1
FFinite Element Methodinite Element Method
FEM FOR TRUSSES
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 4:
2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS INTRODUCTION FEM EQUATIONS
– Shape functions construction– Strain matrix– Element matrices in local coordinate system– Element matrices in global coordinate system– Boundary conditions– Recovering stress and strain
EXAMPLE– Remarks
HIGHER ORDER ELEMENTS
3Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
Truss members are for the analysis of skeletal type systems – planar trusses and space trusses.
A truss element is a straight bar of an arbitrary cross-section, which can deform only in its axis direction when it is subjected to axial forces.
Truss elements are also termed as bar elements. In planar trusses, there are two components in the x and y
directions for the displacement as well as forces at a node. For space trusses, there will be three components in the x,
y and z directions for both displacement and forces at a node.
4Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
In trusses, the truss or bar members are joined together by pins or hinges (not by welding), so that there are only forces (not moments) transmitted between bars.
It is assumed that the element has a uniform cross-section.
5Finite Element Method by G. R. Liu and S. S. Quek
Example of a truss structureExample of a truss structure
6Finite Element Method by G. R. Liu and S. S. Quek
FEM EQUATIONSFEM EQUATIONS
Shape functions constructionStrain matrixElement matrices in local coordinate systemElement matrices in global coordinate
systemBoundary conditionsRecovering stress and strain
7Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
Consider a truss element
D3i - 1
D3i - 2
D3i
D3j - 1
D3j - 2
D3j
le
x
u1
u2
u(x)
fs1
fx
global node j local node 2
global node i local node 1
fs2
X
Y
Z
o
0
8Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
00 1
1
( ) 1h T
T
u x x x
p α
p α
Let
Note: Number of terms of basis function, xn determined by n = nd - 1
At x = 0, u(x=0) = u1
At x = le, u(x=le) = u2
1 0
2 1
1 0
1 e
u
lu
0 1
1 2
1 0
1 1
e e
u
ul l
1 2
1 1
2 2
( ) ( )
1 0
( ) 1 1 ( )1 1
( )
h Te
e ee e N x N x
e
u ux xu x x x
u ul ll l
x
P α N d
dN
9Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
)()()( 21 xNxNx N
1
2
( ) 1
( )
e
e
xN x
l
xN x
l
N1 N2
x
le 0
1 1
1 2
2 11 1 2 2 1( ) ( ) ( )
e
u uu x N x u N x u u x
l
(Linear element)
10Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
2 11 1 2 2 1( ) ( ) ( )
e
u uu x N x u N x u u x
l
2 1x
e
u uu
x l
or
eex Lx
uBdNd
1 11
e e e e
x xL
x l l l l
B Nwhere
11Finite Element Method by G. R. Liu and S. S. Quek
Element Matrices in the Local Coordinate Element Matrices in the Local Coordinate SystemSystem
0
1
1 11 1d d
1 1 1e
e
l eTe
e e eV
e
l AEV A E x
l l l
l
k B cB
Note: ke is symmetrical
Proof: BcB][BcBB]cB TTTTTTT [
12Finite Element Method by G. R. Liu and S. S. Quek
Element Matrices in the Local Coordinate Element Matrices in the Local Coordinate SystemSystem
1 1 1 2
02 1 2 2
2 1d d
1 26
e
e
lT ee e
V
N N N N A lV A l x
N N N N
m N N
Note: me is symmetrical too
111
022
1
2d d d
2
e
e e
x esl sT T
e b s xs x eV S
s
f lffN
f V f S f xfN f l
f
f N N
13Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Perform coordinate transformation
Truss in space (spatial truss) and truss in plane (planar truss)
14Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Spatial truss
ee TDd (Relationship between local DOFs and global DOFs)
whereeijijij
ijijij
nml
nml
000
000T
j
j
j
i
i
i
e
D
D
D
D
D
D
3
13
23
3
13
23
D,
cos( , )
cos( , )
cos( , )
j iij
e
j iij
e
j iij
e
X Xl x X
l
Y Ym x Y
l
Z Zn x Z
l
Direction cosines
(2x1)
(6x1)
15Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Spatial truss (Cont’d)
2 2 2( ) ( ) ( )e j i j i j il X X Y Y Z Z
Transformation applies to force vector as well:
ee TFf where
j
j
j
i
i
i
e
F
F
F
F
F
F
3
13
23
3
13
23
F
16Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Spatial truss (Cont’d)
ee TDd
eeeee fdmdk eeeee fDTmTDk
eT
eeT
eeT fTDTmTDTkT )()(
eeeee FDMDK
17Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Spatial truss (Cont’d)
2 2
2 2
2 2
2 2
2 2
Te e
ij ij ij ij ij ij ij ij ij ij
ij ij ij ij ij ij ij ij ij ij
ij ij ij ij ij ij ij ij ij ij
ij ij ij ij ij ij ij ij ij ije
ij ij ij ij ij ij ij ij ij ij
ij
l l m l n l l m l n
l m m m n l m m m n
l n m n n l n m n nAE
l l m l n l l m l nl
l m m m n l m m m n
l n
K T k T
2 2ij ij ij ij ij ij ij ij ijm n n l n m n n
18Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Spatial truss (Cont’d)
2 2
2 2
2 2
2 2
2 2
2 2 2
2 2 2
2 2 2
2 2 26
2 2 2
Te e
ij ij ij ij ij ij ij ij ij ij
ij ij ij ij ij ij ij ij ij ij
ij ij ij ij ij ij ij ij ij ije
ij ij ij ij ij ij ij ij ij ij
ij ij ij ij ij ij ij ij ij ij
ij
l l m l n l l m l n
l m m m n l m m m n
l n m n n l n m n nA l
l l m l n l l m l n
l m m m n l m m m n
l n
M T m T
2 22 2 2ij ij ij ij ij ij ij ij ijm n n l n m n n
19Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Spatial truss (Cont’d)
1
1
1
1
1
1
( )2
( )2
( )2
( )2
( )2
( )2
x es ij
x es ij
x es ij
Te e
y es ij
y es ij
y es ij
f lf l
f lf m
f lf n
f lf l
f lf m
f lf n
F T f Note:1
1
2
2
x es
ex e
s
f lf
f lf
f
20Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Planar truss
ee TDd
where
ijij
ijij
ml
ml
00
00T ,
j
j
i
i
e
D
D
D
D
2
12
2
12
D
j
j
i
i
e
F
F
F
F
2
12
2
12
FSimilarly (4x1)
21Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Planar truss (Cont’d)
2 2
2 2
2 2
2 2
ij ij ij ij ij ij
ij ij ij ij ij ijTe e
ij ij ij ij ij ije
ij ij ij ij ij ij
l l m l l m
l m m l m mAE
l l m l l ml
l m m l m m
K T k T
22Finite Element Method by G. R. Liu and S. S. Quek
Element matrices in global coordinate Element matrices in global coordinate systemsystem
Planar truss (Cont’d)
2 2
2 2
2 2
2 2
2 2
2 2
2 26
2 2
ij ij ij ij ij ij
ij ij ij ij ij ijT ee e
ij ij ij ij ij ij
ij ij ij ij ij ij
l l m l l m
l m m l m mA l
l l m l l m
l m m l m m
M T m T
23Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditionsBoundary conditionsSingular K matrix rigid body movementConstrained by supportsImpose boundary conditions cancellation
of rows and columns in stiffness matrix, hence K becomes SPD
Recovering stress and strainRecovering stress and strain
x e eE E Bd BTD (Hooke’s law)
x
24Finite Element Method by G. R. Liu and S. S. Quek
EXAMPLEEXAMPLE
Consider a bar of uniform cross-sectional area shown in the figure. The bar is fixed at one end and is subjected to a horizontal load of P at the free end. The dimensions of the bar are shown in the figure and the beam is made of an isotropic material with Young’s modulus E.
P
l
25Finite Element Method by G. R. Liu and S. S. Quek
EXAMPLEEXAMPLE
Exact solution of2
20x
uE f
x
( )
Pu x x
EA , stress: x
P
A :
FEM:
(1 truss element)1 1
1 1e
AE
l
K = k
1 1
2 2
?1 1
1 1
u FAE
u F Pl
1 1
2 2
?1 1
1 1
u FAE
u F Pl
2
Plu
AE 1
2
0( ) ( ) 1 1e
ux x x x Pu x x xPl
ul l l l EAEA
N d
2
01 1x e
PE E
ul l A
Bd
26Finite Element Method by G. R. Liu and S. S. Quek
RemarksRemarks
FE approximation = exact solution in example Exact solution for axial deformation is a first order
polynomial (same as shape functions used) Hamilton’s principle – best possible solution Reproduction property
27Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
1 2 3 4 1 2 3
Quadratic element Cubic element
1
2
3
1( ) (1 )
21
( ) (1 )2
( ) (1 )(1 )
N
N
N
21
22
23
24
1( ) (1 )(1 9 )
161
( ) (1 )(1 9 )16
9( ) (1 3 )(1 )
169
( ) (1 3 )(1 )16
N
N
N
N