Civil Engineering Reference
In-Depth Information
a3
×
3 matrix equation as follows:
⎡
⎤
⎡
⎤
¯
ε
¯
ε
1
¯
¯
⎣
ε
t
γ
t
2
⎦
=
⎣
ε
2
γ
12
2
⎦
[
T
]
−
1
(6.40)
where [
T
]
−
1
is the inverse transformation matrix given in Equation (4.49), Chapter 4.
The two uniaxial strains (¯
ε
t
) obtained from Equation (6.40), or the two equations (6.20)
and (6.21), can then be used to calculate the stresses in the smeared steel bars (
f
and
f
t
)in
the equilibrium equations (6.1)-(6.3). The constitutive relationships connecting the stresses
(
f
,
ε
,¯
f
t
) and the uniaxial strains (¯
ε
,¯
ε
t
) are obtained directly from tests, and are given in
Section 6.1.9.
6.1.5 Experimental Stress-Strain Curves
The experimental stress-strain curves are obtained from testing RC 2-D elements, as shown
in Figure 6.2, in the universal panel tester (UPT) shown in Figure 6.1. All the test specimens
are subjected to a tensile stress
σ
2
in the vertical direction. As such, the horizontal and vertical axes constitute the principal 1-2
coordinate. The mild steel bars in a 2-D element, which define the
σ
1
in the horizontal direction, and to a compressive stress
t
coordinate, could be
oriented in any directions. Figure 6.2(a) and (b) show two typical cases where the angle
−
α
1
is equal to 0
◦
and 45
◦
, respectively. The special case of pure shear is achieved by applying
the stresses
45
◦
(Figure 6.2b). A
σ
1
and
σ
2
in equal magnitude on a 2-D element with
α
1
=
t
coordinate is created in the 45
◦
direction. This shear stress
pure shear stress
τ
t
in the
−
τ
t
could be increased in load stages until the failure of the test panel.
The strains of a test panel were measured by two sets of LVDT (linear voltage differential
transformer) rosettes, one on each face of a panel. Each set of rosette consists of six LVDTs:
two in the horizontal direction to measure the principal tensile strains
ε
1
, two in the vertical
ε
2
, and two oriented at 45
◦
to measure
direction to measure the principal compressive strains
ε
t
, in the case of pure shear panels, Figure 6.2(b). In order to
increase the accuracy of the measured compressive strains
ε
the steel bar strains,
and
ε
2
(which is much smaller than the
other three strains
ε
t
), four additional LVDTs (two on each face) were added in the
vertical directions. In short, a total of 16 LVDTs were installed.
The six LVDTs of a rosette are anchored to the panel at the four corners of a 0.8 m (31.5
in.) square, so that the four horizontal and vertical LVDTs measure the displacements over a
length of 0.8 m and the two diagonal LVDTs over a length of 1.13 m. The principal strains
ε
1
,
ε
and
ε
1
and
ε
2
are calculated from the measured displacements divided by a length of 0.8 m, while the
diagonal strains
ε
t
are divided by a length of 1.13 m. Since the measured lengths of 0.8
and 1.13 m span over several cracks, the measured strains are 'smeared strains', or 'average
strains', that include the gaps of the crack widths.
The pure shear test described above could produce a softened compressive stress-strain
ε
and
c
c
(
σ
2
-¯
ε
2
) curve for concrete. The plotting of this
σ
2
-¯
ε
2
curve from the test results was based on
the following methodology:
