The beam A B is subjected to the uniformly distributed load shown and to two unknown forces P an

The beam A B is subjected to the uniformly distributed load...

Key Concepts

Static Equilibrium

This concept involves ensuring that the sum of all forces and moments acting on a rigid body or beam is zero. It is the foundation for analyzing and solving structures, as it allows for the determination of unknown reaction forces or moments through the application of equilibrium equations.

Free-Body Diagram

A free-body diagram is a simplified representation of a structure or component, isolated from its surroundings, where all external forces, loads, and moments are depicted. This tool is essential in visualizing and setting up the necessary equilibrium equations for analysis.

Uniformly Distributed Load (UDL)

A uniformly distributed load is one in which a constant load is applied per unit length over a beam or structure. It simplifies the analysis by allowing the load to be replaced by an equivalent resultant force, which is applied at the center of the distribution.

Reaction Forces

Reaction forces are the forces exerted by supports or connections that keep a structure in equilibrium. In beam problems, these forces must be determined using the conditions of static equilibrium, and they often include both vertical and horizontal components depending on the loading and support conditions.

Shear Force Diagram

The shear force diagram is a graphical representation that shows how the internal shear force varies along the length of a beam. It is a crucial tool in structural analysis, helping to identify locations where the beam is most likely to experience shear failure or require reinforcement.

Bending Moment Diagram

The bending moment diagram plots the internal bending moment distribution along a beam as a result of external loads and reactions. This diagram is used to determine the locations of maximum moments, which are critical for assessing bending stresses and designing beam sections accordingly.

Transcript

00:01 In this problem, we have a simply supported beam ab with a uniformly distributed load and two unknown loads, p and q.

00:14 We're given the values of the moment at points d and e, and we want to solve for the unknown applied loads p and q and draw the sheer and bending moment diagrams.

00:30 What we're going to do here is we're going to look at three segments of the beam to find sheer forces, unknown forces p and q, and reaction forces a and b.

00:45 And then we can draw the shear and bending moment diagrams.

00:52 So first we're going to look at d .e.

00:56 All right, we are going to have our two moments that we were given.

01:04 So we'll have our 5500 at e and our 6100 at d.

01:16 We're going to have the shears at each point, ve and v at d, and we'll have our resultant.

01:31 Uniform load and that will be at the center.

01:41 Okay, we can sum our moments and set equal to zero.

01:47 So we'll take that at point e and we're going to get a 5 ,500.

01:58 We're going to get a 1000 from the resultant with an arm of two, subtract the 60100 at point d and subtract the shear at d times the arm of four and setting that equal to zero and solving that for v we get the shear at d equals to 350 pounds and now for this segment and we will sum the forces, the vertical direction, and set to zero.

02:47 So we've got a 350 for v sub d minus our 1000 and plus a v sub e equal to zero.

03:06 And we solve that for ve, we give ve equal to a might.

03:12 Minus 650.

03:16 And we can check those on our shear diagram.

03:21 Okay, next we're going to do segment e -b.

03:30 I'm sorry, we're going to do segment a -d.

03:34 Let's do a -d next.

03:36 And then we'll do e -b.

03:40 All right, so looking at our segment a -d.

03:44 So we're going to have our shear at our moment at d, 6100.

03:53 We're going to have our shear at d, 350.

04:00 And we're going to have our reaction at a.