Finite Element Analysis of Steel and Steel-Concrete Composite Bridges - Finite Element Analysis and Design of Steel and Steel-Concrete Composite Bridges - page 506

Stress, s

u
t

σ

u o

Displacement, u

Figure 5.23 Fracture energy cracking model as given in ABAQUS [1.29].

specification often introduces mesh sensitivity in the analysis results in the
sense that the finite element predictions do not converge to a unique solu-
tion as the mesh is refined because mesh refinement leads to narrower crack
bands. This problem typically occurs if only a few discrete cracks form in the
structure and mesh refinement does not result in the formation of additional
cracks. If cracks are evenly distributed (either due to the effect of rebar or due
to the presence of stabilizing elastic material, as in the case of plate bending),
mesh sensitivity is less of a concern.
In practical calculations for reinforced concrete, the mesh is usually such
that each element contains rebars. The interaction between the rebars and
the concrete tends to reduce the mesh sensitivity, provided that a reasonable
amount of tension stiffening is introduced in the concrete model to simulate
this interaction; see Figure 5.22 . The tension-stiffening effect must be esti-
mated. It depends on such factors as the density of reinforcement, the quality
of the bond between the rebar and the concrete, the relative size of the con-
crete aggregate compared to the rebar diameter, and the mesh. A reasonable
starting point for relatively heavily reinforced concrete modeled with a fairly
detailed mesh is to assume that the strain softening after failure reduces the
stress linearly to zero at a total strain of about 10 the strain at failure. The
strain at failure in standard concretes is typically 10 4 , which suggests that
tension stiffening that reduces the stress to zero at a total strain of about
10 3 is reasonable. This parameter should be calibrated to a particular case.
The choice of tension-stiffening parameters is important in ABAQUS [1.29]
since, generally, more tension stiffening makes it easier to obtain numerical
solutions. Too little tension stiffening will cause the local cracking failure in
the concrete to introduce temporarily unstable behavior in the overall
response of the model. Few practical designs exhibit such behavior, so that