Civil Engineering Reference
In-Depth Information
Table 7.3
Results of calculations for Example Problem 7.1
Variables
Eqs
Calculated values
10
−
3
ε
d
selected
−
0.250
−
0.500
−
1.000
−
1.500
−
1.750
10
−
3
ε
ds
[11]
−
0.500
−
1.000
−
2.000
−
3.000
−
3.500
10
−
3
ε
r
last assumed
2.25
4.14
7.34
9.00
9.29
ζ
[14]
0.587
0.482
0.387
0.356
0.351
ε
d
/ζ ε
o
0.426
1.037
2.59
4.21
4.99
k
1
[13]
0.365
0.679
0.842
0.798
0.741
σ
d
MPa
[12]
−
8.87
−
13.51
−
13.48
−
11.73
−
10.76
t
d
mm last assumed
91
96
109
127
140
A
o
10
3
mm
2
[22]
802
793
770
738
717
p
o
mm
[23]
3624
3602
3553
3479
3429
ε
10
−
3
[20]
0.791
1.56
2.28
2.54
2.48
ε
t
10
−
3
[21]
1.207
2.07
4.06
4.96
5.06
ε
r
10
−
3
checked
[21]
2.25
4.13
7.34
9.00
9.29
t
d
mm checked
[19]
91.2
96.0
109.0
127.8
140.2
α
r
degrees
[25]
49.79
48.16
51.16
51.66
51.76
τ
t
MPa
[3]
4.37
6.73
6.58
5.70
5.23
T
KN m
[4]
638
1027
1103
1068
1048
10
−
3
γ
t
[7]
2.47
4.60
8.15
10.21
10.73
10
−
3
θ
rad/m
[8]
5.59
10.43
18.78
24.06
25.67
f
MPa
[15]
158
312
413
413
413
f
t
MPa
[16]
241
413
413
413
413
f
p
MPa
[17]
1158
1311
1379
1441
1405
f
tp
MPa
[18]
0
0
0
0
0
The results of calculations shown above for
ε
d
=−
0.0005 and
−
0.0015 are summarized in
Table 7.3.
Table 7.3 also gives three additional cases of
ε
d
=−
0.00025,
−
0.001 and
−
0.00175.
ε
d
=−
−
−
−
−
Comparison of all five cases (
0.00175)
illustrates clearly the trends of all the variables. The relationship between the torsional moment
T
and the angle of twist
0.00025,
0.0005,
0.001,
0.0015 and
is plotted in Figure 7.9.
Table 7.3 clearly shows that
θ
ε
d
=−
0.0005 closely represents the point of first yield of the
longitudinal mild steel. At
0.001 both the longitudinal and transverse mild steel are
yielding and the torque resisted reaches a maximum shortly thereafter. When
ε
d
=−
ε
d
is increased
further, the torsional resistance decreases. At
ε
d
=−
0.0015 the calculated thickness
t
d
is
equal to the actual wall thickness of 127 mm. When
ε
d
is taken as
−
0.00175, the calculated
thickness
t
d
=
140 mm, i.e. greater than the actual thickness. This situation indicates that the
calculation for
ε
d
=−
0.00175 is actually invalid and that the torque-twist curve would drop
off more quickly for
0.0015 than is shown in Figure 7.9.
The angles of twist at first yield,
ε
d
>
−
θ
y
, is 0.01043 rad
/
m (when
ε
d
=−
0.0005) as given in
Table 7.3. The maximum angle of twist
θ
u
for the given wall thickness of 127 mm is 0.02406
rad/m, when
ε
d
=−
0.0015. Let us define the torsional ductility
µ
t
as the ratio
θ
u
/θ
y
, then the
