E E = E E Substituting into Eq. 9.95 gives:
therefore
= (9.96)
For an element strained in three directions it can be shown in a similar manner that
(9.97) Equation 9.97 is due to Haigh (149) and has been called strain-energy function.
is assumed to occur when the total strain energy is equal to that obtained in simple tension at the yield point:
therefore
Substituting into Eq. 9.94 for the condition at the yield point gives:
1
= 0
therefore
(9.99) Setting Eq. 9.97 equal to Eq. 9.99 gives:
+ + + =
equal to the theoretical maximum value of 0.3 gives a simplified equation
+ = Equation 9.101 is identical to that developed by
Hencky and Von Mises (152) to bring the theory into agreement with the fact that materials can undergo large hydrostatic pressures without yielding.
Equation 9.100 for two-dimensional strain reduces to:
= (9.102)
9.15 GRAPHICAL COMPARISON OF THE FOUR THEORIES
Figure 9.17 graphically shows a comparison of the foul methods of analysis for a two-dimensional stress system
= 0). The upper right quadrant represents tension both and The upper left quadrant represents in
tension and in compression, and the lower right quadrant represents in tension and in compression. The solid lines in the figure represent the locus of the conditions at which yield is assumed to begin according to the four theories. The square represents the maximum stress theory. Point a represents equal tension in both the x and y perpendicular directions, both of which are con-sidered to be equal to the yield-point stress obtained from a simple tensile test.
According to the maximum-stress theory, yielding does not occur inside the square. As is decreased and is held constant, is controlling from point 1 to point 2, which points in compression is taken as equal to in tension.
The irregular hexagonal figure represents the maximum-shear theory and is the same as the
stress theory in the upper right quadrant, l-u-2, and the lower left quadrant, but is more conservative in the other two quadrants where the stresses are of opposite signs.
This can be explained by Eq. 9.85, which is for the three dimensional theory. If and are both positive and is equal to zero, the maximum minus the minimum stress will be equal to either minus zero or minus zero. However, is negative and is positive and is zero, the maximum minus the minimum will be which will give the diagonal line l-4.
The rhombus A-B-C-D represents the maximum-strain theory. By referring to Eqs. 9.87 and 9.88, in which is taken as equal to zero for two-dimensional stress, it may seen that if the stresses and have the same sign, then
181 either or can appreciably exceed the yield point of the
material. This is true because the strain in the y direction reduces the strain in the direction and vice versa when the stresses have the same sign, as is the case at point A or point C.
The strain-energy theory is represented by the ellipse and can be evaluated by means of Eq. 9.102.
9.16 EXAMPLE OF A VESSEL IN WHICH THE FOUR THEORIES ARE COMPARED
Because a tall fractionating tower has inherent resulting from dead weight, pressure stresses, and super-imposed loads such as wind or seismic forces, it is for comparing the four theories discussed in the previous sec-tions. Brummerstedt (153) in 1943 presented an example design of a tower comparing the four theories. His problem concerned an analysis of stresses in a fractionating tower which had a 10 ft inside diameter and a height of 160 ft. The tower was to withstand an internal pressure of 200 psi and a superimposed seismic force equal to of its operating weight. This force was assumed to apply at the center of gravity of the tower.
The tower was to be designed to a maximum allowable stress of 13,750 psi for the material of construction. The welded-joint efficiency was to be 82 This resulted in an allowable stress is 13,750 X 0.82 or 11,300 psi. (These values came from the code restrictions in effect in 1943). The operating weight including steel plate, attachments such as piping platforms, ladders, and ing liquid was estimated to be 620,000 lb.
The seismic force of resulted in a horizontal force of 124,000 lb. This force induced a moment, of ft-lb on the column.
The circumferential- and longitudinal-pressure stresses resulting from internal pressure were computed by merstedt (153) and combined with the dead weight and seismic stresses under the assumption that the radial stress in the tower shell was zero. These stresses were combined by the four theories, and the results tabulated for son. Tables 9.4 and 9.5 summarize the results.
The example tower design is such that the percentage difference in total weights as indicated in Table 9.5 is not very great. This is due to the comparatively high design pressure of 200 psi. A similar comparative analysis made of such a tower operating under a low pressure of under 50 psi often results in a considerably greater variation.
such a case method of analysis becomes of greater
Table 9.4. Summation of Combined Stresses in a Tall Tower (According to Brummerstedt) (153
Resulting Stress, psi Ratio
importance. However, as in earlier section.
tower operating under low pressure may because of elastic instability. To design for such a
appropriate relationships presented must be applied.
Shell and head thicknesses based only upon stress equations provide no allowance for
loads on vertical vessels. However, the shell thicknesses obtained by such equations provide a convenient point for evaluating the thicknesses for vertical vessels since the thicknesses thus obtained may be modified to
structural requirements. In the case of vessels operating under internal pressures of 30 lb per sq in. gage or more it is usually convenient to first check the
tensile stresses from pressure, wind bending moments, and/or seismic moments. The design of tall vessels operation at low internal pressures or the design of vessel under external pressure is controlled by the tive compressive forces. The design of such vessels can determined most rapidly by beginning calculations the cumulative compressive stresses than with
stresses.
Table 9.5. Summation of Thicknesses and Weights Required by the Four Theories (153)
Weight Shellof Bottom Next Next Top Plates Theory 20 20 20 100 ft (lb)
stress in. in. in. in. 100.0
strain in. in. in. in. 285,000 98.2 . shear in. in. in. in. 300,000 5
in. in. 295,000
1. An insulated steel fractionating column located at Oakland, California, is 6 ft, 0 in. in inside diameter and 160 0 in. from tangent to tangent between heads. The heads project 1 ft, 6 in. beyond the point of tangency. The skirt is 8 0 in. from the base to the shell junc-tion at the point of tangency with the bottom head. The vessel is designed to operate at 100 lb per in. gage. It is of SA 285 grade C steel (13,750 psi maximum allowable tensile stress). The effective wind area of is to be 10% of the
182 Design of Tall Vertical Vessels
area of the uninsulated column. The insulation is 3 in. thick and weighs 40 lb per cu ft. The tray spacing is 18 in., and there are 102 trays with an estimated weight of 25 lb per sq ft of column cross section. The top tray is 4 ft, 0 in. below the top tangent line. Calculate the minium shell thickness at the bottom tangent line resulting from wind moment.
2. For the vessel in problem 1, calculate the maximum stress at the bottom tangent line resulting from the seismic moment.
3. If the shell of the vessel in problem 1 is fabricated from 20 plates 96 in. wide, specify the thickness for each course allowing a minimum of in. for corrosion.
4. Redesign the vessel in the example design for full vacuum operation.
5. Redesign the vessel in the example design for 160 lb per sq in. gage operating pressure.
6. Redesign the vessel in the example design for the same conditions but on the basis of the maximum-shear theory.
7. Redesign the vessel in the example design for the same conditions but on the basis of the maximum-strain theory.
8. Redesign the vessel in the example design for the same conditions but on the basis of the maximum-strain-energy theory.
9. A fractionating tower is required to separate styrene from a dilute feed. Preliminary calculations indicate that 70 trays will be required with the feed entering on tray 32 (from the bottom). The reboiler will be separate from the column. Ninety-five per cent recovery of the styrene in the feed is desired. Annual production of styrene from the columns is to be 8000 tons.
Styrene Ethylbenzene Toluene Benzene
Stream Comnosition. Percentages Feed
Mol. Wt (saturated liquid) Bottoms
104.14 3 7 . 0 9 9 . 7
106.16 6 1 . 1 0 . 3
92.13 1 . 1
78.11 0 . 8
Temperatures Top of column 54” c Bottom of column 90” c
Pressures Top of column 30 mm Hg
Bottom of column 310 mm Hg Reflux ratio, L/D = 7 (mol ratio)
The tower is self-sustaining (no guy wires) and is to have a IO-ft skirt extending from the top of the foundation to the tangent line of the bottom dished head. The erected tower is to be located in the Houston, Texas, area. The overhead condenser is to rest on the ground, and the reflux is to be pumped back. The client specifies that the bubble caps are not to be larger than 5 in. but may be smaller if desired. The tower is to be designed for full-vacuum service.
A tray layout and tower design excluding skirt, foundation bolts, nozzles, and bubble-cap details are required.
REFERENCES FOR PROBLEM 9
Bolles, W. L., “Optimum Bubble Cap Tray Design,” Part I, Petroleum Processing, Vol. 11, No.
2 (1956); Part II, No. 3; Part III, No. 4; Part IV, No. 5.
Boundy, Ray, ACS Monograph No. 115, Reinhold Publishing Company, New York, 1952.
Davies, J. A., “Bubble Trays-Design and Layout-Part I,” Petroleum Refiner, Vol. 29, No. 8 (1950); Part II, ibid., No. 9.