Need help using Method of Members to find reaction forces

[dlacombe13]

Homework Statement

Determine the components of the forces acting on each member of the pin-connected frame shown.
(the frame shown is drawn as a free-body diagram in the image provided below (the top-most portion) and is correct)

Homework Equations

Equilibrium equations:
∑M=0
∑Fx=0
∑Fy=0

The Attempt at a Solution

In this section, I am supposed to be using the Method of Members, which I have attempted below. My work is divided into 4 sections (from top to bottom: FBD of entire frame ; FBD of member FCD ; FBD of ABC; FBD of member DBE)
Everything is correct except for the forces Cx and By. Any help?

 

[Valour549]

The correct answer in section 3 is Cx = 0, and By = 4000N upwards, from what I can see. It sounds like you have the answers to the question so you can check if I'm right.

Your FBD for ABC is wrong. You found Cy acting on FCD, to be 2000N upwards, in section 2. This means that the Cy acting on ABC, must be 2000N downwards, in section 3. Once you correct that part, you should find the values of Cx and By I gave above.

Funny enough you wrote ∑Fx = Bx = 0, when the Cx you found was 4000N to the left. In any case, the correct answer for Bx should be 0 (once you've agreed Cx is 0).

 

[dlacombe13]

Ah, so when I find the force of member A acting on Member B, I must make sure that when I draw the FBD of member B, the force is opposite of the force exerted from member A. That basically seemed to correct the problem. And you are correct that the values of Cx = 0 and By is 4000.

However, my book states that the answer to By is -4000 (downwards). I have looked over my math over and over but I can't see how it is -4000N... Any ideas?
Just in case you need to see my math, after correcting the direction of Cy to the downward direction, my math is:

ΣMb = -3Cx + 3(2000) - 3(2000) = 0
Cx=0
ΣFx = Bx = 0
ΣFy = -2000 - 2000 + By = 0
By=2000N (upwards, as assumed...)

 

[Valour549]

ΣFy = -2000 - 2000 + By = 0

By = 4000N upwards (maybe you accidentally typed 2000N... when you meant to type 4000N)

Now you need to understand whether we get 4000N or - 4000N all depends on our original assumption. A positive number means our assumed direction was correct. A negative number means our assumed direction was incorrect. Positive or negative has nothing to do with pointing up or down, since you're already taking into account the direction with pluses and minuses in your equations.

You can assume whatever directions you want, as long as it's consistent. For example when you found Cy in section 2, you arbitrarily assumed it was down, which is totally fine. But you found out the answer was -2000, meaning it is actually acting upwards. In other words, the force Cy on FCD by ABC is upwards.

Now we move into section 3. By Newton's 3rd law, the force Cy on ABC by FCD must be downwards. And you understand that so that's great. But we currently have no clue as to the direction of By. If we assume it's upwards (like it currently is in your diagram), we get:
ΣFy = -2000 - 2000 + By = 0 → By = +4000 (meaning our assumption was correct), therefore By = 4000N upwards.
However you could've also, just as reasonably, assumed By is downwards, in which case we get:
ΣFy = -2000 - 2000 - By = 0 → By = -4000 (meaning our assumption was wrong), and again we get By = 4000N upwards.

Lastly we move into section 4. Here we look at the force By exerted on DBE by ABC. Again by Newton's 3rd law, By = 4000N downwards.

 

[dlacombe13]

Ah okay, I already understood the assumption part, and how if it is negative it is the opposite of what was assumed. I just made the mistake of assuming the negatives in the answer key told the direction, which is obviously wrong.

Any idea then, how I would figure out the direction from the answer? I mean, my book basically states the answers like this:

Cy = 2000N on DF, By = -4000N on DE.

But how can they just leave out the direction or the original assumptions of the directions? Or am I missing some knowledge that would allow me to realize the direction from the answers as given?

 

[Valour549]

How it draws the original assumed arrows I can't see and don't know. But what I do know is your book is obviously using +ve as meaning upwards, and -ve as meaning downwards in the answers. You ask why it can leave out the original assumptions; the answer is simple, because no matter what the original assumptions are, the answers are always the same in terms of the actual directions. It then uses the common system of up = +ve, down = -ve to express its answers.

When you do the question, remember that +ve and -ve in your answer simply allows you confirm or deny your original assumptions. After that you can simply express the directions with an arrow. I would recommend against doing what the book does and use +ve and -ve again to express up or down, that's just not a good idea.

In any case, the book's answers agree with ours.

Force Cy on FCD, or DF if u want to call it that, (by ABC) is 2000N upwards.
Force By on DBE, or DE if u want to call it that, (by ABC) is 4000N downwards.

 

[dlacombe13]

Sorry what do you mean by +ve and -ve? What is "ve"?

 

[Valour549]

+ve (positive), -ve (negative). sorry I assumed everyone's familiar with these shorthands.

 

[dlacombe13]

Oh okay, so all you're saying is that the book does indeed use negative to indicate downward directions, and positive to indicate upward directions? And therefore, it is stating that the force of By acting on member DE is in the downward position (as in the FBD of member DBE). I think I got it, clarify if I am wrong. Thanks for you're help by the way!

 

[Valour549]

You got it! You're welcome :)