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

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.

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

$$R_1 \;=\; V_1 \;=\; R_4 \;=\; V_4 \;=\; 0.400\;w\;L$$

$$R_2 \;=\; R_3 \;=\; 1.100\;w\;L$$

$$V_{2_1} \;=\; V_{3_2} \;=\; 0.500\;w\;L$$

$$V_{2_2} \;=\; V_{3_1} \;=\; 0.600\;w\;L$$

$$M_1 \;=\; M_5 \; (at\; 0.400\;L \; from \; R_1 \;or\; R_4 ) \;=\; 0.080\;w\;L^2$$

$$M_2 \;=\; M_4 \; (at\; R_2 \;or\; R_3 ) \;=\; 0.100\;w\;L^2$$

$$M_3 \; (at \;mid \;center \;span ) \;=\; 0.025\;w\;L^2$$

$$\Delta_{max} \; (at\; 0.446\;L \; from \; R_1 \;or\; R_4 ) \;=\; (0.0069\;w\;L^4) \;/\; (\lambda\; I)$$

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

Tags: Beam Support