Mechanics of Materials -- inclined ladder

[gv3]

Homework Statement

[/B]
The inclined ladder AB supports a house painter (85 kg) at C and the self weight (q = 40 N/m) of the ladder itself. Each ladder rail (tr = 4 mm) is supported by a shoe (ts = 5 mm) which is attached to the ladder rail by a bolt of diameter dp =8 mm.

1) Find support reactions at A and B.
2) Find the reultant force in the shoe bolt at A.
3) Find maximum average shear and bearing stresses in the shoe bolt at A.

(some units in the photo are incorrect. the written question has the correct values)

Homework Equations

T(tau)= V/A

The Attempt at a Solution

I was able to do 1 and 2.
1) B_x = 252.83N
A_x = 252.3N
A_y = 1150.1N
2) Resultant force at A = 1178N

Part 3 is where i am having trouble.
3) i used V = 1150.1N. The radius of the bolt is r = .004m. A = π(.004)² = 5.026e^-5m²
Now when i do V/A i get 22.88MPa but the correct answer is 5.86 MPa.
I suppose my error is in using 1150.1N as the shear force but why wouldn't it be 1150.1N if that's the only force acting in the Y direction at A?

 

[Chestermiller]

Why do you think the force is in the y direction? Did you notice that your answer is off by a factor of 4? Why do you think that is? Did you draw a free body diagram of the bolt?

 

[gv3]

Why do you think the force is in the y direction? Did you notice that your answer is off by a factor of 4? Why do you think that is? Did you draw a free body diagram of the bolt?

I thought shear forced always occurred in the Y direction. When drawing my fbd i have V pointing in the negative Y direction and A_y pointing positively in the y direction. maybe i missed some forces from the ladder shoe? I am not sure where the factor of 4 comes from. is it from the ladder rail?

I did a new fbd to find the two forces in the ladder shoe then recalculated. i still get the wrong answer.

 

[Chestermiller]

I thought shear forced always occurred in the Y direction. When drawing my fbd i have V pointing in the negative Y direction and A_y pointing positively in the y direction. maybe i missed some forces from the ladder shoe? I am not sure where the factor of 4 comes from. is it from the ladder rail?

I did a new fbd to find the two forces in the ladder shoe then recalculated. i still get the wrong answer.

The shear force in the bolt is not in the vertical direction. It is parallel to the ladder. There are two rails containing two bolts, so this gives you one factor of two. Do you know where the other factor of two comes from? Hint: you are going to need to make imaginary cuts through the bolt to get the shear force.

 

[gv3]

The shear force in the bolt is not in the vertical direction. It is parallel to the ladder. There are two rails containing two bolts, so this gives you one factor of two. Do you know where the other factor of two comes from? Hint: you are going to need to make imaginary cuts through the bolt to get the shear force.

Does the other factor of two come from the ladder shoe? since there's two points of contact from the ladder shoe onto the bolt? Since the shear force is parallel to the ladder is it just the resultant force i found in part 2?

Ok so going with what i said above i got the correct answer. I did a cut between the rail and after the first contact from the shoe. which would give me the maximum V. I am still a little confused about the shear force being on an angle though. in class we always drew the V vertically. Is this because we were working with beams?

 

[Chestermiller]

Does the other factor of two come from the ladder shoe? since there's two points of contact from the ladder shoe onto the bolt?

Yes.

Since the shear force is parallel to the ladder is it just the resultant force i found in part 2?

Yes.

Ok so going with what i said above i got the correct answer. I did a cut between the rail and after the first contact from the shoe. which would give me the maximum V. I am still a little confused about the shear force being on an angle though. in class we always drew the V vertically. Is this because we were working with beams?

. The forces exerted by the ladder on the bolt must be equal in magnitude and opposite in direction to the forces the bolt exerts on the ladder. If you draw the free body diagram on the bolt carefully, you will see what's involved.

 

[gv3]

. The forces exerted by the ladder on the bolt must be equal in magnitude and opposite in direction to the forces the bolt exerts on the ladder. If you draw the free body diagram on the bolt carefully, you will see what's involved.

So when you draw the fbd, the force from the rail is on an angle and the two forces from the shoe are vertical. Doing it this way you wouldn't get a force of 294.5N from the shoes. but a force of 294.5N is what gives the right answer.

 

[Chestermiller]

So when you draw the fbd, the force from the rail is on an angle and the two forces from the shoe are vertical. Doing it this way you wouldn't get a force of 294.5N from the shoes. but a force of 294.5N is what gives the right answer.

The forces from the shoe are not vertical.

 

[gv3]

The forces from the shoe are not vertical.

Is that because the shoe is keeping the ladder up and also preventing it from sliding outwards?

 

[Chestermiller]

Is that because the shoe is keeping the ladder up and also preventing it from sliding outwards?

Sure. If you do a force balance on the bolt, you find that the force that the shoe exerts on the bolt has to be equal and opposite to the force the ladder exerts on the bolt. That's why I've been encouraging you to draw a free body diagram of the bolt and to do a force balance.