our instructors send to us the links to see theses videos bcz they are great and easy to understand

did you find some good videos for Frames elastic curves?

You need to examine the shape of the elastic curve in the vicinity of the point. If the curve is concave up in that region then you have compression at the top and tension at the bottom, regardless of the shape of the cross-section. If the curve is concave down the reverse is true (tension at the top and compression at the bottom). The shape of the cross-section however does effect the location of the neutral axis. To make this clear, let's compare a T-shaped section with a rectangular section in a beam made of steel. Say both sections have the same overall height (for example, 12 inches). If the elastic curve is concave up, both sections are going to be in compression at the top, and the bottom of the sections would be in tension. In the case of the rectangular section the neutral axis is at the mid-point of the section, 6 inches from the bottom (or top). That is, the top 6 inches is in compression and the bottom 6 inches is in tension. In the case of the T-shaped section, however, the neutral axis would be closer to the top.

Im so thankful to you, Dr, now its clear for me, thank you again for your precious time and explanation

By definition, the roller support only permits horizontal displacement, no vertical displacement takes place at the support. You are correct to assume that if a beam joint rotates, then it also should has a vertical displacement, but only if it is not resting on a pin or roller. Otherwise, the joint would rotate without any up or down movement.

With regard to visualizing deformation (as opposed to just deflection/bending), think of a large piece of dough. You can make it into almost any shape that you wish by pushing, pulling, twisting, … it in different directions. By applying those force (in different directions) to the dough, you are deforming it. Viewing this from engineering mechanics perspective, think of a cube having six faces/edges. Each edge of the cube is a square. At the center of each square place the origin of a 3D coordinate system. The three axes, x, y, and z are to be placed such that x and y lie on the surface of the square, and z pointing outward away from the surface, the z axis is perpendicular to the square. Now, place a force in the x direction, a force in the y direction, and a force in the z direction on each of these square edges of the cube. Also, place a moment around the x axis, a moment around the y axis, and a moment around the z axis on each square. So, we end up with six forces/moments on each edge of the cube. The cube, therefore, is going to be subjected to a total of 36 forces. These forces, when present, cause the cube to deform. Such a deformation could involve, bending, elongation, shortening, wrapping, twisting,… To determine the general deformed shape of the cube (or a structure, which can be viewed a collection of such cubes), to calculate all the changes in length and angle that results when the cube is subjected to all 36 forces, we can use generalized finite element method. There are many textbook written one the finite element method (FEM), but FEM is an advanced topic best explored as a part of graduate studies in engineering mechanics.

Nice explanation. Now I know that basically the deformation topic is way beyond the deflection and has more advanced topics rather than deflection. Make me more curious and want to learn. Thanks a lot.

It is customary, according to the beam sign convention, to consider the moment that causes compression on top and tension on the bottom of the beam as positive. This means positive moment corresponds to concave up deflection of the beam. The moment diagram can tell us how the beam deflects. The part of the diagram that are above the x-axis suggest a concave up deflection in that beam segment, and the parts that are below the x-axis indicate a concave down deflection of the segment.

Not quite. Here, we are using the beam sign convention. Moment is considered positive when it causes the beam segment to bend concave up. So, the sign of the moment is defined using a pair of moments, one at the left end of the segment and one at the right end of the segment. As shown in the diagram to the left @7:15, a positive moment means a clockwise moment at the left end of the segment and an anti-clockwise moment at the right end of the segment. So, when deciding on the sign of the bending moment in a beam segment, you need to ask: does the moment cause a concave up deflection, or does it cause a concave down deflection? The answer to the question tells you if the moment is positive or negative. The left half of the example beam carries a negative moment. Why? because the downward reaction at the left end of the beam creates an anti-clockwise moment. Per explanation above, that corresponds to a negative moment. On the other hand, the upward reaction at the right end of the beam wants to bend the beam segment in a concave up manner, so it results in a positive moment in the beam. As the moment diagram shows, the negative moment in the left half of the beam starts from zero and goes to M/2 (or, -M/2). The moment at the right half of the beam does the opposite, it starts from zero at the right end and goes to positive M/2. This is obtained by multiplying the reaction force (M/L) by half of the length (L/2). The difference between the left moment (-M/2) and the right moment (M/2) at the mid-point of the beam equals the applied moment (M). That is: (M/2) - (-M/2) = M.

@@DrStructure Thank you so much for the clear explanation! :D

@@limchekhui1792 You're welcome!

Correct, the roller can only move in the horizontal direction, it allows the beam to stretch horizontally at D. In the vertical direction, the roller acts just like a pin, it prevents the beam's displacement.

😮 i thought roller only prevents downward movement of the beam

No, a roller (as a mathematical/modeling construct) prevents both upward and downward movements, meaning, a vertical reaction force does develop at a roller support. That force could be upward or downward.

Thank you so much 😊

No! The curvature of the beam could change sign in other cases too. Consider a continuous beam with two spans supported by a pin/roller at each end and a pin/roller at its midpoint. If we subject one of the spans to a gravity load, we get a downward deflection in that span and an upward deflection in the other span. In such a case, the point of contraflexure is located in the loaded span near the middle support.

@@DrStructure thank you sir..

This particular video was put together using: *An IPad for tracing the text and drawing, * VideoScribe for converting the drawings into video segments, and *Camtesia Studio for synchronizing video and audio.

Yes.

Deformation is a more general term than deflection. Deflection is deformation caused by bending moment. For our purposes here, we use them interchangeably.

Thanks for the enlightenment, I see the way out. hope the best for this channel.

When a structure is subjected to loads, it deforms, and the same time internal forces (such as bending moment) develop in its members. We show such internal forces using diagrams, like bending moment diagrams. There is no such thing as moment diagram in an un-deformed structure, as that implies there are no internal forces present. By definition, bending moments, and by extension moment diagrams, come to existence when the structure is loaded and deformed. The bending moment diagram for the deflected shape of the structure IS the collection of the bending moment diagrams for the individual members of the structure.

Yes, bending moment at B and C is negative, but moment is positive for the most part of BC. So, the elastic curve for the entire beam consists of three parts (from left to right): concave down, concave up, concave down. The left span and a small part of the middle span, about L/10 to the right of B, is concave down as bending moment is negative in that region. The right span and a small part of BC (about L/10 to the left of C) is also concave down as bending moment is negative in that region. The middle part of the beam (about 80% of its length) is concave up as the bending moment in that region is positive. So, there are two inflection points in the elastic curve: one a bit to the right of B and the other a bit to the left of C where the shift in the curvature takes place.

Moment at support B is negative. At span (BC) is positive. At support C is again negative. Therefore, the elastic curve from B to C needs to have three concave (up, down, up). Or you mean that since the most of the part of BC is positive therefore one curve between BC as given in the video

You need to view the entire elastic curve consisting of three parts, not just the part between B and C. The first curve (concave down) start from the left end of the beam and ends just to the right of B. That is where the first inflection point occurs. There is no inflection point at B. The second part of the curve (concave up) starts where the first part ended and continues until just to the left of C. That is the second inflection point occurs. The third curve (concave down) starts where the second one ended and continues until the right end of the beam. Since there are no inflection points at the supports, there is no reason to divide the elastic into parts at B or C. The video does not identify, or clearly show, the inflection points. But these points are not exactly at B and C, they are to the right of B and to left of C.

I agree with to consider entire curve. I want to ask between BC due to changing moment signs. Why does not BC have three curves due to changing moment signs

BC does have three curves. Except that the left curve in BC, which is concave down, is a part of the left curve for the entire beam. And the right curve in BC (also concave down) is a part of the right-most curve for the entire beam. So, at the end you only have three curves. Two concave down curves and one concave up one.

Consider a beam fixed at both ends. If the left support settles, we get a positive moment at the left end of the beam, and a negative moment at the right end of the beam. This means, the left half of the beam bends concave up and the right half of the beam bends concave down, with the point of inflection being at mid span. On the other hand, if the beam has a fixed support and a roller support, and the roller support settles, then we end up with a concave down deflection of the entire beam. If a middle support settles, it generally causes a concave up deflection in its vicinity, eventually turning into a concave down deflection near the adjacent supports.

Deflection is used to denote change in geometry due to bending, like when a beam bends downward. Deformation is a more general term. It is used to refer to any change in geometry, be it due to bending or linear displacement (as in trusses.)

If either AB or CD, or both are loaded, then each segment wants to have a concave up deflection. The amount of deflection depends on the load magnitudes. The larger the magnitude, the larger the deflection in AB/CD. The downward deflection at the outer segments is adversely related to the deflection in the middle segment. The more AB/CD deflect, the less BC deflects down.

Dr. Structure If the loading were downward then wouldn't it be convex and hogging?

yes, the outer segments would be.

You are welcome!

Such a bending moment could be produced by a column that rests on the beam at that point. If we are interested in analyzing the beam only, we can show the effect of the column on the beam (i.e., the bending moment) without showing/drawing the column itself. We can also think of other substructures (i.e., an L-shaped bracket), attached to the beam, that produce a bending moment at that point. Pin and roller supports can often be seen in bridges (supporting the bridge deck). Fixed supports can be found in building, either at the base of the columns or at the beam-column connections. For example, see: theconstructor.org/structural-engg/types-of-supports-reactions-uses-structures/16974/

Got it 👍 Much thank to you and to the people behind this amazing work 💚

+Lyrex Ices Deformation due to bending.

This lecture was done/drawn using an stylus on an ipad, saved as svg file, then converted into video clips.

Whenever bending moment is negative, the beam deflects upward (concave down), and when it is positive, the deflection is downward (concave up). In this problem, the left side of the moment diagram is positive and the right side is negative, meaning the right side has to deflect up while the left side deflects down. Draw two such curves, one deflecting down for the left side of the beam and one deflecting up for the right side of the beam, then place them on the beam observing the fact that deflection at the supports must be zero.

I understood that but at right side bmd is negative and when bmd is negative the beam should bend upwards, so how come you have shown at point where 4kn load is acting(extreme right) the deflection is downwards?

Not quite, when moment is negative, the elastic curve (the deflected shape of the beam) is going to be concave down, meaning the top fiber of the beam is going to be in tension and the bottom fiber in compression. It is not about the direction of deflection at any point, it is about compression vs tension in the extreme fibers of the beam. Think of a cantilever beam subjected to a downward force at its (right) free end. How is it going to deflect under the load? downward (the force is pushing the point down). What is the moment in the beam? negative (just like what we have here at the right end of the beam), what would be the deflection shape of the cantilever beam? concave down. For all practical purposes, you can view the part of the beam from the roller to the free end as a cantilever being pushed down at its free end by the 4 kN force.

Yes I got it thank you but how do u know it will deflect downwards at 4kn?

Here, we are drawing the diagram qualitatively. If in doubt, we would have to do the actual computations in order to determine the direction, and magnitude, of such a deflection. Qualitatively speaking, the downward force of 4 kN is attempting to push the free end of the beam down while the 8 kN force wants to push the same free end up. Which one controls? Examine the moment diagram, note that moment is zero in the middle segment of the beam. More precisely, it goes to zero 2 meters to the left of the roller support. Let's refer to that point at B. Now, isolate the beam segment from B to the free end. What does the free body diagram of the segment look like? Zero moment at B, a downward shear force of 6 kN at B, an upward force of 10 kN at the roller support, and the 4 kN force at the free end of the beam. Think of this as a beam segment sitting on a roller at its mid-point. We are pushing down the left end of the segment with a force of 6 kN, and we are pushing down the right end of it with a force of 4 kN. How would the segment bend? Why would the the right end move up? The beam segment would have the tendency to deflect down at both ends. The right end can actually move up under one scenario, when B is forced to rotate in the counterclockwise direction. If that rotation is large enough then the entire elastic curve could rotate around the roller support forcing the free end of the beam to move above the beam line, resulting in an upward deflection. Although the 8 kN force causes such a rotation at B, but it is not going to be large enough to push the free end of the beam up. The energy needed for that rotation is caused by the positive moment in the beam. But note that total positive area under the moment diagram is significantly smaller than the total negative area. This suggests that the 4 kN force overpowers the 8 kN force in defining the deflection shape.

At the hinge, the elastic curve ends up having a discontinuous slope. Meaning, instead of having a smooth and continuous curve, we get a V-shaped curve where the slope of the curve at the left side of the hinge is different than the slope at the right side of the hinge. For example, suppose we have a beam that is fixed at both ends with a hinge in the middle. What does the elastic curve for the beam look like? Cut the beam at the hinge. This results in having two cantilever beam segments. Draw the elastic curve for each cantilever. They both deflect downward at their free end (the free end being the place that we cut the beam). Now, if we move the two elastic curve so that they touch at their free ends we get the elastic curve for the original beam. Both beam segments have deflected the same amount at the free end, but the left one has a clockwise rotation (at the hinge) and the right one has a counterclockwise rotation. Meaning, they have different slopes at the hinge.

Wow, Thanks Doc! That was a very great explanation. I wish you're one of my instructors.

Yes, the slope of the elastic curve is discontinuous at the hinge. The slope of the curve immediately to the left of the hinge is different than the slope immediately to the right of the hinge.

Thanks. Do you have an example?

Not at the present time.

If you have specific questions about the technical content of the video, it would be best to post it here. Otherwise, you can send email to: Dr.Structure@EducativeTechnologies.net