TECHNICAL NOTE

Upper Bound of Limit Torque for RegularPolygonal Shafts

N. Jones

Impact Research Centre, Department of Engineering, The University of Liverpool, Liverpool L69 3GH, UK

Introduction

D’Souza and Moreton [1] published recently a tech-

nical note in this journal which showed that

TL ¼ 2

3A0RK (1)

is a lower bound estimate (i.e. a statically admissible

calculation) of the limit torque for a regular polygo-

nal shaft having any number of equal sides, n. A0 in

Equation (1) is the cross-sectional area of the shaft, R

is the radius of an inscribed circle drawn within the

shaft cross-section and K is the shear yield stress for

the material. This equation was developed for a shaft

made from a ductile material having perfectly plastic

characteristics.

An upper bound (i.e. kinematically admissible)

value for the same problem studied in Ref. [1] is

presented here so that the exact collapse torque for

an elastic perfectly plastic, or rigid perfectly plastic,

polygonal shaft, is bounded from below by Equation

(1) and above by the present analysis.

Preamble

Equation (1) gives

TL ¼ 2pR3K=3 (2)

for a circular shaft with an outer radius R. This

expression could be used to provide bounds on the

plastic collapse torque of a polygonal shaft by taking

R to inscribe and to circumscribe a polygonal shaft

according to the limit theorems of plasticity and their

corollaries (e.g. [2] or §2.4.4 of [3]). For example,

Equation (2) for a square shaft having a side length 2a

gives 2pa3K/3 and 4�2pa3K/3 for the inscribing and

circumscribing circles, respectively, of the cross-sec-

tion indicated in Figure 1. However, one estimate is

2�2 times larger than the other which is far too large

a difference for practical purposes. Nevertheless, the

two values converge for a polygonal shaft having

many sides and, in the limit, are equal to the exact

limit torque for a circular shaft (n fi 1), as noted in

the Appendix.

Upper Bound Analysis

Now consider the segment of a perfectly plastic

polygonal shaft having n sides each of length 2a, as

shown in Figure 2. An upper bound estimate for the

limit torque Tu of the polygonal shaft is given by

equating the external work rate to the rate of internal

energy dissipation, viz.

Tu _h ¼ 2n

Z p=n

o

Z R= cosu

0

rdu dr Kr _h; (3)

where _h is the rotation rate about the central axis of

the shaft. The original plane transverse sections of

the shaft remain plane and simply rotate relative to

each other with no axial separation (i.e. no warp-

ing). This is an acceptable simplification for an upper

bound calculation and satisfies the requirements of

the limit theorems of plasticity.

Integrating Equation (3) with respect to r gives

Tu ¼ 2nK

3

Z p=n

0

R3 ducos3 u

which can be integrated* again to give

Tu ¼ nR3K

3

sinðp=nÞcos2ðp=nÞ þ loge tan

p4þ p

2n

� �n o� �: (4)

The cross-sectional area A0 in Equation (1) for a

polygonal shaft is

A0 ¼ na2 cotpn

� �; (5a)

*For example, cases 313 and 294 in Ref. [4] can be used for theintegrals.

� 2006 The Author. Journal compilation � 2006 Blackwell Publishing Ltd j Strain (2006) 42, 117–119 117

or

A0 ¼ nR2 tanpn

� �: (5b)

Thus, Equations (1) and (4) give the ratio

Tu

TL¼ 1

2sec

pn

� �þ cot

pn

� �loge tan

p4þ p

2n

� �� �n oh i:

(6)

Discussion

It can be shown for a circular shaft with n fi 1 that

Equation (6) is unity, or, in other words, Equation (4)

reduces to Equation (1), so that the exact limit torque

is predicted for this case. This result also can be

obtained from elementary calculations for a circular

shaft.

Equation (6) for a square shaft gives Tu/TL ¼ 1.1477

which agrees with the value given in Ref. [2] (see

Problem 6.7).

Equation (1) with n ¼ 3 in Equation (5), as well as

Nadai [5], predicts that TL ¼ 2Ka3/3 for an equilateral

triangular shaft having a side length 2a. Equation (6)

predicts that Tu/TL ¼ 1.380 so that the exact collapse

torque (TE) for a shaft with an equilateral tri-

angular shaped cross-section lies within the range

TL £ TE £ 1.380TL.

The exact collapse torque (TE/TL) for a regular

polygonal shaft lies between the curve given by

Equation (6) in Figure 3 and the horizontal line

which represents the lower bound calculation

according to Equation (1). It is evident that the dif-

ference decreases rapidly with increase in n and is less

than 2% for n ‡ 9 approximately.

ACKNOWLEDGEMENTS

The author wishes to record his appreciation to Mrs M.

White for her secretarial assistance and to Mrs I. M.

Arnot for tracing the figures.

REFERENCES

1. D’Souza, A. and Moreton, D. N. (2005) The limit torque for

regular polygonal shafts. Strain 41, 31–32.

2. Calladine, C. R. (2000) Plasticity for Engineers: Theory and

Applications. Horwood Publishing Limited, Chichester, UK.

3. Jones, N. (1997) Structural Impact. Cambridge University

Press, Cambridge, UK.

4. Selby, S. M. (1974) Standard Mathematical Tables, 22nd edn.

CRC Press, Cleveland, OH.

5. Nadai, A. (1931) Plasticity. McGraw-Hill, New York.

1.5

Equation (6)

TL/TL [Equation (A3)]

1.0

T u /

TL

0.5

03 6 9 12

n15

c i

Figure 3: Variation of the dimensionless upper bound to the

limit torque [Equation (6)] with the number of faces (n) in a

polygonal shaft. - - -, dimensionless lower bound [Equation (1)]

a

dr

rR

0

pn

R0

Figure 2: Segment with an included angle of p/n of a regular

polygonal shaft having n faces each of length 2a

Rp4

2a

2a

Inscribingcircle

Circumscribingcircle

Ro

Figure 1: Square shaft with inscribing and circumscribing

circles of radii R ¼ a and R0 ¼ �2R ¼ �2a respectively

118 � 2006 The Author. Journal compilation � 2006 Blackwell Publishing Ltd j Strain (2006) 42, 117–119

Limit Torque for Regular Polygonal Shafts : N. Jones

APPENDIX

Equation (2) for the inscribing and circumscribing

circles of the segment of a regular polygonal shaft in

Figure 2 gives

T iL ¼ 2pKR3

3¼ 2pKa3

3 tan3 pn

� � (A.1)

and

TcL ¼ 2pKR3

0

3¼ 2pKa3

3 sin3 pn

� � (A.2)

respectively. It is evident that for large values of

n, tan(p/n) � sin(p/n) � p/n so that R � R0 giving

TiL �Tc

L.

Equations (A.1) and (A.2) give the ratio

TcL

T iL

¼ sec3 pn

� �(A.3)

which is plotted in Figure 3.

� 2006 The Author. Journal compilation � 2006 Blackwell Publishing Ltd j Strain (2006) 42, 117–119 119

N. Jones : Limit Torque for Regular Polygonal Shafts