why is hollow shaft better than solid shaft?please explain with mathematical proof

Dear punita,

We dont need mathematical proof for this just a simple common sense is enough.The hollow shafts can withstnad the same stress as that of solid shafts.In torsion the stress profile is a linear relationship with diameter.So the outermost surface or the outermost material of the shaft is the place where maximum stress is acting in torsion.To reduce the cost, the manufacturers prefer the hollow shaft rather than the solid shaft.so hollow shaft is better than solid shaft in terms of the cost.A solid shaft in bending or tensile or compressive forces will always be significantly stronger for the same material,dimensions and loading.

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Twisting moment or torque depends upon the distance from neutral axis to the outer most fibre in mm. So taking same weight f material the above dimension of a hollow shaft will be greater than a solid shaft. Now the torque produced in hollow shaft will be greater than the solid shaft. So hollow shaft is more preferable.

It can be shown that torque required to produce unit twist in a solid shaft of radius r length L and made of material of modulus of rigidity G is given by ___ torque =πGr^4/2l also torque` is required to produce unit twist in a hollow shaft of length L internal and external Reddy R1 and R2 respectively and of material of modulus of rigidity G is given by ____ πG(r2)^2-(r1)^2/2l and now if we do torque`/torque it will give some positive value which is greater than 1 and it will show that hollow shaft is stronger then a solid shaft made of same equal material

All the answers here said most of it in terms of solid mechanics theory of pure bending of beams. Since you used the term `shaft` meaning we`re talking about torsion, I`ll just add a layman explanation regarding that:Since the mass is constant, A hollow shaft is necessarily a result of reallocation of material from axis to periphery. This results in decrease of slenderness of the shaft and hence, it`s torsional rigidity increases. Explanation:The strength of a shaft implies the maximum torque it can transmit. It depends on Torsional rigidity of the shaft since it renders the stiffness to the shaft. Torsional rigidity is defined as the torque required to produce a twist of one radian per unit length of the shaft. and by torsional formula:T/(θ/L)=GJ=Torsional rigidity.So, Stiffness and hence strength is proportional to J which is:by maths, less for solid shaft andby intuition, less for most slender of the two (solid shaft).