Finite Element Analysis of Steel and Steel-Concrete Composite Bridges - Finite Element Analysis and Design of Steel and Steel-Concrete Composite Bridges - page 512

refinement leads to narrower crack bands. This problem typically occurs if
cracking failure occurs only at localized regions in the structure and mesh
refinement does not result in the formation of additional cracks. If cracking
failure is distributed evenly (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. This requires an estimate of the tension-stiffening effect,
which 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 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 the presence
of this type of response in the analysis model usually indicates that the tension
stiffening is unreasonably low.
When there is no reinforcement in significant regions of the model, the
tension-stiffening approach described earlier will introduce unreasonable
mesh sensitivity into the results. However, it is generally accepted that
Hillerborg's [ 5.5 ] fracture energy proposal is adequate to allay the concern
for many practical purposes. Hillerborg defines the energy required to open
a unit area of crack, G f , as a material parameter, using brittle fracture con-
cepts. With this approach, the concrete's brittle behavior is characterized by
a stress-displacement response rather than a stress-strain response; see
Figure 5.26 . Under tension, a concrete specimen will crack across some sec-
tion. After it has been pulled apart sufficiently for most of the stress to be
removed (so that the undamaged elastic strain is small), its length will be
determined primarily by the opening at the crack. The opening does not