Simple Beam - Concentrated Load at 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 7D 1

Simple Beam - Concentrated Load at Center formulas

\( R \;=\; V \;=\; P\;/\;2  \)

\( M_{max} \; (at\; point \;of\; load ) \;=\; P\;L\;/\;4  \)

\( M_x   \; [\; x < (L\;/\;2) \;] \;=\; P\;x\;/\;2  \)

\( \Delta_{max} \; (at \;point\; of\; load ) \;=\; P\;L^3\;/\;48\; \lambda\; I  \)

\( \Delta_x  \; (  x < \frac{L}{2} )  \;=\; ( P\;x\;/\;48 \;\lambda\; I ) \; (  3\;L^2 - 4\;x^2 )  \)

S B Concentrated Load at Center - Solve for R

\(\large{ R = V =  \frac{P}{2}  }\)

total concentrated load, P

S B Concentrated Load at Center - Solve for Mmax

\(\large{ M_{max} \; \left(at\; point \;of\; load \right) = \frac{P\;L}{4}  }\)

total concentrated load, P
span length, L

S B Concentrated Load at Center - Solve for Mx

\(\large{ M_x   \;  \left(  x < \frac{L}{2} \right)   =  \frac{ P\;x}{2}    }\)

total concentrated load, P
distance from reaction, x

S B Concentrated Load at Center - Solve for Δmax

\(\large{ \Delta_{max} \; \left(at \;point\; of\; load \right) =  \frac{ P\;L^3}{48\; \lambda\; I}  }\)

total concentrated load, P
span length, L
modulus of elasticity, λ
second moment of area, I

S B Concentrated Load at Center - Solve for Δx

\(\large{ \Delta_x   \;  \left(  x < \frac{L}{2}   \right)  =  \frac{P\;x}{48 \;\lambda\; I}  \; \left(  3\;L^2 - 4\;x^2  \right)     }\)

total concentrated load, P
distance from reaction, x
modulus of elasticity, λ
second moment of area, I
span length, L

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 \(in\) \(mm\)
\( P \) = total concentrated load \(lbf\) \(N\)
\( L \) = 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