What is the formula for maximum deflection of a simply supported beam with a uniform distributed load?

For a simply supported beam of span L with uniform distributed load q, the maximum deflection occurs at mid-span and is given by δ_max = 5qL⁴ / (384EI), where E is Young's modulus and I is the second moment of area. For example, with L=2 m, q=10 kN/m, E=200 GPa, and I=1×10⁻⁵ m⁴, δ_max = 5×10000×16 / (384×200×10⁹×10⁻⁵) ≈ 0.26 mm.

How is the tip deflection of a cantilever beam with a point load calculated?

For a cantilever beam of length L with a tip point load P, the maximum tip deflection is δ_tip = PL³ / (3EI). The deflection curve is w(x) = Px²(3L − x) / (6EI). This is 16 times larger than the mid-span deflection of a simply supported beam under the same span and load: PL³/(48EI).

How can I increase the flexural rigidity EI of a beam?

Flexural rigidity EI is the product of Young's modulus E and the second moment of area I. To increase EI: (1) choose a stiffer material (steel E≈200 GPa vs aluminium E≈70 GPa); (2) increase the section depth—I scales with depth cubed: I = bh³/12 for a rectangular section; (3) use efficient cross-sections such as I-beams. Doubling the section depth increases I by a factor of 8 and reduces deflection to 1/8.

What is the relationship between the shear force diagram and bending moment diagram?

Shear force V(x) and bending moment M(x) are related by the differential equation dM/dx = V(x). The bending moment is the integral of the shear force, so the maximum moment occurs where V=0. For a simply supported beam with UDL, the reaction shear is qL/2 at each support and the maximum moment qL²/8 occurs at mid-span where V=0.

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Structural Analysis Tool

Beam Deflection & Stress Analysis Tool

Calculate and visualize deflection, bending moment, and shear force in real time for simply supported and cantilever beams under point loads and uniform distributed loads.

$$EI\frac{d^4w}{dx^4} = q(x)$$

Parameter Settings

Beam Type

Young's Modulus E 200 GPa

Material stiffness (Steel: 200 GPa, Al: 70 GPa)

Second Moment of Area I 1.0e-5 m⁴

Bending stiffness from cross-section geometry

Beam Length L 2.0 m

Load Intensity q 10.0 kN/m

Uniform distributed load (kN/m)

EI (Bending Stiffness)
—

Max Deflection δ_max

—

Max Moment M_max

—

kN·m

Max Shear V_max

—

Bending Stiffness EI

—

N·m²

Deflection Curve w(x)

Bending Moment M(x)

Shear Force V(x)

Analytical Solutions — Theoretical Formulas

Simply Supported Beam + UDL

$$w(x)=\frac{qx(L^3-2Lx^2+x^3)}{24EI}$$

$$M(x)=\frac{qx(L-x)}{2},\quad V(x)=q\!\left(\frac{L}{2}-x\right)$$

Simply Supported Beam + Center Point Load

$$w(x)=\frac{Px(3L^2-4x^2)}{48EI}\quad\left(x\le\frac{L}{2}\right)$$

$$\delta_{max}=\frac{PL^3}{48EI}$$

Cantilever Beam + UDL

$$w(x)=\frac{qx^2(6L^2-4Lx+x^2)}{24EI}$$

$$\delta_{tip}=\frac{qL^4}{8EI}$$

Cantilever Beam + Tip Point Load

$$w(x)=\frac{Px^2(3L-x)}{6EI}$$

$$\delta_{tip}=\frac{PL^3}{3EI}$$