531:
1965:
92:
51:
194:
1674:) applied parallel to the cross-section, the provided value of rigidity does not have any bearing on the torsional resistance provided to an object as a function of its constituent materials. The rigidity provided by an object's material is a characteristic of its
1519:
2512:
1564:
The polar second moment of area can be insufficient for use to analyze beams and shafts with non-circular cross-sections, due their tendency to warp when twisted, causing out-of-plane deformations. In such cases, a
1007:
2410:
2270:
2650:
517:
is a shaft or beam's resistance to being distorted by torsion, as a function of its shape. The rigidity comes from the object's cross-sectional area only, and does not depend on its material composition or
914:
2200:
1298:
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1857:
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In objects with significant cross-sectional variation (along the axis of the applied torque), which cannot be analyzed in segments, a more complex approach may have to be used. See
1403:
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should be substituted, where an appropriate deformation constant is included to compensate for the warping effect. Within this, there are articles that differentiate between the
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2110:
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As shown, the larger the material's shear modulus and polar second moment of area (i.e. larger cross-sectional area), the greater resistance to torsional deflection.
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693:
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656:(those regarding inertia) are integrated over all the infinitesimal elements of mass in a three-dimensional space occupied by an object. Simply put, the
2925:"What is the difference between the Polar Second Moment of Area ("Polar Moment of Inertia"), IPIP and the torsional constant, JTJT of a cross section?"
918:
2340:
2211:
2578:
840:
271:, also known (incorrectly, colloquially) as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to
2146:
218:
109:
64:
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128:
1241:
2524:
135:
1816:
1080:
1018:
17:
1761:
632:, used to describe a massive object's resistance to rotational motion, not its resistance to torsional deformation. While the
2907:
2037:
Hz. This is assuming that there is a 1:1 correlation between rotational velocity of turbine and the frequency of mains power.
142:
700:
2860:
Ugural AC, Fenster SK. Advanced
Strength and Applied Elasticity. 3rd Ed. Prentice-Hall Inc. Englewood Cliffs, NJ. 1995.
124:
2702:
2865:
254:
236:
175:
78:
2048:
1306:. For objects that have rotational symmetry, such as a cylinder or hollow tube, the equation can be simplified to:
522:. The greater the magnitude of the second polar moment of area, the greater the torsional stiffness of the object.
70:
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311:
applied in a plane perpendicular to the object's central axis (i.e. parallel to the cross-section). Similar to
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664:
second moments of area are an indication of rigidity, and the moment of inertia (second moment of
477:, this designation should be given careful attention so that it does not become confused with the
102:
2827:
1514:{\displaystyle I_{z}=\int _{0}^{2\pi }\int _{0}^{R}r^{2}(r\,dr\,d\phi )={\frac {\pi R^{4}}{2}}}
2940:
2946:
2832:
1721:
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8:
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second moment of area describes an object's resistance to deflection when subjected to a
272:
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2878:
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433:
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Comparing polar second moments of area and moments of inertia (second moments of mass)
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2897:
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672:
640:
second moments of area are integrated over all the infinitesimal elements of a given
629:
204:
31:
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2299:
1953:
1566:
1142:
It is shown that the polar moment of area can be described as the summation of the
479:
308:
300:
283:
and no significant warping or out-of-plane deformation. It is a constituent of the
2507:{\displaystyle I_{c}=I_{z}\rho l={\frac {\pi \rho l\left(D^{4}-d^{4}\right)}{64}}}
430:. While several engineering textbooks and academic publications also denote it as
2080:
2033:; this is the typical frequency in Europe. In North America the frequency is 60
1806:
The polar second moment of area appears in the formulae that describe torsional
2822:
2676:
1655:
1553:
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2837:
1995:
1977:
1969:
1675:
519:
2137:
2001:
1807:
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Though the polar second moment of area is most often used to calculate the
2302:
of 5 and re-calculates the radius with the admissible stress equal to the
530:
1964:
621:
2324:
mm, the approximate size of a turboset shaft in a nuclear power plant.
295:
second moment of area describes an object's resistance to deflection (
1991:
2763:
is the polar second moment of area in meters to the fourth power (m)
91:
2291:
1002:{\displaystyle J=\iint _{A}x^{2}\,dx\,dy+\iint _{A}y^{2}\,dx\,dy}
296:
2405:{\displaystyle I_{z}={\frac {\pi \left(D^{4}-d^{4}\right)}{32}}}
2265:{\displaystyle r={\sqrt{\frac {2T_{\max }}{\pi \tau _{\max }}}}}
2645:{\displaystyle I_{c}=I_{z}\rho l={\frac {\pi \rho lD^{4}}{64}}}
2275:
2122:
2118:
1671:
39:
2030:
1698:. Combining these two features with the length of the shaft,
279:), in objects (or segments of an object) with an invariant
909:{\displaystyle J=\iint _{A}\left(x^{2}+y^{2}\right)dx\,dy}
668:) is the rotational motion resistance of a massive object.
2195:{\displaystyle T_{\max }={\frac {\tau _{\max }J_{z}}{r}}}
1718:, one is able to calculate a shaft's angular deflection,
628:, within physics and mathematics fields, is strictly the
620:
second moments of area, this is primarily a construct of
2952:
2947:
Elastic
Properties and Young Modulus for some Materials
1529:
671:
The equation describing the polar moment of area is a
2793:
2771:
2742:
2713:
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2661:
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2527:
2420:
2343:
2214:
2149:
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1925:
1905:
1885:
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1819:
1764:
1744:
1724:
1704:
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1168:
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1083:
1021:
921:
843:
819:
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773:
753:
703:
681:
584:
564:
544:
489:
456:
436:
409:
375:
348:
321:
538:is calculated for an arbitrary shape about an axis
116:. Unsourced material may be challenged and removed.
2799:
2777:
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2726:
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2667:
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2506:
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2264:
2194:
2104:
2079:The torque carried by the shaft is related to the
2069:
1938:
1911:
1891:
1871:
1851:
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1730:
1710:
1690:
1643:
1623:
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1070:
1001:
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739:
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422:
391:
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334:
2960:
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2155:
27:Quantity of resistance to torsional deformation
2045:can be calculated with the following formula:
1293:{\displaystyle \therefore J=I_{z}=I_{x}+I_{y}}
2879:"Moment Of Inertia; Definition with examples"
2566:{\displaystyle I_{z}={\frac {\pi D^{4}}{32}}}
1651:to describe the polar second moment of area.
608:While it has become common to find the term
1919:is the distance from the central axis, and
1852:{\displaystyle \tau ={\frac {T\,r}{J_{z}}}}
1133:{\displaystyle I_{y}=\iint _{A}x^{2}dx\,dy}
1071:{\displaystyle I_{x}=\iint _{A}y^{2}dx\,dy}
644:in some two-dimensional cross-section, the
79:Learn how and when to remove these messages
403:second moment of area is often denoted as
2114:The angular frequency is therefore 314.16
1832:
1476:
1469:
1123:
1061:
1015:second moments of area equations, where:
992:
985:
955:
948:
899:
730:
299:) when subjected to a force applied to a
255:Learn how and when to remove this message
237:Learn how and when to remove this message
176:Learn how and when to remove this message
1963:
1956:is maximal at the surface of the shaft.
1794:{\displaystyle \theta ={\frac {TL}{JG}}}
529:
2895:
2785:is the length of cylinder in meters (m)
14:
2961:
2208:the following expression is obtained:
578:is the radial distance to the element
2854:
2004:of the steel used to make the shaft (
1959:
740:{\displaystyle J=\iint _{A}r^{2}\,dA}
1946:is the polar second moment of area.
1670:of an object subjected to a moment (
510:, used for non-cylindrical objects.
315:second moment of area calculations (
187:
114:adding citations to reliable sources
85:
44:
1544:), and inches to the fourth power (
1380:For a circular section with radius
24:
2922:
2332:
2316:/5 the result is a radius of 0.343
1987:Power carried by the shaft is 1000
303:parallel to the central axis, the
34:, which characterizes an object's
25:
2990:
2934:
2516:
2026:Electricity has a frequency of 50
1540:, is meters to the fourth power (
60:This article has multiple issues.
2734:is the moment of inertia in kg·m
192:
90:
49:
1879:is the torsional shear stress,
767:is the distance to the element
675:over the cross-sectional area,
101:needs additional citations for
68:or discuss these issues on the
2916:
2889:
2871:
2843:List of second moments of area
2070:{\displaystyle \omega =2\pi f}
1994:; this is typical for a large
1661:
1559:
1483:
1463:
13:
1:
2848:
2337:Polar second moment of area:
1980:shaft radius for a turboset:
1738:, due to the applied torque,
525:
125:"Second polar moment of area"
2953:Material Properties Database
2807:is the specific mass in kg/m
2521:Polar second moment of area
1538:planar second moment of area
534:A schematic showing how the
7:
2811:
2206:polar second moment of area
2083:by the following equation:
1571:polar second moment of area
1534:polar second moment of area
536:second polar moment of area
269:second polar moment of area
214:excessive bold and italics.
212:. The specific problem is:
10:
2995:
2204:After substitution of the
2105:{\displaystyle P=T\omega }
1810:and angular displacement.
1304:perpendicular axis theorem
1302:This is also shown in the
604:
289:perpendicular axis theorem
29:
2896:Obregon, Joaquin (2012).
1952:In a circular shaft, the
2949:- engineeringtoolbox.com
2943:- engineeringtoolbox.com
1371:{\displaystyle J=2I_{y}}
1335:{\displaystyle J=2I_{x}}
30:Not to be confused with
2320:m, or a diameter of 690
2143:The maximal torque is:
1899:is the applied torque,
1731:{\displaystyle \theta }
1523:
18:Polar moment of inertia
2828:Torsion bar suspension
2801:
2779:
2757:
2728:
2695:
2669:
2646:
2567:
2508:
2406:
2266:
2196:
2106:
2071:
1973:
1968:The rotor of a modern
1940:
1913:
1893:
1873:
1853:
1795:
1752:
1732:
1712:
1692:
1645:
1625:
1594:
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1394:
1372:
1336:
1294:
1233:
1206:
1176:
1156:
1134:
1072:
1003:
910:
833:components, using the
827:
807:
784:
761:
741:
689:
602:
595:
572:
552:
504:
471:
444:
424:
393:
392:{\displaystyle I_{xy}}
363:
336:
2974:Mechanical quantities
2833:Second moment of area
2802:
2800:{\displaystyle \rho }
2780:
2758:
2756:{\displaystyle I_{z}}
2729:
2727:{\displaystyle I_{c}}
2696:
2670:
2647:
2568:
2509:
2407:
2267:
2197:
2125:and the torque 3.1831
2107:
2072:
1967:
1941:
1939:{\displaystyle J_{z}}
1914:
1894:
1874:
1872:{\displaystyle \tau }
1854:
1796:
1753:
1733:
1713:
1693:
1646:
1626:
1624:{\displaystyle J_{t}}
1595:
1593:{\displaystyle I_{z}}
1516:
1395:
1373:
1337:
1295:
1234:
1232:{\displaystyle I_{y}}
1207:
1205:{\displaystyle I_{x}}
1177:
1157:
1135:
1073:
1004:
911:
828:
808:
785:
762:
742:
690:
630:second moment of mass
612:used to describe the
596:
573:
571:{\displaystyle \rho }
553:
533:
505:
503:{\displaystyle J_{t}}
472:
470:{\displaystyle J_{z}}
445:
425:
423:{\displaystyle I_{z}}
394:
364:
362:{\displaystyle I_{y}}
337:
335:{\displaystyle I_{x}}
287:, linked through the
285:second moment of area
2791:
2769:
2740:
2711:
2685:
2659:
2579:
2525:
2418:
2341:
2212:
2147:
2087:
2049:
1923:
1903:
1883:
1863:
1817:
1813:Torsional stresses:
1762:
1742:
1722:
1702:
1682:
1668:angular displacement
1635:
1608:
1577:
1550:U.S. Customary units
1404:
1384:
1346:
1310:
1242:
1216:
1189:
1166:
1146:
1081:
1019:
919:
841:
817:
797:
771:
751:
701:
679:
582:
562:
542:
515:polar moment of area
487:
454:
434:
407:
373:
346:
319:
219:improve this article
208:to meet Knowledge's
110:improve this article
36:angular acceleration
2969:Continuum mechanics
2899:Mechanical Simmetry
2414:Moment of inertia:
2298:mm. If one adds a
1976:Calculation of the
1452:
1437:
835:Pythagorean theorem
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2775:
2753:
2724:
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2665:
2642:
2575:Moment of inertia
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2262:
2192:
2102:
2067:
1974:
1960:Sample calculation
1936:
1909:
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1728:
1708:
1688:
1641:
1631:, no longer using
1621:
1602:torsional constant
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1511:
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1420:
1390:
1368:
1332:
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1202:
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906:
823:
803:
783:{\displaystyle dA}
780:
757:
737:
685:
652:second moments of
610:moments of inertia
603:
594:{\displaystyle dA}
591:
568:
548:
500:
467:
440:
420:
389:
359:
332:
2941:Torsion of Shafts
2909:978-1-4772-3372-6
2778:{\displaystyle l}
2694:{\displaystyle D}
2668:{\displaystyle d}
2640:
2561:
2502:
2400:
2260:
2254:
2190:
2043:angular frequency
1912:{\displaystyle r}
1892:{\displaystyle T}
1847:
1789:
1751:{\displaystyle T}
1711:{\displaystyle L}
1691:{\displaystyle G}
1644:{\displaystyle J}
1509:
1393:{\displaystyle R}
1185:moments of area,
1175:{\displaystyle y}
1155:{\displaystyle x}
826:{\displaystyle y}
806:{\displaystyle x}
793:Substituting the
760:{\displaystyle r}
695:, of the object.
688:{\displaystyle A}
673:multiple integral
626:moment of inertia
624:fields. The term
551:{\displaystyle O}
443:{\displaystyle J}
265:
264:
257:
247:
246:
239:
210:quality standards
201:This article may
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185:
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160:
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32:moment of inertia
16:(Redirected from
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2979:Moment (physics)
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2913:
2902:. Author House.
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2818:Torsion constant
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2782:
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2513:
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2505:
2503:
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2323:
2319:
2300:factor of safety
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2222:
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2128:
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2111:
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2103:
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2036:
2029:
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2014:
1990:
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1622:
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1619:
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1589:
1588:
1567:torsion constant
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1505:
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1503:
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1005:
1000:
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983:
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913:
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888:
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832:
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804:
789:
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781:
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729:
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718:
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691:
686:
600:
598:
597:
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577:
575:
574:
569:
557:
555:
554:
549:
513:Simply put, the
509:
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501:
499:
498:
480:torsion constant
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474:
473:
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447:
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235:
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196:
195:
188:
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174:
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159:
118:
94:
86:
75:
53:
52:
45:
21:
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2993:
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2984:
2983:
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2937:
2932:
2921:
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2910:
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2890:
2877:
2876:
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2859:
2855:
2851:
2814:
2792:
2789:
2788:
2770:
2767:
2766:
2747:
2743:
2741:
2738:
2737:
2718:
2714:
2712:
2709:
2708:
2686:
2683:
2682:
2660:
2657:
2656:
2630:
2626:
2616:
2614:
2599:
2595:
2586:
2582:
2580:
2577:
2576:
2551:
2547:
2543:
2541:
2532:
2528:
2526:
2523:
2522:
2519:
2487:
2483:
2474:
2470:
2469:
2465:
2455:
2453:
2438:
2434:
2425:
2421:
2419:
2416:
2415:
2385:
2381:
2372:
2368:
2367:
2363:
2359:
2357:
2348:
2344:
2342:
2339:
2338:
2335:
2333:Hollow Cylinder
2330:
2321:
2317:
2315:
2308:
2295:
2287:
2283:
2255:
2247:
2243:
2239:
2232:
2228:
2224:
2221:
2213:
2210:
2209:
2180:
2176:
2170:
2166:
2165:
2163:
2154:
2150:
2148:
2145:
2144:
2134:
2130:
2126:
2115:
2088:
2085:
2084:
2050:
2047:
2046:
2034:
2027:
2020:
2016:
2012:
2010:
1988:
1962:
1930:
1926:
1924:
1921:
1920:
1904:
1901:
1900:
1884:
1881:
1880:
1864:
1861:
1860:
1841:
1837:
1828:
1826:
1818:
1815:
1814:
1781:
1773:
1771:
1763:
1760:
1759:
1743:
1740:
1739:
1723:
1720:
1719:
1703:
1700:
1699:
1683:
1680:
1679:
1664:
1636:
1633:
1632:
1615:
1611:
1609:
1606:
1605:
1584:
1580:
1578:
1575:
1574:
1562:
1526:
1499:
1495:
1491:
1489:
1457:
1453:
1447:
1442:
1429:
1424:
1411:
1407:
1405:
1402:
1401:
1385:
1382:
1381:
1362:
1358:
1347:
1344:
1343:
1326:
1322:
1311:
1308:
1307:
1284:
1280:
1271:
1267:
1258:
1254:
1243:
1240:
1239:
1223:
1219:
1217:
1214:
1213:
1196:
1192:
1190:
1187:
1186:
1167:
1164:
1163:
1147:
1144:
1143:
1111:
1107:
1101:
1097:
1088:
1084:
1082:
1079:
1078:
1049:
1045:
1039:
1035:
1026:
1022:
1020:
1017:
1016:
979:
975:
969:
965:
942:
938:
932:
928:
920:
917:
916:
882:
878:
869:
865:
864:
860:
854:
850:
842:
839:
838:
818:
815:
814:
798:
795:
794:
772:
769:
768:
752:
749:
748:
724:
720:
714:
710:
702:
699:
698:
680:
677:
676:
669:
583:
580:
579:
563:
560:
559:
543:
540:
539:
528:
494:
490:
488:
485:
484:
461:
457:
455:
452:
451:
435:
432:
431:
414:
410:
408:
405:
404:
380:
376:
374:
371:
370:
353:
349:
347:
344:
343:
326:
322:
320:
317:
316:
261:
250:
249:
248:
243:
232:
226:
223:
216:
197:
193:
182:
171:
165:
162:
119:
117:
107:
95:
54:
50:
43:
28:
23:
22:
15:
12:
11:
5:
2992:
2982:
2981:
2976:
2971:
2957:
2956:
2950:
2944:
2936:
2935:External links
2933:
2931:
2930:
2915:
2908:
2888:
2883:www.efunda.com
2870:
2852:
2850:
2847:
2846:
2845:
2840:
2835:
2830:
2825:
2823:Torsion spring
2820:
2813:
2810:
2809:
2808:
2796:
2786:
2774:
2764:
2750:
2746:
2735:
2721:
2717:
2706:
2703:outer diameter
2690:
2680:
2677:inner diameter
2664:
2639:
2633:
2629:
2625:
2622:
2619:
2613:
2610:
2607:
2602:
2598:
2594:
2589:
2585:
2560:
2554:
2550:
2546:
2540:
2535:
2531:
2518:
2517:Solid cylinder
2515:
2501:
2496:
2490:
2486:
2482:
2477:
2473:
2468:
2464:
2461:
2458:
2452:
2449:
2446:
2441:
2437:
2433:
2428:
2424:
2399:
2394:
2388:
2384:
2380:
2375:
2371:
2366:
2362:
2356:
2351:
2347:
2334:
2331:
2329:
2326:
2313:
2306:
2258:
2250:
2246:
2242:
2235:
2231:
2227:
2220:
2217:
2189:
2183:
2179:
2173:
2169:
2162:
2157:
2153:
2101:
2098:
2095:
2092:
2066:
2063:
2060:
2057:
2054:
2039:
2038:
2024:
2008:
1999:
1961:
1958:
1933:
1929:
1908:
1888:
1868:
1844:
1840:
1835:
1831:
1825:
1822:
1787:
1784:
1779:
1776:
1770:
1767:
1747:
1727:
1707:
1687:
1663:
1660:
1656:3-D elasticity
1640:
1618:
1614:
1587:
1583:
1561:
1558:
1554:imperial units
1525:
1522:
1508:
1502:
1498:
1494:
1488:
1485:
1482:
1479:
1475:
1472:
1468:
1465:
1460:
1456:
1450:
1445:
1441:
1435:
1432:
1427:
1423:
1419:
1414:
1410:
1389:
1365:
1361:
1357:
1354:
1351:
1329:
1325:
1321:
1318:
1315:
1287:
1283:
1279:
1274:
1270:
1266:
1261:
1257:
1253:
1250:
1247:
1226:
1222:
1199:
1195:
1171:
1151:
1129:
1126:
1122:
1119:
1114:
1110:
1104:
1100:
1096:
1091:
1087:
1067:
1064:
1060:
1057:
1052:
1048:
1042:
1038:
1034:
1029:
1025:
998:
995:
991:
988:
982:
978:
972:
968:
964:
961:
958:
954:
951:
945:
941:
935:
931:
927:
924:
905:
902:
898:
895:
891:
885:
881:
877:
872:
868:
863:
857:
853:
849:
846:
822:
802:
779:
776:
756:
736:
733:
727:
723:
717:
713:
709:
706:
684:
590:
587:
567:
547:
527:
524:
497:
493:
464:
460:
439:
417:
413:
386:
383:
379:
356:
352:
329:
325:
263:
262:
245:
244:
200:
198:
191:
184:
183:
98:
96:
89:
84:
58:
57:
55:
48:
26:
9:
6:
4:
3:
2:
2991:
2980:
2977:
2975:
2972:
2970:
2967:
2966:
2964:
2954:
2951:
2948:
2945:
2942:
2939:
2938:
2926:
2919:
2911:
2905:
2901:
2900:
2892:
2884:
2880:
2874:
2867:
2866:0-13-137589-X
2863:
2857:
2853:
2844:
2841:
2839:
2838:Shear modulus
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2815:
2794:
2787:
2772:
2765:
2748:
2744:
2736:
2719:
2715:
2707:
2705:in meters (m)
2704:
2688:
2681:
2679:in meters (m)
2678:
2662:
2655:
2654:
2653:
2637:
2631:
2627:
2623:
2620:
2617:
2611:
2608:
2605:
2600:
2596:
2592:
2587:
2583:
2573:
2558:
2552:
2548:
2544:
2538:
2533:
2529:
2514:
2499:
2494:
2488:
2484:
2480:
2475:
2471:
2466:
2462:
2459:
2456:
2450:
2447:
2444:
2439:
2435:
2431:
2426:
2422:
2412:
2397:
2392:
2386:
2382:
2378:
2373:
2369:
2364:
2360:
2354:
2349:
2345:
2325:
2312:
2305:
2301:
2293:
2281:
2277:
2272:
2256:
2244:
2240:
2229:
2225:
2218:
2215:
2207:
2202:
2187:
2181:
2177:
2167:
2160:
2151:
2141:
2139:
2124:
2120:
2112:
2099:
2096:
2093:
2090:
2082:
2077:
2064:
2061:
2058:
2055:
2052:
2044:
2032:
2025:
2007:
2003:
2000:
1997:
1996:nuclear power
1993:
1986:
1985:
1984:
1983:Assumptions:
1981:
1979:
1978:steam turbine
1971:
1970:steam turbine
1966:
1957:
1955:
1951:
1947:
1931:
1927:
1906:
1886:
1866:
1842:
1838:
1833:
1829:
1823:
1820:
1811:
1809:
1804:
1801:
1785:
1782:
1777:
1774:
1768:
1765:
1745:
1725:
1705:
1685:
1677:
1676:shear modulus
1673:
1669:
1659:
1657:
1652:
1638:
1616:
1612:
1603:
1585:
1581:
1572:
1568:
1557:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1521:
1506:
1500:
1496:
1492:
1486:
1480:
1477:
1473:
1470:
1466:
1458:
1454:
1448:
1443:
1439:
1433:
1430:
1425:
1421:
1417:
1412:
1408:
1387:
1378:
1363:
1359:
1355:
1352:
1349:
1327:
1323:
1319:
1316:
1313:
1305:
1300:
1285:
1281:
1277:
1272:
1268:
1264:
1259:
1255:
1251:
1248:
1245:
1224:
1220:
1197:
1193:
1184:
1169:
1149:
1140:
1127:
1124:
1120:
1117:
1112:
1108:
1102:
1098:
1094:
1089:
1085:
1065:
1062:
1058:
1055:
1050:
1046:
1040:
1036:
1032:
1027:
1023:
1014:
1009:
996:
993:
989:
986:
980:
976:
970:
966:
962:
959:
956:
952:
949:
943:
939:
933:
929:
925:
922:
903:
900:
896:
893:
889:
883:
879:
875:
870:
866:
861:
855:
851:
847:
844:
836:
820:
800:
791:
777:
774:
754:
734:
731:
725:
721:
715:
711:
707:
704:
696:
682:
674:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
588:
585:
565:
545:
537:
532:
523:
521:
520:shear modulus
516:
511:
495:
491:
482:
481:
462:
458:
437:
415:
411:
402:
384:
381:
377:
354:
350:
327:
323:
314:
310:
306:
302:
298:
294:
290:
286:
282:
281:cross-section
278:
275:deformation (
274:
270:
259:
256:
241:
238:
230:
220:
215:
211:
207:
206:
199:
190:
189:
180:
177:
169:
158:
155:
151:
148:
144:
141:
137:
134:
130:
127: –
126:
122:
121:Find sources:
115:
111:
105:
104:
99:This article
97:
93:
88:
87:
82:
80:
73:
72:
67:
66:
61:
56:
47:
46:
41:
37:
33:
19:
2955:- matweb.com
2918:
2898:
2891:
2882:
2873:
2856:
2574:
2520:
2413:
2336:
2310:
2303:
2279:
2273:
2205:
2203:
2142:
2113:
2078:
2040:
2005:
2002:Yield stress
1982:
1975:
1954:shear stress
1949:
1948:
1812:
1805:
1802:
1665:
1653:
1601:
1570:
1563:
1545:
1541:
1537:
1533:
1527:
1379:
1301:
1182:
1141:
1012:
1010:
792:
697:
670:
665:
661:
657:
653:
649:
645:
641:
637:
633:
625:
617:
613:
609:
605:
535:
514:
512:
478:
400:
312:
304:
292:
291:. Where the
268:
266:
251:
233:
227:October 2023
224:
217:Please help
213:
202:
172:
163:
153:
146:
139:
132:
120:
108:Please help
103:verification
100:
76:
69:
63:
62:Please help
59:
1662:Application
1560:Limitations
1536:, like the
622:engineering
221:if you can.
166:August 2019
2963:Categories
2849:References
1600:, and the
1011:Given the
526:Definition
277:deflection
136:newspapers
65:improve it
2795:ρ
2621:ρ
2618:π
2606:ρ
2545:π
2481:−
2460:ρ
2457:π
2445:ρ
2379:−
2361:π
2290:mm, or a
2245:τ
2241:π
2168:τ
2100:ω
2062:π
2053:ω
2011:) is: 250
1867:τ
1821:τ
1766:θ
1726:θ
1532:unit for
1493:π
1481:ϕ
1440:∫
1434:π
1422:∫
1246:∴
1099:∬
1037:∬
967:∬
930:∬
852:∬
712:∬
566:ρ
558:. Where
273:torsional
71:talk page
38:due to a
2923:galtor.
2812:See also
2292:diameter
203:require
2701:is the
2675:is the
2652:where:
2286:m = 200
399:), the
297:bending
205:cleanup
150:scholar
2906:
2864:
2322:
2318:
2296:
2294:of 400
2288:
2284:
2282:=0.200
2276:radius
2135:
2131:
2127:
2116:
2035:
2028:
2021:
2017:
2013:
1998:plant.
1989:
1859:where
1808:stress
1672:torque
1183:planar
1013:planar
747:where
662:planar
650:planar
638:planar
618:planar
369:, and
313:planar
309:moment
293:planar
152:
145:
138:
131:
123:
40:torque
2314:yield
2081:power
2009:yield
1950:Note:
1548:) in
658:polar
646:polar
634:polar
614:polar
606:Note:
401:polar
305:polar
301:plane
157:JSTOR
143:books
2904:ISBN
2862:ISBN
2274:The
2041:The
2023:N/m.
1552:and
1528:The
1524:Unit
1212:and
1162:and
813:and
666:mass
660:and
654:mass
648:and
642:area
636:and
616:and
267:The
129:news
2307:adm
2278:is
2249:max
2234:max
2172:max
2156:max
2138:N·m
2119:rad
1342:or
450:or
112:by
2965::
2881:.
2638:64
2559:32
2500:64
2398:32
2140:.
2133:10
2031:Hz
2019:10
1992:MW
1758::
1678:,
1658:.
1604:,
1573:,
1556:.
1546:in
1530:SI
1400::
837::
790:.
483:,
74:.
2927:.
2912:.
2885:.
2868:.
2773:l
2749:z
2745:I
2720:c
2716:I
2689:D
2663:d
2632:4
2628:D
2624:l
2612:=
2609:l
2601:z
2597:I
2593:=
2588:c
2584:I
2553:4
2549:D
2539:=
2534:z
2530:I
2495:)
2489:4
2485:d
2476:4
2472:D
2467:(
2463:l
2451:=
2448:l
2440:z
2436:I
2432:=
2427:c
2423:I
2393:)
2387:4
2383:d
2374:4
2370:D
2365:(
2355:=
2350:z
2346:I
2311:τ
2309:=
2304:τ
2280:r
2257:3
2230:T
2226:2
2219:=
2216:r
2188:r
2182:z
2178:J
2161:=
2152:T
2129:×
2123:s
2121:/
2097:T
2094:=
2091:P
2065:f
2059:2
2056:=
2015:×
2006:τ
1972:.
1932:z
1928:J
1907:r
1887:T
1843:z
1839:J
1834:r
1830:T
1824:=
1786:G
1783:J
1778:L
1775:T
1769:=
1746:T
1706:L
1686:G
1639:J
1617:t
1613:J
1586:z
1582:I
1542:m
1507:2
1501:4
1497:R
1487:=
1484:)
1478:d
1474:r
1471:d
1467:r
1464:(
1459:2
1455:r
1449:R
1444:0
1431:2
1426:0
1418:=
1413:z
1409:I
1388:R
1364:y
1360:I
1356:2
1353:=
1350:J
1328:x
1324:I
1320:2
1317:=
1314:J
1286:y
1282:I
1278:+
1273:x
1269:I
1265:=
1260:z
1256:I
1252:=
1249:J
1225:y
1221:I
1198:x
1194:I
1170:y
1150:x
1128:y
1125:d
1121:x
1118:d
1113:2
1109:x
1103:A
1095:=
1090:y
1086:I
1066:y
1063:d
1059:x
1056:d
1051:2
1047:y
1041:A
1033:=
1028:x
1024:I
997:y
994:d
990:x
987:d
981:2
977:y
971:A
963:+
960:y
957:d
953:x
950:d
944:2
940:x
934:A
926:=
923:J
904:y
901:d
897:x
894:d
890:)
884:2
880:y
876:+
871:2
867:x
862:(
856:A
848:=
845:J
821:y
801:x
778:A
775:d
755:r
735:A
732:d
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412:I
385:y
382:x
378:I
355:y
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342:,
328:x
324:I
258:)
252:(
240:)
234:(
229:)
225:(
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173:(
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154:·
147:·
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106:.
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