
MECHANICS  EXAMPLE



Example

A Partial Section of a Penstock 

A penstock for a hydro power plant has an inside diameter of 1.5 m and is composed of wooden staves bound together by steel hoops. The crosssectional area for each steel hoop is 300 mm^{2}. If the allowable tensile stress for the steel is 130 MPa, what is the maximum space, L, between the hoop bands under a head of water of 30 m? The mass density of water is 1,000 kg/m^{3}. The water pressure can be assumed to be the same at all interior locations of the penstock. 





Solution

Spacing of the Steel Hoops
in the Penstock 

The pressure corresponding to a head of 30 m water is given by
p = ρgh
= (1,000) (9.81) (30)
= 294 kPa
Circumferential stress in the steel bands is considered as the failure criteria for a safe design. If the maximum spacing between hoops is denoted as L, then each hoop must resist the water pressure over a length L of the penstock. 



Direct Evaluation of Bursting Force F


The bursting force F, acting over the flat surface of the fluid equals the pressure intensity p multiplied by the area, DL, over which it acts.
F = pDL
= (294) (1.5) L
= 441 L
This bursting force will be resisted by the equal forces P acting on each cut surface of the cylindrical wall.
Assuming the whole resisting force will be given by the steel hoops,
P = Aσ
= (300×10^{6}) (130×10^{3}) kN
= 39 kN
Applying the summation of forces,
ΣF = 0
F  2P = 0
F = 2P
441 L = 2 (39)
L = 0.1769 m = 177 mm 





Comments



Please note, this example is a simplistic presentation of an actual penstock. An engineer in Argentina, Rodrigo Salvioli, who worked on real hydro power plant penstocks, correctly mentioned that,
 "In this particular example with wooden staves, the prestress of the staves is fundamental to guarantee tightness/continuity of the walls (no water leak) even under the operating pressure."
 "The prestress is provided by the steel hoops which, in addition, will stand part of the load caused by the water pressure."
 "How much of the extra load will be taken by the hoops will depend on the relative stiffness steel/wood (mostly by the hoops indeed)."




