Uniformly Distributed Load

Problem
A 75 mm × 150 mm beam carries a uniform load w_o over the entire span of 1.2 m. Square notches 25 mm deep are provided at the bottom of the beam at the supports. Calculate the safe value of w_o based on shear alone.

Influence line is the graphical representation of the response function of the structure as the downward unit load moves across the structure. The ordinate of the influence line show the magnitude and character of the function.
 

The most common response functions of our interest are support reaction, shear at a section, bending moment at a section, and force in truss member.
 

With the aid of influence diagram, we can...

1. determine the position of the load to cause maximum response in the function.
2. calculate the maximum value of the function.

Value of the function for any type of load
 

$\displaystyle \text{Function} = \int_{x_1}^{x_2} y_i (y \, dx)$
 

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

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

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

Problem 850
Determine the moment over the supports for the beam loaded as shown in Fig. P-850.
 

Problem 834
Refer to Problem 826.
 

Problem 832
Refer to Problem 824.
 

Problem 829
A uniform load is carried over three equal spans as shown in Fig. P-829.
 

Problem 828 - Reactions of Continuous Beam
A continuous beam carries a uniform load over two equal spans as shown in Fig. P-828.