Step 2: Shear and Moment Functions

A free-body diagram of a beam segment of length x is shown. The intensity of the triangular load at the section is found by proportion, that is,
w/x=(2kN/m)/(3m)
or
𝑤
=
(
2
3

𝑥
)

kN
/
m
. The resultant of the distributed loading is found from the area under the diagram. Thus,

\sum F_y=0=(3\,kN)-\frac{1}{2}\left(\frac{2}{3}x\right)x
[ Select ]
+V
-V
+M
-M

V=\left(3-\frac{1}{3}x^2\right)kN

\sum M_x=0=(6kNm)-(3kN)(x)+\frac{1}{2}(2/3\,x)x(1/3\,x)
[ Select ]
-Vx
+Vx
+M
-M

M=(-6+3x-\frac{1}{9}x^3) kNm

Choose the missing terms in the above equations

Question

Step 2: Shear and Moment Functions

A free-body diagram of a beam segment of length x is shown. The intensity of the triangular load at the section is found by proportion, that is, 
w/x=(2kN/m)/(3m)
or  
𝑤
=
(
2
3
 
𝑥
)
 
kN
/
m
. The resultant of the distributed loading is found from the area under the diagram. Thus,

\sum F_y=0=(3\,kN)-\frac{1}{2}\left(\frac{2}{3}x\right)x
[ Select ]
+V
-V
+M
-M

V=\left(3-\frac{1}{3}x^2\right)kN

\sum M_x=0=(6kNm)-(3kN)(x)+\frac{1}{2}(2/3\,x)x(1/3\,x)
[ Select ]
-Vx
+Vx
+M
-M

M=(-6+3x-\frac{1}{9}x^3) kNm

Choose the missing terms in the above equations

BlackTom AI · Accepted Answer

First, I will restate the problem setup and the two places where a sign choice must be made so the equilibrium equations balance.

- The problem shows a beam segment of length x with a triangular distributed load that yields a resultant downward force, and there is a vertical reaction V at the right support as well as a clockwise external moment M applied at the right end. The task is to choose the correct algebraic signs for the missing terms in two equations: one in the vertical force balance sum F_y = 0, and one in the moment balance sum M_x = 0.

- For the vertical force balance sum F_y = 0, the equation is formed by summing all vertical forces with sign convention (typically upward positive, downward negative). The given partial expression indicates there is a downward resultant from the distributed load, a downward external force of 3 kN at the left or somewhere along the beam, and the vertical reaction V (likely upward) at the right end. To achieve a correct equilibrium expression, the unknown term that must be inserted before the V term must have the opposite sign to the downward components so that the net sum equals zero. Since the downward contributions are positive in the displayed format (or explicitly subtracting them), the missing term that ensures balance with V must be negative. Therefore, the first dropdown should be -V, not +V.

- For the moment balance sum M_x = 0, we choose a sign convention (for example, counterclockwise moments as positive). In the shown setup, the moment M is an external couple applied at the right end with a particular sense. If the diagram uses a positive sense for M (counterclockwise) and the reaction forces create opposing moments about the section, the algebraic term corresponding to the external couple M should be added with the correct sign to satisfy the equilibrium equation. The correct placement of the sign, given the standard convention used in this problem, is to include +M in the sum of moments about the chosen point, rather than -M. Hence, the second dropdown should be +M, not -M.

- Now evaluating the other possible choices:
  - If the first dropdown were +V, the vertical force balance would misalign the contributions from the distributed load and the reaction, leading to an incorrect net sum that does not cancel to zero under the given loading, so +V is incorrect.
  - If the second dropdown were -M, the moment balance would again misrepresent the torque produced by the external couple relative to the chosen sign convention, producing an inconsistent equation with the rest of the setup, so -M is incorrect.

- In summary, the correct sign choices are: the first missing term should be -V to satisfy vertical force equilibrium, and the second missing term should be +M to satisfy the moment equilibrium with the adopted sign convention. The incorrect options fail to balance either the force or the moment equations as dictated by the load, reaction, and moment configuration on the beam.

ENGR20004_2025_SM2 Topic quiz - internal loads of beams

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