Uniform Load on Cantilever Beam Calculator
Calculates the slope at the free end and the deflection at any point along a cantilever beam carrying a uniform load. Enter the applied load, elastic modulus, moment of inertia, beam length and position.
Deflection of Beam with Uniform Load
Cantilever Beam with Uniform Load Formula
Where:
- \(P\) — total applied load (lbs)
- \(E\) — elastic modulus of the beam material (psi)
- \(I\) — moment of inertia of the cross-section (in⁴)
- \(L\) — length of the cantilever, from the fixed end to the free end (in)
- \(x\) — position along the beam, measured from the fixed end, where the deflection is evaluated (in)
- \(P_0\) — equivalent uniform load derived from \(P\), \(L\) and \(x\)
The calculator first converts the applied load into an equivalent uniformly distributed load \(P_0\), then applies the standard cantilever expressions to find the slope at the free end \(\theta\) and the deflection \(\delta\) at the chosen position \(x\).
Cantilever Beam with Uniform Load – Calculation Example
Example: a steel cantilever beam with \(E = 29{,}000{,}000\) psi and \(I = 150\) in⁴ is 120 in long. A total load of 500 lbs is applied, and the deflection is evaluated at \(x = 100\) in from the fixed end.
\[ P_0 = \frac{500 \cdot 120}{120 - 100} = \frac{60{,}000}{20} = 3000 \text{ lbs} \]
\[ \theta = \frac{3000 \cdot 120^3}{6 \cdot 29{,}000{,}000 \cdot 150} = \frac{5.184 \times 10^9}{2.61 \times 10^{10}} \approx 0.1986 \text{ rad} \]
\[ \delta = \frac{3000 \cdot 100^2 \cdot (100^3 + 6 \cdot 120^2 - 4 \cdot 120 \cdot 100)}{24 \cdot 29{,}000{,}000 \cdot 150} \approx 298.39 \text{ in} \]
Result: the slope at the free end is approximately 0.1986 rad and the deflection at 100 in from the fixed end is approximately 298.39 in. A deflection of this magnitude is far beyond typical serviceability limits (such as \(L/240\) or \(L/360\)), which shows immediately that the beam is much too flexible for this loading.
How to Use the Calculator
- Enter the total Applied Load (\(P\)) in pounds acting on the beam.
- Enter the Elastic Modulus (\(E\)) in psi. For structural steel, use 29,000,000 psi; see the reference table below for other materials.
- Enter the Moment of Inertia (\(I\)) of the beam cross-section in in⁴. This value comes from the shape's properties (for a rectangle, \(I = bh^3/12\)).
- Enter the Beam Length (\(L\)) in inches, measured from the fixed end to the free end.
- Enter the Load Position (\(x\)) in inches — the distance from the fixed end at which you want the deflection. This value must be smaller than the beam length.
- Read the results: the Slope at Free End in radians and the Deflection in inches at the selected position.
About Cantilever Beams with Uniform Loads
A cantilever beam is fixed at one end and completely free at the other, so all support reactions — moment, shear and deflection resistance — must come from the single fixed end. Balconies, canopies, overhanging floor joists, brackets and pergola arms are all typical cantilever structures.
A uniform load is spread continuously along the beam rather than concentrated at one point. Common examples include the self-weight of the beam itself, a masonry wall built on top of an overhang, accumulated snow on a canopy, or soil pressure behind a retaining bracket. Because the load is distributed, the internal shear and moment build up gradually from the free end toward the fixed end, where both reach their maximum values.
The two results tell you how the beam deforms. The slope \(\theta\) is the rotation of the beam at the free end in radians; small values (a few thousandths of a radian) are normal in service. The deflection \(\delta\) is the vertical displacement at the position you selected. Deflection is extremely sensitive to length — it grows roughly with the fourth power of the span — and to the stiffness \(EI\). Increasing the beam depth is the most effective fix, since \(I\) grows with the cube of the height: doubling the depth of a rectangular beam multiplies its stiffness by eight.
To judge whether a result is acceptable, compare the deflection with common code limits such as \(L/240\) for total load or \(L/360\) for live load alone. If you are sizing a whole floor or roof system rather than a single member, the Maximum Floor Live Load Capacity calculator and the Rafter Length calculator may also be useful.
Assumptions, Limitations and Practical Tips
- The beam is made of a linear-elastic material, has a constant cross-section, and deflects only slightly (small-deflection theory).
- The fixed end is perfectly rigid; in reality, connection flexibility and support rotation can noticeably increase real deflections.
- Load Position \(x\) must be less than Beam Length \(L\). As \(x\) approaches \(L\), the equivalent load \(P_0\) grows very rapidly and results lose practical meaning.
- Units must be consistent: load in lbs, dimensions in inches, modulus in psi. If you work in metric units (N, mm, MPa), the same formulas apply but every input must be converted.
- This tool checks serviceability (deflection and slope) only. Bending and shear stresses must be verified separately against the material's allowable values before the design is considered safe.
- Only vertical loads are considered; lateral loads, torsion, dynamics and vibration are outside the scope of this calculation.
Typical elastic modulus values for common materials:
| Material | Elastic Modulus \(E\) (psi) |
|---|---|
| Structural steel | 29,000,000 |
| Aluminum | 10,000,000 |
| Reinforced concrete | 3,500,000 |
| Wood (Douglas fir) | 1,600,000 |
When a cantilever supports a wall or slab, remember that the supporting structure below must also carry the load — the Concrete Slab Maximum Wall Load calculator can help check that side of the problem.
Frequently Asked Questions
How do I calculate cantilever beam deflection by hand?
Start by converting your total load into the equivalent uniform load using \(P_0 = \frac{P \cdot L}{L - x}\). Then plug the values into the slope formula \(\theta = \frac{P_0 L^3}{6EI}\) and the deflection formula \(\delta = \frac{P_0 x^2 (x^3 + 6L^2 - 4Lx)}{24EI}\). Work through each step carefully — the cube and fourth powers make arithmetic errors easy, so keep full precision until the end and only round your final answer.
What is a normal or acceptable deflection for a cantilever beam?
Common serviceability limits are \(L/240\) of the span for total load and \(L/360\) for live load alone. For a 120-inch cantilever, that means keeping total deflection under about 0.5 inches. Slopes should stay in the range of a few thousandths of a radian. If your result exceeds these limits, the beam is too flexible for the load — a deeper section is usually the most effective fix because stiffness grows with the cube of the depth.
Can I use this calculator with metric units (mm, N, MPa)?
Yes, the formulas are unit-agnostic as long as everything is consistent. Use load in newtons, dimensions in millimeters, elastic modulus in MPa (N/mm²), and moment of inertia in mm⁴. The results come out in radians and millimeters. Never mix systems — for example, entering inches with a modulus in MPa will give meaningless answers. To convert results, divide an inch value by 25.4 to get millimeters, or multiply a millimeter value by 0.03937.
Why does the result explode when the load position is close to the beam length?
The equivalent load formula divides by \((L - x)\), so as \(x\) approaches \(L\), the denominator shrinks toward zero and \(P_0\) grows without bound. This is why the input must satisfy \(x < L\). Values of \(x\) very close to \(L\) produce enormous, physically meaningless results. If you need the deflection near the free end, evaluate at a position slightly inside the span, such as 95–98% of \(L\).
How do I find the moment of inertia for my beam?
For a solid rectangular section, use \(I = \frac{bh^3}{12}\), where \(b\) is the width and \(h\) is the height — always take \(h\) as the dimension in the bending direction. For standard steel shapes (W-beams, channels, tubes), look up the tabulated \(I\) value in the manufacturer's tables or the AISC manual. Using the wrong axis value is a common mistake, so confirm whether the listed value is about the strong (x-x) or weak (y-y) axis.
What are the most common mistakes when calculating cantilever deflection?
Frequent errors include mixing units (inches with psi is correct, but inches with MPa is not), measuring \(x\) from the free end instead of the fixed end, and confusing the total load with the load per unit length. Also remember that the fixed support is assumed perfectly rigid — in real structures, connection flexibility and rotation at the anchor can add significant deflection, so treat calculated values as a best-case estimate and verify stresses separately.
Does the beam's own weight count as a uniform load?
Yes — self-weight is one of the most common uniformly distributed loads on a cantilever. To include it, calculate the weight per unit length (cross-sectional area times material density) and multiply by the span to get the total load, then add it to any applied load before entering the total in the calculator. For concrete and masonry structures this can be substantial. Always check bending and shear strength separately, since deflection alone does not confirm a safe design.