Civil Engineering Reference
In-Depth Information
5
Rotating Angle Shear Theories
5.1 Stress Equilibrium of RC 2-D Elements
5.1.1 Transformation Type of Equilibrium Equations
In Chapter 4, Section 4.3.3, we derived three equilibrium equations, (4.60)-(4.62), for the
rotating angle theory of a reinforced concrete 2-D element as follows:
σ
=
σ
r
cos
2
α
r
+
σ
d
sin
2
α
r
+
ρ
f
(5.1)
=
σ
r
sin
2
α
r
+
σ
d
cos
2
σ
t
α
r
+
ρ
t
f
t
(5.2)
τ
t
=
(
σ
r
−
σ
d
)sin
α
r
cos
α
r
(5.3)
τ
t
was illustrated
by Mohr stress circles in Figure 4.13. Now we will deal with the general case of an element
subjected not only to shear,
The state of stress in such a RC 2-D element subjected to pure shear
τ
t
, but also to normal stresses
σ
and
σ
t
. The understanding of
the state of stress in a RC element should focus on three aspects:
1. We can look at the reinforced concrete element as a whole. The state of stress due to the
three applied external stresses
τ
t
, is shown by the Mohr circle in Figure 5.1(a).
It can be seen that point A represents the reference
σ
,
σ
t
and
-face with stresses of
σ
and
τ
t
, while
point B represents the
t
-face with stresses of
σ
t
and
−
τ
t
. The principal tensile stress,
σ
1
,
α
1
away from point A. The principal
is denoted as point C which is located at an angle of 2
σ
2
, is denoted as point D located at an angle of 180
◦
away from point C.
2. The axial smeared stresses of the steel grid element are shown in Figure 5.1(c). The
longitudinal and transverse stresses
compressive stress,
ρ
t
f
t
are plotted at the level of points A and B,
respectively. There is no Mohr circle for steel stresses, because the steel bars are assumed
to be incapable of resisting shear (or dowel) stresses.
3. The state of stress in the concrete element is shown in Figure 5.1(b). The stresses in the
ρ
f
and
- and
t
-faces are represented by points A and B, respectively. Point A has a normal stresses
of
c
c
t
σ
=
σ
−
ρ
f
and a shear stress of
τ
t
. Point B has a normal stresses of
σ
=
σ
t
−
ρ
t
f
t
,
and a shear stress of
−
τ
t
. The principal tensile stress
σ
r
is represented by point C located
at an angle of 2
α
r
away from point A. The principal compressive stress
σ
d
, is represented
by point D located at an angle of 180
◦
away from point C.
