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I am studying mechanics of materials and I'm confused with determining internal forces for the sake of stresses.

Consider this diagram: (just for visualization. from Beer Johnston book) enter image description here

when analyzing KH cross section, My professors use to draw the free body diagram of the front part (the part that is not attached to the wall.) because you don't worry about reaction forces of the wall in this case.

But I don't understand how they find the direction of reaction forces. (suppose there is no $P_1$ in the diagram and we want to find the reactions of force $P_2$ in cross section KH for the front part (not attached to the wall). just consider the directions and the written labels are just magnitudes of vectors.) which one is correct?:

  1. they move the force in the way it is, (i.e. if the force is in +x direction, it remains in +x direction) and also the couple caused by that force. enter image description here
  2. they move the force but in opposite direction and put a couple in the opposite direction of the couple caused by main force to cancel out all the forces enter image description here

I would be so thankful if you help me step by step to learn how to do it. I just used this diagram as a sample because didn't find a better one. if you have some better diagrams or so, you are welcome to use it. I just want to learn the easiest method step by step. tell me which part is easier to draw the FBD for? and what steps should I follow? because every time, I get the problem wrong.

thanks in advanced.

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  • $\begingroup$ Just to avoid further confusion: the moment should be bP1, not bP2. Maybe that is (or should be) in the errata. $\endgroup$ Jun 3, 2023 at 5:19

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The greatest error in drawing free-body diagrams and determining reaction forces is arguably to sketch forces on the complete schematic, as is done here.

I see no greater source of confusion when students are learning this material.

As evidenced in your question, severe confusion can arise regarding which direction loads (forces and moments) should point and how loads are replaced by equivalent loads.

The confusion is often related to Newton's third law, which specifies that reaction loads are equal and opposite. If the constraint still appears in the diagram, the appropriate load is ambiguous.

The problem is also usually exacerbated by a desire to draw vectors pointing the "right way," when the direction really doesn't matter; the equations of equilibrium will provide a positive or negative value. (If the value is negative, then sure, the vector was drawn the "wrong way," but this is irrelevant to the actual math of the solution. Nature doesn't care how coordinate systems are assigned.)

Professors can fall into this bad practice as well as students.

If you're going to draw a free-body diagram, draw a free-body diagram; eliminate connections to surrounding structures. Draw the reaction forces and moments as you like; the important aspect isn't that they're drawn the "right way" but that every constraint applied by the connection is represented—and replaced—by a reaction force or moment.

This is as far as we can go with the current question. So now: Draw a free-body diagram of the region of interest. It should be connected to nothing, but all the relevant loads should be included.

Try drawing a free-body diagram of the A–B bar, the B–K region of the cylinder, the K–D region of the cylinder, the B–D region of the cylinder, and the support at D. This provides good practice and preparation for applying the equations of equilibrium.

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  • $\begingroup$ so you mean to draw a free body diagram and balance forces and moments so that the body stays in equilibrium. (kinda option 2.) right? $\endgroup$ Jun 1, 2023 at 14:39
  • $\begingroup$ No; that’s the opposite of what I’m saying. I’m saying that a correct FBD removes the constraint from the diagram (so neither option 1 nor option 2) and replaces it with loads marked in any direction (because it doesn’t matter; the signs of the calculated loads will accommodate whatever choice is made). $\endgroup$ Jun 1, 2023 at 15:49
  • $\begingroup$ Don’t spend a moment thinking about the direction of a drawn vector representing an unknown load; you’ll waste time confusing yourself and get in the bad habit of thinking that vector arrows always point in the positive direction. Just draw them arbitrarily and then work carefully through the math. (Then try drawing them pointing in the opposite directions and gain practice by working through the problem again.) $\endgroup$ Jun 1, 2023 at 15:54

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