Beam Deflection Calculator
For a simply supported beam with a central point load \(P\), the maximum deflection occurs at the center (\(x = L/2\)):
$$ \delta_{max} = \frac{P \cdot L^3}{48 \cdot E \cdot I} \quad | \quad y(x) = \frac{Px}{48EI}(3L^2 – 4x^2) \text{ for } 0 \le x \le L/2 $$
* Note: \(E\) is stiffness, \(I\) is section shape resistance, and \(L\) has a cubic effect on deflection.
1. Structural Computation
2. Holographic Elastic Curve Viewport
Visualization: The curved line represents the exaggerated elastic curve of the beam under load.
Max Deflection (\(\delta\))
0.00 mm
Reaction (\(R\))
0.00 kN
3. Deflection Profile y(x)
Beam Deflection Calculator
Structural Rigidity: Elastic Curve & Compliance Solver V4.0
Quick Answer
Beam deflection ($\delta$) is the vertical displacement of a structural member under load. It is governed by the Euler-Bernoulli equation: $EI \frac{d^2y}{dx^2} = M(x)$. In 2026 engineering, satisfying Span-to-Deflection ratios (e.g., $L/360$ for live loads) is critical for structural serviceability. Our V4.0 engine calculates Elastic Curves, Slope ($\theta$), and integrates Timoshenko shear correction for deep beams.
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By Prof. David Anderson
Structural Metrology & Precision Mechanics Lab
"Strength prevents collapse, but stiffness ensures function. A floor beam may be strong enough not to break, but if its deflection exceeds the L/360 limit, your tiles will crack and the occupants will feel the sag. In the V4.0 lab, we go beyond simple formulas to solve the entire elastic curve, ensuring sub-millimeter precision for even the most complex load arrays."
Deformation Navigation
1. Euler-Bernoulli: The Governing Equation
The deflection of a beam is found by solving the second-order differential equation linking the bending moment ($M$) to the curvature of the beam. By integrating twice, we derive the slope ($\theta$) and the vertical displacement ($y$).
y(x) = ∬ [ M(x) / EI ] dx dx
The double integration method for deriving the elastic curve.
2. Flexural Rigidity: The EI Factor
The product of the material's Young's Modulus ($E$) and the section's Moment of Inertia ($I$) is known as Flexural Rigidity. This value determines the beam's resistance to bending. Higher $EI$ results in lower deflection for the same load.
3. Standard Deflection Formulas
For common loading scenarios, we provide closed-form analytical solutions. These allow for instant verification of maximum displacement at mid-span or cantilever ends.
δ_max = (5wL⁴) / (384EI)
The mid-span deflection for a simply supported beam under UDL.
4. Timoshenko Correction: Deep Beam Shear
For "deep beams" where the span-to-depth ratio is less than 10, bending theory alone underestimates the total sag. V4.0 integrates the shear deformation component, providing a Timoshenko-calibrated result for high-load, short-span members.
5. Compliance: L/360 & L/240 Standards
Modern building codes (IBC 2024 / Eurocode) dictate maximum sag ratios. Our calculator automatically flags "Fail" if the deflection exceeds these serviceability limits, such as $L/360$ for live loads to prevent ceiling cracks.
6. Optimization: Inertia vs. Height
If your beam deflects too much, increasing the height ($h$) of the beam is exponentially more effective than increasing its width ($b$), because the Moment of Inertia ($I$) scales with $h^3$. V4.0's Optimizer calculates the minimum height needed to meet compliance.
🧪 The Stiffness Strategy
Doubling the width of a rectangular beam doubles its stiffness. However, doubling its height increases its stiffness by 8 times. In material-limited designs, vertical geometry is your best tool for deflection control.
7. Elastic Curve FAQs
🚨 The "Creep Underestimate" Trap
Traditional calculators only output "Instantaneous Deflection." For timber or concrete, the deflection will double over time due to Creep. Always factor in long-term coefficients when designing permanent structures.
8. Deflection Engineering Key Takeaways
- 📏 L/Ratio Check: Always compare results against L/360 or L/240 for code compliance.
- 📐 Geometry Dominates: Beam height is the most critical variable for stiffness.
- 🌊 Shear Matters: Use Timoshenko correction for beams with low span-to-depth ratios.
- 🛠️ Boundary Logic: Fixed supports reduce mid-span deflection by 80% vs. simple supports.
Analyze Your Sag
Solve for elastic curves, L/ratio compliance, and stiffness optimization. V4.0 Deformation Lab is active.
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