Chapter 8
8.1 Triangular element
From Section 8.2
N1 = 1–
22
22
3 3 2 4 2x y x xy y
b b bh
bh
2
2xx
2
2yy
8.2
Strains
x
y
xy
=
1
2A
1
1
1 2 3 4 5 6 2
1 2 3 4 5 6 2
1 1 2 2 3 3 4 4 5 5 6 6
6
0 0 0 0 0 0
0 0 0 0 0 0
u
v
u
v
v
(1)
385
4by
3
4 ( )
h
b
b
3
Performing the multiplications in (1)
(After substituting
’s and
’s from (2))
x =
1 2 4 5
4 4 1
3 3 3 3
h h h h
u u u u bh
1
4 4 1
b b b b
3
bh
y =
3
b
[– v1 + v3 + 4v4 – 4v6]
1
bh
1 3 4 6 1 2 4 5
33
bh
11
bh bh
Stresses
{
} = [D] {
}
x
y
xy
2
1
E
v
1
2
10
00 v
v
x
y
xy
x =
2
1
E
v
1 2 4 5 1 3 4 6 1
4 4 4 4
33
hb
u u u u v v v v v bh
E
hb
2(1 )
v
1 3 4 6 1 2 4 5 1
33
8.3
The equation is {fs} =
[]
T
s
sN
{T } ds (1)
{T } =
0
x
y
pp
p
…. is the surface traction (2)
1 2 3 4 5 6
0 0 0 0 0 0
N N N N N N
8.4
{fs} =
s
[Ns]T {T }ds (1)
{T } =
0
0
py
xh
y
p
p
(2)
1 2 3 4 5 6
0 0 0 0 0 0
N N N N N N
0
0
0
0
0
0
1
0
2
3
4
5
6
0
0
0(4)
0
0
at 0
0
hpy
sh
py
h
py
h
py
h
py
h
py
h
dy
ft N
N
N
N
N
x
Nyy
34 0
hp
8.5 (a)
{
} = [B] {d}
1
2A
1 2 3 4 5 6
1 2 3 4 5 6
1 1 2 2 3 3 4 4 5 5 6 6
0 0 0 0 0 0
Element is oriented as in Section 8.2
’s and
’s as in Section 8.2, Eq. (8.2.8)
1 = – 3h +
4
6
hx
+ 4y = 6x + 4y – 18
2 = – h +
4
6
hx
= 6x – 6,
3 = 0
4 = 4y,
5 = – 4y
6 = 4h –
8hx
b
– 4y = – 12x – 4y + 24
4by
8
= 0.001 (6x – 6) + 0.0002 (4y) + 0.0005 (– 12x – 4y + 24)
392
8
8
4 = 4x,
5 = – 4x – 12y + 24,
6 = – 4x
84
3x
16 4 16
3xy
2A
x = – 0.0012y + 0.004
x = – 5 10– 5 y + 1.67 10–4
2A
y = – 0.004 x + 0.0012
84
3x
+ 0.0002 (4x) + 0.0001 (4y) + 0.0001 (– 4y)
16 4 16
3xy
2A
xy = – 0.0012x – 0.001y + 0.005
4
3
(2, )
{}
=
0.0001
0.000284
0.0000527
{
} =
2
1
E
v
1
2
10
10
00 v
v
v
0.0001
0.000284
0.0000527
928
632
8.6
Using Equation (8.1.14) in (8.1.13)
x
y
xy
0 1 0 2 0 0 0 0 0 0 0
0 0 1 0 2 0 1 0 2 0
xy
x y x y
where by Equation (8.1.7) {a} = [X]–1 {d } and
{a} =
1
2
66 3
4
5
6
1
2
3
4
66
5
1 0 0 0 0 0 –1
1 6 0 36 0 0
1 6 6 36 36 36
1 6 3 36 18 9
1 3 3 9 9 9
1 3 0 9 0 0
1 0 0 0 0 0
1 6 0 36 0 0
1 6 6 36 36 36
1 6 3 36 18 9
1 3 3 9 9 9
u
u
Ou
u
u
u
v
v
v
v
O
v
(2)
[X ]–1 {d } =
1 0 0 0 0 0 0 0 0 0 0 0
0.5 0 0.167 0 0 0 0 0 0 0 0.667 0
0 0 0.167 0 0.167 0 0 0 0.667 0 0.667 0
0.056 0 0.056 0 0 0 0 0 0 0 0.111 0
0 0 0.111 0 0 0 0.111 0 0.111 0 0.111 0
0 0 0.056 0 0.056 0 0.111 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 0.5 0 0.167 0 0 0 0 0 0 0 0.667
0 0 0 0.167 0 0.167 0 0 0 0.667 0 0.667
0 0.056 0 0.056 0 0 0 0 0 0 0 0.111
0 0 0 0.111 0 0 0 0.111 0 0.111 0 0.111
0 0 0 0.056 0 0.056 0 0.111 0 0 0 0
1
1
2
6
6
u
v
u
u
v
at centroid (x = 4, y = 2)
x
y
xy
0 1 0 8 2 0 0 0 0 0 0 0
0 0 1 0 4 4 0 1 0 8 2 0
Multiplying matrices in Equation (4) yields
x
y
xy
=
1 2 4 5 6
2 3 5 6
1 2 4 5 6
2 3 5 6
0.052 0.059 0.222 0.222 0.001
0.053 0.391 0.223 0.223
( 0.052 0.059 0.222 0.222 0
0.053 0.391 0.223 0.223 )
u u u u u
v v v v
v v v v v
u u u u
Then
x
y
2
1
E
v
xy
yx
v
8.7
u1 = u(0, 0) = a1 (1)
u2 = u(60, 0) = a1 + 60 a2 + 3600 a4 (2)
u3 = u(60, 60) = a1 + 60 a2 + 60 a3 + 3600 a4 + 3600 a5 + 3600 a6 (3)
By (1) a1 = u1
By (2) – 2 (6) a4 =
2 6 1
2
1800
u u u
6 2 1
43
u u u
1800
(4) – (5) a5 =
2 4 5 6
900
u u u u
(4) a3 =
2 3 5 6
44
60
u u u u
Can verify by substituting all a’s into Equation (3)
u = u1 +
6 2 1
43
60
u u u
x +
2 3 5 6
44
60
u u u u
y
2 6 1
2
u u u
2 4 5 6
u u u u
1800
Shape functions are
2
3
60 1800
xx
22
x y x xy y
60 1800
N4 =
2
2
900 1800
xy y
(From all u4 coefficients)
4
y xy
2
1 = 2A
1
N
x
= 3600
32
60 1800
x
= – 180 + 4x
2 = 3600
12
60 1800 900
xy
= – 60 + 4x – 4y
y
60 1800 900
1 = 2A
1
N
y
= 0,
5 = 3600
4
60 900
x
= 240 – 4x
2 = 3600
12
60 900 1800
xy
= 60 – 4x + 4y
12
y