# Simple Beam - Concentrated Load at Any Point

on . Posted in Structural Engineering

## Simple Beam - Concentrated Load at Any Point formulas

$$\large{ R_1 = V_1 \; \left( max.\; when \;\; a < b \right) \;\;=\;\; \frac{P\;b}{L} }$$

$$\large{ R_2 = V_2 \; \left( max.\; when \;\; a > b \right) \;\;=\;\; \frac{ P\;a}{L} }$$

$$\large{ M_{max} \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac{ P\;a\;b }{ L } }$$

$$\large{ M_x \; \left( x < a \right) \;\;=\;\; \frac{ P\;b\;x }{ L } }$$

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

$$\large{ \Delta_x \; \left( x < a \right) \;\;=\;\; \frac{ P\;b\;x }{ 6\; \lambda \;I \;L } \; \left( L^2 - b^2 - x^2 \right) }$$

$$\large{ \Delta_{max} \; \left(at \; x = \sqrt{ \frac{ a\; \left( a \;+\; 2\;b \right) }{3} } \; when \; a > b \right) \;\;=\;\; \frac{ P\;a\;b \; \left( a \;+\; 2\;b \right) \; \sqrt{ 3\;a \; \left( a \;+\; 2\;b \right) } } { 27\; \lambda \;I \;L } }$$

Symbol English Metric
$$\large{ \Delta }$$ = deflection or deformation $$\large{in}$$ $$\large{mm}$$
$$\large{ a, b }$$ = distance to point load $$\large{in}$$ $$\large{mm}$$
$$\large{ x }$$ = 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.