Civil Engineering Reference
In-Depth Information
Since Equations [1] and [2] involve the steel ratios
ρ
tp
, these two equilibrium
equations are coupled to the compatibility equations through the variable
t
d
. The thickness
t
d
is a geometric variable which has to be determined not only by the equilibrium conditions, but
also by the compatibility conditions. This is similar to the determination of the neutral axis in
the bending theory (see Chapter 3), which requires the plane section compatibility condition.
ρ
,
ρ
t
,
ρ
p
,
7.1.1.2 Bredt's Equilibrium Equation
In Section 7.1.1.1 three equations, [1]-[3], are derived from the equilibrium of a membrane
element in the shear flow zone. To maintain equilibrium of the whole cross-section, however,
a fourth equation must be satisfied. The derivation of this additional equilibrium equation has
been given in Section 2.1.4.1. The resulting equation (2.46) is expressed in terms of the shear
flow
q
. For a shear flow zone thickness of
t
d
the shear stress,
τ
t
,is:
T
2
A
o
t
d
τ
t
=
(7.8) or [4]
This additional equation [4] also introduces an additional variable, the torque
T
.
The thickness of the shear flow zone
t
d
, is strongly involved in Equation [4] not only
explicitly through
t
d
, but also implicitly through
A
o
, which is a function of
t
d
. Consequently,
the three equilibrium equations, [1], [2] and [4], are now coupled, through the variable
t
d
,to
the compatibility equations. These compatibility equations will be derived in Section 7.1.2.
7.1.2 Compatibility Equations
7.1.2.1 2-D elements in shear
As shown in Figure 7.1(a) and (b), the 2-D element A in the shear flow zone is subjected to
a shear stress. The in-plane deformation of this element should satisfy the three compatibility
equations (5.97)-(5.99), in Section 5.4.2:
ε
=
ε
r
cos
2
α
r
+
ε
d
sin
2
α
r
(7.9) or [5]
=
ε
r
sin
2
α
r
+
ε
d
cos
2
ε
t
α
r
(7.10) or [6]
γ
t
2
=
(
ε
r
−
ε
d
)sin
α
r
cos
α
r
(7.11) or [7]
This 2-D element A will also be subjected to out-of-plane deformation. To study this out-
of-plane deformation, we will first relate the shear strain
γ
t
in Equation [7] to the angle of
twist
θ
by the geometric relationship presented in Section 7.1.2.2.
7.1.2.2 Shear Strain due to Twisting
When a tube is subjected to torsion, the relationship between the shear strain
γ
t
in the wall
of the tube and the angle of twist
of the member, can be derived from the compatibility
condition of warping deformation. In Figure 7.2, a longitudinal cut is made in an infinitesimal
length d
θ
of a tube. Progressing from one side of the cut along the circumference to the other
side of the cut, the differential warping displacement (in the
-direction) must be zero when
integrating throughout the whole perimeter.
