SOLID / HOLLOW SHAFTS STRESSES 1

[Adrian77]

Dear friends.

I am interested on getting info about Mise's hypothesys and stresses in solid and hollow shafts. Main interest is about the stress distribution on compressed solid and hollow shafts transmiting torque.

thanks in advance

 

[CESSNA1]

ADRIAN77: Could you please expalin your request in some more detail please.

Regards
Dave

 

[Adrian77]

HI CESSNA, I EXPLAIN

ONE SHAFT FITTED WITH INTERFERENCE WITH AN OUTER SLEEVE BY MEANS OF HYDRAULIC OIL. THE SHAFT IS HOLLOW AND I WANT TO CALCULATE THE STRESSES IN ALL PARTS OF THE COUPLING. THE SYSTEM IS TRANSMITING TORQUE AND THEREFORE I CALCULATED A PRESSURE BETWEEN MATING SURFACES WHICH TOGETHER WITH FRICTION COEFFICIENT ALLOW YOU TO TRANSMIT A CERTAIN TORQUE. I THINK THAT A GOOD APPROACH OWULD BE WITH LAME'S THEORY FOR THICK WALLED CYLINDERS BUT von MISES MUST BE TAKEN INTO ACCOUNT (TANGENTIAL AND RADIAL STRESSES MUST BE COMBINED AND TAKEN INTO ACCOUNT).

ONE MORE THING, THE SHAFT IS HOLLOW AND IS FITTED WITH A REINFORCEMENT SLEEVE INSIDE, THIS SLEEVE HAS MUCH HIGHER YIELD STRESS.

 

[CESSNA1]

Adrian77: You seem to be pretty well along. As long as the cylinder's are truly thick walled, Lame's equation should work. The Von Mises theory requires finding the maximum shear in the assembly. For this you will need to use Mohr's Circle. If the reinforcement sleeve is a significant press fit it will have to be accounted for also. One other thing you may need to consider is buckling. I get the feeling the size of these pipes may minimize the buckling issue. I am thinking thick walled pipes on the order of 2-4 inches in diameter. If the pipes are pressurized that will have to be taken into account. One other thing to consider is at what torque the pipes will slip. Remember that all of this theory only is reasonable if everything is within the elastic limit of the material.

Regards
Dave

 

[desertfox]

Hi Adrian77

If you need formula to calculaate the torque capacity of your cylinders from the interface pressure and friction coefficient they are as follows:-

Here are the formula you need:-

Torque that can be transmitted by an interference fit
without slipping:-

T= f*Pc*3.142*d^2*L/(2)

where f=friction coefficient
Pc=contact pressure between the two members
d=nominal shaft dia
L=length of external member.

to calculate Pc for a given interference use the formula:-

Pc=x/[Dc*[((Dc^2+Di^2)/(Ei(Dc^2-Di^2))+....................
((Do^2+Dc^2)/(Eo*(Do^2-Dc^2))-((Ui/Ei)+
Ui/Eo))]



where x = total interference
Dc=dia of the contact surface
Di = dia of inner member
Do= outside dia of outer member
Uo=poissons ratio for outer member
Ui=poissons ratio for inner member
Eo=modulus of elasticity for outer member
Ei=modulus of elasticity for inner member

This formula for Pc will simplify if the materials are the same.

In addition if your inner hollow shaft is pressurised (acting like a pressure vessel) this would need treating seperatly and would affect the interface pressure at the boundary of the mating cylinders. So if this was the case the pressure at the interface boundary due to the interference between the cylinders and the internal pressure on the inner cylinder would need to be resolved first before working out torque capacity.

However if you are just trying to calculate stresses in each cylinder due to the pressure generated by the interference then Lame's equation should be fine, just remember that the internal cylinder can be treated as a vessel under external pressure and the outer cylinder under internal pressure.

regards desertfox

 

[Adrian77]

I HAVE ALMOST SOLVED THE PROBLEM, MANY THANKS FOR YOUR HELP.

 

[Cockroach]

Adrian77, you can see the Von Mises-Hencky theory in all it's splendor in advance books regarding Pressure Vessel Theory.

Quite simply, Von Mises-Hencky discuss tri-axial states of stress as a result of hoop, radial and longitudinal stresses. The theory mentions the stress vectors as the cross product of vectors defined by cyclic permutation of that tri-axial state. The basis of the theory is that twice the magnitude of stress squared is equal to the gradient of that stress vector formed by hoop, i.e. cross product of the stress vector with itself.

I have worked out the mathematics and resolved the famous equation:

S = sqrt(3) P [R^2 / (R^2 - 1)] for R = D/d

"S" being the stress, "P" as pressure, "D" outer diameter and "d" inner diameter of the pressure vessel. My study needed to account for induced longitudinal stresses by the supports, I was basically crushing the pressure vessel longitudinally from above while pressurized internally. As a result, the hand calculations were very, very close to that measured using strain gauges and correlated well to FEA using COSMOS. For this reason, I have quoted it at many times in the past, almost to the point of nausia to those readers viewing the various postings. Sorry boys!

Good luck with the analysis.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada