Civil Engineering Reference
In-Depth Information
The forces in the members then follow
T
FB
=
t
FB
L
FB
=−
10
×
3
.
46 kN
=−
34
.
6 kN (compression)
Similarly
T
FD
=+
34
.
6 kN (tension)
T
FE
=
0
T
EC
=−
51
.
9 kN (compression)
T
EA
=−
17
.
3 kN (compression)
T
EB
=−
49
.
0 kN (compression)
The solution of Eqs (iv)-(vi) and (x)-(xii) in Ex. 4.6 was relatively straightforward in
that many of the coefficients of the tension coefficients could be reduced to unity. This
is not always the case, so that it is possible that the solution of three simultaneous
equations must be carried out. In this situation an elimination method, described in
standard mathematical texts, may be used.
4.12 A C
OMPUTER-BASED
A
PPROACH
The calculation of the member forces in trusses generally involves, as we have seen, in
the solution of a number of simultaneous equations; clearly, the greater the number
of members the greater the number of equations. For a truss with
N
members and
R
reactions
N
R
equations are required for a solution provided that the truss and the
support systems are both statically determinate. However, in some cases such as in
Ex. 4.6, it is possible to solve for member forces without first calculating the support
reactions. This still could mean that there would be a large number of equations to
solve so that a more mechanical approach, such as the use of a computer, would be
time and labour saving. For this we need to express the equations in matrix form.
+
At the joint F in Ex. 4.6 suppose that, instead of the 40 kN load, there are external
loads
X
F
,
Y
F
and
Z
F
applied in the positive directions of the respective axes. Eqs (i),
(ii) and (iii) are then written as
t
FD
(
x
D
−
x
F
)
+
t
FB
(
x
B
−
x
F
)
+
t
FE
(
x
E
−
x
F
)
+
X
F
=
0
t
FD
(
y
D
−
y
F
)
+
t
FB
(
y
B
−
y
F
)
+
t
FE
(
y
E
−
y
F
)
+
Y
F
=
0
t
FD
(
z
D
−
z
F
)
+
t
FB
(
z
B
−
z
F
)
+
t
FE
(
z
E
−
z
F
)
+
Z
F
=
0
In matrix form these become
=
x
D
−
x
F
x
B
−
x
F
x
E
−
x
F
t
FD
t
FB
t
FE
−
X
F
y
D
−
y
F
y
B
−
y
F
y
E
−
y
F
−
Y
F
z
D
−
z
F
z
B
−
z
F
z
E
−
z
F
−
Z
F
