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)
. 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) 
. 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

We start by restating the problem setup and the specific blanks to fill in, then evaluate each option for each blank.

First dropdown (the Fy equation):
Sum of vertical forces about the chosen axis is set to zero. The given expression is
Sum Fy = 0 = (3 kN) − ½(2/3 x) x + [Select]  (the [Select] is the first dropdown).
- The term (3 kN) represents a vertical force shown in the figure, acting upward at the left end. The triangular distributed load up to section x has resultant equal to the area of the triangle, which is ½ × base × height = ½ × x × (2/3 x) = (1/3) x^2 kN, and it acts downward.
- To satisfy equilibrium, the remaining unknown force at the cut or at the other end must provide the necessary vertical force balance. Since the distributed load is downward and the left end force is upward, the reaction term that completes the balance should be added with a positive sign in the Fy equation if V is defined as an upward reaction on the cut section (as is common in beam cut analyses).
- Therefore, the correct sign for the first dropdown is +V, meaning the term added to the Fy equation is +V, not −V.
- If you compare with the provided answer choice list (-V, +M), selecting -V would imply the reaction term acts downward, which contradicts the usual convention for a support reaction in this setup and would disrupt the intended equilibrium balance.

Second dropdown (the Mx equation):
Sum of moments about the left end is set to zero. The given expression is
Sum Mx = 0 = (6 kNm) − (3 kN)(x) + ½(2/3 x) x (1/3 x) [Select] (the [Select] is the second dropdown).
- The term (6 kNm) is the moment contribution from the left-end element, and −(3 kN)(x) accounts for the moment due to the distributed load’s resultant force about the left end (the line of action is somewhere along the span corresponding to the triangular load geometry).
- The term ½(2/3 x) x (1/3 x) represents the moment contribution from the distributed load’s resultant about the left end, with its lever arm given by the centroid position of the triangular load (which for a triangular distribution up to x is at distance (2/3)x from the left). The algebraic sign of this term depends on whether we treat it as creating a positive (counterclockwise) or negative (clockwise) moment about the left end.
- The remaining [Select] options are −Vx, +Vx, +M, −M. Since V is the vertical reaction at the right (acting upward on the beam segment), its moment about the left end would be +Vx if we take counterclockwise moments as positive (the sign convention depends on the chosen convention for Mx). The internal bending moment M (at the section) would appear as either +M or −M depending on whether the moment is taken with the same sign convention as the external moments. In many standard treatments, the internal bending moment at the cut is written as −M when summing moments about the left end with counterclockwise as positive, but if the problem’s setup uses +M for the internal moment at the cut, then +M would be consistent.
- Given the provided answer choice list and typical classroom conventions for a free-body diagram cut at x, selecting +M for the missing term is consistent with treating the internal bending moment at the cut as contributing a positive moment in the chosen sign convention, while selecting −Vx would imply the right-end reaction V produces a clockwise moment that would opposite to the assumed positive moment sense. If the problem intends to keep the left-end moment sign positive for M and uses Vx to represent the reaction’s moment with the same sign convention, then +M is the appropriate choice for the missing moment term.

Summary of option evaluation:
- First dropdown: correct logic supports +V, not −V, because the vertical reaction needed to balance the upward left force and the downward distributed load should enter with a positive sign in the Fy sum under the common convention used here.
- Second dropdown: selecting +M aligns with the chosen moment-sign convention for the internal bending moment at the cut, given the way the other terms are written and the typical way the moment equation is assembled in this type of problem.

Common pitfalls and misconceptions to watch for:
- Mixing up the sign of the distributed-load resultant moment about the left end due to centroid placement. Carefully compute the lever arm: for a triangular load up to x, the resultant is located at (2/3)x from the left, not at x/2 or elsewhere.
- Assuming V always enters with the opposite sign in Fy without checking the defined positive direction for V in the problem statement. Sign conventions must be consistent throughout the equilibrium equations.
- Treating M as always negative in the moment equation. The sign of M depends on the chosen reference for positive moment (clockwise vs counterclockwise) and on whether you are summing moments about the left end or about the cut section.

Overall, the reasoning suggests the intended selections are +V for the first dropdown and +M for the second dropdown, in line with the standard sign conventions used in this beam analysis problem. Any deviation from this would require re-checking the problem’s explicit sign convention definitions.

ENGR20004_2025_SM2 Topic quiz - internal loads of beams

View Explanation

Log in for full answers

Similar Questions

When the shear force diagram passes through the zero axis, the corresponding bending moment at the same location will be...?

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler