Civil Engineering Reference
In-Depth Information
Figure 8.5
Hyperbolic yield line in concrete element.
During a loading history leading to collapse the principal axes of stress in
the concrete are likely to change directions, and at failure the latest formed
cracks will generally be at an angle to the yield line. This implies that shear
stresses are transferred across the yield line, by friction or aggregate
interlock in old cracks and by crushing zones between cracks.
The transfer of shear in yield lines is expressed by the rate of work dissipated,
which depends upon the inclination a of the displacement rate,
Figure 8.4
.
For
pure separation (
/2) the dissipation reduces to
D
c
=0 reflecting the assumption
of zero tensile concrete strength. However, as soon as tangential deformation is
introduced (
a
=
p
/2) the resistance increases proportionally with the compressive
concrete strength, corresponding to a failure stress
a
<
p
t
=
f
c
*
/2 for pure shearing (
a
=0).
For pure crushing (
=
f
c
*
In the general case a yield line will be a curve AB separating the element
into two rigid parts, the relative movement of which is a rotation about a point
O in the plane of the element (Figure 8.5). By calculus of variation it was
shown by J.F.Jensen (1981, 1982) that the optimal shape of the yield line,
leading to a stationary value of the total dissipation, is a hyperbola with
orthogonal asymptotes through O. The corresponding rate of internal work is:
a
=-
p
/2) the compressive resistance is
s
(8.1)
