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

00:01 Alright, so for this problem, we're going to be looking to find the sheer and bending moment diagrams and the maximum shear and the maximum bending moment of those diagrams.

00:12 So the first step would be to set up your free body diagram, and it's going to look something similar to the one right here, where we have our distributed load w and our two reaction forces, fb of y and f, a of y.

00:28 Now we can note that the beam is symmetrical about the middle, here where that purple line is, and we can find w, the reaction force, the sum of w will act right in the middle where the beam is symmetric about, and that's just going to be equal to the distributed load times the length of the distributed load, which is l over 2, which is then going to be equal to wl over 2.

01:19 So we know that our beam is symmetric.

01:22 Which means that f of a and f of b are going to be equal to each other, and they're going to be equal to half of w, because he's right in the middle of the beam.

01:36 So we can say that fw divided by 2 is equal to f -a -y, f -b -y, which would then be equal to w -l -over -4.

01:57 So that's our reaction forces.

01:59 So now we can go ahead and determine what our shear and moment diagrams are going to look like.

02:07 So we know we're going to have a shear here to counteract our reaction force, right? and then this v is going to travel along this whole segment here.

02:25 So in order to counteract that, we're going to need a shear with a value equal to that of f -a -y.

02:40 Which is w l over 4.

02:46 Now when we go to w, we're going to go along the distributed load.

02:56 And when we hit the middle here, our shear is actually going to be equal to zero because that's equal to one half of w, which is equal to our reaction force, which is this year that we're trying to counteract.

03:09 So our shear is actually going to equal to zero right here at the middle of the beam.

03:13 And we're going to end up with a diagonal line here.

03:22 Okay, so once we end at the end here, we're now going to have a shear that's going to be counteracting the segment of the distributed load, which is then just going to be a straight line here.

03:49 Of course, it goes back down to zero at the end of the beam.

03:57 Okay, so our moment diagram is a little easier.

04:00 We can find that based off of the slopes of the shear diagram.

04:06 So since we have a straight line here on the first segment, we are going to have.

04:16 A slanted line for our moment that's just increasing at that same rate...

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