Civil Engineering Reference
In-Depth Information
a width of cos
α
r
relationship is also shown by the geometry in Figure 2.2(a).
From this force triangle the shear flow
q
can be related to the transverse steel force
n
t
by
the geometry:
α
r
. The cos
q
=
n
t
tan
α
r
(2.5)
σ
d
using either the force
triangle in the longitudinal direction or the force triangle in the transverse direction. From the
geometry of the triangles we obtain:
The shear flow
q
can be related to the diagonal concrete stress
q
=
(
σ
d
h
)sin
α
r
cos
α
r
(2.6)
Assuming that yielding occurs in both the longitudinal and transverse steel, then
n
=
n
y
=
s
t
, where
n
y
and
n
ty
are the longitudinal and transverse yield
force per unit length, respectively, and
f
y
and
f
ty
are the longitudinal and transverse yield
stress, respectively. Combining Equations (2.4) and (2.5) to eliminate
A
f
y
/
s
and
n
t
=
n
ty
=
A
t
f
ty
/
α
r
we obtain:
q
y
=
√
n
y
n
ty
(2.7)
where
q
y
is the shear flow at yielding. Also, combining Equations (2.4) and (2.5) to eliminate
q
,wehave
n
y
n
ty
tan
α
r
=
(2.8)
Equation (2.7) states that the shear flow at yielding
q
y
is the square-root-of-the-product
average of the steel yield forces in the two directions. Equation (2.8) shows that the angle
α
r
at yield depends on the ratio of the longitudinal to transverse steel yield forces,
n
y
/
n
ty
.
In design, the shear flow at yield is usually given. The aim of the design is to find the yield
reinforcement in both directions,
n
y
and
n
ty
, and to check the diagonal concrete stress
σ
d
,
so that the concrete will not crush before the yielding of steel. For this purpose Equations
(2.4)-(2.6) can be written in the following forms:
n
y
=
q
y
tan
α
r
(2.9)
n
ty
=
q
y
cot
α
r
(2.10)
q
y
σ
d
=
(2.11)
α
r
cos
α
r
h
sin
2.1.2.2 Geometric Relationships of Equilibrium Equations
The equations in Section 2.1.2.1. are expressed in terms of
q
,
n
and
n
t
, which represent forces
per unit length. In order to express these equations in terms of stresses, we divide
q
,
n
and
n
t
by
h
, the thickness of the element, and define these three stress terms as follow:
q
h
=
τ
t
=
smeared shear stress
A
f
s
h
=
n
h
=
ρ
f
=
smeared steel stress in longitudinal direction
