Where is the Maximum Bending Stress Located in a Tapering Rod?

[MMCS]

A uniformly tapering cantilever of solid circular cross-section has a length L and
carries a concentrated load at the free end. The diameter at the fixed end is D and
at the free end d. Show that the position of maximum bending stress occurs at a
section

(d/(2(D-d))*L

distance from the free end.

I know that the max bending is when shear = 0

I can't begin to use any formulas because to take moments i need to know where the centre of gravity would be for a tapering rod, which i dont, also how would i use the ∏*d^4/64 to get the second moment of area on a tapering rod?

Thanks

 

[pongo38]

quote: "I know that the max bending is when shear = 0 ". This is true of bending moment but not necessarily of bending stress. You need to look at the bending stress formula.

 

[pongo38]

"how would i use the ∏*d^4/64 to get the second moment of area on a tapering rod?"

Express d in the above formula as a function of x. It's not the same d as given in the question.

 

[SteamKing]

A concentrated load is applied at the free end. You can construct a shear and bending moment diagram without knowing anything about the beam structure except that it is continuous and it is fixed at one end.

Once you have the bending moment calculated as a function of position along the length of the beam, then you can apply what you know about the cross section to identify the location of maximum bending stress.

 

[pongo38]

"i need to know where the centre of gravity would be" Why assume that the cantilever is lying horizontal?

 

[SteamKing]

My point is, if a concentrated load is applied at the free end of the cantilever, knowledge of the center of gravity of the beam is not required in order to calculate the bending moment.

 

[MMCS]

My point is, if a concentrated load is applied at the free end of the cantilever, knowledge of the center of gravity of the beam is not required in order to calculate the bending moment.

In previous questions i have derived functions for unknown forces, one with the weight of the beam acting at its centre of gravity. Should this be excluded in this case or should another method be used? I don't really know where to start with only the reaction force at the fixed end and the concentrated load at the other

 

[pongo38]

Consider a section X distance x from the free end. Can you express the moment and shear as functions of x (ignoring self-weight)? Can you express the internal and external diameters at section X as a function of x? If necessary draw graphs.

 

[MMCS]

Consider a section X distance x from the free end. Can you express the moment and shear as functions of x (ignoring self-weight)? Can you express the internal and external diameters at section X as a function of x? If necessary draw graphs.

Would he shear force not be constant across the beam with a force of the value of the concentrated load? I am really stuck as to how i would i get a function for the diamaters without any values?

 

[SteamKing]

You are given the diameter D at the fixed end, and diameter d at the free end.
You are also told that the diameter of the beam tapers uniformly (what does that suggest?)

How would the bending moment vary as a function of location from the fixed end?

What is the formula for bending stress given the value of a moment M(x)?

There are no numbers to work with. The answer is given in algebraic terms.

You have received several hints about how to work this problem. Now is the time to put pencil to paper and do some work.

 

[MMCS]

You are given the diameter D at the fixed end, and diameter d at the free end.
You are also told that the diameter of the beam tapers uniformly (what does that suggest?)

How would the bending moment vary as a function of location from the fixed end?

What is the formula for bending stress given the value of a moment M(x)?

There are no numbers to work with. The answer is given in algebraic terms.

You have received several hints about how to work this problem. Now is the time to put pencil to paper and do some work.

I have done some work, that's why i am posting on here because i have had no luck. I don't know the equation for the maximum bending position

 

[MMCS]

I have the

shear force at x = concentrated laod at free end(W)
Bending moment = -W*x

diameter equation

D-((D-d)/L)*x

Is this correct?

Where do i go next?

 

[SteamKing]

What is the bending stress, given what you know about the bending moment and the diameter of the beam at a location x?

 

[pongo38]

Your bending moment equation is inconsistent with your diameter equation. From which end do you measure x? It has to be the same in each case. My suggestion to measure it from the free end avoids having to know the reactions at the fixed end (although that isn't a big issue).

 

[MMCS]

I get stress = ((-w*x)*(0.5*(D-(D-d*x)/L))) / pi*(D-(D-d*x)/L)^4/64

 

[MMCS]

Your bending moment equation is inconsistent with your diameter equation. From which end do you measure x? It has to be the same in each case. My suggestion to measure it from the free end avoids having to know the reactions at the fixed end (although that isn't a big issue).

Would it be d+(D-d/L)*x ?

 

[MMCS]

Can anybody help me with the equation for the position of max stress. i don't see how i can get to my answer from the M/I = stress/y equation tihout having terms of W and Pi in my answer

 

[pongo38]

Answer to post #16 is that you can test this proposal for yourself by letting x=0 and x=L to see if you get sensible answers (You don't). But it's nearly right.

 

[pongo38]

answer to post #17 is to use differential calculus. If you had numbers instead of symbols, you could plot the graph. The question doesn't ask for the maximum stress- just its location, which would be independent of W and pi.

 

[MMCS]

answer to post #17 is to use differential calculus. If you had numbers instead of symbols, you could plot the graph. The question doesn't ask for the maximum stress- just its location, which would be independent of W and pi.

Bearing this is mind, i am still unsure of how to get to an equation that defines the position. Is it a form of the equation i gave in post 15 (ecluding the W and pi values) or is it a different equation. Is it possible to give me further assitance in the last part of this question as i have spent a lot of time on this

Thanks for your help

 

[pongo38]

Can you interpret stress=My/I where M, y and I are all functions of x? Then d(stress)/dx=0 will give you an equation in x, at least one of whose solutions will be what you are looking for.

 

[MMCS]

no i can't interperet it. that's what iv been sayin. i can't write what i know as an equation of x without having other terms which arnt included in the equation i need to prove. i don't understand how to do it. id appreciate it if you could just show me.

 

[pongo38]

Let f be the stress at section X
Let e be the diameter at section X
Can you confirm that you agree with M=Wx, and e=d+x*(D-d)/L
Also confirm that you have I=pi*e^4/64 where e is the function of x above.
Putting all that together: f=My/I = (a function of x that you can put together)
Put df/dx=0 and solve for x=