# Beam Fixed at One End - Concentrated Load at Center

on . Posted in Structural Engineering

### diagram Symbols

• Bending moment diagram (BMD)  -  Used to determine the bending moment at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
• Free body diagram (FBD)  -  Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
• Shear force diagram (SFD)  -  Used to determine the shear force at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
• Uniformly distributed load (UDL)  -  A load that is distributed evenly across the entire length of the support area.

### Beam Fixed at One End - Concentrated Load at Center formulas

$$R_1 \;=\; V_1 \;=\; 5\;P\;/\;16$$

$$R_2 \;=\; V_2 max \;=\; 11\;P\;/\;16$$

$$M_{max} \; (at \;fixed \;end ) \;=\; 3\;P\;L\;/\;16$$

$$M_1 \; (at\; point\; of \;load ) \;=\; 5\;P\;L\;/\;32$$

$$M_x \; ( x < \frac {L}{2} ) \;=\; 5\;P\;x\;/\;16$$

$$M_x \; ( x > \frac {L}{2} ) \;=\; P\; [\; ( L\;/\;2) - (11\;x\;/\;16) \;]$$

$$\Delta_x \; (at\; point\; of\; load ) \;=\; 7\;P\;L^3\;/\;768\; \lambda\; I$$

$$\Delta_x \; ( x < \frac {L}{2} ) \;=\; ( P\;x\;/\;96 \;\lambda\; I) \; ( 3\;L^2 - 5\;x^2 )$$

$$\Delta_x \; ( x > \frac {L}{2} ) \;=\; ( P\;/\;96 \;\lambda\; I ) \; ( x - L )^2 \; ( 11\;x - 2\;L )$$

$$\Delta_{max} \; ( at \; x = L \; \left( \frac{1}{5} \right)^{\frac{1}{2}} ) \;=\; P\;L^3\;/\;48\; \lambda\; I \; \left( 5 \right)^{\frac{1}{2}}$$

Symbol English Metric
$$\Delta$$ = deflection or deformation $$in$$ $$mm$$
$$x$$ = horizontal distance from reaction to point on beam $$in$$ $$mm$$
$$M$$ = maximum bending moment $$lbf-in$$ $$N-mm$$
$$V$$ = maximum shear force $$lbf$$ $$N$$
$$\lambda$$   (Greek symbol lambda) = modulus of elasticity $$lbf\;/\;in^2$$ $$Pa$$
$$R$$ = reaction load at bearing point $$lbf$$ $$N$$
$$I$$ = second moment of area (moment of inertia) $$in^4$$ $$mm^4$$
$$L$$ = span length of the bending member $$in$$ $$mm$$
$$P$$ = total concentrated load $$lbf$$ $$N$$

Tags: Beam Support