No forces are imposed in the middle surface. Lines perpendicular to the middle surface before deformation remain perpendicular to the deformed middle surface. These lines are inextensible. These lines remain straight lines. These assumptions form the basis for developing the bending theory of plates and apply to plates where buckling is not a consideration.

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Disregarding higher-order terms, Eq. Equation 7. The shearing force is determined from Eq. Derive the moment expression for a uniformly loaded, simply supported circular plate of radius a. Hence, hum Eq. A plot o! Derive the moment expression for a uniformly loaded circular plate of radius a that is fixed at the edge. A plot of Eqs. In rectangular plates under uniform loads, the shearing stress interacts with the normal stresses in the x- and v-directions and thus cannot be ignored.

This results in a more compli- cated differential equation than that for circular plates. In addition, the solution of the differential equation of rectangular plates is more elaborate and involves the use of Fourier series. Because of this, only the case of a simply supported MRIV. Nonsymmetric loadings and boundary conditions other than simply supported result in quite complicated solutions that are beyond the scope of this book.

The examples given in this section are intended to give the reader a concept of the general behavior of rectangular plates and the difference between them and circular plates. If an infinitesimal section is removed from a rectangular plate, the forces acting on it will occur as shown in Fig. For any given loading and boundary conditions, the deflection w can be obtained from Eq. The bending moments can then be determined from Eq.

Substituting this expression into Eq. Hence, the maximum value of is given by am 3. The solution of the differential equation of a plate on elastic foundation involves Bessel functions. The table also shows the limits for the first derivatives of Z x through Z 4. The relations between the various derivatives of the Z functions are as fol- lows: Table 7.

The differential equation of a circular plate on an elastic foundation can be obtained by modifying Eq. Determine the maximum deflection in a circular plate on an elastic foundation subjected to a concentrated load F in the center of the plate. From Table 7. Therefore, C x and C 2 must be set to zero. But from Table 7. Hence, C 4 must be set to zero. Tiomoshenko, S. McFarland, D. Szilard, R. Courtesy of the Mooter Corporation, St. They are easy to fabricate and install and economical to maintain. The required thickness is generally controlled by internal pressure, although in some instances applied loads and external pressure have control.

Other factors such as thermal stress and discontinuity forces may also influence the required thickness. It is a simplification of Eq. It indicates the wide range of applicability of Eq.

Various forms of Eq. A pressure vessel with an inside diameter of Assume that all circumferential and longitudinal seams are double-welded butt joints and are spot radiographed. A seamless cylindrical shell with an outside diameter of The circumferential seams are not x-rayed. Find the required shell thickness if the allowable stress is 15, psi and the internal design pressure is psi. The head-to-shell seams are partially radiographed. Find the re- quired thickness it Hie allowable stress is 20, psi ami the design pres- sure is psi. The magnitude of these stresses must be kept below a given allowable stress.

The designer has to establish first whether the stress is at a local or a gross structural discontinuity, as defined in Fig. Examples of gross structural discontinuities are head-to-shell and flange-to- shell junctions, nozzles, and Junctions between shells of different diameters or th ic kness.

## Engineering Design and Thermal, Stress, Vibration and Fatigue Analysis

A source of stress or strain intensification which affects a relatively small volume of material and does not have a significant effect on the overall stress or strain pattern or on the structure as a whole. Figure 8. Secondary stress is sel f-1 1mi t1 ng. Local yielding and minor distortions can satisfy the conditions which cause the stress to occur and failure from one application of the stress is not to be expected.

Examples of secondary stress are general thermal stress and bending stress at a gross structural discontinuity. PEAK Peak stress does not cause any noticeable distortion and is objectionable only as a possible source of a fatigue crack or a brittle fracture. Examples of peak stress are: 1 thermal stress in austenitic steel cladding of carbon steel vessels. Once the stress categories are established, the stresses at a vessel's different locations can be classified as in Table 8.

Table 8.

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Application of Table 8. Example 8. Calculate the stress at points A, B, and C of the vessel in Fig.

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The basic characteristic of a primary stress is that it is not self- limiting. Primary stresses which considerably exceed the yield strength will result in failure or at least in gross distortion. A thermal stress is not classified as a primary stress. An example is the stress in a circular cylinder due to internal pressure. Solution Point A. Thermal stress is developed in a solid body whenever a volume of material is prevented from assuming the size and shape that it normally should under a change in temperature. Examples of general thermal stress are: 1 stress produced by an axial temperature distribution in a cyl indrlcal shell.

Examples of local thermal stresses are: 1 stress in a small hot spot i n a vessel wall. From Tables 8. Point B. Stress com- ponent perpendicular to cross section fm External load or moment Bending across full section.

## Process Equipment

Stress com- ponent perpendicular to cross section fm Near nozzle or other opening External load moment, or In- ternal pressure Local membrane Bending Peak fillet or corner fl Q f Any location Temp. Excludes discon- tinuities and concentrations. Produced only by mechanical loads. Average stress across any solid section. Considers dis- continuities but not con- centrations. Bending Component of primary stress proportional to distance from centroid of solid section. Ex- cludes discon- tinuities and concentrations Produced only by mechanical loads. Secondary Membrane plus Bending Self-equilibrating stress necessary to satisfy con- tinuity of structure.

Occurs at struc- tural discontinui- ties. Can be caused by mechan- ical load or by differential ther- mal expansion. Excludes local stress concentra- tions. Peak 1 Increment added to primary or second ary stress by a con- centration notch. Symbol Combination of stress components and allow- able limits of stress intensities. Point C. The discontinuity forces at point c are shown in Fig. This equation is modified to take into consideration nonelastic buckling and 8. These curves are plotted on log-log graphs with a factor of safety of two for stress.

Because the stress-strain curves differ for different temperatures, a number of curves for different temperatures are plotted in Fig.

This reduction occurs because for very thick cylinders, buckling ceases to be a consideration and the allowable values in tension and compression are about the same. A summary of the procedure is shown in Fig. The length of a cylindrical shell is 15 ft, outside diameter 10 ft, and is constructed of carbon steel with minimum yield strength of 36, psi.

The shell is subjected to an external pressure of 10 psi. Find a the required thickness using ASME factor of safety and b the required thickness using a factor of safety of 2. Hence, the above solution of Returning to Fig. A cylindrical shell with length 18 ft and an outside diameter 6 ft is constructed of carbon steel with a yield stress of 38, psi. Determine the thickness needed to resist an external pressure of psi.