Simply supported beam under point load

Formulas

Maximum deflection

if $\alpha < L / 2$

$$\delta_{\mathrm{max}} = \frac{P\ \alpha}{27\ E\ I\ L}\ \sqrt{3\ (L^2 - \alpha^2)^3}$$

otherwise

$$\delta_{\mathrm{max}} = \frac{P\ (L - \alpha)}{27\ E\ I\ L}\ \sqrt{3\ \bigr[\alpha\ (2\ L - \alpha) \bigr]^3}$$

Minimum bending moment

$$M_{\mathrm{min}} = - \frac{P\ \alpha\ (L - \alpha)}{L}$$

Maximum shear force

$$V_{\mathrm{max}} = P\ (1 - \frac{\alpha}{L})$$

Minimum shear force

$$V_{\mathrm{min}} = - \frac{P\ \alpha}{L}$$

Notations

• $E$modulus of elasticity, in GPa
• $I$second moment of area of the cross-section, in cm⁴
• $L$effective span, in cm
• $\alpha$position of the point load, in cm
• $P$point load, in kN