mesh




                      4-node


8-node

    Incompatible Elements…
              Mixed Order Elements

  We can derive a mixed order element for grading



8-node                                4-node


                  7-node

   By blending shape functions appropriately
     Convergence Considerations


For monotonic convergence of solution

         Requirements

Elements (mesh) must be compatible

Elements must be complete
          Element Completeness

    For an element to be complete

 Assumption for displacement field


ux, y   a0  a1 x  a2 y  a3 xy an x   n



         must accommodate

  •RIGID BODY MOTION
  •CONSTANT STRAIN STATE
            Element Completeness

 Consider
 ux, y   a1  a2 x  a3 y



This is not a complete polynomial


However,
             Element Completeness
Assume displacement field
ux, y   a1  a2 x  a3 y
The Computed nodal displacement
corresponding to this field
 ui  a1  a2 xi  a3 yi      i=1,…,#of nodes


Test for ELEMENT completeness
Isoparametric Formulation

u  N1u1  N 2u2  N 3u3   N i ui
          Element Completeness

 u  a1  N i a2  N i xi a3  N i yi
Isoparametric Formulation

 x   N i xi           y   N i yi

Thus, computed displacement field


        u  a1  N i a2 x  a3 y
              Element Completeness


 u  a1  N i a2 x  a3 y       Computed



 ux, y   a1  a2 x  a3 y     Assumed


In order for the computed displacements to be the
assumed ones we must satisfy


N   i   1   Condition for element completeness
      Effects of Element Distortion

Loss of predictive capability of isoparametric
element

                       No Distortion
                               1
                           x        y
                      x2       xy         y2
                        x 2y        xy2

             Behavior accurately predicted
Effects of Element Distortion


                    Angular Distortion
                            1
                        x        y
                   x2       xy         y2
                    x 2y         xy2


 Predictability is lost for all quadratic terms
Effects of Element Distortion

         Quadratic Curved Edge Distortion

                            1
                        x        y
                   x2       xy         y2
                    x 2y         xy2


 Predictability is lost for all quadratic terms
     Effects of Element Distortion




The advantage (reduced #of dof)
of using 8-node higher order element
based on an incomplete polynomial is lost
when high element distortions are present
      Effects of Element Distortion

Loss of predictive capability of isoparametric
element
                       No Distortion
9-node
                              1
                            x        y
                       x2       xy       y2
                        x 2y      xy2
                               x2y2

             Behavior accurately predicted
     Effects of Element Distortion

9-node
                      Angular Distortion
                              1
                          x        y
                     x2       xy         y2
                       x 2y        xy2

         Behavior predicted better than 8-node
     Effects of Element Distortion


9-node         Quadratic Curved Edge Distortion

                                 1
                             x        y
                        x2       xy         y2
                          x 2y        xy2

         Predictability is lost for high order terms
     Effects of Element Distortion

The advantage (reduced #of dof)
of using higher order element
based on an incomplete polynomial is lost
when high element distortions are present

For angular distortion 9-node element
shows better behavior

For Curved edge distortion all elements
give low order prediction
Polynomial Element Predictability
        Tests of Element Quality



Eigenvalue Test
  Identify Element Deficiencies



Patch Test
  Convergence of Solutions
                 Eigenvalue Test

                        4
             1                   3

                        2


Apply loads –{r} in proportion to displacements

                  r  d

                 kd  r
                Eigenvalue Test


 r  d

kd  r
                       k  I d  0

                Eigenproblem
      As many eigenvalues  as dof

     For each  there is a solution for {d}
     Displacement Modes & Stiffness Matrix

                         For all eigenvalues and modes

    Kd  d                          KD  DΛ

     d11   d12   d1n         1         
                                        
    d 21   d 22  d 2 n          2
D                           Λ           
                                    
                                        
    d n1  d n 2  d nn                n 
            Eigenvalue Test

 Scale {d} so that

 d d  1
        T


 then


d k d  d d  
   T                        T


                      2U
                Eigenvalue Test


  d k d  d d  
      T                     T


                     2U
Rigid Body Motion => System is not strained => U=0

System is strained => U=0
             Rigid Body Motion



     Rigid Body Modes


+    Element Straining Modes


    Total Number of Element Displacement Modes
    (=number of degrees of freedom)
 Displacement Modes & Stiffness Matrix
   Consider the 4-node plane stress element
                  1

                                     t=1
       1                             E=1
                                     v=0.3



8 degrees of freedom            8 modes

           Solve Eigenproblem
Displacement Modes & Stiffness Matrix




     1  0
Rigid Body Mode           2  0
                     Rigid Body Mode
 Displacement Modes & Stiffness Matrix




     3  0
Rigid Body Mode
 Displacement Modes & Stiffness Matrix




4  0.495                5  0.495
Flexural Mode             Flexural Mode
Displacement Modes & Stiffness Matrix




         6  0.769
         Shear Mode
Displacement Modes & Stiffness Matrix




7  0.769             8  1.43
                   Uniform Extension Mode
Stretching Mode
                         (breathing)
 Displacement Modes & Stiffness Matrix



The eigenvalues of the stiffness matrix display
directly how stiff the element is in the
corresponding displacement mode


                 2U
                  Patch Test

 Objective
Examine solution convergence for displacements,
  stresses and strains in a particular element
  type with mesh refinement
          Patch Test - Procedure

                          Build a simple FE
                            model

                         Consists of a Patch of
                           Elements

                          At least one internal
                             node

Load by nodal equivalent forces consistent with
  state of constant stress

Internal Node is unloaded and unsupported
         Patch Test - Procedure



                       F  s x Ht 
                          1
                          2
                        Compute results of FE
                          patch


If
(computed sx) = (assumed sx)
test passed

				
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Jun Wang Jun Wang Dr
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