Civil Engineering Reference
In-Depth Information
Equation (5.7) itself can be written as
1
(
σ
r
−
σ
d
)
=
τ
t
(5.20)
sin
α
r
cos
α
r
Equations (5.18)-(5.20) are the second type of expression for the equilibrium condition.
They are convenient for the design of RC 2-D elements.
Equations (5.18)-(5.20) represent the second type of geometric relationship in the Mohr
circle, as shown in Figure 5.2(c). When AE in Figure 5.2(c) is taken as unity, then EC
=
tan
α
r
,ED
=
cot
α
r
, and CD
=
1
/
sin
α
r
cos
α
r
. These three trigonometric values are actually
the ratios of the three stresses (
−
σ
+
ρ
f
+
σ
r
), (
−
σ
t
+
ρ
t
f
t
+
σ
r
) and (
σ
r
−
σ
d
), divided
by the shear stress
τ
t
, respectively.
Multiplying Equation (5.18) by Equation (5.19) gives Equation (5.10), and dividing Equation
(5.18) by Equation (519) produces Equation (5.11).
In design, the small tensile stress of concrete is often neglected, i.e.
σ
r
=
0. Then Equations
(5.18)-(5.20) become
ρ
f
=
σ
+
τ
t
tan
α
r
(5.21)
ρ
t
f
t
=
σ
t
+
τ
t
cot
α
r
(5.22)
1
(
−
σ
d
)
=
τ
t
(5.23)
sin
α
r
cos
α
r
Furthermore, for the case of pure shear,
σ
=
σ
t
=
0. Then
ρ
f
=
τ
t
tan
α
r
(5.24)
ρ
t
f
t
=
τ
t
cot
α
r
(5.25)
1
(
−
σ
d
)
=
τ
t
(5.26)
sin
α
r
cos
α
r
Equations (5.24)-(5.26) are identical to Equations (2.12)-(2.14). When the tensile stress of
concrete is neglected (
σ
r
=
0) and for the case of pure shear elements (
σ
=
σ
t
=
0), Figure
5.2(a) and (c) become the same as Figure 2.3(a) and (c), respectively.
5.1.4 Equilibrium Equations in Terms of Double Angle
The equilibrium condition shown in Figure 5.1(a), (b) and (c) can also be expressed in terms
of the relationships between the fixed angle
α
r
, and an imaginary angle
α
s
to be defined. Subtracting Equation (5.22) from Equation (5.21) gives
α
1
, the rotating angle
(
σ
−
σ
t
)
=−
τ
t
(tan
α
r
−
cot
α
r
)
+
(
ρ
f
−
ρ
t
f
t
)
(5.27)
Dividing Equation (5.27) by 2
τ
t
gives
(
σ
−
σ
t
)
2
1
2
(tan
(
ρ
f
−
ρ
t
f
t
)
2
=−
α
r
−
cot
α
r
)
+
(5.28)
τ
t
τ
t
