Civil Engineering Reference
In-Depth Information
Table 5.3
Results of calculations for Example Problem 5.4.
Variables
Eqs
Calculated values
ε
d
10
−
3
selected
−
0.275
−
0.400
−
0.500
−
0.600
−
0.620
ε
r
10
−
3
last assumed
3.45
7.90
10.80
12.35
12.40
σ
1
MPa last assumed
3.65
4.16
4.26
4.26
4.26
ζ
8
0.514
0.376
0.329
0.3103
0.3098
ε
d
/ζ ε
o
0.268
0.532
0.760
0.967
1.000
σ
d
MPa
7
−
6.57
−
8.09
−
8.54
−
8.54
−
8.53
(
ε
r
−
ε
d
)/(
−
σ
d
)10
−
3
/
MPa
0.567
0.706
1.323
1.515
1.525
10
−
3
ε
13
P
2.07
5.65
7.98
9.11
9.14
ε
t
10
−
3
14
P
1.11
1.85
2.33
2.64
2.63
ε
r
10
−
3
checked
15
3.455
7.90
10.82
12.35
12.39
f
MPa
413
413
413
413
413
9
f
t
MPa
222
369
413
413
413
10
f
p
MPa
0
0
0
0
0
11
f
tp
MPa
0
0
0
0
0
12
σ
1
MPa checked
3.65
4.16
4.26
4.26
4.26
17
α
r
degrees.
37.54
31.38
30.0
30.0
30.0
16
τ
t
MPa
3.17
3.59
3.69
3.69
3.69
3
γ
t
10
-3
3.60
7.38
9.78
11.22
11.28
6
σ
MPa
1.82
2.08
2.13
2.13
2.13
σ
t
MPa
−
1.82
−
2.08
−
2.13
−
2.13
−
2.13
τ
t
MPa
3.16
3.59
3.69
3.69
3.69
5.4.6.3 Discussion
The results of calculations for
0.00062
are summarized in Table 5.3. Comparison of these five cases illustrates clearly the trends of
all the variables. It should be kept in mind that
ε
d
=−
0.000275,
−
0.0004 and
−
0.0005,
−
0.0006 and
−
ε
d
=−
0.000275 is the point of first yield of the
ε
d
=−
longitudinal steel.
0.0004 represents a point after the yielding of longitudinal steel, but
before the yielding of transverse steel.
ε
d
=−
0.0005 gives a point after the yielding of both
the longitudinal and the transverse steel. Computation could not proceed after
0.00062.
Let us compare the Mohr stress circles for concrete and steel at peak load stage in Figure
5.26 with those at first yield in Figure 5.10. The 2
ε
d
=−
75.08
◦
α
r
=
at first yield (Figure 5.10)
60
◦
has gradually reduced to 2
α
r
=
at peak load stage (Figure 5.26). The transverse steel
stress
ρ
t
f
t
=
2.29 MPa at first yield (Figure 5.10) has gradually increased to the yield level of
ρ
t
f
t
=
4.26 MPa at peak load stage. This increase of transverse steel stress is responsible for
the corresponding increase of the applied stresses (
τ
t
).
Let us also compare the Mohr stress circles for applied stresses, concrete stresses and steel
stresses at peak load stage in Figure 5.26 and those in Figure 5.3. It can be seen that they
are identical. In other words, when the steel in both the longitudinal and transverse directions
reaches the yield point, the predictions of RA-STM is the same as the prediction of the
equilibrium (plasticity) truss model. Of course, RA-STM is more powerful theory, because
the incorporation of the Mohr compatibility condition allows RA-STM to produce the Mohr
σ
,
σ
t
,
