Internal stresses of an accelerating body

[Mike J]

Homework Statement

Refer to image attached.

Lets say I have a deformable solid that is being accelerated by a force that is equally distributed along the back face of the Main Body that is drawn in the picture. Attached to this Main Body is a Wing. At high accelerations, there will be inertial stresses that reach a maximum at the base of the wing where it is attached to the Main Body. How can I calculate those stresses? I need to find these stresses so that I can determine how large d has to be in order to prevent material failure. I am kind of lost at where to start.

Homework Equations

The Attempt at a Solution

My attempt:

I tried making a cut at the base of the wing (viewed from bottom of the picture shown above)

From here I can take sum of the moments to find the bending stress caused by the moment, M.

The equation I get is:

-M = m_wingaw/2.

Then I plug into bending stress formula for cantilever beam and compare to the yield stress:

|σ_max|=|M|d / (2I) ≤ σ_yield

where I = 1/12 L d³

If I solve the equations above for d so that σ does not exceed my yield stress, I end up getting unrealistically small values for d. I don't think my approach is correct because I believe it assumes rigid body acceleration when this is not the case. Maybe there's a continuum mechanics approach that involves solving a differential equation to take into account these inertial stresses of a non-rigid accelerating body.

Any help would be appreciated!

 

[PeterDonis]

Moderator's note: Moved from Classical Physics to the homework forum.

 

[haruspex]

I don't think my approach is correct because I believe it assumes rigid body acceleration when this is not the case.

It looks ok to me, provided the deformation of the wing is relatively small. If it flexes hugely then of course the moment is reduced a little.

If I solve the equations above for d so that σ does not exceed my yield stress, I end up getting unrealistically small values for d

Please show your working and result, and say why you think it is unrealistic.

 

[Dr.D]

You have treated the acceleration as a D'Alembert force. This is often very confusing. Instead, I recommend simply treating the wind as a separate body (as you have done), and write F = M*a and the corresponding moment equation. The validity of the whole process should become more evident when you do.

 

[Dr.D]

You have tried to treat the acceleration using a D'Alembert force. This is often confusing.

Instead, treat the wing as a separate body with a required (known, specified) acceleration, write F = m*a and the corresponding moment equation, and solve for the internal moment and shear force.

 

[Chestermiller]

This should be modeled as a plate rather than a beam. In addition, the load should be distributed, rather than concentrated at the center of mass. Also, there is a shear force at the built-in end. Let's see your shear force diagram, and your moment diagram.

 

[haruspex]

the load should be distributed, rather than concentrated at the center of mass

Does that matter if we only want the moment at the point of attachment?

 

[Dr.D]

This should be modeled as a plate rather than a beam. In addition, the load should be distributed, rather than concentrated at the center of mass. Also, there is a shear force at the built-in end. Let's see your shear force diagram, and your moment diagram.

In actual fact, there is no load at the center of mass or anywhere else except at the connection with the main body. The "load" that Chestermiller refers to presumably is the m*a "force" which is no force at all. There is no need to distribute any load, particularly one that is not there. This can all be handled by treating the wing as a single rigid body, and writing F = m*a and the corresponding rotational equation.

 

[haruspex]

In actual fact, there is no load at the center of mass or anywhere else except at the connection with the main body. The "load" that Chestermiller refers to presumably is the m*a "force" which is no force at all. There is no need to distribute any load, particularly one that is not there. This can all be handled by treating the wing as a single rigid body, and writing F = m*a and the corresponding rotational equation.

I prefer inertial frames too, but both methods are perfectly valid

 

[Dr.D]

Done correctly, D'Alembert will get to the answer, but I think that this is the whole reason we are discussing this thread, that is, the confusion it so often causes.

 

[haruspex]

I think that this is the whole reason we are discussing this thread, that is, the confusion it so often causes.

It may be the reason you are posting on this thread but I see no evidence the OP is concerned about that. @Mike J wrote that he didn't believe the answers he got, but as he has so far failed to respond with any data on those answers or why he does not believe them it's hard to take it any further.

 

[Chestermiller]

Does that matter if we only want the moment at the point of attachment?

No. I stand corrected. However, I still contend that the stress should be calculated as for plate bending rather than beam bending (because of the low aspect ratio of the piece).