# Beam Fixed at One End - Concentrated Load at Any Point

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 any point formulas

$$\large{ R_1 = V_1 \;\;=\;\; \frac {P\;b^2} {2\;L^3} \; \left( a + 2\;L \right) }$$

$$\large{ R_2 = V_2 \;\;=\;\; \frac {P\;a} {2\;L^3} \; \left( 3\;L^2 - a^2 \right) }$$

$$\large{ M_1 \; \left(at\; point\; of \;load \right) \;\;=\;\; R_1 \;a }$$

$$\large{ M_2 \; \left(at\; fixed \;end \right) \;\;=\;\; \frac {P\;a\;b} {2\;L^2} \; \left( a +L \right) }$$

$$\large{ M_x \; \left( x < a \right) \;\;=\;\; R_1\; x }$$

$$\large{ M_x \; \left( x > a \right) \;\;=\;\; R_1 \;x - P\; \left( x - a \right) }$$

$$\large{ \Delta_{max} \; \left( at \;x = L \; \frac{ L^2\; +\; a^2 }{ 3\;L^2 \;-\; a^2 } \; when\; a < 0.414 \;L \right) \;\;=\;\; \frac {P\;a} {3\; \lambda\; I }\; \frac { \left( L^2 \;-\; a^2 \right) ^3 } { \left( 3\;L^2 \;- \;a^2 \right) ^2 } }$$

$$\large{ \Delta_{max} \; \left( at \;x = L \;\sqrt{ \frac{ a }{ 2\;L \;+\; a } } \; when\; a > 0.414 \;L \right) \;\;=\;\; \frac {P\;a\;b^2} {6\; \lambda\; I } \; \sqrt{ \frac { a } { 2\;L \;+ \;a } } }$$

$$\large{ \Delta_a \; \left(at\; point\; of\; load \right) \;\;=\;\; \frac { P\;a^3 \;b^2} {12\; \lambda\; I \;L^3}\; \left( 3\;L + b \right) }$$

$$\large{ \Delta_x \; \left( x < a \right) \;\;=\;\; \frac { P\;b^2\; x} {12 \;\lambda\; I \;L^3}\; \left( 3\;a\;L^2 - 2\;L\;x^2 - a\;x^2 \right) }$$

$$\large{ \Delta_x \; \left( x > a \right) \;\;=\;\; \frac { P\;a} {12\; \lambda\; I \;L^3} \; \left( L - x \right)^2 \; \left( 3\;L^2 \;x - a^2 \;x - 2\;a^2 \;L \right) }$$

Symbol English Metric
$$\large{ \Delta }$$ = deflection or deformation $$\large{in}$$ $$\large{m}$$
$$\large{ x }$$ = horizontal distance from reaction to point on beam $$\large{in}$$ $$\large{m}$$
$$\large{ a, b }$$ = length to point load $$\large{in}$$ $$\large{m}$$
$$\large{ M }$$ = 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 $$\large{in}$$ $$\large{m}$$
$$\large{ P }$$ = total concentrated load $$\large{lbf}$$ $$\large{N}$$

Tags: Beam Support