Beam Bending Moment Diagram Calculator
Free Online Structural Analysis Tool for Engineers & Students
Select Beam Type
Add Loads
Beam Visualization
What Is a Bending Moment Diagram?
A bending moment diagram (BMD) is a graphical representation showing how bending moment varies along the length of a beam. The bending moment at any section is the algebraic sum of moments caused by all external forces acting on either side of that section.
Engineers use BMDs to identify the location and magnitude of maximum bending stress, which is critical for designing beams that can safely carry applied loads without failure.
How Does This Beam Calculator Work?
This calculator uses structural analysis principles to compute internal forces and deflections:
Step 1: Define Your Beam
Choose the beam type (simply supported, cantilever, or fixed-fixed) and enter the beam length along with material properties (Elastic Modulus and Moment of Inertia).
Step 2: Add Your Loads
Add point loads, distributed loads, or moments at any position along the beam. You can add multiple loads of different types.
Step 3: Calculate and Analyze
Click “Calculate Diagrams” to generate the shear force diagram, bending moment diagram, and deflection curve. Review the maximum values and reaction forces.
What Is Shear Force in a Beam?
Shear force is the internal force acting perpendicular to the longitudinal axis of a beam. At any cross-section, it represents the sum of all vertical forces on one side of that section.
The shear force diagram shows how this internal force varies along the beam length. Points where shear force equals zero often correspond to maximum bending moment locations.
How to Calculate Beam Deflection?
Beam deflection depends on the loading, support conditions, beam length, and the beam’s flexural rigidity (EI). Common formulas include:
Simply Supported with Center Load P: δmax = PL³/(48EI)
Cantilever with End Load P: δmax = PL³/(3EI)
Simply Supported with UDL w: δmax = 5wL⁴/(384EI)
This calculator computes deflection at multiple points along the beam to draw the complete deflection curve.
What Types of Beams Can Be Analyzed?
Simply Supported Beam
A beam supported by a pin at one end and a roller at the other. It can rotate at both supports and is statically determinate.
Cantilever Beam
A beam fixed at one end and free at the other. All reactions (vertical force, horizontal force, moment) occur at the fixed support.
Fixed-Fixed Beam
A beam with both ends rigidly fixed. This is a statically indeterminate structure requiring additional equations to solve.
What Loads Can Be Applied to a Beam?
Point Loads (Concentrated Loads)
Forces applied at a single point, such as a column bearing on a beam or equipment placed at a specific location.
Distributed Loads (UDL)
Uniformly distributed loads spread over a length, like the weight of a floor slab or storage materials.
Applied Moments
Concentrated couples applied at specific points, often from connected members or eccentric loading.
Why Is Bending Moment Important in Engineering?
Bending moment determines the flexural stress in a beam through the formula σ = My/I. Exceeding the material’s yield strength causes permanent deformation or failure. Engineers must ensure maximum bending moment is within allowable limits.
Proper BMD analysis helps in selecting appropriate beam sizes, materials, and reinforcement to ensure structural safety and economy.
How to Read a Bending Moment Diagram?
Positive values (typically drawn below the beam) indicate sagging moment causing tension on the bottom fiber.
Negative values (drawn above) indicate hogging moment with tension on the top fiber.
Zero crossings mark inflection points where curvature changes direction. These occur at simply supported ends and at points of contraflexure in continuous beams.
What Are Reaction Forces?
Reaction forces are the forces exerted by supports to keep the beam in equilibrium. For equilibrium:
ΣFy = 0: Sum of vertical forces equals zero.
ΣM = 0: Sum of moments about any point equals zero.
This calculator automatically computes reactions at all supports based on the applied loads and beam configuration.
Frequently Asked Questions
Can this calculator handle continuous beams?
The current version supports single-span beams (simply supported, cantilever, fixed-fixed). For multi-span continuous beams, analyze each span separately or use specialized structural analysis software.
What units are used in this calculator?
This calculator uses SI units: meters (m) for length, kilonewtons (kN) for forces, kN/m for distributed loads, and kN·m for moments. Results show deflection in millimeters (mm).
How accurate is this beam calculator?
This calculator uses Euler-Bernoulli beam theory and provides accurate results for slender beams with small deflections. For critical structural applications, verify results with professional engineering software.
Can I apply multiple loads to one beam?
Yes, add as many point loads, distributed loads, and moments as needed. The calculator superimposes effects using the principle of superposition.
What materials can this calculator analyze?
Any material with a known elastic modulus. Enter E = 200 GPa for steel, 30 GPa for concrete, or 12 GPa for timber. The moment of inertia depends on your cross-section geometry.
A civil engineer with a strong love for numbers, teaching, and building practical digital tools. With a Bachelor’s in Civil Engineering, a postgraduate degree in Project Management, and 12 years of teaching experience, he blends technical expertise with creativity.
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