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### Thick-walled Cylinders of Brittle Material, Ends Open or Closed Formula and Calculator

Lamé’s equation is used to determine the wall thickness when cylinders of this type are subjected to internal pressure.

Pressure. In designing a cylinder to withstand internal pressure, the choice of formula to be used depends on:
1) the kind of material of which the cylinder is made (whether brittle or ductile);
2) the construction of the cylinder ends (whether open or closed); and
3) whether the cylinder is classed as a thin- or a thick-walled cylinder.

A cylinder is considered to be thin-walled when the ratio of wall thickness to inside diameter is 0.1 or less and thick-walled when this ratio is greater than 0.1. Materials such as cast iron, hard steel, and cast aluminum are considered to be brittle materials; low-carbon steel, brass, bronze, etc. are considered to be ductile.

Where:

p = internal pressure, psi;
D = inside diameter of cylinder, inches;
t = wall thickness of cylinder, inches;
S = allowable tensile stress, psi.

If metric SI units are used, then:

p = internal pressure in newtons per square meter
D = internal diameter of shell in meter
S = safe tensile stress in newtons per square meter
t = thickness of metal in the shell, in meter

The table Ratio of Outside Radius to Inside Radius, Thick Cylinders below is for convenience in calculating the dimensions of cylinders under high internal pressure without the use of Lames Equation.

… … … … … … … … … … … … … Allowable Stress per Sq. In. of Section Working Pressure in Cylinder, Pounds per Square Inch 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 2000 1.732 2500 1.528 2.000 3000 1.414 1.732 2.236 3500 1.342 1.581 1.915 2.449 4000 1.291 1.483 1.732 2.082 2.646 4500 1.254 1.414 1.612 1.871 2.236 2.828 5000 1.225 1.363 1.528 1.732 2.000 2.380 3.000 5500 1.202 1.323 1.464 1.633 1.844 2.121 2.517 3.162 6000 1.183 1.291 1.414 1.558 1.732 1.949 2.236 2.646 3.317 6500 1.265 1.374 1.500 1.648 1.826 2.049 2.345 2.769 3.464 7000 1.243 1.342 1.453 1.581 1.732 1.915 2.145 2.449 2.887 3.606 7500 1.225 1.314 1.414 1.528 1.658 1.813 2.000 2.236 2.550 3.000 3.742 8000 1.209 1.291 1.382 1.483 1.599 1.732 1.890 2.082 2.324 2.646 3.109 3.873 8500 1.195 1.271 1.354 1.446 1.549 1.667 1.803 1.964 2.160 2.408 2.739 3.215 9000 1.183 1.254 1.330 1.414 1.508 1.612 1.732 1.871 2.035 2.236 2.490 2.828 9500 1.238 1.309 1.387 1.472 1.567 1.673 1.795 1.936 2.104 2.309 2.569 10,000 1.225 1.291 1.363 1.441 1.528 1.624 1.732 1.856 2.000 2.171 2.380 10,500 1.213 1.275 1.342 1.414 1.494 1.581 1.679 1.789 1.915 2.062 2.236 11,000 1.202 1.260 1.323 1.390 1.464 1.544 1.633 1.732 1.844 1.972 2.121 11,500 1.192 1.247 1.306 1.369 1.438 1.512 1.593 1.683 1.784 1.897 2.028 12,000 1.183 1.235 1.291 1.350 1.414 1.483 1.558 1.641 1.732 1.834 1.949 12,500 1.225 1.277 1.333 1.393 1.458 1.528 1.604 1.687 1.780 1.883 13,000 1.215 1.265 1.318 1.374 1.435 1.500 1.571 1.648 1.732 1.826 13,500 1.206 1.254 1.304 1.357 1.414 1.475 1.541 1.612 1.690 1.776 14,000 1.198 1.243 1.291 1.342 1.395 1.453 1.515 1.581 1.653 1.732 14,500 1.190 1.234 1.279 1.327 1.378 1.433 1.491 1.553 1.620 1.693 15,000 1.183 1.225 1.268 1.314 1.363 1.414 1.469 1.528 1.590 1.658 16,000 1.171 1.209 1.249 1.291 1.335 1.382 1.431 1.483 1.539 1.599

Example, Use of the Table: Assume that a cylinder of 10 inches inside diameter is to withstand a pressure of 2500 psi; the material is cast iron and the allowable stress is 6000 psi. To solve the problem, locate the allowable stress per square inch in the left-hand column of the table and the working pressure at the top of the columns. Then find the ratio between the outside and inside radii in the body of the table. In this example, the ratio is 1.558, and hence the outside diameter of the cylinder should be 10 × 1.558, or about 155⁄8 inches. The thickness of the cylinder wall will therefore be (15.58 − 10)/2 = 2.79 inches.

Reference: Machinerys Handbook 30th Edition

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