Civil Engineering Reference
In-Depth Information
A
t
f
t
s
t
h
=
n
t
h
=
ρ
t
f
t
=
smeared steel stress in transverse direction
The three equilibrium equations (2.4)-(2-6) can then be written as:
ρ
f
=
τ
t
tan
α
r
(2.12)
ρ
t
f
t
=
τ
t
cot
α
r
(2.13)
1
σ
d
=
τ
t
α
r
=
τ
t
(tan
α
r
+
cot
α
r
)
(2.14)
sin
α
r
cos
Equations (2.12)-(2,14) can also be expressed in terms of the diagonal concrete stress
σ
d
by substituting
τ
t
from Equation (2.14) into Equations (2.12) and (2.13):
ρ
f
=
σ
d
sin
2
α
r
(2.15)
=
σ
d
cos
2
ρ
t
f
t
α
r
(2.16)
τ
t
=
σ
d
sin
α
r
cos
α
r
(2.17)
These two sets of equations (2.12)-(2.14) and (2.15)-(2.17) each satisfy the Mohr circle,
as illustrated in Figure 2.3. Figure 2.3(a) shows the Mohr circle in a
σ
−
τ
coordinate system,
where
the shear stress. Point A on the Mohr circle represents
the longitudinal face (the direction of a face is defined by its normal axis). Acting on this
face are a shear stress
σ
is the normal stress and
τ
τ
t
, indicated by the vertical coordinate, and a longitudinal steel stress
ρ
f
, indicated by the horizontal coordinate. At 180
◦
from point A, we have point B which
represents the transverse face in the shear element. Acting on this face are a shear stress
−
τ
t
(vertical coordinate) and a transverse steel stress
ρ
t
f
t
, (horizontal coordinate). Point C on the
σ
axis of the Mohr circle represents the principal face in the
r
direction, i.e. the face of cracks.
On this crack face the normal stress is assumed to be zero. The incident angle 2
α
r
between
points A and C in the Mohr circle is twice the angle
axis and the
r
axis in
the shear element. At 180
◦
from point C, we have point D which represents a face on which
only the diagonal concrete stress
α
r
between the
σ
d
is acting. The above explanation of the application of the
Mohr circle shows its importance in the analysis of stresses in various directions. Detailed
derivation and discussion of Mohr circles will be provided in Chapter 4.
The introduction of the Mohr circle at this point is intended to illustrate the geometric
relationships of Equations (2.12)-(2.17). The geometric relationship of the half Mohr circle
defined by ACD are illustrated in Figure 2.3(b) and (c). If CD in Figure 2.3(b) is taken as
unity, then CE
sin
2
cos
2
=
α
r
,ED
=
α
r
, and AE
=
sin
α
r
cos
α
r
. These three trigonometric
values are actually the ratios of the three stresses
ρ
f
,
ρ
t
f
t
and
τ
t
, respectively, divided by
the diagonal concrete stress
σ
d
, in the Mohr circle of Figure 2.3(a). Hence, the set of three
equilibrium equations, (2.15)-(2.17), simply states the geometric relationships illustrated in
Figure 2.3(b). This set of three equilibrium equations will be called the
first type
of expression
for Mohr stress circles in Section 5.1.2.
Similarly, if AE in Figure 2.3(c) is taken as unity, then CE
=
tan
α
r
,ED
=
cot
α
r
, and
CD
=
(tan
α
r
+
cot
α
r
)
=
1
/
sin
α
r
cos
α
r
. These three trigonometric values are actually the
ratios of the three stresses
τ
t
,inthe
Mohr circle of Figure 2.3(a). Hence, the set of three equilibrium equations, (2.12)-(2.14),
simply states the geometric relationships illustrated in Figure 2.3(c). This set of three
ρ
f
,
ρ
t
f
t
and
σ
d
, respectively, divided by the shear stress
