Civil Engineering Reference
In-Depth Information
and compression regions are defined by a series of straight lines. These modifications are
studied in Section 6.3.2.
2. The constitutive relationships of smeared mild steel bars must also include two modifica-
tions: (a) the envelope curves of mild steel are identical to the monotonic curves, except
that a compressive yielding stage should be added; and (b) the unloading and reloading
curves should be defined to express the Bauschinger effect. These modifications are studied
in Section 6.3.3.
3. The Hsu/Zhu ratios need to be adjusted for application to cyclic loading. This subject is
studied in Section 6.3.4.
Once the constitutive relationships of materials under cyclic loading are derived, we can
then discuss the solution procedure in Section 6.3.5.
The results of the solution are presented graphically as hysteretic loops. Section 6.3.6
presents the interesting and crucial problem of a 'pinched shape' in the hysteretic loops. This
is followed by a discussion of the mechanism of pinching and failure under cyclic shear in
Section 6.3.7.
Finally, the hysteretic loops of eight demonstrative panels are generated by CSMM in Section
6.3.8, in order to study the effect of the two variables (steel bar angle and steel percentage) on
the three properties of RC 2-D elements under cyclic shear. The three cyclic properties, which
are shear stiffness, shear ductility, and shear energy dissipation, are discussed in Sections 6.3.9,
6.3.10, and 6.3.11, respectively.
6.3.2 Cyclic Stress-Strain Curves of Concrete
When Equations (6.1)-6.8) are applied to monotonic static loading in SMM, the 1-2 coor-
dinate is defined as follows: 1-axis is the direction of the principal tensile stress and strain,
and 2-axis is the direction of the principal compressive stress and strain. When Equations
(6.1)-(6.8) are applied to reversed cyclic loading in CSMM, however, the 1-2 coordinate
is defined in a more general manner. The 1-2 coordinate is still the coordinate of principal
stress and strain, but both 1-axis and 2-axis are alternately the directions of principal tension
and the principal compression. In other words, when the cyclic load is in the positive direc-
tion, the 1-axis is the principal tension direction and the 2-axis is the principal compression
direction (as in the monotonic loading). When the cyclic load is in the
negative
direction, how-
ever, the 1-axis is the principal
compression
direction and the 2-axis is the principal
tension
direction.
The cyclic uniaxial constitutive relationships of cracked concrete in compression and tension
are summarized in Figure 6.28. In the graph, the vertical axis represents the cyclic stress
c
,
with positive tensile stress above the origin and negative compressive stress below the origin.
The horizontal axis represents the cyclic uniaxial strain ¯
σ
, with positive tensile strain to the
right of origin and negative compressive strain to the left of origin.
The upper right quadrant gives the tensile envelope stress-strain curves T1 and T2. In the
lower left quadrant is the compression envelope stress-strain curves C1 and C2. The unloading
and reloading curves are represented by the series of straight lines C3-C7 in the compressive
strain regions, and T3, T4 in the tensile strain region. Each straight line connects two points
with their coordinates specified in the lower right quadrant.
ε
