Civil Engineering Reference
In-Depth Information
7.2.5 3-D S
tate
oF
S
treSS
C
onCrete
p
laStiCity
m
oDel
Mander (1983) proposed using a multiaxial stress procedure to calculate the ultimate
confined strength (e.g.,
f
cce
) from two given lateral pressures (e.g.,
f
flxe
and
f
flye
). This
numerical procedure is summarized in the following steps:
1. Determine
f
flxe
and
f
flye
using Equations (7.17) and (7.18)
2. Convert the positive sign of
f
flxe
and
f
flye
to negative to represent the major and
intermediate principal stresses (these values are referred to as σ
1
and σ
2
so
that σ
1
> σ
2
)
3. Estimate the confined strength
f
cce
, which is σ
3
as the minor principal stress
4. Calculate the octahedral stress σ
oct
, octahedral shear stress τ
oct
, and lode
angle θ as follows:
1
3
(
)
σ=σ+σ+σ
oct
(7.22)
1
2
3
1
2
1
3
(
)
2
(
)
2
(
)
2
(7.23)
τ= σ−σ+σ−σ+σ−σ
oct
1
2
2
3
3
1
θ=
σ−σ
τ
(7.2 4)
1
oct
cos
2
oct
5. Determining the ultimate strength meridian surfaces
T
and
C
(for θ = 0°
and 60°, respectively) using the following equations derived by Al-Rahmani
and Rasheed (2014) from data by Kupfer, Hilsdorf and Rüsch (1969) while
calibrating the data against the equivalent circular section of Lam and Teng
(20 03b):
0.061898
−
0.62637
σ σ>−
if
0.767
oct
oct
T
=
(7.25)
0.229132
−
0.40824
σ σ≤−
if
0.767
oct
oct
0.107795
−
1.09083
σ σ>−
if
0.333
oct
oct
C
=
(7.26)
0.336883
−
0.40357
σ σ≤−
if
0.333
oct
oct
σ=σ
/
f
(7.27)
oct
oct
c
6. Determining the octahedral shear stress using the interpolation function
found by Willam and Warnke (1975):
1
2
(
)
2
0.5
D
/cos
θ+
T CD T C
DTC
2
−
+
5
−
4
(7.28)
τ=
C
oct
(
)
2
+−
2
Search WWH ::
Custom Search