Simple Beam - Load Increasing Uniformly to Center

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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.

 

sb 3D

Simple Beam - Load Increasing Uniformly to Center formulas

\( R = V_{max} \;=\; W\;/\;2 \)

\( V_x  \;  [ \; x < (L\;/\;2) \;]  \;=\; (W\;/\;2\;L^2)  \; ( L^2 - 4\;x^2 ) \)

\( M_{max}  \; (at \;center) \;=\; W \;L\;/\;6  \)

\( M_x \; [\;  x < (L\;/\;2) \;]   \;=\; W\;x  \; [\; (1/2) - (2\;x^2\;/\;3\;L^2) \;]  \)

\( \Delta_{max} \; (at \;center) \;=\; W \;L^3 \;/\; 60 \; \lambda \;I \)

\( \Delta_x \; [\;  x < (L\;/\;2) \;]  \;=\; (W\; x \;/\;480\; \lambda \;I \;L^2)  \; (  5\;L^2 - 4\;x^2 )^2   \)

S B - Load Increasing Unif to Center - Solve for R

\(\large{ R = \frac{ \frac{w\;L}{2} }{2}  }\)      

load per unit length, w
span length, L

S B - Load Increasing Unif to Center - Solve for Vx

\(\large{ V_x  =  \frac{ \frac{w\;L}{2} }{2\;L^2}  \; \left( L^2 - 4\;x^2    \right) }\)

load per unit length, w
span length, L
dist from reaction, x

S B - Load Increasing Unif to Center - Solve for Mmax

\(\large{ M_{max}  =  \frac{ \frac{w\;L}{2} \;L}{6}  }\)

load per unit length, w
span length, L

S B - Load Increasing Unif to Center - Solve for Mx

\(\large{ M_x  =  \frac{w\;L}{2} \; x  \; \left(  \frac{1}{2} - \frac {2\;x^2}{3\;L^2}  \right)  }\)

load per unit length, w
span length, L
dist from reaction, x

S B - Load Increasing Unif to Center - Solve for Δmax

\(\large{ \Delta_{max}  = \frac{ \frac{w\;L}{2} \;L^3} {60 \; \lambda \;I }  }\)

load per unit length, w
span length, L
modulus of elasticity, λ
second moment of area, I

S B - Load Increasing Unif to Center - Solve for Δx

\(\large{ \Delta_x  =  \frac{ \frac{w\;L}{2} \; x}{480\; \lambda \;I \;L^2}  \; \left(  5\;L^2 - 4\;x^2  \right)^2     }\)

w (load per unit length, w)
L (span length, L)
x (dist from reaction, x)
lambda (modulus of elasticity, λ)
I (second moment of area, I)

Symbol English Metric
\( R \) = reaction load at bearing point \(lbf\) \(N\)
\( V \) = maximum shear force \(lbf\) \(N\)
\( M \) = maximum bending moment \(lbf-in\) \(N-mm\)
\( \Delta \) = deflection or deformation \(in\) \(mm\)
\( W \) = total load or \( w\;L\;/\;2 \) \(lbf\) \(N\)
\( w \) = highest load per unit length of UIL \(lbf\;/\;in\) \(N\;/\;m\)
\( L \) = span length of the bending member \(in\) \(mm\)
\( x \) = horizontal distance from reaction to point on beam \(in\) \(mm\)
\( \lambda  \)   (Greek symbol lambda) = modulus of elasticity \(lbf\;/\;in^2\) \(Pa\)
\( I \) = second moment of area (moment of inertia) \(in^4\) \(mm^4\)

 

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Tags: Beam Support