Recipientes A Presion

PRESSURE VESSELS
David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 August 23, 2001

Introduction
A good deal of the Mechanics of Materials can be introduced entirely within the confines of uniaxially stressed structural elements, and this was the goal of the previous modules. But of course the real world isthree-dimensional, and we need to extend these concepts accordingly. We now take the next step, and consider those structures in which the loading is still simple, but where the stresses and strains now require a second dimension for their description. Both for their value in demonstrating two-dimensional effects and also for their practical use in mechanical design, we turn to a slightly more complicatedstructural type: the thin-walled pressure vessel. Structures such as pipes or bottles capable of holding internal pressure have been very important in the history of science and technology. Although the ancient Romans had developed municipal engineering to a high order in many ways, the very need for their impressive system of large aqueducts for carrying water was due to their not yet having pipes thatcould maintain internal pressure. Water can flow uphill when driven by the hydraulic pressure of the reservoir at a higher elevation, but without a pressure-containing pipe an aqueduct must be constructed so the water can run downhill all the way from the reservoir to the destination. Airplane cabins are another familiar example of pressure-containing structures. They illustrate very dramaticallythe importance of proper design, since the atmosphere in the cabin has enough energy associated with its relative pressurization compared to the thin air outside that catastrophic crack growth is a real possibility. A number of fatal commercial tragedies have resulted from this, particularly famous ones being the Comet aircraft that disintegrated in flight in the 1950’s1 and the loss of a 5-metersection of the roof in the first-class section of an Aloha Airlines B737 in April 19882 In the sections to follow, we will outline the means of determining stresses and deformations in structures such as these, since this is a vital first step in designing against failure.

Stresses
In two dimensions, the state of stress at a point is conveniently illustrated by drawing four perpendicular lines thatwe can view as representing four adjacent planes of atoms taken from an arbitrary position within the material. The planes on this “stress square” shown in Fig. 1 can be identified by the orientations of their normals; the upper horizontal plane is a +y plane, since
1 2

T. Bishop, “Fatigue and the Comet Disasters,” Metal Progress, Vol. 67, pp. 79–85, May 1955. E.E. Murphy, “Aging Aircraft: TooOld to Fly?” IEEE Spectrum, pp. 28–31, June 1989.

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its normal points in the +y direction. The vertical plane on the right is a +x plane. Similarly, the left vertical and lower horizontal planes are −y and −x, respectively.

Figure 1: State of stress in two dimensions: the stress square. The sign convention in common use regards tensile stresses as positive and compressive stresses asnegative. A positive tensile stress acting in the x direction is drawn on the +x face as an arrow pointed in the +x direction. But for the stress square to be in equilibrium, this arrow must be balanced by another acting on the −x face and pointed in the −x direction. Of course, these are not two separate stresses, but simply indicate the stress state is one of uniaxial tension. A positive stress istherefore indicated by a + arrow on a + face, or a − arrow on a − face. Compressive stresses are the reverse: a − arrow on a + face or a + arrow on a − face. A stress state with both positive and negative components is shown in Fig. 2.

Figure 2: The sign convention for normal stresses. Consider now a simple spherical vessel of radius r and wall thickness b, such as a round balloon. An...