Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 188

Figure 5.20 Shear stress vs. shear strain curve for example problem 5.3

5.4.4.3 Concluding Remarks

It can be seen from Table 5.2 and Figure 5.20 that the condition under the strain ε d =− 0.0002

represents very closely the first yield condition , because the transverse steel strain ε t of 0.00208

just exceeds the yield stress of 0.00207 while the longitudinal steel strain is still within the
elastic range. Under this condition the yield shear stress, τ ty , and the yield shear strain γ ty are

3.68 MPa and 0.00400, respectively.
The condition under the strain

ε d =−

0.0004, on the other hand, represents very closely the
ultimate condition . This can be observed as follows: When the strain

ε d is increased negatively

− − ε d /ζ ε o =

from 0.800 to a

level of 1.042, just exceeding the peak stress of unity. In the vicinity of the peak stress, the
compressive stress of

0.0004 to 0.0005, the concrete struts increase from a level of

9.92 MPa in the concrete levels off and the increases of various strains
come to a screeching halt.
It should be mentioned that the rotating angle softened truss model is incapable of pre-
dicting the descending branch of the

−

τ t − γ t curve, because neither the Poisson effect nor
the shear resistance of concrete struts are taken into account. The prediction of the de-
scending branch will be carefully studied in the softened membrane model in Section 6.1
(Chapter 6).

5.4.5 2-D Elements under Proportional Loading
5.4.5.1 Proportional Membrane Loadings

The stress state described by the three applied stresses t coordinate

of a 2-D element is shown in Figure 5.21(a). This stress state can also be expressed in
terms of three principal stress variables

σ , σ t and τ t in the −

σ 1 , S and α 1 ,inthe1 − 2 coordinate as shown in