For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b) determine th

For the beam and loading shown, (a) draw the shear and...

Key Concepts

Shear Force Diagram

A shear force diagram graphically represents the variation of internal shear forces along a beam’s length. As loads are applied to the beam, the internal shear force typically changes abruptly at points of concentrated loads or supports. The diagram is essential for identifying regions with sudden changes in force, which can be critical in design and analysis for assessing structural integrity.

Bending Moment Diagram

A bending moment diagram illustrates how the internal bending moment varies along the length of a beam under applied loads. It shows the moments resulting from the loads and reactions, and it is drawn by integrating the shear force distribution. This diagram is crucial for identifying critical points where moments reach extreme values, which directly influence material selection and structural safety.

Static Equilibrium

Static equilibrium is the fundamental principle that the sum of all forces and moments in a structure must equal zero. This principle is applied when analyzing beams to ensure that the reactions and internal forces balance the externally applied loads. It forms the basis for deriving the shear force and bending moment diagrams, as it allows the determination of internal forces from known external loads.

Sectional Analysis

Sectional analysis involves 'cutting' the beam at various locations to isolate a segment, then applying equilibrium equations to that segment to determine the internal shear force and bending moment at the cut. This method is key in structural analysis as it provides a systematic approach to compute the internal forces and moments at any point along the beam, enabling the construction of accurate diagrams.

Maximum Value Calculation

Determining the maximum absolute values of the shear force and bending moment is important in structural design, as these values indicate the most critical regions within the beam where failure could initiate. This process involves analyzing the diagrams to find the peak values, and often requires solving equations derived from the equilibrium conditions and load distributions.

Transcript

00:02 Alright, so for this question, we are asked to find the shear and bending moment diagrams and the max shear and max bending moments.

00:11 Okay, so to start off, we'll start by creating a free bi diagram like the one shown here with our provided load and our two reaction forces, f, a of y and f c of y, on either end of the beam.

00:31 So the first thing we'll go ahead and do is find our reaction forces.

00:35 So part a, we'll go ahead and sum up reaction forces in the y direction.

00:53 Okay.

00:55 So we'll go ahead.

00:56 We can't solve this just yet because we don't know what a and c are.

01:01 So we'll move on and we'll sum up our moments in the a at point a.

01:08 So f, c of y, times the length from a, which is going to be the full length of the beam, minus p times l over 3 is equal to 0.

01:25 So we don't have a moment created by f of a because we're revolving about point a right here.

01:33 So we're not going to have a moment created by f.

01:36 So if we solve this for f of c, we end up with p over 3.

01:45 Now we can go back to our original equation and solve for f of a.

01:52 F of a is going to be equal to, p minus f of cy, which is p over three, which is equal to 2p over three.

02:09 Okay.

02:10 So now we have our reaction forces.

02:15 Now we can go ahead and make our shear and moment diagrams.

02:20 Okay.

02:22 So as we go along our beam in this first segment right here, our shear is going to be counteracting the internal forces created by f, a of y.

02:41 So we're going to end up with a positive shear along this length.

02:51 And then once we hit p, and then we're going to get on the second segment right here, and we're going to have a shear that will be counteracting fc of y, the internal, let's start over...

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