Finite Element Analysis of Steel and Steel-Concrete Composite Bridges - Finite Element Analysis and Design of Steel and Steel-Concrete Composite Bridges - page 511

s t

s to

E

o

E

o

( 1- d t ) E o

ε t

ck el

ot

ε ε

t

el
t

ε

pl

ε

t

Figure 5.25 Illustration of the definition of the cracking strain E ck used for the definition
of tension-stiffening data as given in ABAQUS [1.29].

some “tension stiffening” into the concrete modeling to simulate load trans-
fer across cracks through the rebar. Defining the rebar can be tedious in
complex problems, but it is important that this be done accurately since it
may cause an analysis to fail due to the lack of reinforcement in key regions
of a model. The postfailure behavior for direct straining is modeled with ten-
sion stiffening, which allows modelers to define the strain-softening behav-
ior for cracked concrete. This behavior also allows for the effects of the
reinforcement interaction with concrete to be simulated in a simple manner.
Tension stiffening is required in the concrete damaged plasticity model.
Modelers can specify tension stiffening by means of a postfailure stress-strain
relation or by applying a fracture energy cracking criterion.
In reinforced concrete, the specification of postfailure behavior generally
means giving the postfailure stress as a function of cracking strain, e ck . The
cracking strain is defined as the total strain minus the elastic strain corre-
sponding to the undamaged material as illustrated in Figure 5.25 . To avoid
potential numerical problems, ABAQUS [1.29] enforces a lower limit on
the postfailure stress equal to 100 of the initial failure stress ( s to /100).
Tension-stiffening data are given in terms of the cracking strain, e ck . When
unloading data are available, the data are provided to ABAQUS in terms of
tensile damage curves, and the software automatically converts the cracking
strain values to plastic-strain values. In cases with little or no reinforcement,
the specification of a postfailure stress-strain relation introduces mesh sensi-
tivity in the results, in the sense that the finite element predictions do not
converge to a unique solution as the mesh is refined because mesh