Civil Engineering Reference
In-Depth Information
(RA-STM), will be treated in Chapter 5. The fixed-angle shear theories are given in Chapter
6, including the fixed-angle softened truss model (FA-STM), the softened membrane model
(SMM), and the cyclic softened membrane model (CSMM). CSMM are used in Chapter 9 and
10 to predict the static, dynamic and earthquake behavior of shear-dominant structures, such
as framed shear walls, low-rise shear walls, large bridge piers, and wall-type buildings.
1.3.2.6 Torsion Theories
Torsion is a more complicated problem than shear because it is a three-dimensional (3-D)
problem involving not only the shear problem of 2-D membrane elements in the tube wall,
but also the equilibrium and compatibility of the whole 3-D member and the warping of tube
walls that causes bending in the concrete struts. The effective thickness of the tube wall was
defined by the shear flow zone (Hsu, 1990) in which the concrete strain varies from zero to a
maximum at the edge, thus creating a strong strain gradient.
By incorporating the two compatibility equations of a member (relating angle of twist to the
shear strain, and to the curvature of concrete struts), as well as the softened stress-strain curve
of concrete, Hsu and Mo (1985a) developed a
rotating-angle softened truss model
(RA-STM)
to predict the post-cracking torsional behavior of reinforced concrete members up to the peak
point. Hsu and Mo's model predicted all the test results available in the literature very well, and
was able to explain why Rausch's model consistently overestimates the experimental ultimate
torque. In essence, the softening of the concrete increases the effective thickness of the shear
flow zone and decreases the lever arm area. This, in turn, reduces the torsional resistance of
the cross-section (Hsu, 1990, 1993).
The
softened membrane model
(SMM) for shear was recently applied to torsion (Jeng and
Hsu, 2009). The SMM model for torsion made two improvements. First, it takes into account
the bending of a 2-D element in the direction of principal tension, as well as the constitutive
relationship of concrete in tension. This allows the pre-cracking torsional response to be
predicted. Second, because the SMM takes into account the Poisson effect (Zhu and Hsu,
2002) of the 2-D elements in the direction of principal compression, the post-peak behavior
of a torsional member can also be accurately predicted. The Poisson effect, however, must be
diluted by 20% to account for the strain gradient caused by the bending of the concrete struts.
As a result of these two improvements, this torsion theory become very powerful, capable
of predicting the entire torque-twist curve, including the pre-cracking and post-cracking
responses, as well as the pre-peak and post-peak behavior.
The rotating-angle softened truss model (RA-STM) for shear will be applied to torsion in
Chapter 7. The softened membrane model (SMM), however, will not be included. Readers
interested in the application of SMM to torsion are referred to the paper by Jeng and
Hsu (2009).
1.4 Struts-and-ties Model
1.4.1 General Description
As discussed in Section 1.2.2, the 'local regions' of a reinforced concrete structure are those
areas where stresses and strains are irregularly distributed. These regions include the knee
joints, corbels and brackets, deep beams, dapped ends of beams, ledgers of spandrel beams,
