# Max bending in tapering rod

1. Feb 20, 2013

### 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 cant 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

2. Feb 20, 2013

### 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.

3. Feb 20, 2013

### 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.

4. Feb 20, 2013

### SteamKing

Staff Emeritus
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.

5. Feb 20, 2013

### pongo38

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

6. Feb 20, 2013

### SteamKing

Staff Emeritus
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.

7. Feb 20, 2013

### MMCS

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 dont really know where to start with only the reaction force at the fixed end and the concentrated load at the other

8. Feb 20, 2013

### 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.

9. Feb 20, 2013

### MMCS

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

10. Feb 20, 2013

### SteamKing

Staff Emeritus
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.

11. Feb 21, 2013

### MMCS

I have done some work, thats why i am posting on here because i have had no luck. I dont know the equation for the maximum bending position

12. Feb 21, 2013

### 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?

13. Feb 21, 2013

### SteamKing

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

14. Feb 21, 2013

### 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).

15. Feb 21, 2013

### MMCS

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

16. Feb 21, 2013

### MMCS

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

17. Feb 22, 2013

### MMCS

Can anybody help me with the equation for the position of max stress. i dont 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

18. Feb 22, 2013

### 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.

19. Feb 22, 2013

### 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.

20. Feb 22, 2013

### MMCS

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 quesiton as i have spent alot of time on this

Thanks for your help