Beam Deflection Calculator

Beam Deflection Solver

Bending Moment Diagram

The bending moment diagram is drawn automatically after each calculation. It shows how the internal moment varies along the beam span. Positive moments cause the beam to sag (concave up) and negative moments cause it to hog (concave down).

The shear force diagram is also drawn, showing the internal shear at each point along the span. Locations where the shear crosses zero correspond to maximum or minimum bending moment.

Beam Support Types

The way a beam is supported determines its reactions, moment distribution, and deflection behavior. This calculator handles three common support conditions.

Simply Supported (Pin-Roller)

The beam rests on a pin support at one end and a roller at the other. Both supports resist vertical force, but neither resists rotation. This is the most common idealization for floor beams, bridge girders, and header beams over openings. The moment is zero at both supports and maximum somewhere along the span.

Cantilever (Fixed-Free)

One end is rigidly clamped (fixed against both translation and rotation) and the other end is free. Loading on the free end produces the largest possible deflections and moments. Cantilevers appear as balconies, diving boards, canopy structures, and overhanging roof eaves.

Fixed-Fixed (Both Ends Clamped)

Both ends are restrained against rotation and translation. This produces negative (hogging) moments at the supports and a reduced positive (sagging) moment at midspan. The result is significantly less deflection than a simply supported beam with the same load. Continuous beams and rigidly welded frames approximate fixed-end conditions.

Load Types and Placement

Point Load (Concentrated)

A single force applied at a specific location along the beam. Real-world examples include a column bearing on a beam, a suspended piece of equipment, or a wheel load. Point loads create a V-shaped bending moment diagram (for simply supported beams) with the peak at the load location.

Uniformly Distributed Load (UDL)

A load spread evenly along the entire beam length, measured in force per unit length (N/mm, kN/m, or lb/ft). Common examples include the self-weight of the beam, a slab bearing on the beam, and snow or wind loads. UDL creates a parabolic bending moment diagram.

For the point load at any position option, the load can be placed anywhere between the left support and the right end. The parameter "a" is the distance from the left support to the load, and "b" is the remaining distance (b = L - a).

Cross-Section Properties

The beam cross-section determines its stiffness and strength. Two key properties are used in beam calculations:

• Moment of inertia (I): Measures the cross-section's resistance to bending. Larger I means less deflection. Units are mm^4 or in^4.
• Section modulus (S = I/c): Relates bending moment to maximum stress. Larger S means lower stress for the same moment.

Formulas for common sections:

• Rectangular: I = bh^3/12, c = h/2, S = bh^2/6
• Circular (solid): I = pi*d^4/64, c = d/2
• Circular (hollow): I = pi*(OD^4 - ID^4)/64, c = OD/2
• I-beam: I = [bf*H^3 - (bf - tw)*(H - 2*tf)^3] / 12, c = H/2

Deflection Formulas

Maximum deflection formulas for common beam and load combinations (all units consistent):

For a simply supported beam with a point load at arbitrary position "a" from the left end (b = L - a), the maximum deflection is not always at midspan. The deflection at the load point is Pa^2*b^2 / (3EIL). The maximum deflection occurs at x = sqrt((L^2 - b^2)/3) from the left support when a > b.

Bending Stress Calculation

The maximum bending stress in a beam occurs at the cross-section where the bending moment is greatest, at the outermost fiber (top or bottom face):

Where M_max is the maximum bending moment, c is the distance from the neutral axis to the extreme fiber, I is the moment of inertia, and S is the section modulus (I/c).

For structural steel (ASTM A992, Fy = 345 MPa), the allowable bending stress under AISC LRFD is 0.9 * Fy = 310.5 MPa for compact sections with full lateral bracing. Under ASD (Allowable Stress Design), the allowable stress is 0.6 * Fy = 207 MPa.

The calculated stress must remain below the material yield stress divided by the appropriate safety factor. This calculator reports the computed stress and compares it against common yield values.

Serviceability Limits

Even if a beam is strong enough not to break, excessive deflection causes practical problems: cracking of ceiling finishes, visible sagging, vibration, and occupant discomfort. Building codes specify maximum allowable deflections.

This calculator automatically checks L/360 (standard floor beam live load limit) and reports whether the computed deflection passes.

Material Properties Table

Common Steel Sections

For reference, here are moment of inertia values for some common AISC W-shapes (strong axis):

Use the "Custom I and c" option in the calculator to enter values directly from the AISC Steel Construction Manual or other section property tables.

Worked Examples

Example 1: Floor Beam Check

A W410x85 steel beam spans 8 meters, simply supported, carrying a uniform distributed load of 15 kN/m (total including self-weight). Check deflection and stress.

I = 316 x 10^6 mm^4, E = 200,000 MPa, w = 15 N/mm, L = 8,000 mm.

Maximum deflection = 5wL^4 / (384EI) = 5 * 15 * 8000^4 / (384 * 200000 * 316e6) = 12.6 mm.

L/360 = 8000/360 = 22.2 mm. Since 12.6 < 22.2, deflection passes.

Maximum moment = wL^2/8 = 15 * 8000^2 / 8 = 120,000,000 N-mm = 120 kN-m.

Maximum stress = M * c / I = 120e6 * 208.5 / 316e6 = 79.2 MPa. Well below 345 MPa yield.

Example 2: Cantilever with Tip Load

A 150x200 mm rectangular aluminum beam (E = 69,000 MPa) extends 2 meters as a cantilever. A 5 kN point load is applied at the free end.

I = 100 * 200^3 / 12 = 66.67 x 10^6 mm^4. Deflection = PL^3 / (3EI) = 5000 * 2000^3 / (3 * 69000 * 66.67e6) = 2.90 mm.

Maximum moment = P * L = 5000 * 2000 = 10,000,000 N-mm. Stress = M * c / I = 10e6 * 100 / 66.67e6 = 15.0 MPa.

Design Considerations

• Lateral-torsional buckling: Long beams without lateral bracing can buckle sideways before reaching their full bending strength. Check unbraced length against L_p and L_r per AISC Chapter F.
• Shear capacity: Short, heavily loaded beams may fail in shear before bending. Check shear stress = V / (A_web) against 0.6 * Fy.
• Web crippling and bearing: Concentrated loads on beam flanges can cause localized failures. Use bearing stiffeners or distribute the load over adequate bearing length.
• Connection design: Beam end connections must transfer the computed reactions safely. Simple shear connections for simply supported beams; moment connections for fixed-end conditions.
• Combined loading: Real beams often carry axial force plus bending. Use interaction equations (AISC Chapter H) to check combined effects.
• adaptable effects: Moving loads, vibration, and impact loads require adaptable amplification factors or detailed adaptable analysis beyond static beam formulas.

References and Standards

• AISC 360 - Specification for Structural Steel Buildings (primary US standard for steel beam design)
• Timoshenko, S.P. - Strength of Materials, 3rd Edition (classic reference for beam deflection and stress theory)
• IBC (International Building Code) - Table 1604.3 for deflection limits
• ASCE 7 - Minimum Design Loads for Buildings (load combinations and load factors)

Frequently Asked Questions

What is beam deflection?

Beam deflection is the vertical displacement of a beam under load. It depends on load magnitude, span length, material stiffness (E), and cross-section geometry (I). Deflection is measured in mm or inches and must be kept within code limits for serviceability.

What is the L/360 deflection limit?

L/360 is a standard maximum deflection for floor beams under live load. For a 6-meter span, the limit is 6000/360 = 16.7 mm. This prevents visible sagging and cracking of finishes. For total load, L/240 is commonly used. These limits originate from IBC Table 1604.3.

What is the difference between simply supported and cantilever beams?

A simply supported beam rests on two supports and is free to rotate. A cantilever is fixed at one end and free at the other. Cantilevers deflect much more for the same load and span. A center-loaded simply supported beam deflects PL^3/(48EI) while a tip-loaded cantilever deflects PL^3/(3EI), about 16 times more.

How do I find the moment of inertia?

For standard steel shapes, use the AISC Steel Construction Manual tables. For rectangular sections, I = bh^3/12. For circles, I = pi*d^4/64. This calculator computes I automatically for rectangular, I-beam, circular, and pipe sections, or you can enter a custom value.

What is bending stress?

Bending stress (sigma = M*c/I) is the internal stress from bending moments. It varies from zero at the neutral axis to maximum at the top and bottom faces. The beam must be sized so that maximum bending stress stays below the allowable stress for the material.

What modulus of elasticity should I use?

Structural steel: 200,000 MPa (29,000 ksi). Aluminum: 69,000 MPa (10,000 ksi). Douglas Fir: 12,400 MPa (1,800 ksi). The modulus controls deflection: a stiffer material (higher E) deflects less under the same load.

What is a bending moment diagram?

A graph showing internal bending moment at every point along the beam span. The peak of the diagram indicates where maximum stress occurs. This calculator draws the moment diagram and shear force diagram on canvas automatically after each calculation.

How does a fixed-fixed beam compare to simply supported?

A fixed-fixed beam has both ends clamped against rotation. This produces negative moments at the supports and reduces midspan deflection to about one-fifth of the simply supported case for uniform load. Fixed ends require moment-resisting connections, which adds construction cost.