the constant strain triangle (cst)
TRANSCRIPT
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MANE 4240 & CIVL 4240Introduction to Finite Elements
Constant Strain Triangle (CST)
Prof. Suvranu De
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Reading assignment:
Logan 6.2-6.5 + Lecture notes
Summary:
• Computation of shape functions for constant strain triangle• Properties of the shape functions • Computation of strain-displacement matrix• Computation of element stiffness matrix• Computation of nodal loads due to body forces• Computation of nodal loads due to traction• Recommendations for use• Example problems
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Finite element formulation for 2D:
Step 1: Divide the body into finite elements connected to each other through special points (“nodes”)
x
y
Su
STu
v
x
px
py
Element ‘e’
3
21
4
y
xvu
1
2
3
4
u1
u2
u3
u4
v4
v3
v2
v1
4
4
3
3
2
2
1
1
vuvuvuvu
d
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44332211
44332211
vy)(x,N vy)(x,N vy)(x,N vy)(x,Ny)(x,vu y)(x,Nu y)(x,Nu y)(x,Nu y)(x,Ny)(x,u
4
4
3
3
2
2
1
1
4321
4321
vuvuvuvu
N0N0N0N00N0N0N0N
y)(x,vy)(x,u
u
dNu
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TASK 2: APPROXIMATE THE STRAIN and STRESS WITHIN EACH ELEMENT
...... vy)(x,N
uy)(x,Ny)(x,vy)(x,u
vy)(x,N
vy)(x,N
v y)(x,N
vy)(x,Ny)(x,v
u y)(x,N
u y)(x,N
u y)(x,N
u y)(x,Ny)(x,u
11
11
xy
44
33
22
11
y
44
33
22
11
x
xyxy
yyyyy
xxxxx
Approximation of the strain in element ‘e’
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4
4
3
3
2
2
1
1
B
44332211
4321
4321
xy
vuvuvuvu
y)(x,Ny)(x,Ny)(x,Ny)(x,N y)(x,N y)(x,Ny)(x,Ny)(x,N
y)(x,N0
y)(x,N0
y)(x,N0
y)(x,N0
0y)(x,N
0y)(x,N
0 y)(x,N
0y)(x,N
xyxyxyxy
yyyy
xxxx
y
x
dBε
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Displacement approximation in terms of shape functions
Strain approximation in terms of strain-displacement matrix
Stress approximation
Summary: For each element
Element stiffness matrix
Element nodal load vector
dNu
dBD
dBε
eVk dVBDBT
S
eT
b
e
f
S ST
f
V
T dSTdVXf NN
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Constant Strain Triangle (CST) : Simplest 2D finite element
• 3 nodes per element• 2 dofs per node (each node can move in x- and y- directions)• Hence 6 dofs per element
x
yu3
v3
v1
u1
u2
v2
2
3
1
(x,y)
vu
(x1,y1)
(x2,y2)
(x3,y3)
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166212 dNu
3
3
2
2
1
1
321
321
vuvuvu
N0N0N00N0N0N
y)(x,vy)(x,u
u
The displacement approximation in terms of shape functions is
321
321
N0N0N00N0N0N
N
1 1 2 2 3 3u (x,y) u u uN N N
1 1 2 2 3 3v(x,y) v v vN N N
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Formula for the shape functions are
Aycxba
NA
ycxbaN
Aycxba
N
2
2
2
3333
2222
1111
12321312213
31213231132
23132123321
33
22
11
x1x1x1
det21
xxcyybyxyxaxxcyybyxyxaxxcyybyxyxa
yyy
triangleofareaA
where
x
yu3
v3
v1
u1
u2
v2
2
3
1
(x,y)
vu
(x1,y1)
(x2,y2)
(x3,y3)
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Properties of the shape functions:
1. The shape functions N1, N2 and N3 are linear functions of x and y
x
y
2
3
1
1
N1
2
3
1
N2
12
3
1
1
N3
nodesotheratinodeat
0''1
N i
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2. At every point in the domain
yy
xx
i
i
3
1ii
3
1ii
3
1ii
N
N
1N
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3. Geometric interpretation of the shape functionsAt any point P(x,y) that the shape functions are evaluated,
x
y
2
3
1P (x,y)
A1A3
A2
AAAAAA
33
22
11
N
N
N
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Approximation of the strains
xy
u
v
u v
x
y
x
Bdy
y x
332211
321
321
332211
321
321
000000
21
y)(x,Ny)(x,N y)(x,N y)(x,Ny)(x,Ny)(x,N
y)(x,N0
y)(x,N0
y)(x,N0
0y)(x,N
0 y)(x,N
0y)(x,N
bcbcbcccc
bbb
A
xyxyxy
yyy
xxx
B
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Element stresses (constant inside each element)
dBD
Inside each element, all components of strain are constant: hence the name Constant Strain Triangle
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IMPORTANT NOTE:1. The displacement field is continuous across element boundaries2. The strains and stresses are NOT continuous across element boundaries
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Element stiffness matrix
eVk dVBDBT
AtkeV
BDBdVBDB TT t=thickness of the elementA=surface area of the element
Since B is constant
t
A
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S
eT
b
e
f
S ST
f
V
T dSTdVXf NN
Element nodal load vector
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Element nodal load vector due to body forces
ee A
T
V
T
bdAXtdVXf NN
e
e
e
e
e
e
A b
A a
A b
A a
A b
A a
yb
xb
yb
xb
yb
xb
b
dAXNt
dAXNt
dAXNt
dAXNt
dAXNt
dAXNt
ffffff
f
3
3
2
2
1
1
3
3
2
2
1
1
x
yfb3x
fb3y
fb1y
fb1x
fb2x
fb2y
2
3
1
(x,y)
Xb Xa
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EXAMPLE:
If Xa=1 and Xb=0
03
03
03
0
0
0
3
2
1
3
3
2
2
1
1
3
3
2
2
1
1
tA
tA
tA
dANt
dANt
dANt
dAXNt
dAXNt
dAXNt
dAXNt
dAXNt
dAXNt
ffffff
f
e
e
e
e
e
e
e
e
e
A
A
A
A b
A a
A b
A a
A b
A a
yb
xb
yb
xb
yb
xb
b
![Page 21: The Constant Strain Triangle (CST)](https://reader031.vdocuments.mx/reader031/viewer/2022021418/58690d8b1a28ab3f7c8b9741/html5/thumbnails/21.jpg)
Element nodal load vector due to traction
eTS
ST
SdSTf N
EXAMPLE:
x
yfS3x
fS3y
fS1y
fS1x
2
3
1
el Salong
T
SdSTtf
31 31N
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Element nodal load vector due to traction
EXAMPLE:
x
y
fS3x
2
31
el Salong
T
SdSTtf
32 32N
fS3y
fS2x
fS2y
(2,0)
(2,2)
(0,0)
01
STtt
dyNtf ex l alongS
1221
)1(32
2 322
0
0
3
3
2
y
x
y
S
S
S
f
tf
f
Similarly, compute
1
2
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Recommendations for use of CST
1. Use in areas where strain gradients are small
2. Use in mesh transition areas (fine mesh to coarse mesh)
3. Avoid CST in critical areas of structures (e.g., stress concentrations, edges of holes, corners)
4. In general CSTs are not recommended for general analysis purposes as a very large number of these elements are required for reasonable accuracy.
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Example
x
y
El 1
El 2
1
23
4
300 psi1000 lb
3 in
2 inThickness (t) = 0.5 inE= 30×106 psi=0.25
(a) Compute the unknown nodal displacements.(b) Compute the stresses in the two elements.
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Realize that this is a plane stress problem and therefore we need to use
psiED 72 10
2.10002.38.008.02.3
2100
0101
1
Step 1: Node-element connectivity chart
ELEMENT Node 1 Node 2 Node 3 Area (sqin)
1 1 2 4 3
2 3 4 2 3
Node x y
1 3 0
2 3 2
3 0 2
4 0 0
Nodal coordinates
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Step 2: Compute strain-displacement matrices for the elements
332211
321
321
000000
21
bcbcbcccc
bbb
AB
Recall
123312231
213132321
xxcxxcxxcyybyybyyb
with
For Element #1:
1(1)
2(2)
4(3)(local numbers within brackets)
0;3;30;2;0
321
321
xxxyyy
Hence
033202
321
321
cccbbb
200323003030020002
61)1(B
Therefore
For Element #2:
200323003030020002
61)2(B
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Step 3: Compute element stiffness matrices
7
)1(T)1()1(T)1()1(
10
2.005333.002.02.1
3.00045.02.02.02.13.04.1
3.05333.02.045.05.09833.0BDB)5.0)(3(BDB
Atk
u1 u2 u4 v4v2v1
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7
)2(T)2()2(T)2()2(
10
2.005333.002.02.1
3.00045.02.02.02.13.04.1
3.05333.02.045.05.09833.0BDB)5.0)(3(BDB
Atk
u3 u4 u2 v2v4v3
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Step 4: Assemble the global stiffness matrix corresponding to the nonzero degrees of freedom
014433 vvuvuNotice that
Hence we need to calculate only a small (3x3) stiffness matrix
7104.102.0
0983.045.02.045.0983.0
K
u1 u2v2
u1
u2
v2
![Page 30: The Constant Strain Triangle (CST)](https://reader031.vdocuments.mx/reader031/viewer/2022021418/58690d8b1a28ab3f7c8b9741/html5/thumbnails/30.jpg)
Step 5: Compute consistent nodal loads
y
x
x
fff
f
2
2
1
yf2
00
ySy ff2
10002
The consistent nodal load due to traction on the edge 3-2
lbx
dxx
dxN
tdxNf
x
x
xS y
2252950
250
3150
)5.0)(300(
)300(
3
0
2
3
0
3
0 233
3
0 2332
3 2
3232xN
![Page 31: The Constant Strain Triangle (CST)](https://reader031.vdocuments.mx/reader031/viewer/2022021418/58690d8b1a28ab3f7c8b9741/html5/thumbnails/31.jpg)
lb
ffySy
1225
100022
Hence
Step 6: Solve the system equations to obtain the unknown nodal loads
fdK
122500
4.102.00983.045.02.045.0983.0
10
2
2
17
vuu
Solve to get
ininin
vuu
4
4
4
2
2
1
109084.0101069.0102337.0
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Step 7: Compute the stresses in the elements
)1()1()1( dBD
With
00109084.0101069.00102337.0
d444
442211)1(
vuvuvuT
Calculate
psi
1.761.13911.114
)1(
In Element #1
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)2()2()2( dBD
With
44
224433)2(
109084.0101069.00000
d
vuvuvuT
Calculate
psi
35.36352.281.114
)2(
In Element #2
Notice that the stresses are constant in each element