Civil Engineering Reference
In-Depth Information
l
0
ε
Ouj
dx
;
l
0
ψ
uj
x dx
;
l
0
ψ
uj
dx
with
j
f
1
j
=
−
f
2
j
=
−
f
3
j
=
=
1, 2, 3
(13.8)
The integrals in this equation are evaluated numerically (Section 13.8) using
values of
, determined by Equation (2.19) at a number of sections for
which the geometry and cross-sectional area of reinforcement are known. For
cracked sections,
ε
O
and
ψ
represent mean values determined by Equations
(8.43) and (8.44). This requires that the depth
c
of the compression zone and
the interpolation coe
ε
O
and
ψ
ned in Section 8.3) be known. The two
parameters depend upon the stresses existing before introducing the incre-
ments in the forces at the ends. Thus, the tangent sti
cient
ζ
(de
fi
ff
ness depends upon the
stress level and the state of cracking of the member.
In order to generate the tangent sti
ff
ness matrix for the structure, the tan-
gent sti
ness matrices, [
S
*] of individual members must be transformed from
the local coordinate systems to the global system:
ff
[
S
member
]
=
[
T
]
T
[
S
*] [
T
]
(13.9)
where [
S
member
] is the member sti
ff
ness matrix in global coordinates; [
T
] is a
transformation matrix given by:
c
−
s
c
0
0
0
1
[
t
]
[0]
[0]
[
t
]
[
T
]
=
;
[
t
]
=
s
(13.10)
0
where
c
being the angle between the global
x
-
direction and the local
x
*-axis (Fig. 13.2(a) ). The matrix [
T
] can be used for
transformation of member end forces and displacements from local to global
or vice versa:
=
cos
α
and
s
=
sin
α
, with
α
{
D
*}
=
[
T
]{
D
}
;
{
F
}
global
=
[
T
]
T
{
F
*}
(13.11)
13.5
Examples of stiffness matrices
Example 13.1 Stiffness matrix of an uncracked
prismatic cantilever
Derive the sti
ness matrix with respect to non-centroidal coordinates,
shown in Fig. 13.4(a), for an uncracked cantilever having a constant
cross-section with properties,
A
,
B
and
I
. What are the displacements at
the three coordinates due to a downward force
P
applied at the free end?
ff
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