Simple Beam - Load Increasing Uniformly to One End

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.

 

sb 2D

Simple Beam - Load Increasing Uniformly to One End formulas

\( R_1 \;=\ V_1 \;=\; W \;/\;3\)

\( R_2 \;=\ V_2 \;=\; 2\;W \;/\;3\)

\( V_x \;=\ (W\;/\;3)  - (W\;x^2\;/\;L^2)\)

\( M_{max} \; (at \;  x = L\;/\; \sqrt{3}\; ) \;=\; 2\;W\;L \;/\; 9\; \sqrt{3}   \)

\( M_x \;=\; (W \;x\;/\;3\;L^2)  \; ( L^2  - x^2 )   \)

\( \Delta_{max} \; ( \;at \; x = L\; \sqrt{1 - ( 8/15 )^{\frac{1}{2} } } \;)  \;=\; 0.01304 \; ( W \;L^3\;/\; \lambda \;I ) \)

\( \Delta_x \;=\; (W \;x\;/\; 180\; \lambda \;I \;L^2 ) \; ( 3\;x^4 - 10\;L^2\;x^2 + 7\;L^4 )  \)

S B Load Increasing Unif to One End - Solve for R1

\(\large{ R_1 = \frac{ \frac{w\;L}{2} }{3}  }\)

load per unit length, w
span length, L

S B Load Increasing Unif to One End - Solve for R2

\(\large{ R_2 = \frac{2\; \frac{w\;L}{2} }{3}  }\)

load per unit length, w
span length, L

S B Load Increasing Unif to One End - Solve for Vx

\(\large{ V_x =  \frac{ \frac{w\;L}{2} }{3}  - \frac{ \frac{w\;L}{2} \;x^2}{L^2} }\)

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

S B Load Increasing Unif to One End - Solve for Mmax

\(\large{ M_{max} = \frac{ 2\; \frac{w\;L}{2} \;L }{ 9\; \sqrt{3} }  }\)

load per unit length, w
span length, L

S B Load Increasing Unif to One End - Solve for Mx

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

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

S B Load Increasing Unif to One End - Solve for Δmax

\(\large{ \Delta_{max}  =  0.01304 \; \frac{ \frac{w\;L}{2} \;L^3}{ \lambda \;I}  }\)

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

S B Load Increasing Unif to One End - Solve for Δx

\(\large{ \Delta_x = \frac{ \frac{w\;L}{2} \;x}{ 180\; \lambda \;I \;L^2  } \; \left( 3\;x^4 - 10\;L^2\;x^2 + 7\;L^4  \right)  }\)

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