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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
 Dynamics deals with the mechanics of solids and
 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
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
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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