Civil Engineering Reference
In-Depth Information
Later, Vecchio and Collins (1986) proposed the modified compression field theory (MCFT)
which included a constitutive relationship for concrete in tension to better model the post-
cracking shear stiffness.
In 1988, a universal panel tester was built at the University of Houston (Hsu, Belarbi
and Pang, 1995) to perform biaxial tests on large 2-D elements of 1.4
×
1.4
×
0.179 m
(55
7 in.). By confirming and establishing the softening coefficient as a function of
principal tensile strain
×
55
×
ε
1
, Pang and Hsu (1995) and Belarbi and Hsu (1994, 1995) developed
the
rotating-angle softened truss model
(RA-STM). This model made two improvements over
the CFT: (1) the tensile stress of concrete was taken into account so that the deformations
could be correctly predicted; and (2) the smeared (or average) stress-strain curve of steel bars
embedded in concrete was derived on the 'smeared crack level' so that it could be correctly
used in the equilibrium and compatibility equations which are based on continuous materials.
Shortly after the development of the rotating-angle model, Pang and Hsu (1996) and Hsu
and Zhang (1997) reported the
fixed-angle softened truss model
(FA-STM) that is capable of
predicting the 'concrete contribution'
V
c
by assuming the cracks to be oriented at the fixed
angle, rather than the rotating angle. Zhu, Hsu and Lee (2001) derived a rational shear modulus
that is a function of the compressive and the tensile stress-strain curves of concrete. Using this
simple shear modulus, the solution algorithm of fixed-angle model became greatly simplified.
In 1995, a servo-control system (Hsu, Zhang and Gomez 1995) was installed on the universal
panel tester at the University of Houston, so that it could perform strain control tests. Using this
new capability, Zhang and Hsu (1998) studied high-strength concrete 2-D elements up to 100
MPa. They found that the softening coefficient was not only a function of the perpendicular
tensile strain
ε
1
, but also a function of the compressive strength of concrete
f
c
. More recently,
Wang (2006) and Chintrakarn (2001) tested 2-D shear elements with large longitudinal to
transverse steel ratios. These tests showed that the softening coefficient was a function of the
deviation angle
β
. Summarizing all three variables, the softening coefficient become a function
ε
1
,
f
c
and
of
.
All the above shear theories, based either on rotating-angle or fixed-angle, could predict
only the pre-peak branch of the shear stress vs shear strain curve, but not the post-peak branch
of the curves, because the Poisson effect of cracked reinforced concrete was neglected. Using
the strain-control feature of the universal panel tester, Zhu and Hsu (2002) quantified the
Poisson effect and characterized this property by two Hsu/Zhu ratios. Taking into account the
Poisson effect, Hsu and Zhu (2002) developed the
softened membrane model
(SMM) which
could satisfactorily predict the entire monotonic response of the load-deformation curves,
including both the pre-cracking and the post-cracking responses, as well as the ascending and
the descending branches.
Mansour and Hsu (2005a,b) extended the SMM for application to reversed cyclic loading.
This powerful theory, called the
cyclic softened membrane model
(CSMM), includes new
constitutive relationships of concrete and mild steel bars in compressive and tensile directions
of cyclic loading, as well as in the unloading and reloading stages. Consequently, CSMM is
capable of predicting the hysteretic loops of RC 2-D elements subjected to cyclic loading,
particularly their pinching characteristics. Furthermore, CSMM could be used to evaluate the
shear stiffness, the shear ductility and the shear energy dissipation of structures subjected to
dominant shear (Hsu and Mansour, 2005).
The fundamentals of shear are presented in Chapters 4. The rotating-angle shear theories,
including the Mohr compatibility truss model and the rotating-angle softened truss model
β
