Shear Force & Bending Moment Calculator - Free Beam Analysis Tool
Calculate shear force, bending moment, and support reactions for simply supported beams with point and distributed loads
Shear Force & Bending Moment Calculator
Results
Note: Values are calculated for a simply supported beam under static loading conditions.
What is a Shear Force & Bending Moment Calculator?
A Shear Force & Bending Moment Calculator is a free engineering tool that analyzes beams under various loading conditions to determine support reactions, shear force distribution, and bending moment distribution. These diagrams are fundamental for structural design and analysis, helping engineers identify critical sections where failure might occur.
This calculator is essential for:
- Structural Engineering - Design of beams, bridges, and building frames
- Academic Learning - Understanding beam mechanics and structural behavior
- Construction Planning - Verifying load-carrying capacity of structural members
- Safety Analysis - Identifying maximum stress locations and ensuring code compliance
- Design Optimization - Determining efficient beam sizes and support locations
For stress analysis, try our Beam Bending Stress Calculator.
For spring mechanics, use our Spring Constant & Deflection Calculator.
How Shear Force & Bending Moment are Calculated
The calculator uses fundamental equilibrium equations and beam theory formulas (2025 standards):
1. Support Reactions (Equilibrium):
ΣMA = 0: RB·L = P·a + w·L²/2
2. Shear Force at any section x:
3. Bending Moment at any section x:
Where:
- RA, RB = Support reactions at left and right supports (kN)
- P = Point load magnitude (kN)
- a = Distance of point load from left support (m)
- w = Uniformly distributed load intensity (kN/m)
- L = Total beam length (m)
- V(x) = Shear force at distance x from left support (kN)
- M(x) = Bending moment at distance x from left support (kN·m)
Key Relationships:
- dV/dx = -w (slope of shear diagram equals negative of distributed load)
- dM/dx = V (slope of moment diagram equals shear force)
- Maximum moment occurs where shear force equals zero
Key Structural Analysis Concepts
Shear Force (V)
The algebraic sum of all vertical forces on either side of a section. Causes shear stress that can lead to diagonal tension failure in beams.
Bending Moment (M)
The algebraic sum of moments about a section. Causes normal stress (tension/compression) and beam curvature. Maximum at zero shear points.
Support Reactions
Forces exerted by supports to maintain equilibrium. Calculated using statics equations: ΣF = 0 and ΣM = 0.
Simply Supported Beam
Beam resting on two supports (pin and roller) at ends. Most common configuration with zero moments at supports.
Point Load
Concentrated force acting at a specific point. Causes sudden change (jump) in shear force diagram and slope change in moment diagram.
Distributed Load
Load spread over beam length (kN/m). Causes linear variation in shear force and parabolic variation in bending moment.
How to Use This Calculator
- Enter Beam Length (L): Input the total span of the simply supported beam in meters
- Enter Point Load (P): Input the magnitude of concentrated load in kN (enter 0 if no point load)
- Enter Load Position (a): Input the distance from left support to point load in meters
- Enter Distributed Load (w): Input uniformly distributed load in kN/m (enter 0 if none)
- Select Beam Type: Choose beam configuration (currently supports simply supported beams)
- Calculate: Click "Calculate Diagrams" to compute reactions, shear forces, and bending moments
- Review Results: Analyze support reactions, maximum values, and critical locations
Example Calculation:
Beam: L = 6 m, P = 10 kN at a = 2 m, w = 5 kN/m
Step 1: Total distributed load = 5 × 6 = 30 kN
Step 2: ΣMA = 0: RB × 6 = 10 × 2 + 30 × 3 = 110
RB = 18.33 kN
Step 3: RA = (10 + 30) - 18.33 = 21.67 kN
Step 4: Mmax occurs where V = 0
Benefits of Using This Calculator
- Instant Analysis: Calculate support reactions and critical values in seconds
- Engineering Accuracy: Uses standard beam theory equations from mechanics of materials
- Combined Loading: Handles both point loads and distributed loads simultaneously
- Educational Tool: Perfect for civil engineering students learning structural analysis
- Design Validation: Verify beam designs meet structural requirements
- Critical Locations: Identifies where maximum shear and moment occur
- Free & Accessible: No registration required, works on any device
- 2025 Standards: Complies with current engineering calculation methods
- Professional Use: Suitable for preliminary design and quick checks
Factors Affecting Shear Force & Bending Moment
- Load Magnitude: Higher loads directly increase shear forces and bending moments proportionally
- Load Position: Point load location significantly affects moment distribution; center loading maximizes moment
- Beam Span: Longer beams increase bending moments; M varies with L² for distributed loads
- Support Conditions: Simply supported, cantilever, and fixed beams have different moment distributions
- Load Distribution: Uniform loads create parabolic moments; point loads create linear segments
- Multiple Loads: Superposition principle allows combining effects of different loads
- Beam Continuity: Continuous beams over multiple supports reduce maximum moments
- Material Properties: While not affecting diagrams, material strength limits allowable stresses
Frequently Asked Questions (FAQ)
What is a shear force diagram?
A shear force diagram (SFD) is a graphical representation showing the variation of shear force along the length of a beam. It helps engineers identify critical sections where shear failure might occur and is essential for structural design and analysis.
How is the maximum bending moment calculated for a simply supported beam?
For a simply supported beam with a point load P at distance 'a' from the left support and 'b' from the right support, the maximum bending moment is M_max = (P × a × b) / L, where L is the total beam length. For a uniformly distributed load w, M_max = (w × L²) / 8 at the center.
What are support reactions in beam analysis?
Support reactions are the forces exerted by supports (such as pins, rollers, or fixed supports) on a beam to maintain equilibrium. For a simply supported beam, reactions are calculated using equilibrium equations: ΣF_y = 0 and ΣM = 0.
What is the relationship between shear force and bending moment?
The shear force and bending moment are related by the differential equation: dM/dx = V, where V is the shear force and M is the bending moment. This means the slope of the bending moment diagram at any point equals the shear force at that point, and maximum bending moment occurs where shear force is zero.