Shear Moment Diagram Calculator
Free Shear moment diagram Calculator for statics. Enter variables to compute results with formulas and detailed steps.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Shear Moment Diagram Calculator
Calculator
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Formula: V(x) = RA - Sum of loads to left of x | M(x) = Integral of V(x) dx
Worked example — RA = RB = 5 kN | Max Shear: 5 kN | Max Moment: 15 kN-m at center
Formula
V(x) = RA - Sum of loads to left of x | M(x) = Integral of V(x) dx
The shear force at any section is the algebraic sum of all vertical forces to one side of that section. The bending moment is the algebraic sum of moments of all forces to one side. For a point load P at distance a: RA = Pb/L, RB = Pa/L. For UDL w: RA = RB = wL/2 (full span). Max moment occurs where shear equals zero.
Worked Examples
Example 1: Simply Supported Beam with Central Point Load
Problem:A 6 m simply supported beam carries a 10 kN point load at the center (3 m from each support). Determine the reactions, maximum shear, and maximum bending moment.
Solution:Reactions: RA = RB = P/2 = 10/2 = 5 kN (symmetric loading) Shear: V = +5 kN from left support to load, V = -5 kN from load to right support Max Shear = 5 kN (at supports) Max Moment = PL/4 = 10 x 6/4 = 15 kN-m (at center) Zero shear occurs at x = 3 m (load point)
Result:RA = RB = 5 kN | Max Shear: 5 kN | Max Moment: 15 kN-m at center
Example 2: Simply Supported Beam with Full UDL
Problem:A 6 m beam carries a uniformly distributed load of 5 kN/m over its entire length. Find the maximum shear and moment.
Solution:Total load = 5 x 6 = 30 kN Reactions: RA = RB = 30/2 = 15 kN Max Shear = wL/2 = 5 x 6/2 = 15 kN (at supports) Max Moment = wL^2/8 = 5 x 36/8 = 22.5 kN-m (at midspan) Shear is zero at midspan (x = 3 m)
Result:RA = RB = 15 kN | Max Shear: 15 kN | Max Moment: 22.5 kN-m at midspan
Frequently Asked Questions
What are shear force and bending moment diagrams used for in structural analysis?
Shear force diagrams (SFD) and bending moment diagrams (BMD) are fundamental tools in structural engineering that graphically represent the internal forces and moments acting along the length of a beam. The shear force diagram shows how the internal vertical force varies at each cross-section, which is critical for designing against shear failure and determining required web thickness in steel beams or stirrup spacing in reinforced concrete. The bending moment diagram reveals the internal moment distribution, which determines the maximum bending stress and governs the required section modulus or reinforcement for the beam. Together, these diagrams enable engineers to identify critical sections, size structural members, and verify that designs meet safety requirements.
How are support reactions calculated for a simply supported beam?
For a simply supported beam, the support reactions are determined using the equations of static equilibrium: the sum of vertical forces equals zero and the sum of moments about any point equals zero. For a point load P located at distance a from the left support on a beam of length L, taking moments about the left support gives RB = P times a divided by L, and from vertical equilibrium, RA = P minus RB. For a uniformly distributed load w over the entire span, each reaction equals wL divided by 2 due to symmetry. For partial distributed loads, the resultant force acts at the centroid of the loaded region, and moments are taken about one support to find the other. These reactions serve as the starting values for constructing shear and moment diagrams.
What is the relationship between shear force and bending moment?
Shear force and bending moment are mathematically related through calculus. The bending moment at any point is the integral (area under the curve) of the shear force diagram up to that point. Conversely, the shear force is the derivative (rate of change) of the bending moment. This means the bending moment reaches its maximum or minimum value where the shear force equals zero or changes sign. The distributed load intensity equals the negative derivative of the shear force. These relationships provide important shortcuts: the change in moment between two points equals the area under the shear diagram between those points, and the slope of the moment diagram at any point equals the shear force at that point.
How does a cantilever beam differ from a simply supported beam in analysis?
A cantilever beam is fixed at one end and free at the other, creating fundamentally different shear and moment distributions compared to a simply supported beam. The fixed support provides both a vertical reaction force and a reaction moment, whereas simple supports only provide vertical forces. In a cantilever, the maximum bending moment always occurs at the fixed support and decreases toward the free end. For a point load at the free end, the moment varies linearly from maximum at the fixed end to zero at the load. The shear force is constant along the beam. Cantilevers experience greater deflections than simply supported beams of the same span and load because there is no intermediate support to limit displacement.
Where does maximum bending stress occur in a beam and how is it calculated?
Maximum bending stress occurs at the cross-section where the bending moment is largest, specifically at the outermost fibers (top and bottom) of that cross-section. The bending stress formula is sigma equals M times y divided by I, where M is the bending moment, y is the distance from the neutral axis to the point of interest, and I is the second moment of area (moment of inertia) of the cross-section. The maximum stress occurs when both M and y are at their maximum values. For a rectangular cross-section of width b and height h, the section modulus S equals I divided by y-max equals bh-squared divided by 6, simplifying the calculation to sigma-max equals M divided by S. This stress must remain below the allowable stress determined by the factor of safety.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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