1918:
115:
2287:
267:
1862:
Fracture is governed by the hoop stress in the absence of other external loads since it is the largest principal stress. Note that a hoop experiences the greatest stress at its inside (the outside and inside experience the same total strain, which is distributed over different circumferences); hence
1848:
In thick-walled cylinders, the maximum shear stress at any point is given by half of the algebraic difference between the maximum and minimum stresses, which is, therefore, equal to half the difference between the hoop and radial stresses. The shearing stress reaches a maximum at the inner surface,
100:
In thick-walled pressure vessels, construction techniques allowing for favorable initial stress patterns can be utilized. These compressive stresses at the inner surface reduce the overall hoop stress in pressurized cylinders. Cylindrical vessels of this nature are generally constructed from
1708:
1844:
In pressure vessel theory, any given element of the wall is evaluated in a tri-axial stress system, with the three principal stresses being hoop, longitudinal, and radial. Therefore, by definition, there exist no shear stresses on the transverse, tangential, or radial planes.
1840:
is less than 10, the radial stress, in proportion to the other stresses, becomes non-negligible (i.e. P is no longer much, much less than Pr/t and Pr/2t), and so the thickness of the wall becomes a major consideration for design (Harvey, 1974, pp. 57).
1867:
the pipe. This is why pipe inspections after earthquakes usually involve sending a camera inside a pipe to inspect for cracks. Yielding is governed by an equivalent stress that includes hoop stress and the longitudinal or radial stress when absent.
1900:
forms in a blood vessel wall, the radius of the vessel has increased. This means that the inward force on the vessel decreases, and therefore the aneurysm will continue to expand until it ruptures. A similar logic applies to the formation of
322:
For the thin-walled assumption to be valid, the vessel must have a wall thickness of no more than about one-tenth (often cited as
Diameter / t > 20) of its radius. This allows for treating the wall as a surface, and subsequently using the
656:
1972:. The vertical, longitudinal force is a compressive force, which cast iron is well able to resist. The hoop stress is tensile, and so wrought iron, a material with better tensile strength than cast iron, is added.
1491:
1381:
1503:
122:
The hoop stress is the force over area exerted circumferentially (perpendicular to the axis and the radius of the object) in both directions on every particle in the cylinder wall. It can be described as:
907:
2077:
1042:
437:
546:
When the vessel has closed ends, the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.
378:
174:
983:
869:
717:
101:
concentric cylinders shrunk over (or expanded into) one another, i.e., built-up shrink-fit cylinders, but can also be performed to singular cylinders though autofrettage of thick cylinders.
490:
261:
794:
1838:
829:
89:
differential will ultimately give rise to hoop stresses. Similarly, if this pipe has flat end caps, any force applied to them by static pressure will induce a perpendicular
750:
1245:
1168:
1803:
1744:
1272:
1195:
500:
may reach several atmospheres. In practical engineering applications for cylinders (pipes and tubes), hoop stress is often re-arranged for pressure, and is called
1770:
1110:
1088:
1068:
1952:. Fairbairn realized that the hoop stress was twice the longitudinal stress, an important factor in the assembly of boiler shells from rolled sheets joined by
1849:
which is significant because it serves as a criterion for failure since it correlates well with actual rupture tests of thick cylinders (Harvey, 1974, p. 57).
2084:
552:
70:
These three principal stresses- hoop, longitudinal, and radial can be calculated analytically using a mutually perpendicular tri-axial stress system.
496:
The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal
1994:
283:
1388:
1703:{\displaystyle \sigma _{\theta }={\dfrac {P_{a}a^{2}-P_{b}b^{2}}{b^{2}-a^{2}}}+{\dfrac {a^{2}b^{2}(P_{a}-P_{b})}{(b^{2}-a^{2})r^{2}}}\ }
1284:
2370:
2216:
2304:
31:
distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.
874:
989:
2351:
909:) the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and
2308:
2323:
387:
333:
129:
930:
842:
512:
309:. These components of force induce corresponding stresses: radial stress, axial stress, and hoop stress, respectively.
2330:
1893:
667:
324:
1917:
2251:
2297:
2337:
1984:
831:
is large, so in most cases this component is considered negligible compared to the hoop and axial stresses.
327:
for estimating the hoop stress created by an internal pressure on a thin-walled cylindrical pressure vessel:
2319:
1961:
1925:
1885:
222:), which usually is defined as the total circumferential force exerted along the entire radial thickness:
1936:
The first theoretical analysis of the stress in cylinders was developed by the mid-19th century engineer
467:
228:
758:
2236:
1811:
802:
190:
exerted circumferentially on an area of the cylinder wall that has the following two lengths as sides:
2004:
290:
2129:"Pressure Vessel, Thin Wall Hoop and Longitudinal Stresses Equation and Calculator - Engineers Edge"
725:
501:
294:
97:, but accurate models of thicker-walled cylindrical shells require such stresses to be considered.
74:
1202:
1125:
2392:
2261:
2238:
Two
Lectures: The Construction of Boilers, and On Boiler Explosions, with the means of prevention
2052:
1889:
752:
that is developed perpendicular to the surface and may be estimated in thin walled cylinders as:
2128:
2232:
2212:
1949:
1937:
2344:
2118:
By R Nave. Department of
Physics and Astronomy, Georgia State University. Retrieved June 2011
1775:
1716:
2009:
1989:
1945:
1941:
279:
78:
28:
1250:
1173:
2273:
2034:
82:
2115:
1749:
921:
2102:
2019:
1095:
1073:
1053:
266:
114:
2142:
2386:
2039:
1906:
528:
66:, a normal stress in directions coplanar with but perpendicular to the symmetry axis.
62:
1969:
1929:
1881:
910:
2211:. Vol. II: Communications and Coal. Stroud: The History Press. p. 247.
651:{\displaystyle \sigma _{z}={\dfrac {F}{A}}={\dfrac {Pd^{2}}{(d+2t)^{2}-d^{2}}}\ }
2286:
1902:
2169:
1957:
1112:
is the radius at the point of interest (e.g., at the inside or outside walls).
1090:
are constants of integration, which may be found from the boundary conditions,
1965:
1921:
1877:
20:
1497:
Using these constants, the following equation for hoop stress is obtained:
2046:
2014:
1953:
1897:
86:
34:
49:
2170:"Mechanics of Materials - Part 35 (Thick cylinder - Lame's equation)"
2024:
1956:. Later work was applied to bridge-building and the invention of the
497:
1932:
hoops (stronger in tension than cast iron) resist the hoop stresses.
1486:{\displaystyle B={\dfrac {a^{2}b^{2}(P_{a}-P_{b})}{b^{2}-a^{2}}}\ }
1916:
265:
187:
93:
on the same pipe wall. Thin sections often have negligibly small
1376:{\displaystyle A={\dfrac {P_{a}a^{2}-P_{b}b^{2}}{b^{2}-a^{2}}}\ }
58:, a normal stress parallel to the axis of cylindrical symmetry.
2280:
73:
The classical example (and namesake) of hoop stress is the
902:{\displaystyle {\text{diameter}}/{\text{thickness}}<20}
1772:
and a solid cylinder cannot have an internal pressure so
1037:{\displaystyle \sigma _{\theta }=A+{\dfrac {B}{r^{2}}}\ }
297:
into components parallel to the cylindrical coordinates
920:, a set of equations developed by French mathematician
916:
These stresses and strains can be calculated using the
1944:. Their first interest was in studying the design and
913:
through the cross section can no longer be neglected.
85:, any force applied to the cylindrical pipe wall by a
2083:. Swansea University. 2020. p. 8. Archived from
1816:
1814:
1778:
1752:
1719:
1600:
1521:
1506:
1399:
1391:
1295:
1287:
1253:
1205:
1176:
1128:
1098:
1076:
1056:
1013:
992:
954:
933:
877:
845:
807:
805:
761:
728:
685:
670:
585:
570:
555:
470:
432:{\displaystyle \sigma _{\theta }={\dfrac {Pr}{2t}}\ }
405:
390:
351:
336:
239:
231:
147:
132:
373:{\displaystyle \sigma _{\theta }={\dfrac {Pr}{t}}\ }
169:{\displaystyle \sigma _{\theta }={\dfrac {F}{tl}}\ }
2311:. Unsourced material may be challenged and removed.
978:{\displaystyle \sigma _{r}=A-{\dfrac {B}{r^{2}}}\ }
1832:
1797:
1764:
1738:
1702:
1485:
1375:
1266:
1239:
1189:
1162:
1104:
1082:
1062:
1036:
977:
901:
864:{\displaystyle {\text{radius}}/{\text{thickness}}}
863:
823:
788:
744:
711:
650:
484:
431:
372:
255:
168:
1808:Being that for thick-walled cylinders, the ratio
481:
77:applied to the iron bands, or hoops, of a wooden
1863:cracks in pipes should theoretically start from
2072:
2070:
712:{\displaystyle \sigma _{z}={\dfrac {Pr}{2t}}\ }
278:, circumferential stress is a component of the
2103:"Theory and Design of Modern Pressure Vessels"
1968:pillars are strengthened by external bands of
1892:on the wall of the vessel. As a result of the
8:
16:Rotationally symmetric stress distribution
2371:Learn how and when to remove this message
1815:
1813:
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1718:
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760:
733:
727:
684:
675:
669:
635:
622:
595:
584:
569:
560:
554:
507:Inch-pound-second system (IPS) units for
475:
469:
404:
395:
389:
350:
341:
335:
238:
230:
146:
137:
131:
2105:. Van Nostrand Reinhold. pp. 60–61.
1995:Boiling liquid expanding vapor explosion
799:In the thin-walled assumption the ratio
210:in describing circumferential stress is
113:
2235:(1851). "The Construction of Boilers".
2066:
2020:Blood pressure#Relation_to_wall_tension
1940:, assisted by his mathematical analyst
1119:For cylinder with boundary conditions:
196:is the radial thickness of the cylinder
2269:
2259:
2031:Designs very affected by this stress:
1278:the following constants are obtained:
871:ratio of less than 10 (often cited as
839:When the cylinder to be studied has a
48:, a normal stress in the tangential (
7:
2309:adding citations to reliable sources
2197:Mosby Elsevier, Rapid Review Series.
293:any force applied to an object with
202:is the axial length of the cylinder.
661:Though this may be approximated to
485:{\displaystyle \sigma _{\theta }\!}
256:{\displaystyle T={\dfrac {F}{l}}\ }
1981:Can be caused by cylinder stress:
1888:, the wall tension represents the
789:{\displaystyle \sigma _{r}={-P}\ }
462:is the mean radius of the cylinder
14:
1833:{\displaystyle {\dfrac {r}{t}}\ }
824:{\displaystyle {\dfrac {r}{t}}\ }
2285:
2296:needs additional citations for
2078:"Advanced Structural Analysis"
1680:
1654:
1649:
1623:
1448:
1422:
1221:
1209:
1144:
1132:
722:There is also a radial stress
619:
603:
523:are inches (in). SI units for
1:
745:{\displaystyle \sigma _{r}\ }
313:Relation to internal pressure
2253:Thin-walled Pressure Vessels
2001:Related engineering topics:
1240:{\displaystyle p(r=b)=P_{b}}
1163:{\displaystyle p(r=a)=P_{a}}
513:pounds-force per square inch
274:Along with axial stress and
2409:
2207:Jones, Stephen K. (2009).
2116:Tension in Arterial Walls
118:Components of hoop stress
37:stress patterns include:
2101:Harvey, John F. (1974).
1985:Boston Molasses Disaster
1247:(i.e. external pressure
1170:(i.e. internal pressure
450:is the internal pressure
289:It is usually useful to
81:. In a straight, closed
1962:Chepstow Railway Bridge
1926:Chepstow Railway Bridge
1798:{\displaystyle A=P_{o}}
1739:{\displaystyle R_{i}=0}
270:Cylindrical coordinates
2025:Piping#Stress_analysis
1933:
1886:gastrointestinal walls
1834:
1799:
1766:
1740:
1713:For a solid cylinder:
1704:
1487:
1377:
1268:
1241:
1191:
1164:
1106:
1084:
1064:
1038:
979:
903:
865:
825:
790:
746:
713:
652:
486:
433:
374:
325:Young–Laplace equation
318:Thin-walled assumption
271:
257:
170:
119:
42:circumferential stress
2209:Brunel in South Wales
1920:
1835:
1800:
1767:
1741:
1705:
1488:
1378:
1269:
1267:{\displaystyle P_{b}}
1242:
1192:
1190:{\displaystyle P_{a}}
1165:
1107:
1085:
1065:
1039:
980:
904:
866:
826:
791:
747:
714:
653:
487:
456:is the wall thickness
434:
375:
269:
258:
171:
117:
2305:improve this article
2005:Stress concentration
1928:, 1852. Pin-jointed
1812:
1776:
1750:
1717:
1504:
1389:
1285:
1251:
1203:
1174:
1126:
1096:
1074:
1054:
990:
931:
875:
843:
835:Thick-walled vessels
803:
759:
726:
668:
553:
468:
388:
334:
229:
130:
1765:{\displaystyle B=0}
543:are in meters (m).
492:is the hoop stress.
295:rotational symmetry
2233:Fairbairn, William
2195:Pathology, 2nd ed.
2143:"Pressure Vessels"
2090:on 19 August 2019.
2053:Florence Cathedral
1934:
1913:Theory development
1830:
1825:
1795:
1762:
1736:
1700:
1695:
1594:
1483:
1478:
1373:
1368:
1274:at outer surface),
1264:
1237:
1197:at inner surface),
1187:
1160:
1102:
1080:
1060:
1034:
1029:
975:
970:
899:
861:
821:
816:
786:
742:
709:
704:
648:
643:
579:
515:(psi). Units for
482:
429:
424:
370:
365:
272:
253:
248:
206:An alternative to
166:
161:
120:
2381:
2380:
2373:
2355:
2320:"Cylinder stress"
1938:William Fairbairn
1853:Practical effects
1829:
1824:
1699:
1694:
1593:
1482:
1477:
1372:
1367:
1105:{\displaystyle r}
1083:{\displaystyle B}
1063:{\displaystyle A}
1033:
1028:
974:
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891:
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859:
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820:
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785:
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703:
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428:
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252:
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165:
160:
2400:
2376:
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2362:
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2354:
2313:
2289:
2281:
2277:
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2267:
2265:
2257:
2243:
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2229:
2223:
2222:
2204:
2198:
2191:
2185:
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2182:
2180:
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2160:
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2157:
2156:
2147:
2139:
2133:
2132:
2125:
2119:
2113:
2107:
2106:
2098:
2092:
2091:
2089:
2082:
2074:
2010:Hydrostatic test
1990:Boiler explosion
1942:Eaton Hodgkinson
1890:muscular tension
1839:
1837:
1836:
1831:
1827:
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1817:
1804:
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1801:
1796:
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1793:
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1737:
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1709:
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1706:
1701:
1697:
1696:
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1692:
1691:
1679:
1678:
1666:
1665:
1652:
1648:
1647:
1635:
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1622:
1621:
1612:
1611:
1601:
1595:
1592:
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1590:
1578:
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1567:
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1565:
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1516:
1515:
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1462:
1461:
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1447:
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1434:
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1420:
1411:
1410:
1400:
1382:
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1379:
1374:
1370:
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1365:
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1352:
1351:
1341:
1340:
1339:
1330:
1329:
1317:
1316:
1307:
1306:
1296:
1273:
1271:
1270:
1265:
1263:
1262:
1246:
1244:
1243:
1238:
1236:
1235:
1196:
1194:
1193:
1188:
1186:
1185:
1169:
1167:
1166:
1161:
1159:
1158:
1111:
1109:
1108:
1103:
1089:
1087:
1086:
1081:
1069:
1067:
1066:
1061:
1043:
1041:
1040:
1035:
1031:
1030:
1027:
1026:
1014:
1002:
1001:
984:
982:
981:
976:
972:
971:
968:
967:
955:
943:
942:
908:
906:
905:
900:
892:
889:
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882:
879:
870:
868:
867:
862:
860:
857:
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850:
847:
830:
828:
827:
822:
818:
817:
808:
795:
793:
792:
787:
783:
782:
771:
770:
751:
749:
748:
743:
739:
738:
737:
718:
716:
715:
710:
706:
705:
702:
694:
686:
680:
679:
657:
655:
654:
649:
645:
644:
641:
640:
639:
627:
626:
601:
600:
599:
586:
580:
571:
565:
564:
502:Barlow's formula
491:
489:
488:
483:
480:
479:
438:
436:
435:
430:
426:
425:
422:
414:
406:
400:
399:
380:(for a cylinder)
379:
377:
376:
371:
367:
366:
360:
352:
346:
345:
262:
260:
259:
254:
250:
249:
240:
175:
173:
172:
167:
163:
162:
159:
148:
142:
141:
2408:
2407:
2403:
2402:
2401:
2399:
2398:
2397:
2383:
2382:
2377:
2366:
2360:
2357:
2314:
2312:
2302:
2290:
2268:
2258:
2256:. 19 June 2008.
2250:
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2246:
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2230:
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2201:
2192:
2188:
2178:
2176:
2168:
2167:
2163:
2154:
2152:
2145:
2141:
2140:
2136:
2127:
2126:
2122:
2114:
2110:
2100:
2099:
2095:
2087:
2080:
2076:
2075:
2068:
2063:
2035:Pressure vessel
1978:
1915:
1874:
1860:
1855:
1810:
1809:
1785:
1774:
1773:
1748:
1747:
1720:
1715:
1714:
1683:
1670:
1657:
1653:
1639:
1626:
1613:
1603:
1602:
1582:
1569:
1568:
1557:
1547:
1534:
1524:
1523:
1507:
1502:
1501:
1466:
1453:
1452:
1438:
1425:
1412:
1402:
1401:
1387:
1386:
1356:
1343:
1342:
1331:
1321:
1308:
1298:
1297:
1283:
1282:
1254:
1249:
1248:
1227:
1201:
1200:
1177:
1172:
1171:
1150:
1124:
1123:
1116:
1094:
1093:
1072:
1071:
1052:
1051:
1018:
993:
988:
987:
959:
934:
929:
928:
873:
872:
841:
840:
837:
801:
800:
762:
757:
756:
729:
724:
723:
695:
687:
671:
666:
665:
631:
618:
602:
591:
587:
556:
551:
550:
498:turgor pressure
471:
466:
465:
415:
407:
391:
386:
385:
353:
337:
332:
331:
320:
315:
282:in cylindrical
227:
226:
152:
133:
128:
127:
112:
107:
25:cylinder stress
17:
12:
11:
5:
2406:
2404:
2396:
2395:
2385:
2384:
2379:
2378:
2293:
2291:
2284:
2279:
2278:
2245:
2244:
2224:
2217:
2199:
2186:
2161:
2134:
2120:
2108:
2093:
2065:
2064:
2062:
2059:
2058:
2057:
2056:
2055:
2049:
2044:
2043:
2042:
2029:
2028:
2027:
2022:
2017:
2012:
2007:
1999:
1998:
1997:
1992:
1987:
1977:
1974:
1914:
1911:
1894:Law of Laplace
1873:
1870:
1859:
1856:
1854:
1851:
1823:
1820:
1792:
1788:
1784:
1781:
1761:
1758:
1755:
1735:
1732:
1727:
1723:
1711:
1710:
1690:
1686:
1682:
1677:
1673:
1669:
1664:
1660:
1656:
1651:
1646:
1642:
1638:
1633:
1629:
1625:
1620:
1616:
1610:
1606:
1598:
1589:
1585:
1581:
1576:
1572:
1564:
1560:
1554:
1550:
1546:
1541:
1537:
1531:
1527:
1519:
1514:
1510:
1495:
1494:
1473:
1469:
1465:
1460:
1456:
1450:
1445:
1441:
1437:
1432:
1428:
1424:
1419:
1415:
1409:
1405:
1397:
1394:
1384:
1363:
1359:
1355:
1350:
1346:
1338:
1334:
1328:
1324:
1320:
1315:
1311:
1305:
1301:
1293:
1290:
1276:
1275:
1261:
1257:
1234:
1230:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1198:
1184:
1180:
1157:
1153:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1114:
1113:
1101:
1091:
1079:
1059:
1045:
1044:
1025:
1021:
1017:
1011:
1008:
1005:
1000:
996:
985:
966:
962:
958:
952:
949:
946:
941:
937:
918:Lamé equations
898:
895:
886:
854:
836:
833:
814:
811:
797:
796:
781:
778:
774:
769:
765:
736:
732:
720:
719:
701:
698:
693:
690:
683:
678:
674:
659:
658:
638:
634:
630:
625:
621:
617:
614:
611:
608:
605:
598:
594:
590:
583:
577:
574:
568:
563:
559:
494:
493:
478:
474:
463:
457:
451:
441:
440:
439:(for a sphere)
421:
418:
413:
410:
403:
398:
394:
382:
381:
363:
359:
356:
349:
344:
340:
319:
316:
314:
311:
264:
263:
246:
243:
237:
234:
204:
203:
197:
191:
177:
176:
158:
155:
151:
145:
140:
136:
111:
108:
106:
103:
68:
67:
59:
53:
15:
13:
10:
9:
6:
4:
3:
2:
2405:
2394:
2391:
2390:
2388:
2375:
2372:
2364:
2353:
2350:
2346:
2343:
2339:
2336:
2332:
2329:
2325:
2322: –
2321:
2317:
2316:Find sources:
2310:
2306:
2300:
2299:
2294:This article
2292:
2288:
2283:
2282:
2275:
2263:
2255:
2254:
2249:
2248:
2240:
2239:
2234:
2228:
2225:
2220:
2218:9780752449128
2214:
2210:
2203:
2200:
2196:
2190:
2187:
2175:
2171:
2165:
2162:
2151:
2144:
2138:
2135:
2130:
2124:
2121:
2117:
2112:
2109:
2104:
2097:
2094:
2086:
2079:
2073:
2071:
2067:
2060:
2054:
2050:
2048:
2045:
2041:
2040:Rocket engine
2038:
2037:
2036:
2033:
2032:
2030:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2002:
2000:
1996:
1993:
1991:
1988:
1986:
1983:
1982:
1980:
1979:
1975:
1973:
1971:
1967:
1963:
1959:
1955:
1951:
1950:steam boilers
1947:
1943:
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1931:
1927:
1923:
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1912:
1910:
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1658:
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1636:
1631:
1627:
1618:
1614:
1608:
1604:
1596:
1587:
1583:
1579:
1574:
1570:
1562:
1558:
1552:
1548:
1544:
1539:
1535:
1529:
1525:
1517:
1512:
1508:
1500:
1499:
1498:
1471:
1467:
1463:
1458:
1454:
1443:
1439:
1435:
1430:
1426:
1417:
1413:
1407:
1403:
1395:
1392:
1385:
1361:
1357:
1353:
1348:
1344:
1336:
1332:
1326:
1322:
1318:
1313:
1309:
1303:
1299:
1291:
1288:
1281:
1280:
1279:
1259:
1255:
1232:
1228:
1224:
1218:
1215:
1212:
1206:
1199:
1182:
1178:
1155:
1151:
1147:
1141:
1138:
1135:
1129:
1122:
1121:
1120:
1117:
1099:
1092:
1077:
1057:
1050:
1049:
1048:
1023:
1019:
1015:
1009:
1006:
1003:
998:
994:
986:
964:
960:
956:
950:
947:
944:
939:
935:
927:
926:
925:
923:
919:
914:
912:
896:
893:
884:
852:
834:
832:
812:
809:
779:
776:
772:
767:
763:
755:
754:
753:
734:
730:
699:
696:
691:
688:
681:
676:
672:
664:
663:
662:
636:
632:
628:
623:
615:
612:
609:
606:
596:
592:
588:
581:
575:
572:
566:
561:
557:
549:
548:
547:
544:
542:
538:
534:
530:
526:
522:
518:
514:
510:
505:
503:
499:
476:
472:
464:
461:
458:
455:
452:
449:
446:
445:
444:
419:
416:
411:
408:
401:
396:
392:
384:
383:
361:
357:
354:
347:
342:
338:
330:
329:
328:
326:
317:
312:
310:
308:
304:
300:
296:
292:
287:
285:
281:
280:stress tensor
277:
276:radial stress
268:
244:
241:
235:
232:
225:
224:
223:
221:
217:
213:
209:
201:
198:
195:
192:
189:
185:
182:
181:
180:
156:
153:
149:
143:
138:
134:
126:
125:
124:
116:
109:
104:
102:
98:
96:
95:radial stress
92:
88:
84:
80:
76:
71:
65:
64:
63:radial stress
60:
57:
54:
51:
47:
43:
40:
39:
38:
36:
32:
30:
26:
22:
2367:
2358:
2348:
2341:
2334:
2327:
2315:
2303:Please help
2298:verification
2295:
2252:
2241:. p. 6.
2237:
2227:
2208:
2202:
2194:
2189:
2177:. Retrieved
2173:
2164:
2153:. Retrieved
2149:
2137:
2123:
2111:
2096:
2085:the original
2051:The dome of
1970:wrought iron
1935:
1930:wrought iron
1875:
1864:
1861:
1847:
1843:
1807:
1712:
1496:
1277:
1118:
1115:
1046:
922:Gabriel Lamé
917:
915:
911:shear stress
838:
798:
721:
660:
545:
540:
536:
532:
531:(Pa), while
524:
520:
516:
508:
506:
495:
459:
453:
447:
442:
321:
306:
302:
298:
288:
275:
273:
219:
216:wall tension
215:
211:
207:
205:
199:
193:
183:
178:
121:
99:
94:
91:axial stress
90:
72:
69:
61:
56:axial stress
55:
52:) direction.
45:
41:
33:
24:
18:
2270:|work=
2193:E. Goljan,
2174:youtube.com
2150:web.mit.edu
1903:diverticuli
1858:Engineering
284:coordinates
212:wall stress
208:hoop stress
110:Hoop stress
105:Definitions
46:hoop stress
2361:March 2012
2331:newspapers
2179:23 October
2155:2020-06-12
2061:References
1958:box girder
1924:pillar of
2393:Mechanics
2272:ignored (
2262:cite book
1966:cast iron
1960:. In the
1922:Cast iron
1878:pathology
1668:−
1637:−
1580:−
1545:−
1513:θ
1509:σ
1464:−
1436:−
1354:−
1319:−
999:θ
995:σ
951:−
936:σ
890:thickness
858:thickness
777:−
764:σ
731:σ
673:σ
629:−
558:σ
477:θ
473:σ
397:θ
393:σ
343:θ
339:σ
291:decompose
139:θ
135:σ
21:mechanics
2387:Category
2047:Flywheel
2015:Buckling
1976:See also
1954:riveting
1946:failures
1898:aneurysm
1896:, if an
1882:vascular
1872:Medicine
880:diameter
87:pressure
35:Cylinder
2345:scholar
1905:in the
1876:In the
1047:where:
529:pascals
186:is the
179:where:
75:tension
50:azimuth
2347:
2340:
2333:
2326:
2318:
2215:
1964:, the
1865:inside
1828:
1698:
1481:
1371:
1032:
973:
848:radius
819:
784:
740:
707:
646:
519:, and
443:where
427:
368:
305:, and
251:
164:
79:barrel
29:stress
2352:JSTOR
2338:books
2146:(PDF)
2088:(PDF)
2081:(PDF)
1746:then
1070:and
188:force
44:, or
27:is a
2324:news
2274:help
2213:ISBN
2181:2022
894:<
535:and
527:are
511:are
83:pipe
23:, a
2307:by
1948:of
1907:gut
1884:or
1880:of
214:or
19:In
2389::
2266::
2264:}}
2260:{{
2172:.
2148:.
2069:^
1909:.
1805:.
924:.
897:20
539:=2
504:.
301:,
286:.
2374:)
2368:(
2363:)
2359:(
2349:·
2342:·
2335:·
2328:·
2301:.
2276:)
2221:.
2183:.
2158:.
2131:.
1822:t
1819:r
1791:o
1787:P
1783:=
1780:A
1760:0
1757:=
1754:B
1734:0
1731:=
1726:i
1722:R
1689:2
1685:r
1681:)
1676:2
1672:a
1663:2
1659:b
1655:(
1650:)
1645:b
1641:P
1632:a
1628:P
1624:(
1619:2
1615:b
1609:2
1605:a
1597:+
1588:2
1584:a
1575:2
1571:b
1563:2
1559:b
1553:b
1549:P
1540:2
1536:a
1530:a
1526:P
1518:=
1493:.
1472:2
1468:a
1459:2
1455:b
1449:)
1444:b
1440:P
1431:a
1427:P
1423:(
1418:2
1414:b
1408:2
1404:a
1396:=
1393:B
1383:,
1362:2
1358:a
1349:2
1345:b
1337:2
1333:b
1327:b
1323:P
1314:2
1310:a
1304:a
1300:P
1292:=
1289:A
1260:b
1256:P
1233:b
1229:P
1225:=
1222:)
1219:b
1216:=
1213:r
1210:(
1207:p
1183:a
1179:P
1156:a
1152:P
1148:=
1145:)
1142:a
1139:=
1136:r
1133:(
1130:p
1100:r
1078:B
1058:A
1024:2
1020:r
1016:B
1010:+
1007:A
1004:=
965:2
961:r
957:B
948:A
945:=
940:r
885:/
853:/
813:t
810:r
780:P
773:=
768:r
735:r
700:t
697:2
692:r
689:P
682:=
677:z
637:2
633:d
624:2
620:)
616:t
613:2
610:+
607:d
604:(
597:2
593:d
589:P
582:=
576:A
573:F
567:=
562:z
541:r
537:d
533:t
525:P
521:d
517:t
509:P
460:r
454:t
448:P
420:t
417:2
412:r
409:P
402:=
362:t
358:r
355:P
348:=
307:θ
303:z
299:r
245:l
242:F
236:=
233:T
220:T
218:(
200:l
194:t
184:F
157:l
154:t
150:F
144:=
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