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Finite Element Method_1_

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									 The Finite Element Method
                 A Practical Course




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 3D SOLIDS
 EQUATIONS FOR 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.
 The stresses lead to strains, which can be
  observed as a deformation or displacement.
 Solid mechanics and structural mechanics
Statics and dynamics

 Forces can be static and/or dynamic.
 Statics deals with the mechanics of solids and
  structures subjected to static loads.
 Dynamics deals with the mechanics of solids and
  structures subjected 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
                        z          Sd            fs2

                                                Sf
         fs1
                                        V
           Sf
                            fb1   fb2                  Sd
                                            y


                   Sd
               x
                                                      z                Sd               fs2
Different structural                                                                Sf
                                     fs1
components                                                                  V
                                       Sf
                                                          fb1         fb2                     Sd
                                                                                y

   Truss and beam
                                                 Sd
    structures                             x




                      y                                         fy2

          z                                    fy1
                             x                                                      x
                                 z
                 fx

              Truss member
                                                     Beam member
          Different structural components
                                                                                     z

             Plate and shell                                                                y
                                                                                 x
              structures
                                                                                                     Neutral surface
                                                                                                 h

                                                                                     Plate
                                                       Neutral surface

                   z          Sd            fs2

                                           Sf                                z
fs1
                                   V                                                 y
  Sf                                                                     x
                       fb1   fb2                  Sd
                                       y

                                                                                                         Neutral surface
              Sd                                                                                     h
      x                                                                      Shell
                                                       Neutral surface
    EQUATIONS FOR 3D SOLIDS
 Stress and strain
 Constitutive equations
 Dynamic and static equilibrium equations
                          z          Sd            fs2

                                                  Sf
           fs1
                                          V
             Sf
                              fb1   fb2                  Sd
                                              y


                     Sd
                 x
Stress and strain

   Stresses at a point in a 3D solid:
                                                  z

 xz   zx                                                  zz
                                                 zx                zy


 zy   yz                   yy         yx
                                                  xy
                                                  xz xz
                                                                    xx yz
                                                                                yy



 xy   yx
                                    yz                 xy               yx         y
                                          xx
                                                zy           zx




                                                                               
                                                       zz


    xx  yy  zz  yz  xz  xy
     T                    x
Stress and strain

   Strains
      T
            
       xx  yy  zz  yz  xz  xy                         
           u             v           w
     xx      ,  yy        ,  zz 
            x            y            z
           u v                u w               v w
     xy             ,  xz            ,  yz     
            y  x               z  x              z  y
Stress and strain

   Strains in matrix form
      LU
    where
         x  0    0 
        0     y      
                   0           u 
        0          z          
               0             U  v 
     L                         w
        0     z  y          
         z  0    x 
                       
        y  x
                   0  
Constitutive equations
=c
or

 xx   c11   c12   c13   c14   c15   c16   xx 
            c22   c23   c24   c25         
                                        c26  yy 
 yy                                      
 zz               c33   c34   c35   c36   zz 
                                          
 yz                     c44   c45   c46   yz 
 xz         sy.               c55   c56   xz 
                                          
 xy                                 c66   xy 
Constitutive equations
   For isotropic materials
       c11         c12       c12          0          0           0      
                   c11       c12          0          0           0      
                                                                        
                             c11          0          0           0      
                                      c11  c12                         
c                                                   0            0 
                                          2                             
                                                  c11  c12
                    sy.                                            0 
                                                      2                 
                                                              c11  c12 
  
                                                                  2    
                  E (1   )                  E
      c11                      , c12 
              (1  2 )(1   )         (1  2 )(1   )
      c11  c12                        E
                G ,         G
          2                         2(1   )
Dynamic equilibrium equations

   Consider stresses on an infinitely small
    block                   +d               zz      zz

                                  zx+dzx           zy+dzy

                                                     xx yz+dyz
                                       xy
                  yy                                         yy+dyy
                               yx xz+dxz
              z         yz                   xz       yx+dyx
                                         xy+dxy
                  dz          xx+dxx
                                   zy         zx
                                                            dx
                                  dy     zz

                              y
          x
    Dynamic equilibrium equations

       Equilibrium of forces in x direction
        including the inertia forces
( xx  d xx )dydz   xx dydz  ( yx  d yx )dxdz   yx dxdz 
     ( zx  d zx )dxdy   zx dxdy  f x dxdydz  udxdydz
                                                     
                                                                               zz+dzz
                                                                  zx+dzx            zy+dzy

                         xx                                                         xx yz+dyz
          Note: d xx        dx ,                                    xy
                         x                         yy                                        yy+dyy
                                                                 yx xz+dxz
                         yx                                                   xz
                d yx        dy ,                        yz              xy+dxy      yx+dyx
                         y                         dz          xx+dxx
                         zx                                       zy         zx
                d zx        dz                                                          dx
                         z
                                                                   dy zz
Dynamic equilibrium equations

   Hence, equilibrium equation in x direction
   xx  yx  zx
                    f x  u
                              
   x      y    z
 Equilibrium equations in y and z directions
   xy  yy  zy
                    f y  v
   x      y    z
   xz  yz  zz
                    f z  w
   x      y    z
Dynamic and static equilibrium equations

   In matrix form                   fx 
                                     
    LT   fb   U    Note:   fb   f y 
                                    f 
    or                               z
    L cLU  fb   U
         T


   For static case
     L cLU  fb  0
             T
Boundary Condition
   Displacement (essential) boundary condition

        u u vv ww
   Force (natural) boundary condition
                    nσ  t
              nx    0   0    0    nz   ny 
                                          
            n0    ny   0    nz   0    nx 
              0    0    nz   ny   nx   0
                                          
EQUATIONS FOR 2D SOLIDS
                           x




                       z
                   x



y
                                        y


    Plane stress
                               Plane strain
Stress and strain

   xx  yy  zz  yz  xz  xy 
  T
                                                 (3D)



                xx                   xx 
                                      
              yy                 yy 
                                     
                xy                    xy 
             u           v           u v
       xx      ,  yy      ,  xy        
              x           y           y  x
Stress and strain

   Strains in matrix form
    ε  LU
    where
                       
                      0 
                x      
                          ,     u 
             L 0              U 
                      y         v 
                      
               
                y
                      x
                         
                         
Constitutive equations
=c
           1            0     
      E                        
 c      2 
             1          0      
                                            (For plane stress)
    1 
           0 0
                    1   / 2
                                

                                            
                       1     1 
                                        0     
                                             
        E (1  )  
                                         0 
                                                        (For plane strain)
 c                            1
    (1   )(1  2 ) 1                    
                                             
                       0             1  2 
                               0
                      
                                    2(1  ) 
                                              
Dynamic equilibrium equations

  xx  yx  zx
                  f x  u
                                     (3D)
  x    y    z

       xx  yx
                  f x  u
                           
       x    y
      xy        yy
                         f y  v
                                  
      x          y
Dynamic and static equilibrium equations

   In matrix form
                                          fx 
         L   fb   U
          T                 Note:   fb   
    or                                    fy 

         L cLU  fb   U
          T


   For static case
         LT cLU  fb  0
     EQUATIONS FOR TRUSS
          MEMBERS

     xx 
     
σ   yy       xx
     
     xy                    y
                         z
                                 x

      u
 x 
       x      fx
Constitutive equations

   Hooke’s law in 1D
    =E
Dynamic and static equilibrium equations
     x
          f x  u
                  
     x
     x
          fx  0      (Static)
     x
    EQUATIONS FOR BEAMS
 Stress and strain
 Constitutive equations
 Moments and shear forces
 Dynamic and static equilibrium equations
                             fy2
                                             x
                      fy1


             y
Stress and strain
    Euler–Bernoulli theory
                          y

                              x


             Centroidal
             axis


                          
Stress and strain
   xy  0      Assumption of thin beam

  u   y      Sections remain normal

     v
              Slope of the deflection curve
      x
         u        v2
                                                    2
   xx       y         yLv     where       L
          x       x 2
                                                    x2

  xx = E xx            xx   yELv
Constitutive equations
xx = E xx

Moments and shear forces
 Consider isolated beam cell of length dx
        y
                 (fy(x)-   Av ) dx
                             

                                  Mz + dMz
            Mz

                              Q + dQ
                 Q
                      dx
                                             x
Moments and shear forces

   The stress and moment

                xx




        M                       M
            y




                      dx
                            x
                                             xx
Moments and shear forces
                                    M                                  M
                                         y
   Since    xx   yELv
                                                     dx
   Therefore,                                                      x


                                                             2v
     M z     xx ydA  E (    y 2dA) Lv  EI z Lv  EI z 2
                A            A
                                                            x
   Where

     I z   y dA   2    (Second moment of area about z
                         axis – dependent on shape and
           A
                         dimensions of cross-section)
Dynamic and static equilibrium equations
Forces in the x direction
dQ  ( f y x   Av )dx  0
                                       (fy(x)-   Av ) dx
                                                     


    f y x   Av
dQ                                                         Mz + dMz
                                Mz
dx                                                 A
                                                       Q + dQ
Moments about point A                    Q
                                              dx
            1
dM z  Qdx  ( f y - Av )(dx) 2  0
                       
            2
      dM z                         3v
          Q       Q   EI z 3
       dx                         x
Dynamic and static equilibrium equations
Therefore,

    f y x   Av
dQ
                  
dx
           4v
     EI z 4  Av  f y
                 
          x

           4v
      EI z 4  f y     (Static)
          x
    EQUATIONS FOR PLATES
   Stress and strain
   Constitutive equations
   Moments and shear forces
   Dynamic and static equilibrium equations
   Reissner-Mindlin platey, v  f
                                z
                   z, w




                                    h          x, u
Stress and strain

   Thin plate theory or Classical Plate Theory (CPT

                            zy



                                       x


               Centroidal
              Neutral
               axis
              surface



                            
Stress and strain
Assumes that xz = 0, yz = 0

            w                 w
     u  z         ,   v  z
             x                 y
 Therefore,
         u       2w                  v       2w
   xx       z           ,    yy       z
          x       x 2
                                        y       y 2


                  u v           2w
            xy          2 z
                   y  x         x y
Stress and strain
   Strains in matrix form
     = z Lw
    where           2 
                        2 
                     x 
                    2 
                 L     2 
                     y 
                    2 
                           
                     x y 
                           
Constitutive equations

   =c
    where c has the same form for the plane
    stress case of 2D solids

               1        0      
          E                     
     c      2 
                 1       0      
        1 
               0 0
                     1   / 2
                                 
Moments and shear forces
    Stresses on isolated plate cell
                         z




                                                         y
         h         O


                             fz                    yz

                                             yy


                                       yx


             xz   xx       xy

 x
Moments and shear forces

   Moments and shear forces on a plate cell dx x dy
                         z


                                             Qx
                                                  Mx
                                     Mxy
              Qy
                         O                                        Qy+dQy            y
         My        Myx

                                                                           My+dMy
                                                       Myx+dMyx

                         Qx+dQx
                                                                             dx

                                  Mxy+dMxy
               Mx+dMx



x
                    dy
Moments and shear forces
=c              =  c z Lw

Like beams,
       Mx 
                                         h3
M p   M y    zdz  c(  z dz )Lw   cLw
                                2

      M  A                 A
                                           12
       xy 
Note that         Qx                    Qy
            dQx      dx   ,     dQy          dy
                   x                    y
Moments and shear forces
Therefore, equilibrium of forces in z direction
  Qx           Qy
(     dx)dy  (     dy)dx  ( f z  hw)dxdy  0
                                      
   x            y
 or                                               x


 Qx Qy
                                                                      Qx

          f z  hw
                                 Qy

  x   y                                                                        Qy+dQy        y
                              My        Myx   o

                                                  Qx+dQx              Myx+dMyx       My+dMy
                                    A
 Moments about A-A                                         Mxy+dMxy
                                                                                 A        dx
                                        Mx+dMx


      M x M xy
                          x
                                                      dy

 Qx      
       x   y
Dynamic and static equilibrium equations
                                          2 
                                               2 
                             Mx          x 
                                    h3   2 
                             My    c       2 
                                                    w
                             M      12     y
                              xy               
                                            2 
                                           x y 
                                                 
                      M x M xy
                 Qx      
                       x   y
 Qx Qy
          f z  hw
                    
  x   y
Dynamic and static equilibrium equations

       4w    4w  4w
    D( 4  2 2 2  4 )  hw  f z
                            
       x    x y y

       4w    4w  4w
     D( 4  2 2 2  4 )  f z   (Static)
       x    x y y

    where         Eh3
            D
               12(1  2 )
Reissner-Mindlin plate




                         Neutral
                         surface
Reissner-Mindlin plate

u  z y    ,   v   z x
Therefore, in-plane strains    = z L
where                    
                      0 
             x           
                                    y 
         L 0              ,       
                       y            x 
                        
                         
             x
                        y
                           
Reissner-Mindlin plate
                                            w 
                              xz    y  x 
                                                
Transverse shear strains γ               w 
                              yz    x  
                                     
                                             y 
                                                 

Transverse shear stress


     xz  G 0 
              [Ds ]
     yz   0 G

                            = p2/12 or 5/6

								
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