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Calculate bending moments, support reactions, and stresses in structural beams. Supports multiple beam configurations, load cases, and provides moment diagrams for structural analysis.
@ 10.00 ft from left
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Bending moment is the internal moment (rotational force) that develops within a structural beam when external loads are applied. It represents the sum of moments about a section of the beam and causes it to bend or flex. Understanding bending moments is critical for:
Positive moment: Causes tension in the bottom fiber and compression in the top (sagging)
Negative moment: Causes compression in the bottom fiber and tension in the top (hogging)
Zero moment: Occurs at supports (pinned) or where shear force crosses zero
Beam with pin supports at both ends, free to rotate. Most common configuration for spanning across two supports. Maximum positive moment occurs near the center.
Beam fixed at one end, unsupported at the other (like a diving board). Maximum moment occurs at the fixed end. Creates large negative moments and deflections.
Beam fixed at both ends, no rotation allowed. Develops negative moments at supports and positive moment in the span. More rigid than simply supported.
Beam extending beyond one or both supports. Develops both positive and negative moments. Used for architectural effects and extended applications.
Common maximum bending moment formulas for different load configurations:
| Configuration | Load Type | Max Moment Formula | Location |
|---|---|---|---|
| Simply Supported | Point Load (center) | PL/4 | Center (L/2) |
| Simply Supported | Point Load (any) | Pab/L | At load location |
| Simply Supported | UDL (full span) | wL²/8 | Center (L/2) |
| Cantilever | Point Load | PL | At fixed end |
| Cantilever | UDL (full span) | wL²/2 | At fixed end |
| Fixed-Fixed | Point Load (center) | PL/8 (span) | Center |
| Fixed-Fixed | UDL (full span) | wL²/12 (supports) | At supports |
Notation: P = point load (lbs), w = distributed load (lbs/ft), L = span (ft), a = distance from left support, b = distance from right support
A bending moment diagram is a graphical representation showing how the internal moment varies along the length of a beam. The vertical axis represents moment magnitude, and the horizontal axis represents distance along the beam. The area under the shear force diagram equals the change in moment.
The moment diagram shows a triangular distribution with zero moment at both supports and maximum positive moment at the load location. The diagram is below center if the load is off-center.
The moment diagram shows a parabolic curve starting at zero at the free end and reaching maximum (negative) moment at the fixed support. The shape is characteristic of uniformly distributed loads.
The point of contraflexure (or inflection point) is the location along a beam where the bending moment changes sign - from positive (sagging) to negative (hogging) or vice versa. At this point, the moment equals zero.
In reinforced concrete design, tensile reinforcement must be present wherever the moment diagram crosses zero. The point of contraflexure marks where top reinforcement transitions to bottom reinforcement.
Ignoring this point can lead to inadequate reinforcement and structural failure under loading.
Solve the moment equation M(x) = 0 for the location x. In a fixed-fixed beam with point load, it occurs where the sum of positive and negative moments equals zero.
Solve: M(x) = 0 → x = location of contraflexure
Shear force is the internal force perpendicular to the beam that causes sliding between sections. Bending moment is the internal rotational force that causes bending. Shear force causes diagonal cracking, while bending moment causes vertical sagging or hogging. The slope of a shear force diagram equals the load intensity, and the slope of a moment diagram equals the shear force.
Calculate the maximum bending stress using σ = M/S (moment divided by section modulus). Compare this to the allowable stress for your material. For steel, typical allowable stress is 0.6 × Fy. For wood, it depends on grade and species. The beam is safe if calculated stress is less than allowable stress with an appropriate safety factor (typically 1.5-2.0 for structural design).
Only in the theoretical case of a beam with zero load. Any applied load creates bending moments somewhere in the beam. Even a cantilever with no load has zero moment, but once you apply any load, moments develop. In practical structures, bending moments are unavoidable and must be properly designed for.
A negative bending moment (or hogging moment) creates compression in the bottom fibers and tension in the top fibers - the opposite of sagging. This occurs in cantilever beams and overhanging beams. The negative sign is just a convention; it doesn't mean the moment is less important. Both positive and negative moments must be designed for in structural elements.
Bending stress is directly proportional to moment and inversely proportional to section modulus (σ = M/S). Increasing the section modulus (by using a larger or stronger section) reduces bending stress proportionally. This is why I-beams and wide-flange beams are efficient - they concentrate material farther from the neutral axis, increasing section modulus without adding excessive weight.
For a simply supported beam with uniformly distributed load: δmax = (5 × w × L⁴) / (384 × E × I). The deflection increases with the fourth power of span, so doubling the span increases deflection 16 times. To limit deflection, typical limits are L/360 for floors and L/240 for roofs, where L is span in inches.
Use moment equilibrium: sum of moments about one support equals zero, which lets you solve for the reaction at the other support. Then use vertical force equilibrium (sum of vertical forces = 0) to find the remaining reaction. For a point load P at distance 'a' from left support on a span L: RA = P(L-a)/L and RB = Pa/L.
Yes, in beams with both positive and negative moments (like fixed-fixed or overhanging beams). This location is the point of contraflexure. At this point, the moment changes from positive to negative, and the moment value is exactly zero. This is critical for reinforcement design in concrete beams where reinforcement must change from one face to the other.
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