# Beam Fixed at One End - Concentrated Load at Center

on . Posted in Structural Engineering

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

$$\large{ R_1 = V_1 \;\;=\;\; \frac {5\;P} {16} }$$

$$\large{ R_2 = V_2 max \;\;=\;\; \frac {11\;P} {16} }$$

$$\large{ M_{max} \; \left(at \;fixed \;end \right) \;\;=\;\; \frac {3\;P\;L} {16} }$$

$$\large{ M_1 \; \left(at\; point\; of \;load \right) \;\;=\;\; \frac {5\;P\;L} {32} }$$

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

$$\large{ M_x \; \left( x > \frac {L}{2} \right) \;\;=\;\; P\; \left( \frac { L} {2} - \frac { 11\;x} {16} \right) }$$

$$\large{ \Delta_x \; \left(at\; point\; of\; load \right) \;\;=\;\; \frac { 7\;P\;L^3} {768\; \lambda\; I} }$$

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

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

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

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

## diagrams

• 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. 