Solve Cantilever Question: Mechanical Engineering Student

[LostStudent5]

Hey guys, first-year Mechanical Engineering student here and a new forumer of this site too. I got a static question that I need help with.

Question:

Relevant Equations:

Sum of all forces in the y direction = 0
Sum of all moments = 0

Attempted Working:

I just need help finding the y-reaction force and moment reaction at A. I believe I can solve the rest of the question once I figure this out. What I did was:

Ray + P = 0 by considering equilibrium of beam AB in the y direction
RA = -P

Ray + P + Rb/c = 0 by considering equilibrum of the entire beam AC (where Rb/c is the reaction force of the roller at 3L/2 from the LHS of the beam)

Rb/c = -Ray - P = -P - P = -2P

Sum of moments at A (anticlockwise is positive):

Rb/c(3L/2) + PL - M1 - M2 - Mr = 0 where Mr is the moment reaction at A (assumed to be clockwise)
Mr = Rb/c(3L/2) + PL - M1 - M2 = -2P(3L/2) + PL - M1 - M2 = -2PL - M1 - M2

This yielded me the wrong final answer. I apologise for not providing you guys with some free-body diagrams. I hope you can see what I've tried to do here.

Any feedback is appreciated! Thanks!

 

[Mech_Engineer]

Hi LostStudent5, thanks for posting your work in a logical manner, its a big help for us. Have you considered the fact that since point B is a hinge, it cannot transfer a moment load to the cantilever beam element?

 

[LostStudent5]

Hi LostStudent5, thanks for posting your work in a logical manner, its a big help for us. Have you considered the fact that since point B is a hinge, it cannot transfer a moment load to the cantilever beam element?

Sorry, I don't understand what you mean by that. Does that mean the moment load M2 cannot affect the bending moments of the cantilever beam, so we can just ignore the second beam entirely when considering bending moments of the cantilever beam? Or does that mean the bending moments around point B are zero?

Thanks for responding.

 

[Mech_Engineer]

Sorry for the late response, this thread dropped off my radar. Regarding your question, the hinge cannot transfer moment so if you visualized a bending moment diagram the bending moment at the hinge would be zero, but the shear force will still transmit through the hinge. Does this make sense?

 

[LostStudent5]

Sorry for the late response, this thread dropped off my radar. Regarding your question, the hinge cannot transfer moment so if you visualized a bending moment diagram the bending moment at the hinge would be zero, but the shear force will still transmit through the hinge. Does this make sense?

No problems.

Yes, it makes sense now. However, I still got the wrong answer. This is my working out:

I don't know where I went wrong. All y-reactions are intially assumed upwards and the moment reaction at A is assumed clockwise. The actual answer is 35 kN/m.

Thanks for your help so far. I really appreciate it.

 

[Mech_Engineer]

It seems to me your "given answer" of 35 kNm is in fact incorrect.

My work (attached in PDF) shows the maximum bending moment to be 60 kNm (at point A). Your analysis method seems correct (analyze the hinged portion of the beam first, the cantilever portion next). Analyzing in this direction finds a reaction force of 7.5 kN at point B (and a normal force of 2.5 kN at the roller). When you carry this reaction force into the cantilever beam section, the resulting bending moment at point A is 60 kNm CCW, and the reaction force at point A is 2.5 kN.

Secondly if we "work forward" from the given answer of 35 kNm (pg. 3 of PDF), Rb ends up being 1.25 kN, resulting in a non-compression force at the roller and conflicting solution results between the force balance and moment balance due to unbalanced forces. Where did your given answer come from?