Shear Moment Diagram Calculator
Free Shear moment diagram Calculator for statics. Enter variables to compute results with formulas and detailed steps.
Reviewed by Manoj Kumar, Mathematics Educator
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)\nShear: V = +5 kN from left support to load, V = -5 kN from load to right support\nMax Shear = 5 kN (at supports)\nMax Moment = PL/4 = 10 x 6/4 = 15 kN-m (at center)\nZero 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\nReactions: RA = RB = 30/2 = 15 kN\nMax Shear = wL/2 = 5 x 6/2 = 15 kN (at supports)\nMax Moment = wL^2/8 = 5 x 36/8 = 22.5 kN-m (at midspan)\nShear 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.
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy