Strut-and-tie modeling - Computational Analysis and Design of Bridge Structures - page 416

The truss model and the results obtained from CAST are presented in
Figure 13.11c. Based on the calculation by the CAST program, maximum
compression in the diagonal strut is 101.87 kip (453.14 kN) and in the
vertical strut is 107.61 kip (478.67 kN). Maximum tension in the top tie
is 31.76 kip (141.28 kN) and in the bottom tie is 50.87 kip (226.28 kN).
Size of the upper nodes is determined by the size of bearing, and the size
of the lower nodes is decided by the sizes of piles. Rebar sizes and arrange-
ments are finalized after a few iterations. Bearing reinforcement details
in the width direction can be determined by a simple truss model in the
horizontal direction. The abutment is 914.4 mm (3′) wide, and the strut
section 914.4 mm × 152.4 mm (36″ × 6″) provides the required strength
for the struts. For ties, three no. 6 bars can provide the required strength.
However, code-specified minimum reinforcement must be provided to pre-
vent temperature-, creep-, and shrinkage-related issues.
13.4 2d illuStrated exaMPle 2—Walled Pier
Another common structure found in the transportation field is a solid shaft
bridge pier on a mat foundation shown in Figure 13.12a (Fu et al. 2005).
This case study is done for a 5.49-m (18′) high by 0.91-m (3′) wide wall on a
mat foundation. Four girders are resting on the wall, and each girder reac-
tion is 215.22 kip (957.35 kN). St. Venant's principle states: “The localized
effects caused by any load acting on the body will dissipate or smooth out
within regions that are sufficiently away from the location of the load.”
Elevation of the structure is shown in Figure 13.12b.
Based on the same principle, an STM model is developed for the walled
pier and presented in Figure 13.12c. The inclined angle q can either be
obtained from a stress trajectory plot or be assumed to vary from 65° for
l / d = 1°-55° for l / d = 2.0, where l is the wall length and d is the height. A
reasonable path at a 2-to-1 slope is created here to flow the concentrated
loads from the top of the wall toward the mat foundation. Maximum
strut force is 128.9 kip (573.38 kN), and maximum tie force is 50.22 kip
(223.39 kN), which are in the same range of Case Study 1, and a similar
strut width and reinforcement will be sufficient. Again, for this case, mini-
mum steel per code provisions applicable to the wall have to be provided.
13.5 2d illuStrated exaMPle 3— crane BeaM
A conservative estimate of the resistance of a concrete structure may be
obtained by the application of the lower-bound theorem of plasticity. If suf-
ficient ductility is present in the system, a STM fulfills the conditions for the
application of the lower-bound theory. The lower-bound theorem requires