----------------------------------------- | ---------------- | ---------- | ------------------------------- | ---------------------- | | Point load P at midspan | PL/4 | P/2 | PL^3/(48EI) | Midspan | | Point load P at distance a from left (a <= L/2) | Pab/L | max(RA,RB) | Pb(L^2-b^2)^1.5 / (9sqrt(3)LEI) | At load point | | Two equal point loads at L/3 from each end | PL/3 | P | 23PL^3/(648EI) | Constant between loads | | UDL w full span | wL^2/8 | wL/2 | 5wL^4/(384EI) | Midspan | | Partial UDL w from x=a to x=b | Varies | Varies | Requires integration | Varies with a,b | | Triangular load 0 to w0 | w0L^2/(9sqrt(3)) | w0L/3 | 0.00652w0L^4/(EI) | x = L/sqrt(3) |

Cantilever Beam

Fixed-Fixed Beam

The midspan moment for a fixed-fixed beam under UDL is exactly half the simply supported value. The support moment is twice the midspan moment. If the section is symmetric, the support moments govern.

Propped Cantilever

The propped cantilever has one-quarter to one-third less deflection than a pure cantilever for the same load, depending on the load distribution.

Worked Example 1: Simply Supported W18x55 Under UDL

This is the most common beam design scenario: a floor beam carrying a uniform distributed load, simply supported at both ends.

Given:

• Beam: W18x55, ASTM A992 (Fy = 50 ksi)
• From AISC Table 1-1: Ix = 890 in^4, Sx = 98.3 in^3, Zx = 112 in^3, d = 18.11 in, tw = 0.390 in
• Span L = 30 ft = 360 in
• Factored load w_u = 1.5 kips/ft (includes load factors and self-weight)
• Service load w_s = 1.0 kips/ft (unfactored, for deflection check)

Step 1: Reactions

RA = RB = w_u L / 2 = 1.5 x 30 / 2 = 22.5 kips

Step 2: Maximum factored moment

Mu = w_u L^2 / 8 = 1.5 x 30^2 / 8 = 1.5 x 900 / 8 = 168.75 kip-ft = 2025 kip-in

Step 3: Flexural capacity check (LRFD)

Check compactness: bf/(2tf) = 7.53/(2x0.630) = 5.98. lambda_p = 0.38 sqrt(E/Fy) = 0.38 sqrt(29000/50) = 9.15. Flange is compact (5.98 < 9.15). Web: h/tw = (18.11-2x0.630)/0.390 = 16.85/0.390 = 43.2. lambda_p = 3.76 sqrt(E/Fy) = 90.6. Web is compact (43.2 < 90.6).

Since both flange and web are compact, use plastic moment:

phi Mn = 0.9 x Fy x Zx = 0.9 x 50 x 112 = 5040 kip-in = 420 kip-ft

Mu = 168.75 kip-ft < phi Mn = 420 kip-ft. Flexure OK. Demand/capacity ratio = 0.40.

Step 4: Shear check (LRFD)

Vu = w_u L / 2 = 22.5 kips

Web shear area: Aw = d x tw = 18.11 x 0.390 = 7.06 in^2

Check web shear buckling: h/tw = 43.2 < 2.24 sqrt(E/Fy) = 2.24 x 24.08 = 53.9. Shear buckling does not govern.

phi Vn = 0.9 x 0.6 x Fy x Aw = 0.9 x 0.6 x 50 x 7.06 = 190.6 kips

Vu = 22.5 kips < phi Vn = 190.6 kips. Shear OK. D/C = 0.12.

Step 5: Deflection check (service load)

delta_max = 5 w_s L^4 / (384 E Ix) = 5 x (1.0/12) x 360^4 / (384 x 29000 x 890)

Numerator: 5 x 0.08333 x 16,796,160,000 = 6,998,400,000

Denominator: 384 x 29,000 x 890 = 9,908,160,000

delta_max = 6,998,400,000 / 9,908,160,000 = 0.706 in

L/360 = 360/360 = 1.0 in. delta_max = 0.706 in < 1.0 in. Deflection OK. D/C = 0.71.

Step 6: Lateral-torsional buckling check

Assume the compression flange is braced at the supports only: Lb = 30 ft = 360 in.

Lp = 1.76 ry sqrt(E/Fy) = 1.76 x 1.57 x 24.08 = 66.6 in = 5.55 ft

Lr = pi ry sqrt(E/(0.7Fy)) x sqrt(...) per AISC Eq F2-6. For W18x55, Lr is approximately 14.2 ft per AISC Table 3-2.

Lb = 30 ft > Lr = 14.2 ft. The beam is in the elastic LTB range.

phi Mn for elastic LTB must be calculated per AISC Eq F2-3. This typically governs over the plastic moment for long unbraced spans. At Lb = 30 ft, phi Mn is substantially less than 420 kip-ft.

This is why simply supported beams at long spans require intermediate bracing of the compression flange (joists, deck attachment, or kicker bracing). Adding bracing at 10 ft on center (Lb = 10 ft < Lp = 5.55 ft... actually Lb is still > Lp, in inelastic range) would increase phi Mn significantly.

Summary for this beam:

• Flexure (compact, braced): D/C = 0.40
• Shear: D/C = 0.12
• Deflection: D/C = 0.71 (this is the governing limit state for this span)
• LTB (unbraced 30 ft): would govern -- bracing required

Use the beam capacity calculator to run all seven AISC limit states automatically.

Worked Example 2: Cantilever Beam Under Tip Load

A cantilever beam supports a hoist at its free end. The hoist applies a factored point load of 5 kips at the tip.

Given:

• Beam: W12x40, ASTM A992
• From AISC: Ix = 310 in^4, Sx = 51.9 in^3, Zx = 57.0 in^3, d = 11.94 in
• Cantilever length L = 8 ft = 96 in
• Factored tip load Pu = 5 kips (vertical, at free end)

Step 1: Reactions at fixed support

Vertical reaction: R = Pu = 5 kips Moment reaction: M_fixed = Pu x L = 5 x 8 = 40 kip-ft = 480 kip-in

Step 2: Maximum moment (at fixed support)

Mu = 40 kip-ft = 480 kip-in

Step 3: Flexural capacity

phi Mn = 0.9 x 50 x 57.0 = 2565 kip-in = 213.8 kip-ft

Mu = 40 kip-ft < phi Mn = 213.8 kip-ft. Flexure OK. D/C = 0.19.

Step 4: Deflection at free tip (service load, assume P_service = 3.5 kips)

delta_max = P L^3 / (3 E I) = 3.5 x 96^3 / (3 x 29000 x 310) = 3.5 x 884,736 / (26,970,000) = 3,096,576 / 26,970,000 = 0.115 in

For a cantilever supporting a hoist, a typical deflection limit is L/360 = 96/360 = 0.267 in, but for sensitive equipment or overhead cranes, L/600 or L/800 may apply. At L/360, the beam is adequate (0.115 < 0.267).

Step 5: LTB for cantilever

For a cantilever, the compression (bottom) flange is braced at the fixed support only. The unbraced length is the full 8 ft = 96 in.

Check Lp: ry for W12x40 = 1.94 in. Lp = 1.76 x 1.94 x 24.08 = 82.2 in = 6.85 ft.

Lb = 8 ft > Lp = 6.85 ft. The beam is in the inelastic LTB range (Lp < Lb < Lr). LTB reduces phi Mn below the plastic moment.

For detailed LTB verification, use the beam capacity calculator which applies the correct AISC Chapter F equations based on your unbraced length and Cb factor.

How to Use the Beam Calculator Tool

The beam calculator at SteelCalculator.app automates the entire analysis. Here is the workflow:

1. Select support condition: simply supported, cantilever, fixed-fixed, propped cantilever, or continuous (2-10 spans).
2. Enter span length in feet or meters.
3. Select beam section from the database (500+ W, UB, IPE, HEA, HSS, channel, angle shapes) or enter custom Ix and E values.
4. Add loads: point loads (magnitude and position), UDL (magnitude and start/end positions), triangular loads, and concentrated moments. Multiple loads can be superimposed.
5. Run the analysis: the calculator computes reactions, shear and moment at 200+ points along the beam, and maximum deflection.
6. View results: the SFD and BMD are plotted graphically. Numerical values can be exported as CSV or printed.
7. Check capacity (optional): click through to the beam capacity calculator to run AISC 360, AS 4100, EN 1993, or CSA S16 design checks on the same section and loading.

For determinate beams, the calculator uses direct equilibrium (no numerical approximation). For indeterminate beams, it uses the matrix stiffness method (the same algorithm as commercial structural analysis software).

SFD and BMD: How to Read the Diagrams

The shear force and bending moment diagrams are the engineer's primary tools for understanding beam behavior. Reading them correctly is essential for design.

Shear Force Diagram:

• The SFD value at any point x is the algebraic sum of all vertical forces to the left (or right) of x. The sign convention is typically positive upward on the left face.
• Where the SFD crosses zero, the BMD reaches a local maximum or minimum. This is a direct consequence of dM/dx = V: when V = 0, the slope of the moment diagram is zero.
• Point loads create vertical jumps in the SFD equal to the load magnitude.
• A UDL creates a linear (sloping) SFD because the accumulated load increases linearly with distance.
• The area under the SFD between two points equals the change in moment between those points.

Bending Moment Diagram:

• The BMD value at any point x is the sum of moments of all forces to the left (or right) about x.
• Positive moment (sagging) means tension in the bottom fiber, compression in the top. Negative moment (hogging) means the reverse.
• For simply supported beams, moments are positive everywhere unless an overhang or cantilever is present.
• For continuous beams, moments are negative over interior supports and positive between them.
• A concentrated moment applied at a point creates a vertical jump in the BMD.

Differential relationships:

• dV/dx = -w(x): the slope of the shear diagram equals the negative of the load intensity.
• dM/dx = V(x): the slope of the moment diagram equals the shear.
• d^2M/dx^2 = -w(x): the curvature of the moment diagram equals the negative load. For UDL, the moment diagram is parabolic.

These relationships provide a powerful verification: if your computed shear diagram does not have slopes consistent with the applied loads, or if the moment diagram slope does not match the shear ordinate, there is an error in the analysis.

Input Parameters Explained

Every beam analysis requires the following inputs. Getting any of them wrong makes the output unreliable.

Span length (L): The center-to-center distance between supports, not the clear span. For steel beams bearing on concrete or masonry walls, the span is typically taken as the clear span plus the bearing length at one end, or conservatively the center-to-center distance. For beams framing into girder webs, use center-to-center of the supporting members.

Modulus of elasticity (E): 29,000 ksi for structural steel (all grades). 10,000-12,000 ksi for 6061-T6 aluminum. 1,400-1,800 ksi for sawn lumber (species-dependent). 3,600-5,800 ksi for normal-weight concrete (strength-dependent). Using the wrong E produces proportionally wrong deflections.

Moment of inertia (I): Must match the axis of bending. For a beam loaded vertically and spanning horizontally, use Ix (strong axis). For a beam bending about its weak axis (e.g., a spandrel beam resisting wind load on the facade), use Iy. I-values come from AISC Table 1-1 for standard shapes or are computed from geometry for custom sections.

Load magnitude: Must be entered in consistent units. If the span is in feet, enter distributed loads in kips per foot and point loads in kips. If using metric, keep everything in kN and meters. Mixed units are the most common source of analysis errors.

Load position: For point loads, enter the distance from the left support. For partial UDL, enter the start and end positions. Positions are measured from the left support along the beam.

Support fixity: This is where engineering judgment enters. Is the connection really a pin? Does it truly prevent rotation? A shear tab connection is modeled as a pin. A fully welded moment connection is modeled as fixed. A double-angle or end-plate shear connection is typically modeled as a pin, but can develop 10-20% of the fixed-end moment -- this partial restraint is ignored conservatively for strength design but can matter for deflection.

Beam Design Checklist

After running the beam calculator, verify all design limit states:

1. Flexure: Mu <= phi Mn. For compact sections in LRFD, phi Mn = 0.9 Fy Zx. For noncompact sections, use the effective section modulus. For slender elements, use AISC Chapter E.
2. Shear: Vu <= phi Vn. phi Vn = 0.9 x 0.6 Fy Aw for webs without stiffeners when h/tw <= 2.24 sqrt(E/Fy).
3. Lateral-torsional buckling: Check whether Lb <= Lp (no LTB reduction), Lp < Lb <= Lr (inelastic LTB), or Lb > Lr (elastic LTB). Use AISC Chapter F equations.
4. Deflection: delta_max <= delta_allowable. Floors: L/360 live load, L/240 total load. Roofs: L/360 snow or live, L/180 total. Crane runways: L/800 or stricter per AISC DG7.
5. Web local crippling and yielding: concentrated loads at supports and load points. Check per AISC Chapter J10.
6. Web sidesway buckling: for compression flanges not laterally braced.
7. Connection design: verify the connection can deliver the assumed support reactions and, if a moment connection, the required moment.

All seven checks are automated in the beam capacity calculator, which supports AISC 360, AS 4100, EN 1993, and CSA S16.

Common Beam Design Mistakes

Using the wrong I value: Iy instead of Ix for a beam loaded in the strong-axis direction. A W12x26 has Ix = 204 in^4 but Iy = 17.3 in^4 -- a factor of 12 difference. Using Iy for strong-axis bending produces absurdly conservative deflections.

Forgetting self-weight: A W18x55 weighs 55 lb/ft. On a 30-ft span, that is 1,650 lb = 1.65 kips of additional load, which increases midspan moment by (0.055 x 30^2 / 8) = 6.2 kip-ft. Small, but not negligible when added to other dead loads.

Neglecting LTB: Simply supported beams with unbraced compression flanges can fail by lateral-torsional buckling at a fraction of their plastic moment. A long, unbraced W18x55 at 30 ft may only achieve 30-40% of its plastic moment capacity. Always check LTB.

Incorrect support modeling: Modeling a shear tab connection as fixed produces unrealistically low midspan moments and deflections. The analysis must match the actual connection detail.

Not checking bearing at supports: A W12x26 with a 22.5 kip reaction requires adequate bearing area at the support. Without a bearing plate, the web may cripple under the concentrated reaction. Check web local yielding and web crippling per AISC J10.

Mixed units: kips and feet for moments produce kip-ft. kips and inches produce kip-in. The ratio is 12. Confusing the two produces either 12x-conservative or 12x-unconservative results. AISC Table 1-1 provides section properties in inch units; convert moments to kip-in before checking against Sx and Zx.

When to Use Each Type of Beam Analysis

Frequently Asked Questions

What is the difference between the beam calculator and the beam capacity calculator? The beam calculator performs structural analysis: it computes reactions, SFD, BMD, and deflection given the span, supports, and loads. The beam capacity calculator performs design checks: given a section, material, unbraced length, and moments, it checks flexure, shear, LTB, and deflection per the selected design code. They are complementary -- run the analysis first, then check capacity.

Can the calculator handle beams with overhangs? Yes. The continuous beam calculator supports overhangs by modeling the beam extending past the exterior supports. For a simple beam with a single overhang, model it as a propped cantilever or a two-span continuous beam where the overhang is the second span with zero load on the overhang portion if appropriate.

Does the calculator account for shear deformation? For standard steel beams with span-to-depth ratios above 10, shear deformation is negligible (less than 2% of bending deflection). The Euler-Bernoulli beam theory used by the calculator neglects shear deformation, which is conservative and standard practice in steel design. For very short, deep beams (L/d < 8), shear deformation becomes significant and Timoshenko beam theory is required.

How do I model a beam with varying cross-section? Divide the beam into segments with constant I and run the continuous beam analysis with segment properties. Alternatively, run separate analyses for the critical segments and envelope the results. The continuous beam calculator supports up to 10 segments with independent section properties.

What is Cb and when does it matter? Cb is the lateral-torsional buckling modification factor that accounts for the shape of the moment diagram between brace points. A uniform moment (Cb = 1.0) is the worst case. A moment diagram that reverses curvature (end moments of opposite sign) can have Cb up to 2.3 or higher, effectively tripling the LTB capacity. Cb is calculated automatically by the beam capacity calculator.

Can I use the beam calculator for wood or concrete beams? The analysis portion (SFD, BMD, deflection) is material-independent and works for any material. Enter the appropriate E and I for your material. However, the capacity checks are specific to structural steel per AISC 360, AS 4100, EN 1993, and CSA S16.

How do I know if my beam is adequately braced for LTB? If the compression flange is continuously connected to a concrete slab through shear studs (composite construction) or to a steel deck through puddle welds or screws at close spacing (typically 12 in or less), it is considered continuously braced and LTB does not govern. For unbraced top flanges, check Lb against Lp and Lr from AISC Table 3-2.

What is the maximum point load I can apply at midspan? This depends on the section, span, and unbraced length. As a rough approximation for preliminary sizing only: a compact W-shape can carry a midspan point load P such that PL/4 <= 0.9 x Fy x Zx, or P <= 3.6 Fy Zx / L. For a W12x26 at 20 ft: P <= 3.6 x 50 x 37.2 / (20 x 12) = 27.9 kips (factored). This ignores LTB and deflection -- always run the full check.

How many spans can the continuous beam calculator handle? Up to 10 spans. For analyses beyond 10 spans, use a dedicated structural analysis program. Most practical floor systems have 3-6 spans, so 10 is more than sufficient for hand-verification and preliminary design.

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

Related Pages

• Beam Calculator Tool -- Full beam analysis: reactions, SFD, BMD, and deflection
• Beam Capacity Calculator -- Code checks per AISC 360, AS 4100, EN 1993, CSA S16
• Beam Deflection Calculator -- Deflection checks against L/360, L/240 limits
• Continuous Beam Calculator -- Multi-span analysis with up to 10 spans
• Steel Beam Sizes Chart -- Dimensions and section properties for W, UB, IPE, HEA
• Cantilever Beam Reference -- Cantilever moment and deflection formulas
• Moment of Inertia Calculator -- Ix, Iy, and J for any cross-section
• Beam Capacity Worked Example -- Step-by-step AISC 360 flexure check

Disclaimer

SteelCalculator.app is a calculation tool, not a substitute for professional engineering certification. All beam analysis results and design checks are for educational and preliminary design reference only. All results must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in construction, fabrication, or permit documents. The user bears full responsibility for confirming all inputs, verifying all outputs against applicable code provisions, and obtaining professional engineering review. See our full disclaimer and how to verify calculations.