Continuous Beams

Problem 877
By means of moment-distribution method, solve the moment at R₂ and R₃ of the continuous beam shown in Fig. P-815.
 

Problem 871
The continuous beam in Figure P-871 is supported at its left end by a spring whose constant is 300 lb/in. For the beam, E = 1.5 × 10⁶ psi and I = 115.2 in.⁴. Compute the load on the spring and its deflection.
 

Problem 869
Find the value of EIδ at the center of the first span of the continuous beam in Figure P-869 if it is known that M₂ = -980 lb·ft and M₃ = -1082 lb·ft.
 

Problem 448
A beam carrying the loads shown in Figure P-448 is composed of three segments. It is supported by four vertical reactions and joined by two frictionless hinges. Determine the values of the reactions.
 

Moment distribution is based on the method of successive approximation developed by Hardy Cross (1885–1959) in his stay at the University of Illinois at Urbana-Champaign (UIUC). This method is applicable to all types of rigid frame analysis.
 

Problem 855
If the distributed load in Prob. 854 is replaced by a concentrated load P at midspan, determine the moments over the supports.
 

Answers:
$M_1 = \dfrac{PL}{8} \cdot \dfrac{1}{\alpha + 2} = M_4$

$M_2 = -\dfrac{PL}{8} \cdot \dfrac{2}{\alpha + 2} = M_3$
 

Problem 854
Solve for the moment over the supports in the beam loaded as shown in Fig. P-854.
 

Answers:
$M_1 = \dfrac{w_o L^2}{12} \cdot \dfrac{1}{\alpha + 2} = M_4$

$M_2 = -\dfrac{w_o L^2}{12} \cdot \dfrac{2}{\alpha + 2} = M_3$
 

Problem 853
For the continuous beam shown in Fig. P-853, determine the moment over the supports. Also draw the shear diagram and compute the maximum positive bending moment. (Hint: Take advantage of symmetry.)
 

Problem 852
Find the moments over the supports for the continuous beam in Figure P-852. Use the results of Problems 850 and 851.
 

Answers
$M_1 = -146.43 ~ \text{N}\cdot\text{m}$

$M_2 = -307.14 ~ \text{N}\cdot\text{m}$

$M_3 = -521.43 ~ \text{N}\cdot\text{m}$