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Three Span Continuous Beam - Equal Spans, Uniformly Distributed Load

on . Posted in Structural Engineering

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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.

 

Three Span Continuous Beam - Equal Spans, Uniformly Distributed Load formulas

\(\large{ R_1 = V_1 = R_4 = V_4    \;\;=\;\; 0.400\;w\;L    }\) 

\(\large{ R_2 = R_3   \;\;=\;\;  1.100\;w\;L    }\) 

\(\large{ V_{2_1} =  V_{3_2}    \;\;=\;\; 0.500\;w\;L    }\) 

\(\large{ V_{2_2} =  V_{3_1}    \;\;=\;\; 0.600\;w\;L    }\)

\(\large{ M_1 =  M_5 \;  \left(at\; 0.400\;L \; from  \; R_1  \;or\; R_4 \right)    \;\;=\;\; 0.080\;w\;L^2    }\)

\(\large{ M_2 =  M_4 \;  \left(at\; R_2  \;or\;  R_3 \right)    \;\;=\;\; 0.100\;w\;L^2    }\)

\(\large{ M_3  \;  \left(at \;mid \;center \;span \right)  \;\;=\;\; 0.025\;w\;L^2    }\)

\(\large{ \Delta_{max}  \; \left(at\; 0.446\;L \; from  \; R_1  \;or\; R_4 \right)    \;\;=\;\;  \frac{0.0069\;w\;L^4}{\lambda\; I}    }\)

3 S C B - E S, Unif Dist Load - Solve for R1

\(\large{ R_1 =  0.400 \; w \; L   }\) 

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for R2

\(\large{ R_2  = 1.100\;w\;L    }\) 

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for V21

\(\large{ V_{2_1} =   0.500\;w\;L    }\) 

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for V22

\(\large{ V_{2_2} =  0.600\;w\;L    }\)

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for M1

\(\large{ M_1 =   0.080\;w\;L^2    }\)

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for M2

\(\large{ M_2 =  0.100\;w\;L^2    }\)

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for M3

\(\large{ M_3  = 0.025\;w\;L^2    }\)

load per unit length, w
span length under consideration, L

3 S C B - E S, Unif Dist Load - Solve for Δmax

\(\large{ \Delta_{max}   =  \frac{0.0069\;w\;L^4}{\lambda\; I}    }\)

w (load per unit length, w)
L (span length under consideration, L)
lambda (modulus of elasticity, λ)
I (second moment of area, I)

Symbol English Metric
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ V }\) = maximum shear force \(\large{lbf}\) \(\large{N}\)
\(\large{ M }\) = maximum bending moment \(\large{lbf-ft}\) \(\large{N-m}\)
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ w }\) = load per unit length \(\large{\frac{lbf}{in}}\) \(\large{\frac{N}{m}}\)
\(\large{ L }\) = span length under consideration \(\large{in}\) \(\large{mm}\)
\(\large{ \lambda  }\)   (Greek symbol lambda) = modulus of elasticity \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ I }\) = second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)

 

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Tags: Beam Support Equations

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