Chapter 6
6.1 For sketch of Ni see Figure 6.8. Others follow similarly
By Equation (6.2.18)
1
2A
By Equation (6.2.10)
i +
j +
m = xj ym – yj xm + yi xm – xi ym + xi yj – yi xj
= 2A (by Equation (6.2.9)) (2)
i +
j +
m = yj – ym + ym – yi + yi – yj = 0 (3)
i +
j +
m = xm – xj + xi – xm + xj – xi = 0 (4)
By using (2)–(4) in (1), we obtain
Ni + Nj + Nm = 1 identically
6.2 By Equation (6.2.47)
p =
1
2
{d}T
v
[B]T [D] [B] dV{d} – {d}T { f }
0 0 0
10 0 0
2
i j m
i j m
i i j j m m
A
1
E
10
v
i
i
j
u
v
u
2
0
2
0
0
vjj
mm
mm
A
2
1
2
1 0 0 0 0
1
1 0 0 0 0
2
1
00
i i m
i j m
v
i i j j m m
v
EvA
v
dV
i
i
j
j
m
m
u
v
u
v
u
v
i
i
j
j
m
m
u
v
u
v
u
v
1
1
2
2
3
3
x
y
x
y
x
y
f
f
f
f
f
f
From Equation (3.10.27) or Equation (6.2.48)
10
v
Ev
002
4 0.9375 0 0 0.375
1 0 2
0 2 1
6
1 0.25 0.75
0.5 2 0.375
2 0.5 0 10 10
0 0 0.75 4 0.9375
–1 0.25 0.75
0.5 2 0.375
[k] = t A[B]T [D] [B]
6
1 0.25 0.75
0.5 2 0.375
2 0.5 0
10 10
42 1 0 2 2 1
[k] = 1.333 106
1 2 3
2.5 1.25 2 1.5 0.5 0.25
1.25 4.375 1 0.75 0.25 3.625
2 1 4 0 2 1
1.5 0.75 0 1.5 1.5 0.75
0.5 0.25 2 1.5 2.5 1.25
0.25 3.625 1 0.75 1.25 4.375
i j m
(b) xi = 1.2, yi = 0, xj = 2.4, yj = 0, xm = 1.2, ym = 1
A =
1
2
(1.2) (1) = 0.6 in.2
6
10 10
1 0 1.2
0 1.2 1 1 0.25 0
1 0 0 0.25 1 0
1 0 1 0 0 0
0 1.2 0 0 0 1.2
1.2 1 0 1 1.2 0
6
1 2 3
1.54 0.75 1 0.45 0.54 0.3
0.75 1.815 0.3 0.375 0.45 1.44
i j m
0.5 0 0.5 0 0 0
x
6
1 0.25 0
10 10 0.25 1 0
1 0 2 0 1 0
10 2 0 0 0 2
10667 psi
8333 psi
x
y
xy
1 =
1
22
2
10667 2667 10667 2667 8333
22
1 = 15910 psi
1
22
2
10667 2667 10667 2667 8333
p =
1
2
tan–1
10667 2667
2
8333
(c) For third element we have
0.5 0 0.5 0 0 0 1
x
3
3
6.4 10
lb
i = yj – ym = 30 – 120 = – 90
i = xm – xi = 50 – 80 = – 30
2
1 (0.25) 0 0 0.375
0 0 0.375
[k] = t A [B]T [D] [B]
[k] = (0.01)
–3
3
90 0 30
0 30 90
90 0 30
5.4 10 1
0 30 90
25.4 10
0 0 60
0 60 0
(1.12 1011)
1 0.25 0
0.25 1 0
0 0 0.375
[B]
[k] = 0.56 109
90 22.5 11.25
7.5 30 33.75
90 22.5 11.25
7.5 30 33.75
0 0 22.5
15 60 0
3
90 0 90 0 0 0
10 30 0 30 0 60
5.4 10 30 90 30 90 60 0
[k] = 1.037 105
8437.5 1687.5 7762.5 337.5 675 1350
1687.5 3937.5 337.5 2137.5 2025 1800
7762.5 337.5 8437.5 1687.5 675 1350
337.5 2137.5 1687.5 3937.5 2025 1800
675 2025 675 2025 1350 0
1350 1800 1350 1800 0 3600
(b) Similarly
i = – 5
i = 0
m = 2.5
m = 5
25.0 0 12.5 6.25 12.5 6.25
9.375 9.375 4.6875 9.375 4.6875
15.625 7.8125 3.125 1.5625
2
2
2.645 ( 0.078) 2.645 0.078 0.1165
x
yy
xy
=
11
6
1 0.25 0
1.12 10 0.25 1 0
25 10 0 0 0.375
5 0 2.5 0 2.5 0
0 0 0 5 0 5
0 5 5 2.5 5 2.5
0.002
0.001 0
0.0005
m21.0GPa
A = 2.5 10–5 m2
{
} = [D] [B] {d}
10
v
Ev
Equation (6.1.8) [D] =
11 10
10 11
10
1.12 10 2.8 10 0
2.8 10 1.12 10 0
0 0 4.2 10
Equation (6.2.32) combined
[B] =
1 2 3
1 2 3
1 1 2 2 2 3
0 0 0
10 0 0
TA
[B] =
–100 0 100 0 0 0
0 – 50 0 – 50 0 100
– 50 –100 – 50 100 100 0
N
1
8
0.001
0.005
4 04000 0.001
0 200020.0025
2 40420 0
0
5000
6000
1,2 =
2
2
5000 ( 15000) 5000 ( 15000) ( 6000)
22
= –10000 ± 7813
1 = – 2187 psi
2 = –17810 psi
tan 2
p =
2( 6000)
5000 ( 15000)
p = – 25.1°
(b)
i = 2 – 4 = – 2
i = 4 – 4 = 0
y
xy
4
0 0 0.25 0 2 2 2 2 0
332
0.001
0.005
0.001
5,000
(c) Given displacements (in.)
u1 = 0.001 v1 = 0.005
u2 = 0.001 v2 = 0.0025
u3 = 0 v3 = 0
1
1
2
2
3
3
u
v
u
v
u
v
Material definition
E = 10 106 psi v = 25
Geometry description
i = 0 – 2 (yj – ym)
j = 2 – 0 (ym – yi)
m = 0 – 0 (yi – yj)
2
0 0 0
10 0 0
i j m
334
(d)
y
xy
0 0 0.25
80 4 2 2 2 2
0.001
0.005
0.001
5.5
(e)
1 1 1
12 3 2.25
23 1 3.25