🚀 Beam Deflection Calculator & Simulator
- Select Beam Type: Click the tab or button at the top to specify the structure to analyze—either a Cantilever or a Simply Supported beam.
- Adjust Design Parameters: Move the sliders or enter values to determine the beam length (L), load (P), modulus of elasticity (E), and moment of inertia (I).
- Set Concentrated Load and Location: Define the point (a) where the load acts on the beam to map the deflection according to the load position on the graph in real-time.
- Review Structural Analysis Results: Instantly evaluate cross-sectional safety by checking the real-time calculated support reactions (Reaction Forces), slope at supports (Slope), deflection at the load point, and maximum deflection (Max Deflection).
📚 View Detailed Structural Mechanics Explanation & Beam Deflection Design Theory (Euler-Bernoulli Beam) ▼
1. Overview of Beam Deflection and Engineering Safety Standards
In structural engineering and mechanical design, a beam is the most critical structural member that supports lateral loads. When a load is applied to a beam, bending deformation perpendicular to the axial direction occurs, and the vertical displacement of this deformation is called deflection.
Controlling beam deflection is essential not only for ensuring structural stability but also for preventing cracks in finishes (such as glass windows or drywall installed above the beam) and satisfying the Serviceability Limit State (SLS). Most architectural and mechanical design standards (AISC, KS, etc.) strictly limit the deflection-to-span ratio (L) of beams:
- General Floor Beams: L/360 or less (under live load conditions)
- Roof Beams: L/240 or less
- Cantilever Beams: L/180 or less
2. Euler-Bernoulli Beam Theory and the Differential Equation of Deflection
The Euler-Bernoulli Beam Theory, which assumes that a plane perpendicular to the neutral axis remains plane after deformation, is universally applied to the analysis of beam deflection. The governing differential equation of beam deflection is derived from the relationship between curvature and bending moment as follows.
EI × (d²y / dx²) = M(x)
Here, E is the modulus of elasticity (Young's Modulus), I is the second moment of area (Moment of Inertia), and the product EI is called the flexural rigidity. M(x) is the bending moment function at position x. By integrating this equation twice with respect to x and applying the boundary conditions of the supports, the deflection curve equation y(x) for the entire beam can be completely derived.
3. Key Concentrated Load Deflection Formulas for Simply Supported and Cantilever Beams
The formulas for the two representative boundary conditions most frequently used in design practice are as follows.
① Cantilever Beam (Concentrated Load P at Free End): Deflection equation at distance x from the fixed support
y(x) = (P × x²) / (6EI) × (3L - x)
Accordingly, the maximum deflection formula at the free end (x=L) is simplified as follows:
δ_max = (P × L³) / (3EI) [mm]
② Simply Supported Beam (Concentrated Load P at an Arbitrary Point x=a): Maximum deflection formula when the load acts at the center of the beam (a=L/2)
δ_max = (P × L³) / (48EI) [mm]
As shown in these formulas, the most critical strategies to effectively reduce beam deflection are either exponentially increasing the moment of inertia I by increasing the depth (height) of the beam, or reducing the span length L of the beam.