Fixed-fixed beam under point load

Formulas

Maximum deflection

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

Maximum bending moment

if $\alpha < L / 2$

$$M_{\mathrm{max}} = \frac{P\ \alpha\ (L - \alpha)^2}{L^2}$$

otherwise

$$M_{\mathrm{max}} = \frac{P\ \alpha^2\ (L - \alpha)}{L^2}$$

Minimum bending moment

$$M_{\mathrm{min}} = - \frac{2\ P\ \alpha^2\ (L - \alpha)^2}{L^3}$$

Maximum shear force

$$V_{\mathrm{max}} = \frac{P\ (L - \alpha)^2\ (L + 2\ \alpha)}{L^3}$$

Minimum shear force

$$V_{\mathrm{min}} = - \frac{P\ \alpha^2\ (3\ L - 2\ \alpha)}{L^3}$$

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