Beam Deflection Calculator
Compute elastic deflection profiles v(x), slope distributions θ(x), and peak displacement coordinates using Euler-Bernoulli double integration and Macaulay singularity brackets.
EI · v(x) = ∬ M(x) dx dx + C₁x + C₂
Beam Master handles any combination of point loads, uniform distributed loads (UDL), varying loads (UVL), and applied moments with instant SFD, BMD, and deflection curves.
How Deflection is Derived
Euler-Bernoulli beam theory governs the relationship between transverse loading $q(x)$, shear force $V(x)$, bending moment $M(x)$, slope $\theta(x)$, and vertical deflection $v(x)$:
1. Load: q(x)
2. Shear: V(x) = -∫ q(x) dx + C_v
3. Moment: M(x) = ∫ V(x) dx + C_m
4. Slope: EI·θ(x) = ∫ M(x) dx + C₁
5. Deflection: EI·v(x) = ∫ EI·θ(x) dx + C₂
Engineering Formulas & Governing Equations
Euler-Bernoulli Beam Equation
Equation 01The curvature of the elastic beam is directly proportional to internal bending moment and inversely proportional to flexural rigidity EI.
Slope Equation (First Integration)
Equation 02Integrating bending moment yields the rotational slope equation with constant of integration C₁.
Deflection Equation (Second Integration)
Equation 03Second integration yields vertical displacement v(x) with boundary constants C₁ and C₂.
Macaulay Bracket Singularity Rule
Equation 04Enables expressing discontinuous point and distributed loads across the entire span in a single continuous equation.
Calculation Details & Clarifications
Solve in Beam Master
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