- See Article - Beam Design Formulas
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.
Simple Beam - Uniform Load Partially Distributed at Each End formulas |
||
|
\( R_1 = V_1 \;=\; (\;w_1 \cdot a \cdot ( 2 \cdot L - a ) \;) + \dfrac{ w_2 \cdot c^2 }{ 2 \cdot L } \) \( R_2 = V_2 \;=\; (\;w_2 \cdot c \cdot ( 2 \cdot L - c ) \;) + \dfrac{ w_1 \cdot a^2 }{ 2 \cdot L } \) \( V_x \; ( x < a ) \;=\; R_1 - w_1 \cdot x \) \( V_x \; [ \; a < x < ( a + b) \;] \;=\; R_1 - w_1 \cdot a \) \( V_x \; [ \; x > ( a + b ) \;] \;=\; R_2 - (\; w_2 \cdot ( 1 - x ) \;) \) \( M_{max} \; [\; at \; x = (R_1\;/\;w_1) \; when \; R_1 < w_1 \;a \;] \;=\; \dfrac{ R_{1}{^2} }{ 2\cdot w_1 } \) \( M_{max} \; [\; at \; x = L - (R_2\;/\;w_2) \; when \; R_2 < w_2 \;c \;] \;=\; \dfrac{ R_{2}{^2} }{ 2 \cdot w_2 }\) \( M_x \; ( w < a ) \;=\; (R_1 \cdot x) - \dfrac{ w_1 \cdot x^2 }{ 2 } \) \( M_x \; [\; a < x < ( a + b ) \;] \;=\; (R_1 \cdot x) - \left( \dfrac{ w_1 \cdot a}{ 2 } \cdot ( 2 \cdot x - a ) \right) \) \( M_x \; [\; x > ( a + b ) \;] \;=\; (\; R_2 \cdot ( L - x ) \;) - \left( \dfrac{ w_2 \cdot ( L - x )^2 }{ 2 } \right) \) |
||
| Symbol | English | Metric |
| \( R \) = reaction load at bearing point | \(lbf\) | \(N\) |
| \( V \) = maximum shear force | \(lbf\) | \(N\) |
| \( M \) = maximum bending moment | \(lbf - in\) | \(N - mm\) |
| \( w_1, w_2 \) = load per unit length | \(lbf\;/\;in\) | \(N\;/\;m\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( a, b, c \) = width and seperation of UDL | \(in\) | \(mm\) |
