Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

120 kN ⋅ m concentrated moment: From case 1 of Table 7.2, the 120 kN ⋅ m concentrated
moment acting at x = 2 m is represented by the singularity function
30 kN/m
wx ( ) =−120 kN⋅ m〈 x − 2 m〉 − 2
(b)
Note that the negative sign is included to account for the counterclockwise moment rotation
shown on this beam.
45 kN concentrated load: From case 2 of Table 7.2, the 45 kN concentrated load acting at
x = 4 m is represented by the singularity function
wx ( ) =− 45 kN〈 x − 4m〉 − 1
(c)
Note that the negative sign is included to account for the downward direction of the
45 kN concentrated load shown on the beam.
30 kN/m uniformly distributed load: The uniformly distributed load requires the use of
two terms. Term 1 applies the 30 kN/m downward load at point D, where x = 6 m:
wx ( ) =− 30 kN/m 〈 x − 6m〉
0
The uniformly distributed load represented by this term continues to act on the beam for
values of x greater than x = 6 m. For the beam and loading considered here, the distributed
load should act only within the interval 6 m ≤ x ≤ 9 m. To terminate the downward distributed
load at x = 9 m requires the superposition of a second term. This second term applies
an equal-magnitude upward uniformly distributed load that begins at E, where x = 9 m:
0 0
wx ( ) =− 30 kN/m 〈 x − 6m〉 + 30 kN/m 〈 x − 9m〉 (d)
The addition of these two terms produces a downward 30 kN/m distributed load that
begins at x = 6 m and terminates at x = 9 m.
30 kN/m
30 kN/m
x
D E F
+
x
D E F
=
x
D E F
6 m
3 m
3 m
6 m
3 m
3 m
6 m
3 m
3 m
F y
F y
F y
term 1 : Downward uniformly
term 2: upward uniformly
distributed load beginning at D distributed load beginning at E
–(30 kN/m) x – 6 m 0 + (30 kN/m) x – 9 m 0
30 kN/m distributed load
beginning at D and ending at E
Reaction force F y : The upward reaction force at F is expressed by
−1 − 1
wx ( ) = F 〈 x − 12 m〉 = 61.25 kN 〈 x − 12 m〉
(e)
y
As a practical matter, this term has no effect, since the value of Equation (e) is zero for all
values of x ≤ 12 m. Because the beam is only 12 m long, values of x > 12 m make no sense
in this situation. However, the term will be retained here for completeness and clarity.
Complete-beam loading expression: The sum of Equations (a) through (e) gives the load
expression w(x) for the entire beam:
wx ( ) = 73.75 kN 〈 x − 0m〉 −120 kN⋅ m〈 x − 2 m〉 − 45kN〈 x − 4 m〉
−1 −2 −1
− 30 kN/m 〈 x − 6m〉 + 30 kN/m 〈 x − 9m〉 + 61.25 kN 〈 x − 12 m〉
0 0 − 1
(f)
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