Beam Deflection Angle Calculator

Formula used

This tool computes the beam slope angle, also called the deflection angle.

• Cantilever, end point load: θ = P L² / (2 E I)
• Cantilever, uniform load: θ = w L³ / (6 E I)
• Simply supported, midspan point load: θ = P L² / (16 E I)
• Simply supported, uniform load: θ = w L³ / (24 E I)

Angles are small in most service designs. Use radians for calculations.

How to use this calculator

1. Select the load case that matches your support condition.
2. Enter span length and elastic modulus for the material.
3. Provide the load magnitude and choose correct load units.
4. Enter I directly, or compute it from section geometry.
5. Press Calculate to view slope in rad, mrad, and degrees.

Professional article

1) What the deflection angle represents

The deflection angle (slope) is the rotation of the beam’s elastic curve at a point. In construction practice it is used to assess serviceability, finish performance, and compatibility with connections. Small rotations can still crack finishes, misalign cladding, or increase secondary stresses in rigidly connected members.

2) Where the equations come from

The calculator applies classic Euler–Bernoulli beam theory for slender members, linear elasticity, and small deflections. The closed‑form slopes shown correspond to standard boundary conditions and load patterns. Results are most reliable when span-to-depth ratios are large and shear deformation is not dominant.

3) Typical material stiffness values

Elastic modulus strongly controls slope. Structural steel is commonly about 200 GPa, aluminum about 69 GPa, and normal‑weight concrete often 25–35 GPa depending on mix and age. Engineered timber varies widely (often 8–14 GPa parallel to grain). Use project specifications when available.

4) Why section inertia matters

The second moment of area, I, amplifies stiffness with geometry. For rectangles, I = b h³/12, so increasing depth is far more effective than increasing width. Doubling depth increases I by eight, reducing slope by roughly the same factor under identical loading and span.

5) Understanding units and outputs

Internally the tool converts inputs to SI units (N, m, Pa, m⁴) and returns slope in radians, milliradians, and degrees. For most service designs, milliradians are convenient: 1 mrad ≈ 0.0573°. A rotation of 5 mrad is about 0.286°.

6) Interpreting results for construction tolerances

Serviceability checks often reference deflection limits (for example L/360 to L/240 depending on occupancy and finish), but rotation can be the governing driver for brittle finishes or sensitive partitions. Compare computed slope near supports or free ends to connection detailing allowances and expected joint movements.

7) Common modeling cautions

Real beams may have partial fixity, composite action, cracked concrete stiffness, or time-dependent effects such as creep. Uniform loads from floor systems can include superimposed dead load and live load patterns. If rotations approach construction tolerances, consider a refined analysis or verify assumptions with a structural engineer.

8) Recommended workflow on site

Start with conservative loads and a realistic stiffness estimate, then run sensitivity checks by varying E and I within expected ranges. Use the exported CSV/PDF to document inputs, assumptions, and results in design notes. When results change, capture the case to maintain traceability.

FAQs

1) Is slope the same as deflection?

No. Deflection is vertical displacement, while slope is rotation of the deflected shape. They are related through the beam’s curvature and boundary conditions.

2) Which I should I use for a rectangle?

Use the I about the bending axis. For a rectangular section bending about its strong axis, use I = b h³/12 where h is the depth in the bending direction.

3) When should I use radians instead of degrees?

Use radians for calculations and comparisons in beam formulas. Degrees are mainly for interpretation. Small-angle beam relations assume radians.

4) Why are my angles extremely small?

Most beams are designed to be stiff in service. High E, large I, or short spans can make rotations tiny, often below 1 mrad.

5) Can I use this for deep beams?

Be cautious. Deep beams may require shear deformation and strut‑and‑tie methods. This calculator assumes slender-beam behavior and small deflections.

6) How do I include multiple loads?

Superposition applies for linear elastic behavior. Compute slope for each load case separately (same E and I) and add the angles, keeping consistent units.

7) What if my support is not perfectly fixed or pinned?

Partial fixity can change slope significantly. Use the closest case for quick checks, then validate with a structural model or engineering judgment when fixity is uncertain.

Example data table

Example angles are rounded and for illustration.