Shear Force & Bending Moment Diagrams: Complete Interactive Guide

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Introduction

The load has to go somewhere, and the shear force and bending moment diagrams show you exactly where it goes. When a truck crosses a bridge, its weight doesn't magically teleport to the supports. Instead, internal forces develop throughout the beam: shear forces that try to slice the beam apart, and bending moments that try to snap it like a twig. Experienced designers know that finding the maximum values and locations of these internal forces is the essential first step in any beam design.

This is where structures fail: at the point of maximum bending moment where tensile and compressive stresses reach their peaks, or at locations of high shear where diagonal tension can split the concrete. Any structural engineer will tell you that drawing the shear and moment diagrams isn't just academic exercise. It's how we trace the load path from application point to support reaction, revealing every critical section along the way. Once you know the maximum bending moment, you can calculate beam deflection to ensure serviceability limits are met.

The relationship between load, shear, and moment follows elegant mathematical rules (the shear diagram is the integral of the load; the moment diagram is the integral of shear), but the real power comes from visual intuition. When you see how a point load creates a sudden jump in shear, or how a distributed load produces a parabolic moment curve, you develop the instincts that let you spot problems in designs before they become failures in reality. This interactive simulation lets you build that intuition by constructing and analyzing beam loading scenarios in real time.

How to Use This Simulation

This interactive shear and moment diagram builder constructs V and M diagrams in real time as you add loads and modify support conditions. Any structural engineer will tell you that these diagrams are the foundation of every beam design, revealing exactly where maximum stresses and critical sections occur.

Main Controls

Input Parameters

Results Display

The simulation presents three diagrams with key values:

• Beam Diagram: Shows the applied loads, support conditions, and reaction forces. The load has to go somewhere, and the reactions at each support tell you how much goes where.
• Shear Force Diagram (SFD): Vertical axis shows V in kN. Jumps occur at point loads; linear slopes under UDLs; parabolic curves under triangular loads. This is where structures fail in diagonal tension if shear capacity is insufficient.
• Bending Moment Diagram (BMD): Vertical axis shows M in kN·m. Kinks occur at point loads; parabolic under UDLs. Maximum positive moment indicates maximum tensile stress in the bottom fiber.
• Key Values Panel: Reactions Ra and Rb, maximum positive/negative shear Vmax, maximum positive/negative moment Mmax, and maximum deflection δmax.

Tips for Exploration

1. Watch the jump at point loads. Add a 50 kN point load and observe the instant vertical drop in the shear diagram. The magnitude of this jump equals the load magnitude exactly. Any structural engineer will tell you that sketching these jumps is the first step in hand-drawing shear diagrams.

2. Trace the slope-area relationships. Under a UDL, the shear diagram has constant slope equal to -w. The change in shear between two points equals the area under the load diagram. The load has to go somewhere, and this relationship lets you calculate shear at any point by integration.

3. Find moment from shear area. The change in moment between two points equals the area under the shear diagram. Start at a support where you know M = 0 (for simply-supported beams), then accumulate area to find moment at any section. This is where structures fail: at the location of maximum moment area accumulation.

4. Compare support conditions. Apply the same UDL to simply-supported, cantilever, and fixed-fixed beams. Watch how the moment diagram changes shape completely. Fixed supports develop negative (hogging) moments that reduce midspan positive moments. Experienced designers exploit this to reduce beam depth.

5. Add multiple loads. Build up a complex loading scenario with point loads, UDLs, and moments. Watch how the diagrams superimpose. Each load contribution adds to the previous, but the overall shape follows the same rules. This superposition principle is fundamental to beam analysis.

Types of Loading and Support Conditions

1. Point Loads (Concentrated Forces)

Point loads represent forces applied at a single location on the beam, such as a column bearing on a beam or a piece of equipment. These create discontinuities (jumps) in the shear force diagram at the point of application, with the magnitude of the jump equal to the load magnitude.

2. Uniformly Distributed Loads (UDL)

UDLs represent loads spread evenly over a portion of the beam, measured in force per unit length (kN/m). Common examples include the weight of a concrete slab, snow accumulation, or stored materials on a floor. Under a UDL, the shear force diagram shows a linear (sloped) section, and the bending moment diagram shows a parabolic (curved) section.

3. Triangular (Linearly Varying) Loads

Triangular loads vary linearly from zero to a maximum value (or vice versa) over a length of beam. These commonly occur in retaining walls due to earth pressure, in liquid storage tanks, or in wind loading on tall structures.

4. Applied Moments (Couples)

Applied moments are rotational forces applied at a point, such as the torque from a rigidly connected member or an eccentric connection. Unlike forces, moments do not contribute to the vertical force equilibrium but create a discontinuity (jump) in the bending moment diagram.

Support Types

• Simply-Supported: Pin support at one end, roller at the other. Provides two reaction forces (one vertical at each end) but no moment resistance.
• Cantilever: Fixed support at one end, free at the other. The fixed support provides both a vertical reaction force and a moment reaction.
• Fixed-Pinned: Fixed support at one end (provides vertical force and moment), pin at the other (provides vertical force only).
• Overhanging: Pin support at one end with a roller support before the other end, allowing the beam to extend beyond the last support.

Key Parameters

Fundamental Formulas

1. Shear-Load Relationship

dV/dx = -w(x)

The slope of the shear force diagram at any point equals the negative of the distributed load intensity at that point. This means:

• Under zero load (no distributed load), the shear diagram is horizontal
• Under uniform load w, the shear diagram has constant slope -w (linear)
• Under linearly varying load, the shear diagram is parabolic
• Point loads create vertical jumps in the shear diagram

2. Moment-Shear Relationship

dM/dx = V(x)

The slope of the bending moment diagram equals the shear force. This leads to several important observations:

• Where shear is positive, moment is increasing (positive slope)
• Where shear is negative, moment is decreasing (negative slope)
• Where shear crosses zero, moment reaches a local maximum or minimum (critical for design)
• The change in moment between two points equals the area under the shear curve between those points

3. Area Method for Shear

ΔV = V₂ - V₁ = -∫[x₁ to x₂] w(x)dx

The change in shear force between two points equals the negative of the area under the loading curve between those points.

4. Area Method for Moment

ΔM = M₂ - M₁ = ∫[x₁ to x₂] V(x)dx

The change in bending moment between two points equals the area under the shear force curve between those points.

5. Equilibrium Equations for Simply-Supported Beam

ΣF_y = 0: Ra + Rb - ΣP - Σ(wL) = 0

ΣM_A = 0: Rb·L - Σ(P·x_P) - Σ(wL·x_centroid) = 0

6. Maximum Moment Location

For simply-supported beams: M_max occurs where V = 0

To find the location of maximum moment, set the shear force equation equal to zero and solve for x.

Learning Objectives

By engaging with this interactive simulation, you will:

1. Construct SFD and BMD by hand: Develop the ability to sketch accurate shear and moment diagrams for beams with multiple load types and support conditions
2. Identify critical design points: Recognize where maximum shear forces and bending moments occur under various loading patterns
3. Apply equilibrium principles: Calculate reaction forces and moments at supports using the three equations of static equilibrium
4. Predict diagram shapes: Understand the mathematical relationships between load types and diagram shapes
5. Interpret physical meaning: Translate diagram features into real structural behavior
6. Optimize load placement: Explore how changing the position and magnitude of loads affects internal forces

Exploration Activities

Activity 1: Single Point Load Analysis

Objective: Understand the fundamental relationships between a point load and resulting diagrams

Steps:

1. Select "Simply-Supported" support type
2. Add a single point load of 100 kN at x = 5m (midpoint)
3. Observe that Ra = Rb = 50 kN due to symmetry
4. Note the V-shaped shear diagram: V = +50 kN from 0-5m, then jumps to -50 kN from 5-10m
5. Identify the triangular moment diagram with M_max = 125 kN·m at x = 5m (where V = 0)
6. Verify M_max = Ra × L/2 = 50 × 5 = 250/2 = 125 kN·m

Observe: Maximum moment occurs where shear crosses zero

Activity 2: Moving Load Effect

Objective: Explore how load position affects maximum moment location and magnitude

Steps:

1. Start with a 100 kN point load at x = 2m
2. Note M_max location and value
3. Move the load to x = 3m, then x = 4m, then x = 5m
4. Record M_max for each position and observe the pattern
5. Confirm that M_max increases as the load moves toward the center
6. Verify the formula: For load P at position a from left support with span L, M_max = Pab/L where b = L-a

Observe: Maximum moment is greatest when load is at midspan (a = b = L/2)

Activity 3: UDL vs. Equivalent Point Load

Objective: Compare a distributed load to its statically equivalent point load

Steps:

1. Apply a 20 kN/m UDL over the entire 10m beam (total = 200 kN)
2. Note the reactions: Ra = Rb = 100 kN (due to symmetry)
3. Observe the linear shear diagram (starts at +100, ends at -100, crossing zero at midspan)
4. Note the parabolic moment diagram with M_max at midspan
5. Calculate M_max = wL²/8 = 20 × 10² / 8 = 250 kN·m
6. Clear and add a single 200 kN point load at x = 5m
7. Compare: Point load gives M_max = PL/4 = 200 × 10 / 4 = 500 kN·m
8. Note that distributed load produces HALF the maximum moment of an equivalent point load at midspan

Observe: Load distribution significantly affects maximum moment magnitude

Activity 4: Combined Loading Analysis

Objective: Analyze complex real-world loading scenarios with multiple load types

Steps:

1. Load the "Combined Loading" preset (point load + UDL + moment)
2. Identify each load contribution visually on the beam
3. Sketch predicted SFD and BMD shapes before viewing the calculated diagrams
4. Reveal the actual diagrams and compare to your predictions
5. Identify where the shear force is maximum
6. Find where the bending moment is maximum (where shear crosses zero between loads)
7. Modify one load at a time and observe how each affects the overall diagrams
8. Remove the applied moment and note how the BMD changes (jump disappears)

Observe: Superposition principle: each load contributes independently to final diagrams

Real-World Applications

• Building Floor Systems: Structural engineers use SFD and BMD to design floor beams supporting distributed loads from slabs, partitions, and occupancy loads. Typical office floors experience 2.4-4.8 kPa live load plus 1.0-2.5 kPa dead load.

• Bridge Design: Highway bridge girders are designed for moving vehicle loads using influence lines combined with shear and moment diagrams. Engineers place design truck loads at critical positions that maximize shear at supports and moment at midspan.

• Crane Runway Beams: Overhead crane runway beams experience moving concentrated loads (wheel loads) that create varying shear and moment diagrams as the crane travels.

• Aircraft Wing Spars: Aircraft structural engineers model wing spars as beams with aerodynamic distributed lift loads (upward) and concentrated loads from engines, landing gear, and fuel tanks (downward).

• Balconies and Canopies: Cantilever structures are analyzed as fixed-end beams with the critical section (maximum moment) occurring at the support connection.

• Elevator Guide Rails: Building elevator guide rails are designed as continuous beams with bracket supports every 3-5 meters.

Reference Data: Common Beam Formulas

Simply-Supported Beam Formulas

Cantilever Beam Formulas (Fixed at Left, Free at Right)

Challenge Questions

1. Conceptual: Why does the bending moment diagram for a simply-supported beam with uniform load have a parabolic shape, while the shear diagram is linear? Explain this using the differential relationship dM/dx = V and the shape of the shear curve.

2. Calculation: A 12-meter simply-supported beam has a 10 kN/m UDL from x = 0 to x = 8m, and a point load of 60 kN at x = 10m. Calculate the reactions at both supports and determine the location and magnitude of the maximum bending moment.

3. Analysis: You notice that a beam's bending moment diagram shows two peaks of nearly equal magnitude, one positive and one negative. What type of support condition or loading pattern could create this?

4. Application: A warehouse floor beam spans 8 meters and must support a uniform load of 15 kN/m (from slab dead load + storage live load) plus a 40 kN concentrated load at midspan (from a column above). Calculate the maximum bending stress.

5. Design Challenge: You need to design a 6-meter cantilever canopy to support snow load of 2.5 kN/m² on a 3-meter wide canopy (giving w = 7.5 kN/m). What is the maximum bending moment at the support, and what minimum section modulus is required?

Common Mistakes to Avoid

• Sign Convention Errors: Different textbooks use different sign conventions for shear and moment. Always establish and label your sign convention clearly before beginning analysis, and be consistent throughout the problem.

• Forgetting to Include Reactions: When constructing shear diagrams starting from the left end, students often forget to include the upward reaction force at the left support, causing the entire diagram to be shifted.

• Discontinuity Location Errors: Point loads and applied moments cause discontinuities (jumps) in the diagrams, but students sometimes place these jumps at the wrong location or with incorrect magnitude.

• Incorrect Area Calculations: When using the area method to construct moment diagrams from shear diagrams, remember that the area is algebraic; regions below the x-axis contribute negatively.

• Misidentifying Maximum Moment Location: For beams with multiple loads, students often assume the maximum moment occurs at midspan or directly under a point load. The actual maximum always occurs where the shear force equals zero.

References

1. MIT OpenCourseWare — Solid Mechanics (Course 2.001). Available at: ocw.mit.edu — CC BY-NC-SA License

2. OpenStax — University Physics Volume 1, Chapter 12: Static Equilibrium. Available at: openstax.org — CC BY License

3. Engineering Toolbox — Beam Stress and Deflection. Available at: engineeringtoolbox.com — Free educational resource

4. HyperPhysics — Bending and Shear in Beams. Georgia State University. Available at: hyperphysics.gsu.edu — Free educational resource

5. eFunda — Beam Calculator and Formula Reference. Available at: efunda.com — Free engineering reference

6. Khan Academy — Mechanical Engineering: Beams. Available at: khanacademy.org — Free educational videos

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How to Cite

If you use this simulation in educational materials or research, please cite as:

Simulations4All (2025). "Shear Force & Bending Moment Diagram Calculator: Interactive Structural Analysis Tool." Available at: https://simulations4all.com/simulations/shear-moment-diagrams

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Verification Log

All scientific claims, formulas, and data have been verified against authoritative sources.

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