Determine (a) the distance a for which the maximum absolute value of the bending moment in beam

Determine (a) the distance a for which the maximum absolute...

Key Concepts

Bending Moment

The bending moment is an internal moment that develops in a structural element, such as a beam, due to applied external loads. It quantifies the tendency of the load to induce bending in the element and is crucial for determining the resulting stresses and deflections. Understanding its distribution along the member is key to safe and efficient design.

Static Equilibrium

Static equilibrium is a foundational principle in mechanics stating that for a structure or a body at rest, the sum of all forces and the sum of all moments about any point must be zero. This concept forms the basis for deriving the equations used to calculate internal forces like reactions and bending moments in beams.

Calculus-Based Optimization

Calculus-based optimization involves using techniques such as differentiation to identify the minimum or maximum values of a function that represents a physical quantity. In structural analysis, this method is applied to determine design parameters that minimize adverse effects, such as reducing the maximum bending moment, ensuring efficient use of materials and overall safety.

Moment Distribution Analysis

Moment distribution analysis is a method used to determine the variation of bending moments along a beam. By constructing bending moment diagrams, engineers can identify the locations of maximum and minimum moments, which is essential for verifying that the designed structure meets performance criteria and remains within safe limits under loading.

Transcript

00:03 This problem we have a beam that is simply supported on its two ends here.

00:12 And then there is some additions to the beam and then some cables connected and there is a 80 newton load on it and with these two here.

00:26 So what the question that we want to answer is that these distances here are can be very very very and we'd like to minimize the maximum absolute bending moment in the beam.

00:44 Figure out how long these should be to do that.

00:48 So what we can do is we can figure out that, well, because of symmetry, both of these, the y components have to be 40 newtons each, and their x components have to be equal and opposite.

01:02 And because of geometry, the tangent of theta is 2, and so we wind up with this would be 80 newtons.

01:15 So we have 80 newtons acting this way and this way, and obviously those are acting to bend the beam.

01:24 And so we have, if we draw through bi -diagram of our beam now, with the forces and bending moments on it, this one will be, yeah, this one will be positive.

01:45 If this one will be why not being negative.

01:51 So we have the, we have these, this whole situation here and we can figure out, again, we can do a force and moment balance on this, figure out what f -a -y and f -b -y are.

02:10 I'm not going to go through all of that in detail, but what we find out is if we draw, if we make our cuts, so let's say we make our cuts here, here, here.

02:22 So we have shear forces and bending moment in each of these.

02:29 We'll see that what we'll have for sheer forces, their shear forces are constant in each of these regions.

02:41 And what we have is a sheer force in this region that's constant.

02:48 So our bending moment is going to start at zero and it's going to decrease less than zero.

02:57 And then at this point, shear force will jump, but the bending moment will also jump by whatever the magnitude of this.

03:06 And then at this point, again, the shear force is constant in this area.

03:14 And in fact, it turns out it's zero in that area.

03:18 And then it's positive in this area.

03:22 And so we'll have a positive slope in the base.

03:24 Bending moment.

03:27 So if you play around with this a little bit, what you'll find is that the peak moment's negative moment occurs at either one of these points here...

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