Beam Deflection and Moment Formulas — Simply Supported, Cantilever & Fixed

This reference covers the most commonly used beam formulas for deflection (δ), slope (θ), shear (V), and bending moment (M) for standard loading conditions. All formulas assume linear elastic behavior, prismatic sections, and small deflections.

Variables

Simply Supported Beam — Midspan Point Load

P applied at midspan (x = L/2)

Max moment:   M_max = PL/4        (at midspan)
Max shear:    V_max = P/2         (at supports)
Max deflection: δ_max = PL³/48EI  (at midspan)
End slope:    θ = PL²/16EI

Example: W18x55, L=20ft, P=30 kips, EI=200×10³ kip·ft² δ_max = 30×20³ / (48×200,000) = 0.050 in

Simply Supported Beam — Uniform Distributed Load (UDL)

Total load W = wL

Max moment:   M_max = wL²/8       (at midspan)
Max shear:    V_max = wL/2        (at supports)
Max deflection: δ_max = 5wL⁴/384EI (at midspan)
End slope:    θ = wL³/24EI

This is the most common formula for floor and roof beams.

Simply Supported Beam — Off-Center Point Load

Load P at distance 'a' from left, 'b' from right (a < b)

Left reaction:  R_A = Pb/L
Right reaction: R_B = Pa/L
Max moment:    M_max = Pab/L     (at load point)

Deflection under load:
δ_P = Pa²b²/3EIL

Max deflection (if a < b, occurs at x = √((L²-b²)/3)):
δ_max = Pb(L²-b²)^(3/2) / (9√3·EI·L)

Cantilever Beam — Point Load at Free End

P at free end

Max moment:   M_max = PL         (at fixed support)
Max shear:    V_max = P          (constant throughout)
Max deflection: δ_max = PL³/3EI  (at free end)
Free end slope: θ = PL²/2EI

Cantilever Beam — Uniform Distributed Load

Total load W = wL

Max moment:   M_max = wL²/2      (at fixed support)
Max shear:    V_max = wL         (at fixed support)
Max deflection: δ_max = wL⁴/8EI  (at free end)
Free end slope: θ = wL³/6EI

Cantilever Beam — Point Load at Any Location

Load P at distance 'a' from fixed end (a < L)

Deflection at free end:
δ_tip = Pa²(3L - a) / 6EI

Deflection under load:
δ_a = Pa³/3EI

Fixed-Fixed Beam — Midspan Point Load

P at midspan, both ends fully fixed

Left reaction:   R_A = P/2
Right reaction:  R_B = P/2
End moment:      M_end = PL/8     (each end, hogging)
Midspan moment:  M_mid = PL/8     (sagging)
Max shear:       V_max = P/2      (at supports)
Max deflection:  δ_max = PL³/192EI (at midspan)
End slope:       θ = 0            (fixed ends)

Deflection comparison: A fixed-fixed beam deflects 4× less than a simply supported beam under the same midspan point load (PL³/192EI vs PL³/48EI). This stiffness advantage makes fixed-fixed beams attractive for deflection-critical applications, but only if the connections can develop full fixity.

Fixed-Fixed Beam — UDL

Total load W = wL

Left reaction:   R_A = wL/2
Right reaction:  R_B = wL/2
Max end moment:  M_end = wL²/12  (each end, hogging)
Max span moment: M_mid = wL²/24  (sagging, at midspan)
Max shear:       V_max = wL/2    (at supports)
Max deflection:  δ_max = wL⁴/384EI (at midspan)
End slope:       θ = 0           (fixed ends)

Note the end moment for fixed-fixed (wL²/12) vs simply supported (wL²/8): fixity reduces midspan moment by 33% but introduces support moments. The total design moment envelope includes both the hogging moment at supports (wL²/12) and the sagging moment at midspan (wL²/24). For practical design, check both locations.

Fixed-Fixed Beam — Point Load at Any Location

Load P at distance 'a' from left end, 'b' from right end (a + b = L)

Fixed-end moments (from AISC/CRC tables):
  M_FE_left  = Pab² / L²   (hogging)
  M_FE_right = Pa²b / L²   (hogging)

Reactions:
  R_A = Pb/L + (M_FE_left - M_FE_right)/L
  R_B = Pa/L + (M_FE_right - M_FE_left)/L

Simplifying:
  R_A = Pb²(3a + b) / L³
  R_B = Pa²(3b + a) / L³

Deflection under load point:
  δ_P = Pa³b³ / (3EI·L³)

Max deflection (occurs between load and nearer support):
  Use numerical evaluation or structural analysis for exact location

For the general off-center point load case, engineers typically use structural analysis software or the AISC beam diagrams and formulas tables (Part 3 of the Steel Construction Manual). The fixed-end moments above are the standard starting point for moment distribution or direct stiffness analysis.

Key takeaway: When the point load moves off center, the fixed-end moment increases at the nearer support and decreases at the farther support. The sum M_FE_left + M_FE_right equals Pab/L. When a = b = L/2, both moments equal PL/8, recovering the symmetric case.

Fixed-Fixed Beam Formulas — Summary Table

Fixed-Pinned (Propped Cantilever) Beam — Point Load at Midspan

Fixed at left, pinned (simply supported) at right
P at midspan (x = L/2)

Left reaction:   R_A = 11P/16
Right reaction:  R_B = 5P/16
Left end moment: M_A = 3PL/16  (hogging, at fixed support)
Max + moment:    M_mid = 5PL/32 (sagging, at midspan)
Max shear:       V_max = 11P/16 (at left support)
Max deflection:  δ_max = 7PL³/768EI (at x ≈ 0.447L from left)
End slope left:  θ_A = 0  (fixed)
End slope right: θ_B = PL²/32EI

Comparison to simply supported: The propped cantilever reduces midspan moment from PL/4 to 5PL/32 — a 37.5% reduction — at the cost of a fixed-end moment of 3PL/16 that must be designed for.

Propped Cantilever — UDL

Fixed at left, pinned at right
Total load W = wL

Left reaction:   R_A = 5wL/8
Right reaction:  R_B = 3wL/8
Left end moment: M_A = wL²/8   (hogging, at fixed support)
Max + moment:    M_max = 9wL²/128 (sagging, at x = 5L/8 from fixed end)
Max shear:       V_max = 5wL/8  (at fixed support)
Max deflection:  δ_max = wL⁴/185EI (at x ≈ 0.422L from pinned end)
End slope left:  θ_A = 0  (fixed)
End slope right: θ_B = wL³/48EI

Design significance: The propped cantilever is one of the most practical beam configurations. It provides significant moment reduction compared to a simply supported beam, yet requires fixity at only one end — easier to achieve in practice than a fully fixed-fixed beam. The fixed-end moment (wL²/8) equals the simply supported midspan moment, and the positive moment (9wL²/128) is only 56% of the simply supported value.

Propped Cantilever Formulas — Summary Table

Simply Supported Beam — Partial UDL

A uniformly distributed load of intensity w applied over a portion of the span, from x = a to x = b, where the total loaded length is c = b − a.

Simply supported beam, span L
UDL w from x = a to x = b (measured from left support)
Loaded length: c = b − a

Left reaction:
  R_A = w·c·(2L − a − b) / (2L)

Right reaction:
  R_B = w·c·(a + b) / (2L)

Moment at distance x from left support (for a ≤ x ≤ b):
  M_x = R_A·x − w·(x − a)² / 2

Max moment (at midspan of loaded region if symmetric, otherwise at x where shear = 0):
  For UDL centered on span (a = (L−c)/2, b = (L+c)/2):
  M_max = w·c·(2L − c) / 8

Shear at left edge of load:
  V_a = R_A

Shear at right edge of load:
  V_b = R_A − w·c

Common special case — UDL over center half of span (c = L/2, centered):

a = L/4,  b = 3L/4,  c = L/2

R_A = R_B = wL/4
M_max = 3wL²/32  (at midspan)
δ_max ≈ 11wL⁴ / (768EI)  (at midspan, approximate)

Common special case — UDL from left support to distance c:

a = 0,  b = c

R_A = wc(2L − c) / (2L)
R_B = wc² / (2L)
M_max at x = R_A/w:  M_max = R_A² / (2w)

Partial UDLs are common in practice: equipment loads, storage areas, and partial floor loading all produce non-full-span distributed loads. For complex partial loading patterns, superposition of multiple partial UDL cases gives accurate results.

Simply Supported Beam — Two Equal Symmetric Point Loads

Two equal concentrated loads P, each located at distance 'a' from the nearest support. This is the classic case for beams carrying two concentrated loads from framing members or equipment supports.

Simply supported beam, span L
Load P at x = a from left support
Load P at x = L − a from right support

Left reaction:   R_A = P
Right reaction:  R_B = P

Shear diagram:
  V = 0        for 0 < x < a
  V = +P       for a < x < L−a  (between the loads)
  V = 0        for x > L−a

Wait — let me correct the shear convention:

Left reaction:   R_A = P
Right reaction:  R_B = P
Total load = 2P

Shear:
  From left to first load (0 < x < a):       V_x = P
  Between the two loads (a < x < L−a):       V_x = 0
  From second load to right (L−a < x < L):   V_x = −P

Moment diagram:
  Between supports and first load (0 ≤ x ≤ a):
    M_x = Px

  Between the two loads (a ≤ x ≤ L−a):
    M_x = Pa  (constant — pure bending region)

  Max moment:  M_max = Pa  (anywhere between the two loads)

Max deflection (at midspan):
  δ_max = Pa(3L² − 4a²) / (24EI)

Special case — loads at third points (a = L/3):

M_max = PL/3
δ_max = 23PL³ / (648EI)  ... let me recalculate:

δ_max = P·(L/3)·(3L² − 4(L/3)²) / (24EI)
      = PL(3L² − 4L²/9) / (72EI)
      = PL(27L² − 4L²) / (9·72EI)
      = PL·23L² / (648EI)
      = 23PL³ / 648EI

Approximately: δ_max ≈ 0.0355 PL³/EI

Comparison to single midspan point load:

The two-load case produces a uniform moment region between the loads, which is the loading configuration used in standard beam bending tests (ASTM A992 coupon testing uses a similar four-point bending setup).

Comprehensive Beam Formula Comparison Table

This master comparison table covers all four boundary conditions for both point load and UDL, enabling rapid selection of the correct formula for any standard case.

Point Load at Center — All Beam Types

UDL (w over full span) — All Beam Types

Deflection Stiffness Ranking (least to most deflection for same P and L)

Fixed-fixed:     δ = PL³/192EI    →  baseline (stiffest)
Propped:         δ = 7PL³/768EI   →  1.56× fixed-fixed deflection
Simply supported: δ = PL³/48EI    →  4.0× fixed-fixed deflection
Cantilever:      δ = PL³/3EI      →  64× fixed-fixed deflection

For UDL stiffness ranking:

Fixed-fixed:      δ = wL⁴/384EI     →  baseline (stiffest)
Propped:          δ = wL⁴/185EI     →  ~2.1× fixed-fixed deflection
Simply supported: δ = 5wL⁴/384EI    →  5.0× fixed-fixed deflection
Cantilever:       δ = wL⁴/8EI       →  48× fixed-fixed deflection

Worked Example — Simply Supported W16x36 Under UDL

Problem: A W16x36 steel beam (ASTM A992, Fy = 50 ksi) spans 20 ft and carries a uniformly distributed service live load of 2.0 klf. Check the beam for moment, shear, and live load deflection against L/360.

Given Data

Section:    W16x36
Span:       L = 20 ft
Steel:      ASTM A992, Fy = 50 ksi
Live load:  w_L = 2.0 kip/ft (service)
Deflection limit: L/360 (floor beam, live load only)

Section properties (from AISC Manual Table 1-1):
  I_x  = 448 in⁴
  S_x  = 56.5 in³
  Z_x  = 64.0 in³
  d    = 15.86 in
  t_w  = 0.295 in
  A    = 10.6 in²

Step 1 — Calculate Maximum Bending Moment

M_max = wL² / 8

M_max = (2.0 klf)(20 ft)² / 8
      = (2.0)(400) / 8
      = 800 / 8
      = 100 kip·ft

Required moment strength (LRFD): Apply load factor 1.6 for live load.

M_u = 1.6 × 100 = 160 kip·ft

Nominal moment capacity:

M_n = F_y × Z_x = 50 ksi × 64.0 in³ = 3200 kip·in = 266.7 kip·ft

φM_n = 0.90 × 266.7 = 240.0 kip·ft

Check: φM_n = 240.0 kip·ft > M_u = 160 kip·ft — OK (demand/capacity = 0.667)

Step 2 — Calculate Maximum Shear

V_max = wL / 2

V_max = (2.0 klf)(20 ft) / 2
      = 40 / 2
      = 20.0 kips (service)

V_u = 1.6 × 20.0 = 32.0 kips (factored)

Nominal shear capacity (AISC Chapter G):

For W16x36: h/t_w = (d − 2k)/t_w ≈ 15.86/(0.295) ≈ 48.8

Since h/t_w < 1.10√(k_v·E/F_y) = 1.10√(5×29000/50) = 59.2:

C_v = 1.0 (no web shear buckling)

φV_n = 0.90 × 0.6 × F_y × A_w
     = 0.90 × 0.6 × 50 × d × t_w
     = 0.90 × 0.6 × 50 × 15.86 × 0.295
     = 0.90 × 0.6 × 50 × 4.679
     = 0.90 × 140.4
     = 126.3 kips

Check: φV_n = 126.3 kips > V_u = 32.0 kips — OK (demand/capacity = 0.253)

Step 3 — Calculate Maximum Live Load Deflection

δ_max = 5wL⁴ / (384EI)

Where:
  w = 2.0 klf = 2.0/12 kip/in = 0.1667 kip/in
  L = 20 ft = 240 in
  E = 29,000 ksi
  I = 448 in⁴

δ_max = 5 × 0.1667 × (240)⁴ / (384 × 29,000 × 448)

Numerator:
  5 × 0.1667 = 0.8335
  240⁴ = 3,317,760,000
  0.8335 × 3,317,760,000 = 2,766,423,360

Denominator:
  384 × 29,000 × 448 = 384 × 12,992,000 = 4,988,928,000

δ_max = 2,766,423,360 / 4,988,928,000
      = 0.555 in

Step 4 — Check Against L/360 Deflection Limit

L/360 = 240 in / 360 = 0.667 in

δ_max = 0.555 in < L/360 = 0.667 in — OK

Deflection ratio: δ_max / L = 0.555/240 = 1/432 — well within L/360.

Summary of Results

The W16x36 is adequate for the given loading. Moment governs at 67% capacity utilization, while deflection is at 83% of the allowed limit — a typical result for medium-span floor beams where deflection often controls the design.

Quick Reference — Maximum Deflection Summary

Note: Cantilever deflection is 16× greater than a simply supported beam for the same point load and span — a critical comparison for overhanging structures. Fixed-fixed beams are 4× stiffer than simply supported for point loads and 5× stiffer for UDL.

Serviceability Deflection Limits

Frequently Asked Questions

Which beam formula should I use for a uniform floor load? Use the simply supported UDL formula: max moment M_max = wL²/8 at midspan, max deflection δ_max = 5wL⁴/384EI at midspan. This is the most common formula for floor and roof framing. Convert your area load (psf) to linear load (kip/ft) by multiplying by the tributary width before applying the formula.

How does a fixed-fixed beam compare to a simply supported beam? For a fixed-fixed beam under UDL, the midspan moment is wL²/24 — one-third of the simply supported value (wL²/8). However, this comes with fixed-end moments of wL²/12 at each support that must also be designed for. For deflection, fixed-fixed gives wL⁴/384EI versus 5wL⁴/384EI for simply supported — five times stiffer. Real connections rarely achieve full fixity, so designers often use intermediate assumptions or analyze both cases.

Why is a cantilever deflection so much larger than a simply supported beam? For a point load at the free end, cantilever deflection = PL³/3EI. For a simply supported beam with the same load at midspan: PL³/48EI. The cantilever is 16× more flexible. The key reason is boundary conditions: a cantilever has only one support, so the entire beam must curve to carry the load, while a simply supported beam distributes curvature across the span with two supports restraining movement at both ends.

What is the deflection limit for a steel floor beam? AISC Design Guide and IBC typically limit live load deflection to L/360 for floor beams supporting brittle finishes (tile, plaster). For beams supporting flexible finishes, L/240 is common. Total load deflection (dead + live) is often limited to L/240 for floor beams. Always check both live load deflection (for cracking concerns) and total load deflection (for visual appearance).

How do I apply these formulas for metric units? All formulas work in any consistent unit system. For SI: use w in kN/m, L in m, E in kPa (or GPa × 10⁶), and I in m⁴. This gives M in kN·m and δ in m. In practice, engineers use E = 200 GPa = 200 × 10⁶ kN/m², convert I from mm⁴ to m⁴ (divide by 10¹²), and express results in mm by multiplying δ in m by 1000.

How do I handle a partial uniformly distributed load? For a UDL of intensity w applied over only part of the span (from x = a to x = b), the reactions are R_A = wc(2L − a − b)/(2L) and R_B = wc(a + b)/(2L), where c = b − a is the loaded length. The moment diagram is parabolic within the loaded region and linear outside it. The maximum moment occurs where the shear passes through zero. For complex partial loading, superpose multiple partial UDL cases or use the Beam Calculator for automatic analysis.

What is the difference between a propped cantilever and a fixed-fixed beam? A propped cantilever (fixed-pinned) has full fixity at one end and a pin (zero moment) at the other. A fixed-fixed beam has full fixity at both ends. For UDL, the propped cantilever has a fixed-end moment of wL²/8 and maximum positive moment of 9wL²/128, while the fixed-fixed beam has end moments of wL²/12 and midspan moment of wL²/24. The propped cantilever is easier to construct (only one rigid connection) but deflects more. Both are significantly stiffer than a simply supported beam.

How do two equal symmetric point loads differ from a single midspan point load? Two equal loads P at distance 'a' from each support produce a constant moment region of M = Pa between the loads, unlike the triangular moment diagram from a single midspan load. For loads at third points (a = L/3), the maximum moment is PL/3 versus PL/4 for a single center load — 33% higher. However, the total load is 2P versus P, so the moment per unit of applied load is actually lower. This loading pattern is common for beams supporting joists or equipment at two locations.

When should I assume fixed-end versus pinned-end conditions? True fixed ends require rigid connections that resist rotation — fully welded connections, heavy moment connections with stiffeners, or concrete-encased beams. In practice, most steel connections are somewhere between pinned and fixed. AISC recommends designing for the worst case of both assumptions: check the beam for the higher of (a) simply supported moment and deflection, and (b) fixed-end moment at supports. For preliminary design, assume pinned supports unless the connection is clearly a moment-resisting connection.

Run This Calculation

→ Beam Calculator — beam reactions and moment/shear diagrams for any span and loading condition.

→ Beam Deflection Calculator — compute deflections and reactions automatically for common loading cases, checking L/360 and L/240 limits.

→ Continuous Beam Calculator — reactions, moments, and deflections for multi-span beams with any loading pattern.

See Also

• W-Shape Beam Sizes — Section Properties (Ix, rx, ry)
• Beam Bending Moment Formulas — Reference Table
• Steel Beam Load Tables — W-Shape UDL Capacity
• Steel Beam Span Guide — W-Shape Span Ranges by Depth
• Lateral-Torsional Buckling — Lp, Lr, Cb
• Steel Fy & Fu Reference — Yield and Tensile Strength by Grade
• beam capacity calculator
• deflection limits reference
• Composite Beam
• Floor Vibration

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Design Resources

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• Steel Beam Capacity Calculator
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• Beam Calculator — SFD, BMD & Reactions
• Beam Optimizer — Find Most Efficient Section
• Beam Span Table Tool

Design guides

• Steel Beam Calculator Guide
• Beam Capacity Worked Example
• Beam Deflection Worked Example
• Steel Beam Design Workflow
• Steel Beam Span Reference