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

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.

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.

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 Okay, so for this problem, we need to find the shear and bending moment diagrams, as well as the maximum shear and the maximum bending moment.

00:09 So we go ahead and look at our distribution of forces, we should end up with a free by diagram that looks similar to this one up on the top here.

00:18 We'll have our force that's given to us, p.

00:22 We'll have our moment m0.

00:25 Now we are told that m0 is equal to pl.

00:32 It's important to note that this is separate from our other force p right here.

00:38 So these are two separate things.

00:40 They're not the same thing.

00:42 And then we have our reaction forces on the other end of the beam where it's fixed.

00:46 We have our f, b of y, and our m, our moment around b.

00:52 So the first step here is going to be taking this free body diagram and then finding our forces, our reaction forces.

01:02 So we sum our forces in the y direction.

01:04 We get a negative p plus fb y is equal to zero.

01:12 This we can solve, and we find fb y is just going to be equal to p.