Steel Beam End Reactions — Shear & Moment Diagrams

Complete beam end reaction formulas for simply supported, cantilever, fixed-end, and continuous beams. Shear and moment diagrams, sign conventions per AISC 360, and worked examples for steel connection design.

Overview

PRELIMINARY — NOT FOR CONSTRUCTION. All formulas and examples are for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.

Beam end reactions are the forces and moments that develop at supports to maintain equilibrium. They are the fundamental input for connection design: every bolted or welded connection is sized to resist the reaction it must transfer. In AISC 360-22, connection design begins with the factored reaction from the structural analysis (LRFD) or service-level reaction (ASD).

This reference covers the four primary beam support conditions used in steel design, with reaction formulas, shear and moment diagrams, sign conventions, and worked examples.

Support Conditions

Steel beams fall into one of four support categories, each producing different reaction distributions and internal force diagrams:

Condition Supports Statically Typical Use
Simply supported Pin + roller Determinate Floor beams, roof beams, girders
Cantilever Fixed at one end, free at other Determinate Balconies, canopies, crane runway stops
Fixed-end Fixed at both ends Indeterminate (2nd degree) Moment frames, rigid portal frames
Continuous 3+ supports (pin/roller or fixed) Indeterminate Multi-span girders, runway beams

Simply Supported Beam — Reaction Formulas

The simply supported beam is the most common configuration in steel construction. One end provides horizontal and vertical restraint (pin), the other only vertical (roller), allowing thermal expansion.

Uniform Load w (kips/ft or plf)

Parameter LRFD Formula ASD Formula
Left reaction, R_A wL/2 wL/2
Right reaction, R_B wL/2 wL/2
Max shear, V_max wL/2 at supports wL/2 at supports
Max moment, M_max wL²/8 at midspan wL²/8 at midspan
Deflection at midspan 5wL⁴/(384EI) 5wL⁴/(384EI)
End slope wL³/(24EI) wL³/(24EI)

Shear diagram: Linear from +wL/2 at left to 0 at midspan to -wL/2 at right.

Moment diagram: Parabolic, zero at supports, maximum wL²/8 at midspan.

Concentrated Load P at Midspan

Parameter Formula
Left reaction, R_A P/2
Right reaction, R_B P/2
Max moment, M_max PL/4 at midspan
Deflection at midspan PL³/(48EI)

Concentrated Load P at Distance a From Left

Parameter Formula
Left reaction, R_A Pb/L where b = L - a
Right reaction, R_B Pa/L
Max moment, M_max Pab/L at load point
Deflection (at load if a > b) Pa²b²/(3EIL)

Two Equal Concentrated Loads P at Third Points

Parameter Formula
Each reaction P
Max moment PL/3 (constant between loads)
Deflection at midspan 23PL³/(648EI)

Worked Example — Simply Supported Floor Beam

Given: W18x50 floor beam, L = 30 ft, D = 720 plf, L = 480 plf (office occupancy).

LRFD factored load: w_u = 1.2(720) + 1.6(480) = 864 + 768 = 1,632 plf = 1.632 kips/ft

Reaction: R_u = 1.632 × 30 / 2 = 24.5 kips at each support

Shear connection required: 24.5 kips. A single-plate shear tab with 3 bolts (3/4 in. A325-N, single shear) provides phi*Rn = 3 × 17.9 = 53.7 kips > 24.5 kips. OK.

Max moment: M_u = 1.632 × 30² / 8 = 183.6 kip-ft

ASD service load: w_a = 720 + 480 = 1,200 plf = 1.200 kips/ft

ASD reaction: R_a = 1.200 × 30 / 2 = 18.0 kips

Cantilever Beam — Reaction Formulas

A cantilever is fixed at one end and free at the other. The fixed support must resist both vertical reaction and moment.

Uniform Load w Over Full Length

Parameter Formula
Reaction at fixed end, R wL
Moment at fixed end, M_fixed wL²/2
Shear at free end 0
Deflection at free end wL⁴/(8EI)

Shear diagram: Linear from wL at support to 0 at free end.

Moment diagram: Parabolic from wL²/2 at support to 0 at free end. Moment is negative (tension top).

Concentrated Load P at Free End

Parameter Formula
Reaction at fixed end, R P
Moment at fixed end, M_fixed PL
Deflection at free end PL³/(3EI)

Worked Example — Canopy Beam

Given: W12x26 cantilever, L = 6 ft, D = 150 plf, roof live L_r = 300 plf.

LRFD: w_u = 1.2(150) + 1.6(300) = 180 + 480 = 660 plf = 0.660 kips/ft

Fixed-end reaction: R_u = 0.660 × 6 = 3.96 kips

Fixed-end moment: M_u = 0.660 × 6² / 2 = 11.9 kip-ft

Check W12x26 capacity: phiM_n = 131 kip-ft (AISC Table 3-2). M_u << phiM_n. Connection must transfer 3.96 kips vertical + 11.9 kip-ft moment at the fixed support. Use a stiffened seated connection or end-plate moment connection.

Fixed-End Beam — Reaction Formulas

Both ends are fully restrained against rotation. The beam is statically indeterminate (2nd degree). End moments develop that reduce the mid-span moment.

Uniform Load w

Parameter Formula
End reaction, each wL/2
End moment, each wL²/12
Mid-span moment wL²/24
Point of zero moment (from end) 0.211L
Deflection at midspan wL⁴/(384EI)

Shear diagram: Linear from +wL/2 to -wL/2, same as simply supported.

Moment diagram: Parabolic. Negative wL²/12 at both ends, positive wL²/24 at midspan. Inflection points at 0.211L from each end.

Concentrated Load P at Midspan

Parameter Formula
End reaction, each P/2
End moment, each PL/8
Mid-span moment PL/8 (same magnitude as end)

Worked Example — Fixed-End Spandrel Beam

Given: W21x44 spandrel beam, L = 24 ft, D = 500 plf + wall load 300 plf = 800 plf, L = 400 plf.

LRFD: w_u = 1.2(800) + 1.6(400) = 960 + 640 = 1,600 plf = 1.600 kips/ft

End reaction: R_u = 1.600 × 24 / 2 = 19.2 kips

End moment: M_u_end = 1.600 × 24² / 12 = 76.8 kip-ft

Mid-span moment: M_u_mid = 1.600 × 24² / 24 = 38.4 kip-ft

Connection design: Each end connection must resist 19.2 kips shear AND 76.8 kip-ft moment. An extended end plate (4E) moment connection is required. A shear tab alone cannot resist the end moment.

Continuous Beam — Reaction Formulas

Continuous beams span over 3 or more supports. They are statically indeterminate to (n-2) degrees for n supports. Reactions and moments depend on span ratios and loading patterns.

Two Equal Spans — Uniform Load w

Parameter Span AB Span BC
Exterior reaction, R_A or R_C 3wL/8 3wL/8
Interior reaction, R_B 10wL/8 = 1.25wL 10wL/8 = 1.25wL
Exterior end moment 0 0
Interior support moment -wL²/8 -wL²/8
Max positive moment 9wL²/128 ≈ 0.0703wL² 9wL²/128

Three Equal Spans — Uniform Load w

Parameter Span AB / CD (exterior) Span BC (interior)
Exterior reaction, R_A or R_D 4wL/10 = 0.40wL —
Interior reaction, R_B or R_C 11wL/10 = 1.10wL 11wL/10 = 1.10wL
Exterior end moment 0 —
First interior moment -wL²/10 -wL²/11
Interior interior moment — -wL²/11
Max positive moment, exterior span wL²/14 —
Max positive moment, interior span — wL²/16

Pattern Loading (Critical Concept)

For continuous beams, loading ALL spans simultaneously does NOT produce the maximum moments everywhere. Pattern live loading (alternate spans loaded) is required per AISC 360 Section B3.6:

Worked Example — Three-Span Continuous Girder

Given: W24x55 continuous girder, 3 spans at 28 ft each, D = 600 plf, L = 500 plf.

LRFD: w_u = 1.2(600) + 1.6(500) = 720 + 800 = 1,520 plf = 1.520 kips/ft

Exterior reaction: R_A = R_D = 0.40 × 1.520 × 28 = 17.0 kips

Interior reaction: R_B = R_C = 1.10 × 1.520 × 28 = 46.8 kips

Note: Interior reactions are 2.75× the exterior reactions. Interior connections must be designed for substantially higher shear than exterior connections.

Interior negative moment: M_u = -1.520 × 28² / 10 = -119 kip-ft (at first interior support)

Sign Conventions per AISC 360

AISC 360-22 does not mandate a specific sign convention for internal forces, but U.S. structural engineering practice follows these conventions:

Shear Sign Convention

Moment Sign Convention

Reaction Sign Convention

AISC Steel Construction Manual Resources

AISC Manual Reference Content
Table 3-22a Continuous beam moment and reaction coefficients
Table 3-22b Fixed-end beam moment coefficients
Table 3-23 Shear, moment, and deflection formulas for 35+ loading cases
Part 10 Connection design — sizing connections for beam end reactions
Table 10-10 Single-plate shear tab capacities
Table 10-1 Double-angle connection capacities

Common Mistakes in Beam Reaction Calculations

  1. Forgetting pattern loading for continuous beams. Loading all spans simultaneously underestimates positive moments and overestimates negative moments at some supports. Always check pattern loading per AISC 360 Section B3.6.

  2. Applying simply supported formulas to beams with moment connections. A beam connected with extended end plates (moment connection) develops end moments. Treating it as simply supported overestimates mid-span moment and ignores the moment that the connection must resist.

  3. Using the wrong load combination. LRFD reactions (using load factors 1.2D + 1.6L) are typically 1.4-1.5× ASD reactions (using D + L). Connections designed for ASD reactions may be undersized for LRFD. Always match the connection design to the design method.

  4. Neglecting eccentricity on shear connections. Single-plate shear connections have an eccentricity between the bolt line and the weld line. AISC Manual Table 10-10 accounts for this eccentricity, but hand calculations must include the moment from reaction × eccentricity when checking bolt shear and bearing.

  5. Ignoring thermal effects. Simply supported beams with one pin and one roller accommodate thermal expansion. If both ends are laterally restrained (e.g., both ends bolted to rigid walls), thermal expansion can produce unintended axial forces of significant magnitude.

Calculator Tools

Try it now: Check your beam end reactions with our free Beam Calculator calculator →

Related References

Design Resources

Calculator tools

Reference pages

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All reaction values must be verified by a licensed Professional Engineer for the specific loading, support conditions, and design code applicable to your project. The site operator disclaims liability for any loss arising from the use of this information.