Civil Engineering Reference
In-Depth Information
deep beams sections that are plane before bending do not remain so after
bending. The stresses on the beam cross section therefore do not vary
linearly with depth. Generally, the nonlinearity is of more interest when
reinforcement is designed by the working stress method than when
reinforcement is designed by the strength method. Much of the early work
on deep beams emphasised elastic analysis and many elastic solutions can
be found in the literature. The Portland Cement Association (1980) still
provides information on the elastic stress distribution in deep beams. It
covers simple span and continuous beams.
Leonhardt and Walther (1966) found that until cracking develops, the
stresses approximate those predicted by elastic theory. After cracking, the
stresses deviate significantly from the elastic distribution. Beam capacity
cannot be predicted by elastic analysis.
If reinforcement is proportioned solely in accordance with an elastic
analysis, main reinforcement would for example be curtailed in regions of
low bending moment. In real beams that demonstrate marked strut and tie
action the curtailed reinforcement is ineffective as a tie and much of the
potential post-cracking strength is lost.
One useful insight which can be drawn from the elastic solutions is the
estimation of the depth of the tension zones. The main flexural reinforcement
should be distributed over most of the tension zone to control cracks. The
CEB and CIRIA recommendations recognise this in their reinforcement
detailing requirements. The amount of reinforcement is determined by a
strength design, but the reinforcement is distributed in general accordance
with elastic analysis. Shear strength analysis was largely ignored because the
beams of interest at the time (i.e. deep enough to have non-linear but elastic
behaviour) were generally deep enough for shear strength not to be critical.
4.3.2
Finite element analysis
Finite element analysis is the subject of Chapter 9 so the comments here will
be brief and pertain to continuous deep beams. Finite element programs are
now available which can with reasonable accuracy predict the capacity of
reinforced concrete beams. The literature now contains the results of such an
analysis for continuous deep beams.
Cook and Mitchell (1988) report a non-linear finite element analysis of two
of Rogowsky
et al.
's test specimens. The beams had shear span to depth ratios
of 1.5 and 2.0 respectively, and contained heavy vertical stirrups. The analysis
predicted deformations, principal stresses, principal strains and ultimate loads.
All four (two failures per beam) test results were within 4% of the predicted
values. The failure resulted from yielding of the transverse reinforcement
followed by crushing of the concrete. The high negative moment near the central
support produced large tensile strains in the adjoining shear spans thus softening
the concrete and reducing its compressive strength. The softening was gradual as
demonstrated by the ductile experimental load deflection curve.
