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1Finite Element Method by G. R. Liu and S. S. Quek
FFinite Element Methodinite Element Method
INTRODUCTION TO MECHANICS
FOR SOLIDS AND STRUCTURES
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 2:
2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS
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
3Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
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
4Finite Element Method by G. R. Liu and S. S. Quek
Statics and dynamicsStatics 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.
5Finite Element Method by G. R. Liu and S. S. Quek
Elasticity and Elasticity and pplasticitylasticity
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.
6Finite Element Method by G. R. Liu and S. S. Quek
Isotropy and Isotropy and aanisotropynisotropy
Anisotropic: the material property varies with direction.
Composite materials: anisotropic, many material constants.
Isotropic material: property is not direction dependent, two independent material constants.
7Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditionsBoundary conditions
Displacement (essential) boundary conditions
Force (natural) boundary conditions
8Finite Element Method by G. R. Liu and S. S. Quek
Different structural componentsDifferent structural components
Truss and beam structures
x
fy1
z
fy2 y
x
fx
z
Truss member Beam member
9Finite Element Method by G. R. Liu and S. S. Quek
Different structural componentsDifferent structural components
Plate and shell structures
Plate Neutral surface
z
y x
Neutral surface
z
y x
h
h Shell
Neutral surface
Neutral surface
10Finite Element Method by G. R. Liu and S. S. Quek
EQUATIONS FOR 3D SOLIDSEQUATIONS FOR 3D SOLIDSStress and strainConstitutive equationsDynamic and static equilibrium equations
y
x
z
V
Sd
fs1
fs2
fb2 fb1
Sf
Sf Sd
Sd
11Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Stresses at a point in a 3D solid:
zxxz
yzzy
yxxy
Txx yy zz yz xz xy
xy
xz
xx
yy
yz
yx
zy
zz
zx
y
x
z
yy
yz
yx
zy
zz
zx
xy
xz
xx
12Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Strains
Txx yy zz yz xz xy
y
w
z
v
x
w
z
u
x
v
y
u
z
w
y
v
x
u
yzxzxy
zzyyxx
,
,
,
,
13Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Strains in matrix form
LUwhere
0 0
0 0
0 0
0
0
0
x
y
z
z y
z x
y x
L
w
v
u
U
14Finite Element Method by G. R. Liu and S. S. Quek
Constitutive equationsConstitutive equations
= c or
xy
xz
yz
zz
yy
xx
xy
xz
yz
zz
yy
xx
c
ccsy
ccc
cccc
ccccc
cccccc
66
5655
464544
36353433
2625242322
161514131211
.
15Finite Element Method by G. R. Liu and S. S. Quek
Constitutive equationsConstitutive equations
For isotropic materials
2
02
.
002
000
000
000
1211
1211
1211
11
1211
121211
cc
ccsy
ccc
cc
ccc
c
)1)(21(
)1(11
Ec
)1)(21(12
E
c
Gcc
21211
)1(2 E
G
,
,
16Finite Element Method by G. R. Liu and S. S. Quek
Dynamic equilibrium equationsDynamic equilibrium equations
Consider stresses on an infinitely small block
xy +d xy
xz +d xz
xx +d xx
yy +d yy
yz +d yz
yx +d yx
zy +d zy
zz +d zz zx +d zx
y x
z
yy yz
yx
zy zz
zx
xy
xz
xx
d x d y
d z
17Finite Element Method by G. R. Liu and S. S. Quek
Dynamic equilibrium equationsDynamic equilibrium equations
Equilibrium of forces in x direction including the inertia forces
Note: d d ,
d d ,
d d
xxxx
yxyx
zxzx
xx
yy
zz
xy+dxy
xz+dxz
xx+dxx
yy+dyy
yz+dyz
yx+dyx
zy+dzy
zz+dzz
zx+dzx
yy
yz
yx
zy
zz
zx
xy
xz
xx
dx dy
dz
external force inertial force
( d )d d d d ( d )d d d d
( d )d d d d d d d
xx xx xx yx yx yx
zx zx zx x
y z y z x z x z
x y x y f u x y z
18Finite Element Method by G. R. Liu and S. S. Quek
Dynamic equilibrium equationsDynamic equilibrium equations
Hence, equilibrium equation in x direction
ufzyx xzxyxxx
Equilibrium equations in y and z directions
vfzyx yzyyyxy
wfzyx zzzyzxz
19Finite Element Method by G. R. Liu and S. S. Quek
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
In matrix formT
b L f Uor
Tb L cLU f U
For static case
0Tb L cLU f
Note:
x
b y
z
f
f
f
f
20Finite Element Method by G. R. Liu and S. S. Quek
EQUATIONS FOR 2D SOLIDSEQUATIONS FOR 2D SOLIDS
Plane stressPlane strain
x
y
z
y
x
21Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Txx yy zz yz xz xy
xx
yy
xy
xx
yy
xy
x
v
y
u
y
v
x
uxyyyxx
,,
(3D)
22Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Strains in matrix form
ε LU
where
0
0
x
y
y x
L, u
v
U
23Finite Element Method by G. R. Liu and S. S. Quek
Constitutive equationsConstitutive equations
= c
2
1 0
1 01
0 0 1 / 2
E
c (For plane stress)
1 01
(1 )1 0
(1 )(1 2 ) 11 2
0 02(1 )
E
c(For plane strain)
24Finite Element Method by G. R. Liu and S. S. Quek
Dynamic equilibrium equationsDynamic equilibrium equations
ufzyx xzxyxxx
ufyx xyxxx
vfyx yyyxy
(3D)
25Finite Element Method by G. R. Liu and S. S. Quek
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
In matrix formT
b L f Uor
Tb L cLU f U
For static case
0Tb L cLU f
Note: x
by
f
f
f
26Finite Element Method by G. R. Liu and S. S. Quek
EQUATIONS FOR TRUSS EQUATIONS FOR TRUSS MEMBERSMEMBERS
fx
y
x
z
xy
yy
xx
σ xx
x
ux
27Finite Element Method by G. R. Liu and S. S. Quek
Constitutive equationsConstitutive equations
Hooke’s law in 1D
= E
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
ufx x
x
0
xx f
x
(Static)
28Finite Element Method by G. R. Liu and S. S. Quek
EQUATIONS FOR BEAMSEQUATIONS FOR BEAMS Stress and strainConstitutive equationsMoments and shear forcesDynamic and static equilibrium equations
x
fy1
y
fy2
29Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Euler–Bernoulli theory
Centroidal axis
x
y
30Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
0xy
yu
x
v
Assumption of thin beam
Sections remain normal
Slope of the deflection curve
yLvx
vy
x
uxx
2
2
2
2
x L
where
xx = E xx yELvxx
31Finite Element Method by G. R. Liu and S. S. Quek
Constitutive equationsConstitutive equations
xx = E xx
Moments and shear forcesMoments and shear forcesConsider isolated beam cell of length dx
dx
Mz
Mz + dMz
Q Q + dQ
(fy(x)- vA ) dx
x
y
32Finite Element Method by G. R. Liu and S. S. Quek
Moments and shear forcesMoments and shear forces
The stress and moment
M M
dx
y
x
xx
33Finite Element Method by G. R. Liu and S. S. Quek
Moments and shear Moments and shear forcesforces
Since yELvxx
Therefore,2
22
d ( d )z xx z z
A A
vM y A E y A Lv EI Lv EI
x
Where
2dz
A
I y A (Second moment of area about z axis – dependent on shape and dimensions of cross-section)
M M
dx
y
x
xx
34Finite Element Method by G. R. Liu and S. S. Quek
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
Forces in the x direction
0)( dxvAxfdQ y
vAxfdx
dQy
Moments about point A
0))(2
1 2 x(dvA-fxdQdM yz
dx
M z
M z + dM z
Q Q + dQ
(fy(x)- vA ) dx
A
Qdx
dM z 3
3
x
vEIQ z
35Finite Element Method by G. R. Liu and S. S. Quek
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
vAxfdx
dQy
Therefore,
yz fvAx
vEI
4
4
yz fx
vEI
4
4
(Static)
36Finite Element Method by G. R. Liu and S. S. Quek
EQUATIONS FOR PLATESEQUATIONS FOR PLATES Stress and strainConstitutive equationsMoments and shear forcesDynamic and static equilibrium equationsMindlin plate
z, w
h
fzy, v
x, u
37Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Thin plate theory or Classical Plate Theory (CPT)
Centroidal axis
x
y
38Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Assumes that xz = 0, yz = 0
x
wzu
y
wzv
,
Therefore,
2
2
x
wz
x
uxx
2
2
y
wz
y
vyy
y x
wz
x
v
y
uxy
2
2
,
39Finite Element Method by G. R. Liu and S. S. Quek
Stress and strainStress and strain
Strains in matrix form
= z Lw
where 2
2
2
2
2
x
y
x y
L
40Finite Element Method by G. R. Liu and S. S. Quek
Constitutive equationsConstitutive equations
= c where c has the same form for the plane stress case of 2D solids
2
1 0
1 01
0 0 1 / 2
E
c
41Finite Element Method by G. R. Liu and S. S. Quek
Moments and shear forcesMoments and shear forces
Stresses on isolated plate cellz
x
y
fz
h
xyxxxz
yx
yy
yz
O
42Finite Element Method by G. R. Liu and S. S. Quek
Moments and shear forcesMoments and shear forces
Moments and shear forces on a plate cell dx x dy
z
x
yO
dx
dy
Qy
MyMyx
Qy+dQy
Myx+dMyx
My+dMy
Qx
MxMxy
Qx+dQx
Mxy+dMxyMx+dMx
43Finite Element Method by G. R. Liu and S. S. Quek
Moments and shear forcesMoments and shear forces
= c = c z Lw
Like beams,
32d ( d )
12
x
p y
A Axy
Mh
M z z z z w w
M
M c L cL
Note thatxd
x
QQd x
x
ydy
QQd y
y
,
44Finite Element Method by G. R. Liu and S. S. Quek
Moments and shear forcesMoments and shear forces
Therefore, equilibrium of forces in z direction
0)()()(
ydxdwhfxdydy
Qydxd
x
Qz
yx
or
whfy
Q
x
Qz
yx
Qy
Qx
Mx+dMx
Qx+dQx My+dMy
My
Myx+dMyx
Mxy+dMxy
Qy+dQy Myx o
x
y
x
dy
dx
A A Moments about A-A
y
M
x
MQ xyx
x
45Finite Element Method by G. R. Liu and S. S. Quek
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
y
M
x
MQ xyx
x
whfy
Q
x
Qz
yx
w
yx
y
xh
M
M
M
xy
y
x
12
2
2
2
2
2
3
c
46Finite Element Method by G. R. Liu and S. S. Quek
Dynamic and static equilibrium equationsDynamic and static equilibrium equations
zfwhy
w
yx
w
x
wD
)2(
4
4
22
4
4
4
zfy
w
yx
w
x
wD
)2(4
4
22
4
4
4
where
)1(12 2
3
EhD
(Static)
48Finite Element Method by G. R. Liu and S. S. Quek
Mindlin plateMindlin plate
yzu xzv ,
Therefore, in-plane strains = z L
where0
0
x
y
x y
L ,y
x