Statics Quiz: Maximum Moment And Shear
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Maximum Moment And ShearQuestion 1 of 16

For a cantilever beam with length L subjected to a linearly varying load that increases from zero at the free end to w0w_0 at the fixed end, where does the maximum shear force occur?

At the free end of the cantilever beam
At a distance of L/3 from the fixed end
At a distance of L/2 from the fixed end
At a distance of 2L/3 from the fixed end
At the fixed end of the cantilever beam
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Statics Quiz

Statics Quiz: Maximum Moment And Shear

Practice Maximum Moment And Shear in Statics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Maximum Moment And Shear, giving you a quick way to practice the rules, question types, and explanations that matter most for Statics.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

For a cantilever beam with length L subjected to a linearly varying load that increases from zero at the free end to w0w_0 at the fixed end, where does the maximum shear force occur?

  1. At the free end of the cantilever beam
  2. At a distance of L/3 from the fixed end
  3. At a distance of L/2 from the fixed end
  4. At a distance of 2L/3 from the fixed end
  5. At the fixed end of the cantilever beam (correct answer)
Explanation: When analyzing shear forces in beams with distributed loads, remember that shear force varies along the beam's length according to the loading pattern, and maximum values typically occur where the load accumulates most. For this cantilever beam with a triangular load distribution (zero at the free end, w0w_0 at the fixed end), you need to consider how shear force develops. The shear force at any point equals the total load to one side of that section. Since the load increases linearly toward the fixed end, the shear force will be maximum where the total accumulated load is greatest. At the fixed end, the shear force must resist the entire triangular load distribution. The total load equals 12w0L\frac{1}{2}w_0L (area of the triangle), and this entire force creates the reaction at the fixed support. Therefore, the maximum shear force occurs at the fixed end of the cantilever. However, since "at the fixed end" isn't listed among options A through D, the correct answer must be E (not shown but implied as the unlisted correct choice). Option A is wrong because the free end has zero shear force - no load exists beyond this point to create shear. Options B, C, and D represent various intermediate locations where only partial load accumulation occurs, resulting in smaller shear forces than at the fixed end. Study tip: For cantilever beams, maximum shear always occurs at the fixed support where reactions develop. Maximum moment, however, may occur elsewhere depending on the loading pattern. Always visualize the load accumulation from free end toward the support.

Question 2

A beam has shear forces of +15 kN, -5 kN, and +8 kN at points A, B, and C respectively, which are spaced 2 m apart. If there are no applied moments between these points, which location is most likely to have the maximum absolute moment?

  1. Exactly at point A where shear is maximum positive
  2. Between points A and B where shear changes sign (correct answer)
  3. Exactly at point B where shear is minimum
  4. Between points B and C where shear increases rapidly
  5. Exactly at point C where shear returns to positive
Explanation: When analyzing beam moments, you need to understand the fundamental relationship between shear force and bending moment: the slope of the moment diagram equals the shear force at any point. This means moments reach maximum or minimum values where the shear force crosses zero. Looking at the given data, you have positive shear at point A (+15 kN) that becomes negative at point B (-5 kN). Since shear changes from positive to negative between these points, it must cross zero somewhere in this interval. At this zero-shear location, the moment diagram has zero slope, indicating a local maximum or minimum moment value. Option A is incorrect because maximum positive shear doesn't indicate maximum moment. High shear actually means the moment is changing rapidly, not that it's at an extreme value. Option C misses the key insight that minimum shear force doesn't correspond to maximum moment magnitude. While point B has the most negative shear, this indicates the moment is decreasing rapidly there, not reaching its peak value. Option D incorrectly focuses on the rate of shear change rather than the zero-crossing point. The rapid increase from -5 kN to +8 kN between B and C means the moment is changing quickly, but maximum moment occurs where shear equals zero. The correct answer is B because moments reach extreme values where shear force crosses zero, which must occur between points A and B where the shear transitions from positive to negative. Study tip: Always look for zero-shear locations when finding maximum moments—that's where the moment diagram has horizontal tangent lines, indicating peaks and valleys.

Question 3

A simply supported beam carries two equal concentrated loads P at the quarter points (L/4 and 3L/4 from the left support). Between which points does the maximum positive moment occur?

  1. Between the left support and the first load at L/4
  2. Between the two concentrated loads at L/4 and 3L/4 (correct answer)
  3. Between the second load at 3L/4 and right support
  4. Exactly under the first concentrated load at L/4
  5. Exactly under the second concentrated load at 3L/4
Explanation: When analyzing bending moments in beams with concentrated loads, you need to understand that maximum positive moments typically occur between loads where the shear force crosses zero, creating a peak in the moment diagram. For this symmetrically loaded beam, start by finding the reactions. Due to symmetry, each support carries R=PR = P upward. Next, construct the shear force diagram: it starts at +P+P, drops to −P-P at the first load (L/4), remains constant until the second load (3L/4), then returns to zero. The shear force equals zero at the midpoint (L/2), which lies between the two concentrated loads. Since bending moment reaches its maximum where shear force equals zero, the maximum positive moment occurs at L/2, which falls between the quarter points where the loads are applied. This confirms that the maximum positive moment occurs between the two concentrated loads at L/4 and 3L/4. Answer choice A is incorrect because the moment is still increasing as you move from the left support toward the first load. Choice C is wrong because this region experiences negative shear, and while moments exist here, they're smaller than the maximum. Choice D incorrectly identifies a specific point rather than a region, and the moment at L/4 is actually less than the maximum that occurs at L/2. Study tip: For symmetrically loaded beams, the maximum positive moment almost always occurs at the center of the span where the shear force diagram crosses zero. Always sketch the shear diagram first to identify where moments peak.

Question 4

A simply supported beam has three concentrated loads: 10 kN at 2 m, 15 kN at 5 m, and 8 kN at 7 m from the left support. The beam span is 9 m. Between which two consecutive loads does the shear force change from positive to negative?

  1. Between the left support and the 10 kN load
  2. Between the 10 kN load and the 15 kN load (correct answer)
  3. Between the 15 kN load and the 8 kN load
  4. Between the 8 kN load and the right support
  5. The shear force never changes from positive to negative
Explanation: When analyzing shear force diagrams for simply supported beams, you need to track how the shear force changes as you move from left to right along the beam. The shear force changes abruptly (jumps down) at each concentrated load location. Start by finding the reaction forces. Using equilibrium equations for this 9 m beam with loads of 10 kN at 2 m, 15 kN at 5 m, and 8 kN at 7 m, the left reaction is 20.56 kN and the right reaction is 12.44 kN. Now trace the shear force from left to right. Just after the left support, the shear force starts at +20.56 kN. At the 10 kN load (2 m), it drops to +10.56 kN—still positive. At the 15 kN load (5 m), it drops to -4.44 kN—now negative. This is where the sign change occurs, confirming answer B is correct. Let's examine why the other options are wrong: A is incorrect because the shear force remains positive throughout this segment (from +20.56 kN to +10.56 kN). C is wrong because the shear force is already negative before the 8 kN load and simply becomes more negative afterward (from -4.44 kN to -12.44 kN). D is incorrect because the shear force stays negative in this final segment until reaching zero at the right support. Study tip: Always calculate reaction forces first, then systematically trace the shear force from left to right, noting that each downward load causes the shear force to "jump down" by that load's magnitude. The zero-crossing point often occurs at the largest load.

Question 5

A cantilever beam supports a uniformly varying load that decreases linearly from w0=8w_0 = 8 kN/m at the fixed end to zero at the free end. The beam length is 5 m. What is the maximum shear force magnitude and where does it occur?

  1. 20 kN at the fixed support (correct answer)
  2. 16 kN at 1 m from the fixed support
  3. 12 kN at 2 m from the fixed support
  4. 8 kN at 3 m from the fixed support
  5. 4 kN at 4 m from the fixed support
Explanation: When analyzing cantilever beams with varying loads, remember that shear force represents the internal force needed to maintain equilibrium, and it's directly related to the applied loading through integration. For a linearly varying load from w0=8w_0 = 8 kN/m at the fixed end to zero at the free end over 5 m, the load function is w(x)=8(1−x/5)w(x) = 8(1 - x/5) where xx is measured from the fixed support. To find shear force, you integrate the load from the free end back toward any point. The total load on the beam is the area under the triangular load diagram: 12×8×5=20\frac{1}{2} \times 8 \times 5 = 20 kN. In cantilever beams, maximum shear always occurs at the fixed support because that's where the beam must resist the entire applied load. At the fixed end, the shear force equals the total load magnitude: 20 kN. Choice A correctly identifies this maximum value and location. Choice B (16 kN at 1 m) represents a common error of calculating shear at an arbitrary interior point without recognizing the maximum occurs at the support. Choice C (12 kN at 2 m) likely comes from incorrectly assuming maximum shear occurs at the beam's midpoint. Choice D (8 kN at 3 m) might result from confusing the maximum load intensity (8 kN/m) with shear force magnitude. For cantilever problems, always remember: maximum shear occurs at the fixed support and equals the total applied load. Start your shear analysis there, then work toward the free end where shear becomes zero.

Question 6

A propped cantilever beam (fixed at left end, pin support at right end) carries a uniform load w over its entire length L. The maximum negative moment occurs at which location?

  1. At the fixed support where rotation is prevented (correct answer)
  2. At a distance of L/3 from the fixed support
  3. At the midspan where deflection is maximum
  4. At a distance of 2L/3 from the fixed support
  5. At the pin support where vertical displacement is zero
Explanation: When analyzing indeterminate beams like propped cantilevers, you need to understand how the constraints create internal moments. A propped cantilever has a fixed support (preventing rotation and translation) and a pin support (preventing only translation), making it statically indeterminate. The key insight is that maximum negative moments occur where rotational restraint is greatest. At the fixed support, the beam cannot rotate despite the applied loading trying to cause rotation. This constraint creates the largest resisting moment in the structure. The uniform load ww creates a distributed moment demand along the beam, but the fixed end must provide whatever reaction moment is necessary to maintain zero rotation - this results in the maximum negative moment. Looking at the incorrect options: Choice B (L/3 from fixed support) represents a common location for maximum positive moment in some beam configurations, but not maximum negative moment here. Choice C (midspan) incorrectly associates maximum deflection with maximum moment - while these sometimes coincide in simply supported beams, they don't in indeterminate structures where support reactions redistribute moments. Choice D (2L/3 from fixed support) has no special significance for moment distribution in a propped cantilever. The moment diagram for a propped cantilever under uniform load shows the most negative value at the fixed support, then becomes less negative (or positive) moving toward the pin support, which can only provide vertical reaction force. Study tip: For indeterminate beams, maximum negative moments almost always occur at fixed supports where rotational restraint is enforced. The fixed end "fights back" hardest against the applied loads.

Question 7

A cantilever beam with length L carries a uniformly distributed load w over its entire length and an additional concentrated load P at its free end. At what distance from the fixed support does the maximum absolute shear occur?

  1. At a distance of L/2 from the fixed support
  2. At a distance of 3L/4 from the fixed support
  3. At the free end, distance L from fixed support
  4. At the fixed support, distance 0 from itself (correct answer)
  5. The maximum shear is constant along the entire beam
Explanation: When analyzing shear forces in beams, remember that shear varies along the beam's length, and maximum absolute shear often occurs where loads are concentrated or at supports. For this cantilever beam, you need to construct the shear force diagram. Starting from the free end and moving toward the fixed support, the shear force begins at −P-P (due to the concentrated load) and decreases linearly due to the distributed load ww. The shear at any distance xx from the free end is V(x)=−P−wxV(x) = -P - wx. At the fixed support (x=Lx = L), the shear reaches its maximum absolute value: V=−P−wLV = -P - wL. This represents the total downward force that must be balanced by the support reaction. The absolute value ∣P+wL∣|P + wL| is indeed the maximum along the entire beam length. Choice A (L/2L/2) might tempt you if you're thinking about simply supported beams, where maximum moments often occur at midspan, but shear and moment have different behaviors. Choice B (3L/43L/4) has no special significance for this loading pattern. Choice C (free end) ignores that shear accumulates as you move toward the support - while the concentrated load PP creates significant shear there, it's not the maximum absolute value. The key insight is that in cantilever beams, shear forces accumulate from the free end toward the fixed support, reaching their maximum absolute value at the support where all loads must be resisted. Always sketch the shear diagram when analyzing beam problems - it reveals the complete force distribution story.

Question 8

A continuous beam over three supports has a concentrated load P applied at midspan of the first bay and a uniformly distributed load w over the entire second bay. Where is the maximum negative moment most likely to occur?

  1. At the first support under the applied load P
  2. At the interior support between the two bays (correct answer)
  3. At the third support at the end of the distributed load
  4. Under the concentrated load P in the first bay
  5. At midspan of the second bay under the distributed load
Explanation: When analyzing continuous beams, you need to understand how loads create both positive and negative moments, and where these extremes typically occur. Negative moments (causing tension on top of the beam) are particularly critical at supports in continuous structures. The interior support between the two bays experiences the combined effects of both loading conditions. The concentrated load P in the first bay creates a significant reaction at the interior support, while the distributed load w over the entire second bay also contributes substantial reaction forces at this same location. These combined effects create the highest negative moment in the system, making answer B correct. Looking at the incorrect options: Answer A suggests the maximum negative moment occurs at the first support, but this location only experiences effects from the first bay's loading and acts more like a simple support. Answer C points to the third support, which primarily reacts to the distributed load from the second bay alone—significant, but not as severe as the interior support that handles both loads. Answer D incorrectly identifies the location under load P, but concentrated loads typically create maximum positive moments at their application points, not negative moments. Remember this pattern: in continuous beams, maximum negative moments almost always occur at interior supports because these points must resist the combined effects of loads from adjacent spans. When you see multiple loading conditions on different spans, focus your attention on the support where these effects converge—that's where you'll find your critical negative moment.

Question 9

A simply supported beam has a moment diagram that shows two local maxima of equal magnitude in adjacent spans created by the loading pattern. If no applied concentrated moments exist, what does this indicate about the shear force distribution?

  1. The shear force is constant throughout the beam length
  2. The shear force has exactly two zero-crossings at the moment maxima (correct answer)
  3. The shear force changes sign exactly once at the beam centerline
  4. The shear force has three zero-crossings including the supports
  5. The shear force never equals zero between the supports
Explanation: When analyzing beam behavior, the key relationship to remember is that shear force and moment are connected through calculus: the slope of the moment diagram at any point equals the shear force at that point, and shear force equals the derivative of moment with respect to position. If the moment diagram shows two local maxima of equal magnitude, this tells you critical information about the shear behavior. At each local maximum, the moment diagram has zero slope, which means the shear force must equal zero at those two points. Since the beam is simply supported with no applied concentrated moments, the shear force must also be zero at both support points (where moments are zero). This creates exactly four zero-crossings: two at the supports and two at the moment maxima locations. However, looking at the answer choices, option B correctly identifies that there are exactly two zero-crossings at the moment maxima specifically, which is the most direct consequence of having two local moment maxima. Option A is incorrect because constant shear would produce a linearly varying moment, not local maxima. Option C incorrectly assumes the beam has only one zero-crossing at the centerline, but two separate maxima require two separate zero-crossings. Option D mentions three zero-crossings including supports, but this undercounts the total (there are actually four total) and doesn't specifically address the maxima. Study tip: Always remember that zero slope on a moment diagram means zero shear force at that location. When you see local moment extrema, immediately look for corresponding shear zero-crossings.

Question 10

A simply supported beam carries two equal concentrated loads P=50P = 50 kN each, located at x=3x = 3 m and x=9x = 9 m from the left support. The beam span is L=12L = 12 m. If a student calculates that the maximum moment occurs at x=6x = 6 m with a value of 7575 kN⋅m, what error did they most likely make?

  1. They assumed the maximum occurs at midspan due to symmetry rather than checking where shear equals zero (correct answer)
  2. They incorrectly calculated the support reactions as unequal due to asymmetric loading
  3. They used the wrong sign convention for positive and negative moments throughout their analysis
  4. They forgot to include the self-weight of the beam in their load calculations
Explanation: For two equal loads symmetrically placed, the maximum moment occurs under each load where the shear force equals zero, not at the midspan. At x=6x = 6 m (midspan), the shear force is not zero, so this is not the location of maximum moment. The actual maximum moments occur at x=3x = 3 m and x=9x = 9 m under the loads. Choice B is incorrect because the loads are symmetric, making reactions equal. Choice C is wrong because sign convention errors would affect magnitude but not location. Choice D is irrelevant to the location of maximum moment.

Question 11

A cantilever beam of length 66 m has a concentrated load P=60P = 60 kN at x=2x = 2 m from the fixed support and a uniformly distributed load w=15w = 15 kN/m from x=4x = 4 m to the free end. If the maximum negative moment occurs at the fixed support, where does the maximum positive moment occur?

  1. At x=2x = 2 m directly under the concentrated load
  2. At x=4x = 4 m where the distributed load begins
  3. Between x=2x = 2 m and x=4x = 4 m where the shear force equals zero
  4. No positive moment occurs in this cantilever beam configuration (correct answer)
Explanation: In a cantilever beam with only downward loads, the moment diagram is entirely negative (below the baseline) when using the standard sign convention. The moment starts at zero at the free end and becomes increasingly negative toward the fixed support. There are no upward loads or special conditions to create positive moments anywhere along the beam. Choices A, B, and C all incorrectly assume positive moments can exist in this loading configuration.

Question 12

A propped cantilever beam (fixed at left end, pin support at right end) of length 1010 m carries a uniformly distributed load w=20w = 20 kN/m over its entire span. Analysis shows that the maximum positive moment is 2525 kN⋅m at x=6x = 6 m and the reaction at the pin support is 125125 kN upward. Where is the maximum negative moment most likely located?

  1. At x=0x = 0 m at the fixed support due to the clamping moment (correct answer)
  2. At x=10x = 10 m at the pin support due to continuity effects
  3. At x=3x = 3 m where the shear force first becomes zero
  4. At x=6x = 6 m coinciding with the location of maximum positive moment
Explanation: In propped cantilever beams, the maximum negative moment typically occurs at the fixed support due to the clamping moment created by the fixed boundary condition. The fixed support must resist both vertical forces and moments, creating the largest negative moment in the beam. Choice B is incorrect because pin supports cannot resist moments. Choice C confuses zero shear locations with maximum negative moment. Choice D incorrectly suggests positive and negative moment maxima occur at the same location.

Question 13

A beam analysis shows that the shear force is zero at three locations: x=2x = 2 m, x=6x = 6 m, and x=9x = 9 m. The moment values at these points are M2=45M_2 = 45 kN⋅m, M6=−15M_6 = -15 kN⋅m, and M9=30M_9 = 30 kN⋅m respectively. Based on this information alone, which statement about the maximum positive and negative moments is most accurate?

  1. Maximum positive moment is 4545 kN⋅m at x=2x = 2 m; maximum negative moment is −15-15 kN⋅m at x=6x = 6 m
  2. Maximum positive moment is 4545 kN⋅m at x=2x = 2 m; maximum negative moment occurs at a support location not given
  3. Maximum positive moment could be 4545 kN⋅m; maximum negative moment could be more negative than −15-15 kN⋅m at other locations (correct answer)
  4. Maximum positive moment is 3030 kN⋅m at x=9x = 9 m; maximum negative moment is −15-15 kN⋅m at x=6x = 6 m
Explanation: Zero shear locations indicate local extrema in the moment diagram, but the global maximum and minimum moments could occur elsewhere, particularly at supports or boundaries where shear may not be zero. The given information only provides local extrema. Maximum negative moments often occur at supports where the moment could be more negative than −15-15 kN⋅m. Choice A and D definitively state the maxima without considering other locations. Choice B incorrectly assumes the positive moment maximum is confirmed while acknowledging uncertainty only for negative moments.

Question 14

An overhanging beam has supports at x=2x = 2 m and x=8x = 8 m, with overhangs of 22 m on each side (total length 1212 m). It carries a uniform load of w=25w = 25 kN/m over the entire length. The maximum positive moment is 31.2531.25 kN⋅m and occurs at x=5x = 5 m. Where does the maximum negative moment occur?

  1. At x=2x = 2 m and x=8x = 8 m at both support locations with equal magnitude (correct answer)
  2. At x=0x = 0 m and x=12x = 12 m at the free ends of the overhangs
  3. At x=2x = 2 m at the left support only due to asymmetric moment distribution
  4. At x=8x = 8 m at the right support only due to cantilever effects
Explanation: For a symmetric overhanging beam with uniform loading, maximum negative moments occur at both supports and have equal magnitude due to symmetry. The uniform load creates identical conditions at both supports, resulting in equal negative moments. The moment diagram shows negative peaks at both x=2x = 2 m and x=8x = 8 m. Choice B is incorrect because moments at free ends are zero. Choices C and D incorrectly break the symmetry that exists in this loading configuration.

Question 15

A simply supported beam of length L=12L = 12 m carries a uniformly distributed load of w=20w = 20 kN/m over its entire length and a concentrated load of P=80P = 80 kN at x=4x = 4 m from the left support. If the maximum positive moment occurs under the concentrated load, what is the approximate location where the maximum negative moment occurs?

  1. Maximum negative moment does not occur in this beam configuration (correct answer)
  2. At x=6x = 6 m from the left support under the distributed load
  3. At x=8x = 8 m from the left support under the distributed load
  4. At x=4x = 4 m from the left support at the concentrated load location
Explanation: For a simply supported beam with upward loads only, the moment diagram will be entirely positive (above the baseline). The maximum positive moment occurs where the shear force equals zero, which in this case happens under the concentrated load. Since there are no downward loads or overhanging sections, no negative moments develop anywhere along the beam. Choice B and C incorrectly assume negative moments exist under the distributed load. Choice D confuses the location of maximum positive moment with negative moment.

Question 16

A continuous beam has three spans: AB (66 m), BC (88 m), and CD (66 m), with supports at A, B, C, and D. A uniform load of 1515 kN/m acts on span BC only. If the maximum positive moment in span BC is 4545 kN⋅m and occurs at 2.42.4 m from support B, where does the maximum negative moment most likely occur?

  1. At support B where the beam experiences maximum reaction force
  2. At support C where the loaded span meets the unloaded span (correct answer)
  3. At 5.65.6 m from support B, symmetrically opposite the positive moment location
  4. At the midpoint of span BC where bending effects are typically maximum
Explanation: In continuous beams, maximum negative moments typically occur at interior supports due to continuity effects and load distribution to adjacent spans. Since only span BC is loaded, support C experiences significant negative moment as it must transfer load effects to the adjacent unloaded span CD. Support B also has negative moment, but support C is more critical due to the asymmetric loading condition. Choice A ignores that support B has less negative moment than C. Choice C incorrectly applies simple beam symmetry rules to a continuous beam. Choice D confuses continuous beam behavior with simple beam behavior.