# Simple Beam - Central Point Load and Variable End Moments

on . Posted in Structural Engineering

## Simple Beam - Central Point Load and Variable End Moments formulas

$$\large{ R_1 = V_1 \;\;=\;\; \frac { P } { 2 } + \frac { M_1 \;- \;M_2 } { L } }$$

$$\large{ R_2 = V_2 \;\;=\;\; \frac { P } { 2 } - \frac { M_1 \;-\; M_2 } { L } }$$

$$\large{ M_3 \; \left(at\; center \right) \;\;=\;\; \frac { P\;L } { 4 } - \frac { M_1 \;+\; M_2 } { L } }$$

$$\large{ M_x \left( x < \frac{L} {2} \right) \;\;=\;\; \left( \frac { P } { 2 } + \frac { M_1\; -\; M_2 } { L } \right) x - M_1 }$$

$$\large{ M_x \left( > \frac{L} {2} \right) \;\;=\;\; \frac {P}{2} \; \left( L - x \right) + \frac { \left( M_1 \;-\; M_2 \right) \;x } { L } \; -\; M_1 }$$

$$\large{ \Delta_x \left( x < \frac{L} {2} \right) \;\;=\;\; \frac { P\;x } { 48\; \lambda\; I } \; \left[ 3\;L^2 - 4\;x^2 - \frac { 8\; \left( L\; -\; x \right) } { P\;L } \; \left[ M_1 \left( 2\;L - x \right) + M_2\; \left( L + x \right) \right] \right] }$$

$$\large{ x \; \left(first \;point \;of \;contraflexure \right) \;\;=\;\; \frac { 2\;L\; M_1 } { L\;P \;+ \;2\;M_1\; - \;2\;M_2 } }$$

$$\large{ x \; \left(second\; point \;of\; contraflexure \right) \;\;=\;\; \frac { L \; \left( L\;P\; - \;2\;M_1 \right) } { L\;P \;- \;2\;M_1 \;+\; 2\;M_2 } }$$

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. 