Civil Engineering Reference
In-Depth Information
y
B (
x
B
,
y
B
)
T
AB
a
A (
x
A
,
y
A
)
x
F
IGURE
4.19
Method of tension coefficients
4.8 M
ETHOD OF
T
ENSION
C
OEFFICIENTS
An alternative form of the method of joints which is particularly useful in the analysis
of space trusses is the
method of tension coefficients.
Consider thememberAB, shown inFig. 4.19, which connects two pinned jointsAandB
whose coordinates, referred to arbitrary
xy
axes, are (
x
A
,
y
A
) and (
x
B
,
y
B
) respectively;
the member carries a
tensile
force,
T
AB
, is of length
L
AB
and is inclined at an angle
α
to the
x
axis. The component of
T
AB
parallel to the
x
axis at A is given by
(
x
B
−
x
A
)
L
AB
T
AB
L
AB
T
AB
cos
α
=
T
AB
=
(
x
B
−
x
A
)
Similarly the component of
T
AB
at A parallel to the
y
axis is
T
AB
L
AB
T
AB
sin
α
=
(
y
B
−
y
A
)
We now define a
tension coefficient t
AB
=
T
AB
/
L
AB
so that the above components of
T
AB
become
parallel to the
x
axis:
t
AB
(
x
B
−
x
A
)
(4.3)
parallel to the
y
axis:
t
AB
(
y
B
−
y
A
)
(4.4)
Equilibrium equations may be written down for each joint in turn in terms of tension
coefficients and joint coordinates referred to some convenient axis system. The solu-
tionof these equations gives
t
AB
, etc, whence
T
AB
=
t
AB
L
AB
in
which
L
AB
, unless given
,
may be calculated using Pythagoras' theorem, i.e.
L
AB
=
(
x
B
−
y
A
)
2
.
Again the initial assumption of tension in a member results in negative values
corresponding to compression. Note the order of suffixes in Eqs (4.3) and (4.4).
x
A
)
2
+
(
y
B
−
E
XAMPLE
4.4
Determine the forces in the members of the pin-jointed truss shown
in Fig. 4.20.
The support reactions are first calculated and are as shown in Fig. 4.20.
