Civil Engineering Reference
In-Depth Information
The constitutive matrices of concrete and steel shown in Equations (6.7) and (6.8) will
be carefully studied in Sections 6.1.2-6.1.10. Section 6.1.2 provides a historical overview of
research in RC 2-D elements. Section 6.1.3 studies the Poisson effect in RC 2-D elements,
including the concept of biaxial strains versus uniaxial strains, and the constitutive matrices
of smeared concrete and smeared steel expressed in terms of Hsu/Zhu ratios. Section 6.1.4
carefully studies the Hsu/Zhu ratios
ν
21
, including the test methods, the experimental
formulas, the effect of Hsu/Zhu ratios on the post-peak behavior, and the strain conversion
matrix [
V
].
The stress-strain curves of concrete and steel in Equations (6.7) and (6.8) must be 'smeared'
or 'averaged', because the stress equilibrium equations (6.1)-(6.3), and the strain compatibility
equations (6.4)-(6.6), are derived for
continuous
materials. The smeared stress-strain curves
of concrete are studied in Sections 6.1.5-6.1.8. The smeared stress-strain curves of mild steel
bars are studied in Section 6.1.9.
ν
12
and
6.1.2 Research in RC 2-D Elements
The truss model had been applied to treat shear (Ritter, 1899; Morsch, 1902) and torsion
(Rausch, 1929) of reinforced concrete since the turn of the 20th century. However, the predic-
tion based on truss model consistently overestimated the shear and torsional strengths of tested
specimens. The overestimation might exceed 50% in the case of low-rise shear walls and 30%
in the case of torsional members. This nagging mystery had plagued the researchers for over
half a century. The source of this difficulty was first understood by Robinson and Demorieux
(1972). They realized that a reinforced concrete membrane element subjected to shear stresses
is actually subjected to biaxial stresses (principal compression and principal tension). Viewing
the shear action as a two-dimensional problem, they discovered that the compressive strength
in the direction of principal compression was reduced by cracking due to principal tension
in the perpendicular direction. Applying this softened effect of concrete struts to the thin
webs of eight I-section test beams, they were able to explain the equilibrium of stresses in
the webs according to the truss model. Apparently, the mistake in applying the truss model
theory before 1972 was the use of the compressive stress-strain relationship of concrete ob-
tained from the uniaxial tests of standard cylinders without considering this two-dimensional
softening effect.
The tests of Robinson and Demorieux, unfortunately, could not delineate the variables
that govern the softening coefficient, because of the technical difficulties in the biaxial test-
ing of 2-D elements. The quantification of the softening phenomenon, therefore, had to
wait for a decade until a unique 'shear rig' test facility was built by Vecchio and Collins
(1981) at the University of Toronto. Based on their tests of 19 panels, 0.89 m square
and 70 mm thick, they proposed a softening coefficient that was a function of the tensile
principal strains.
The softening coefficient was significantly improved by Hsu and his colleagues at the
University of Houston from 1988 to 2009 using the Universal Panel Tester (UPT) (Hsu,
Belarbi and Pang, 1995). This large test facility is shown in Figure 6.1. It stands 5 m tall,
weighs nearly 40 tons and contains more than a mile of pipes to transport oil pressure to its 40
jacks. Because each jack has a high force capacity of 100 tons, the UPT is capable of testing
full-size RC 2-D elements of 1.4 m (55 in.) square and 178 mm (7 in.) thick, as shown in
Figure 6.2. Tests of 39 such large specimens by Belarbi and Hsu (1994, 1995) and Pang and
