Engineering Optimization: Theory and Practice, Fourth Edition

Problems 297

5.5 The shear stress induced along thez-axis when two cylinders are in contact with each
other is given by

τzy

pmax

= −

1

2







−

1

√

1 +

(z

b

) 2 +







2 −

1

1 +

(z

b

) 2







×

√

1 +

(z

b

) 2

− 2

(z

b

)







 (1)

where 2bis the width of the contact area andpmaxis the maximum pressure developed

at the center of the contact area (Fig. 5.21):

b=







2 F

π l

1 −v 12

E 1

+

1 −v^22

E 2

1

d 1

+

1

d 2







1 / 2

(2)

pmax=

2 F

π bl

(3)

Fis the contact force;E 1 andE 2 are Young’s moduli of the two cylinders;ν 1 andν 2 are

Poisson’s ratios of the two cylinders;d 1 andd 2 the diameters of the two cylinders, andl

the axial length of contact (length of the shorter cylinder). In many practical applications,

Figure 5.21 Contact stress between two cylinders.