Tema 4: Torsión en barras y en tubos no circulares 4.1. Torsión de elementos no circulares Denotando con L la longitud de la barra, con a el lado más ancho y con b el lado más angosto de su sección transversal y con T la magnitud de los momentos torsionantes de los pares aplicados a la barra de la Figura, el esfuerzo cortante máximo, que ocurre a lo largo de la línea centran de la cara más ancha de la barra, es igual a: T c1 ab2 (4.1) TL c2 ab2 G (4.2) τ= El ángulo de giro se calcula como: θ= Los coeficientes c1 y c2 dependen de la razón a/b, dados en la tabla, estas ecuaciones son válidas dentro del rango elástico. a/b c1 c2 1.0 0.208 0.1406 1.2 0.219 0.1661 1.5 0.231 0.1958 2.0 0.246 0.229 2.5 0.258 0.249 3.0 0.267 0.263 4.0 0.282 0.281 5.0 0.291 0.291 10.0 0.312 0.312 ∞ 0.333 0.333 En la tabla anterior, los coeficientes c1 y c2 son iguales para la razón a/b > 5. Para tales valores: c1 = c2 = 4.2. 1 (1 − 0,630b/a) 3 (4.3) Torsión de elementos huecos de pares delgada El esfuerzo cortante τ en cualquier punto de un elemento hueco de paredes delgadas se determina con la siguiente expresión τ= T 2tã (4.4) Donde T es la magnitud de los momentos torsionantes, t el espesor del elemento y ã es el área bordeada por la línea central. Para calcular el ángulo θ utilice la tabla anexa. 6 SEC. 10.7 Tables Formulas for torsional deformation and stress GENERAL FORMULAS: y ¼ TL=KG and t ¼ T =Q, where y ¼ angle of twist (radians); T ¼ twisting moment (force-length); L ¼ length, t ¼ unit shear stress (force per unit area); G ¼ modulus of 10.7] TABLE 10.1 rigidity (force per unit area); K (length to the fourth) and Q (length cubed) are functions of the cross section Form and dimensions of cross sections, other quantities involved, and case no. Formula for K in y ¼ 1. Solid circular section K¼ 1 4 2 pr 2. Solid elliptical section K¼ pa3 b3 a2 þ b2 3. Solid square section K ¼ 2:25a4 4. Solid rectangular section K ¼ ab3 TL KG tmax ¼ 2T pr3 tmax ¼ 2T pab2 tmax ¼ 0:601T a3 tmax ¼ at boundary at ends of minor axis at midpoint of each side " 2 3 4 # 3T b b b b 1 þ 0:6095 þ 0:8865 1:8023 þ 0:9100 2 8ab a a a a at the midpoint of each longer side for a 5 b Torsion 16 b b4 for a 5 b 3:36 1 3 a 12a4 Formula for shear stress 401 402 TABLE 10.1 Formulas for torsional deformation and stress (Continued) Form and dimensions of cross sections, Formula for K in y ¼ 5. Solid triangular section (equilaterial) pffiffiffi a4 3 K¼ 80 6. Isosceles triangle For 2 3 K¼ a3 b3 15a2 þ 20b2 (Note: See also Ref. 21 for graphs of stress magnitudes and locations and stiffness factors) TL KG Formula for shear stress tmax ¼ < a=b < pffiffiffi 3 ð39 < a < 82 Þ at midpoint of each side For 39 < a < 120 Q¼ approximate formula which is exact at a ¼ 60 where K ¼ 0:02165c4 . pffiffiffi pffiffiffi For 3 < a=b < a < 120 Þ a < 2 3 ð82 K ¼ 0:0915b4 0:8592 b 20T a3 K b½0:200 þ 0:309a=b 0:0418ða=bÞ2 approximate formula which is exact at a ¼ 60 and a ¼ 90 For a ¼ 60 For a ¼ 90 Formulas for Stress and Strain other quantities involved, and case no. Q ¼ 0:0768b3 ¼ 0:0500c3 Q ¼ 0:1604b3 ¼ 0:0567c3 tmax at center of longest side approximate formula which is exact at a ¼ 90 where K ¼ 0:0261c4 (errors < 4%) (Ref. 20) 7. Circular segmental section K ¼ 2Cr4 where C varies with For 0 4 h as follows. r h 4 1:0: r 2 h h C ¼ 0:7854 0:0333 2:6183 r r [Note: h ¼ rð1 cos aÞ 3 4 5 h h h þ 4:1595 3:0769 þ0:9299 r r r tmax ¼ TB h where B varies with r3 r as follows. For 0 4 h 4 1:0 : r 2 h h B ¼ 0:6366 þ 1:7598 5:4897 r r 3 4 5 h h h þ14:062 14:510 þ 6:434 r r r [CHAP. 10 (Data from Refs. 12 and 13) SEC. TABLE 10.1 Formulas for torsional deformation and stress (Continued) 10.7] 8. Circular sector K ¼ Cr4 where C varies with For 0:1 4 a as follows: p a 4 2:0: p with a2 a C ¼ 0:0034 0:0697 þ 0:5825 p p 0:2950 (Note: See also Ref. 21) a 3 p tmax ¼ þ 0:0874 a 4 p 0:0111 T on a radial boundary. B varies Br3 a a as follows. For 0:1 4 4 1:0: p p a2 a B ¼ 0:0117 0:2137 þ 2:2475 p p a5 a3 a4 a5 4:6709 þ 5:1764 2:2000 p p p p ðData from Ref. 17) 9. Circular shaft with opposite sides flattened h as follows: K ¼ 2Cr where C varies with r 4 TB h as follows. For two flat sides where where B varies with r3 r h 4 0:6: r h 4 0:8: r 2 h h C ¼ 0:7854 0:4053 3:5810 r r 3 4 h h þ 5:2708 2:0772 r r 2 3 h h h B ¼ 0:6366 þ 2:5303 11:157 þ 49:568 r r r For four flat sides where h 0 4 4 0:293 : r For four flat sides where 0 4 For two flat sides where 0 4 (Note: h ¼ r wÞ tmax ¼ C ¼ 0:7854 0:7000 4 5 h h 85:886 þ 69:849 r r h 4 0:293: r 2 3 h h h þ 30:853 B ¼ 0:6366 þ 2:6298 5:6147 r r r (Data from Refs. 12 and 13) Torsion 2 3 h h h þ 14:578 7:7982 r r r 04 403 TABLE 10.1 Formulas for torsional deformation and stress (Continued) other quantities involved, and case no. 11. Eccentric hollow circular section TL KG K ¼ 12 pðr40 r4i Þ K¼ pðD4 d4 Þ 32C where Formula for shear stress tmax ¼ 2Tro at outer boundary pðr4o r4i Þ tmax ¼ 16TDF pðD4 d4 Þ F ¼1þ 16n2 384n4 l2 þ l4 C ¼ 1þ ð1 n2 Þð1 n4 Þ ð1 n2 Þ2 ð1 n4 Þ4 4n2 32n2 48n2 ð1 þ 2n2 þ 3n4 þ 2n6 Þ 3 lþ l2 þ l ð1 n2 Þð1 n4 Þð1 n6 Þ 1 n2 ð1 n2 Þð1 n4 Þ þ 12. Hollow elliptical section, outer and inner boundaries similar ellipses K¼ pa3 b3 ð1 q4 Þ a2 þ b2 tmax ¼ 64n2 ð2 þ 12n2 þ 19n4 þ 28n6 þ 18n8 þ 14n10 þ 3n12 Þ 4 l ð1 n2 Þð1 n4 Þð1 n6 Þð1 n8 Þ 2T pab2 ð1 q4 Þ (Ref. 10) Formulas for Stress and Strain 10. Hollow concentric circular section Formula for K in y ¼ 404 Form and dimensions of cross sections, at ends of minor axis on outer surface where q¼ ao bo ¼ a b (Note: The wall thickness is not constant) 13. Hollow, thin-walled section of uniform thickness; U ¼ length of elliptical median boundary, shown dashed: " U ¼ pða þ b tÞ 1 þ 0:258 4p2 t½ða 12 tÞ2 ðb 12 tÞ2 U taverage ¼ T 2ptða 12 tÞðb 12 tÞ (stress is nearly uniform if t is small) # ða bÞ2 2 ða þ b tÞ [CHAP. 10 ðapproximatelyÞ K¼ SEC. TABLE 10.1 Formulas for torsional deformation and stress (Continued) 15. Any thin tube. U and A as for case 14; t ¼ thickness at any point 16. Hollow rectangle, thin-walled K¼ 4A2 t U K ¼Ð K¼ 4A2 dU =t 2tt1 ða tÞ2 ðb t1 Þ2 at þ bt1 t2 t21 taverage ¼ T 2tA 10.7] 14. Any thin tube of uniform thickness; U ¼ length of median boundary; A ¼ mean of areas enclosed by outer and inner boundaries, or (approximate) area within median boundary (stress is nearly uniform if t is small) taverage on any thickness AB ¼ taverage ¼ T 2tA ðtmax where t is a minimum) 8 > > > < T 2tða tÞðb t1 Þ near midlength of short sides > > > : T 2t1 ða tÞðb t1 Þ near midlength of long sides (There will be higher stresses at inner corners unless fillets of fairly large radius are provided) Torsion (Note: For thick-walled hollow rectangles see Refs. 16 and 25. Reference 25 illustrates how to extend the work presented to cases with more than one enclosed region.) 405 TABLE 10.1 Formulas for torsional deformation and stress (Continued ) Form and dimensions of cross sections, 17. Thin circular open tube of uniform thickness; r ¼ mean radius Formula for K in y ¼ K ¼ 23 prt3 TL KG Formula for shear stress tmax ¼ T ð6pr þ 1:8tÞ 4p2 r2 t2 18. Any thin open tube of uniform thickness; U ¼ length of median line, shown dashed K¼ 19. Any elongated section with axis of symmetry OX; U ¼ length, A ¼ area of section, Ix ¼ moment of inertia about axis of symmetry K¼ 1 3 Ut 3 tmax ¼ T ð3U þ 1:8tÞ U 2 t2 along both edges remote from ends (this assumes t small compared wtih least radius of curvature of median line; otherwise use the formulas given for cases 19–26) 4Ix 1 þ 16Ix =AU 2 For all solid sections of irregular form (cases 19–26 inclusive) the maximum shear stress occurs at or very near one of the points where the largest inscribed circle touches the boundary,* and of these, at the one where the curvature of the boundary is algebraically least. (Convexity represents positive and concavity negative curvature of the boundary.) At a point where the curvature is positive (boundary of section straight or convex) this maximum stress is given approximately by tmax ¼ G 20. Any elongated section or thin open tube; dU ¼ elementary length along median line, t ¼ thickness normal to median line, A ¼ area of section F 3 þ 4F=AU 2 where F ¼ 0 t3 dU A4 40J or tmax ¼ T C K where C¼ K¼ y C L 2 4 D p D D 1 þ 0:15 16A2 2r p2 D4 1þ 16A2 D ¼ diameter of largest inscribed circle r ¼ radius of curvature of boundary at the point (positive for this case) A ¼ area of the section *Unless at some point on the boundary there is a sharp reentant angle, causing high local stress. [CHAP. 10 21. Any solid, fairly compact section without reentrant angles, J ¼ polar moment of inertia about centroid axis, A ¼ area of section K¼ ðU Formulas for Stress and Strain along both edges remote from ends (this assumes t is small comopared with mean radius) 406 other quantities involved, and case no. SEC. TABLE 10.1 Formulas for torsional deformation and stress (Continued) 1 K ¼ 12 bðm þ nÞðm2 þ n2 Þ VL m4 Vs n4 where VL ¼ 0:10504 0:10s þ 0:0848s2 0:06746s3 þ 0:0515s4 Vs ¼ 0:10504 þ 0:10s þ 0:0848s2 þ 0:06746s3 þ 0:0515s4 mn s¼ b (Ref. 11) 23. T-section, flange thickness uniform. For definitions of r; D; t; and t1 , see case 26. K ¼ K1 þ K2 þ aD4 1 b b4 0:21 1 12a4 3 a 1 d d4 1 K2 ¼ cd3 0:105 3 c 192c4 t r a¼ 0:15 þ 0:10 t1 b where K1 ¼ ab3 10.7] 22. Trapezoid At a point where the curvature is negative (boundary of section concave or reentrant), this maximum stress is given approximately by T C K D D D 2f 0:238 tanh where C ¼ 1 þ 0:118 ln 1 2r 2r p p2 D4 1þ 16A2 y tmax ¼ G C L or tmax ¼ and D; A, and r have the same meaning as before and f ¼ a positive angle through which a tangent to the boundary rotates in turning or traveling around the reentrant portion, measured in radians (here r is negative). The preceding formulas should also be used for cases 17 and 18 when t is relatively large compared with radius of median line. ðb þ rÞ2 þ rd þ d2 =4 ð2r þ bÞ for d < 2ðb þ rÞ D¼ 24. L-section; b 5 d. For definitions of r and D, see case 26. Torsion K ¼ K1 þ K2 þ aD4 1 b b4 where K1 ¼ ab3 0:21 1 12a4 3 a 1 d d4 1 K2 ¼ cd3 0:105 192c4 3 c d r a¼ 0:07 þ 0:076 b b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ 2½d þ b þ 3r 2ð2r þ bÞð2r þ d for b < 2ðd þ rÞ 407 TABLE 10.1 Formulas for torsional deformation and stress (Continued ) other quantities involved, and case no. Formula for K in y ¼ TL KG K ¼ sum of K’s of constituent L-sections computed as for case 24 26. I-section, flange thickness uniform; r ¼ fillet radius, D ¼ diameter largest inscribed circle, t ¼ b if b < d; t ¼ d if d < b; t1 ¼ b if b > d; t1 ¼ d if d > b K ¼ 2K1 þ K2 þ 2aD4 Formulas for Stress and Strain 25. U- or Z-section Formula for shear stress 1 b b4 0:21 1 4 3 a 12a K2 ¼ 13 cd3 t r a¼ 0:15 þ 0:1 t1 b where K1 ¼ ab3 Use expression for D from case 23 27. Split hollow shaft K ¼ 2Cr4o where C varies with For 0:2 4 ri as follows: ro ri 4 0:6: ro C ¼ K1 þ K2 2 3 ri r r þ K3 i þ K4 i ro ro ro where for 0:1 4 h=ri 4 1:0, K1 K2 K4 At M ; t ¼ TB r where B varies with i as follows. r3o ro For 0:2 4 ri 4 0:6: ro B ¼ K1 þ K2 2 3 ri r r þ K3 i þ K4 i ro ro ro where fore 0:1 4 h=ri 4 1:0, 3 h h 0:3231 ri ri 2 h h K2 ¼ 2:9047 þ 3:0069 þ 4:0500 ri ri 2 h h K3 ¼ 15:721 6:5077 12:496 ri ri 2 h h K4 ¼ 29:553 þ 4:1115 þ 18:845 ri ri K1 ¼ 2:0014 0:1400 (Data from Refs. 12 and 13) [CHAP. 10 K3 2 h h ¼ 0:4427 þ 0:0064 0:0201 ri ri 2 h h ¼ 0:8071 0:4047 þ 0:1051 ri ri 2 h h ¼ 0:0469 þ 1:2063 0:3538 ri ri 2 h h ¼ 0:5023 0:9618 þ 0:3639 ri ri 408 Form and dimensions of cross sections, SEC. TABLE 10.1 Formulas for torsional deformation and stress (Continued) K ¼ 2Cr4 where C varies with For 0 4 b as follows. r b 4 0:5: r C ¼ K1 þ K2 At M ; t ¼ TB b b where B varies with as follows. For 0:2 4 4 0:5 : r3 r r B ¼ K1 þ K2 2 3 b b b þ K4 þ K3 r r r 10.7] 28. Shaft with one keyway 2 3 b b b þ K3 þ K4 r r r where for 0:5 4 a=b 4 1:5; a2 a 1:1690 0:3168 þ 0:0490 b b a2 a K2 ¼ 0:43490 1:5096 þ 0:8677 b b a2 a K3 ¼ 1:1830 þ 4:2764 1:7024 b b a2 a K4 ¼ 0:8812 0:2627 0:1897 b b K1 ¼ where for 0:3 4 a=b 4 1:5; K1 ¼ 0:7854 a2 a K2 ¼ 0:0848 þ 0:1234 0:0847 b b a2 a K3 ¼ 0:4318 2:2000 þ 0:7633 b b a2 a K4 ¼ 0:0780 þ 2:0618 0:5234 b b 29. Shaft with two keyways K ¼ 2Cr4 where C varies with For 0 4 b as follows. r b 4 0:5: r C ¼ K1 þ K2 2 3 b b b þ K3 þ K4 r r r where for 0:3 4 a=b 4 1:5; K1 ¼ At M ; t ¼ TB b b where B varies with as follows. For 0:2 4 4 0:5 : r3 r r B ¼ K1 þ K2 2 3 b b b þ K4 þ K3 r r r where for 0:5 4 a=b 4 1:5; a2 a 1:2512 0:5406 þ 0:0387 b b a a2 K2 ¼ 0:9385 þ 2:3450 þ 0:3256 b b a2 a K3 ¼ 7:2650 15:338 þ 3:1138 b b a2 a K4 ¼ 11:152 þ 33:710 10:007 b b K1 ¼ Torsion 0:7854 a2 a K2 ¼ 0:0795 þ 0:1286 0:1169 b b a2 a K3 ¼ 1:4126 3:8589 þ 1:3292 b b a2 a K4 ¼ 0:7098 þ 4:1936 1:1053 b b (Data from Refs. 12 and 13) (Data from Refs. 12 and 13) 409 Formulas for torsional deformation and stress (Continued) Form and dimensions of cross sections, 30. Shaft with four keyways Formula for K in y ¼ K ¼ 2Cr4 where C varies with For 0 4 TL KG b as follows. r b 4 0:4: r C ¼ K1 þ K2 2 3 b b b þ K4 þ K3 r r r 0:7854 a2 a K2 ¼ 0:1496 þ 0:2773 0:2110 b b a2 a K3 ¼ 2:9138 8:2354 þ 2:5782 b b a2 a K4 ¼ 2:2991 þ 12:097 2:2838 b b For 0 4 b as follows. r b 4 0:5: r C ¼ K1 þ K2 2 3 b b b þ K4 þ K3 r r r where for 0:2 4 a=b 4 1:4; K1 ¼ 0:7854 2 3 b b b þ K3 þ K4 r r r where for 0:5 4 a=b 4 1:2; a2 a 1:0434 þ 1:0449 0:2977 b b a2 a K2 ¼ 0:0958 9:8401 þ 1:6847 b b a2 a K3 ¼ 15:749 6:9650 þ 14:222 b b a2 a K4 ¼ 35:878 þ 88:696 47:545 b b (Data from Refs. 12 and 13) At M ; t ¼ TB b b where B varies with as follows. For 0 4 4 0:5; r3 r r B ¼ K1 þ K2 2 3 b b b þ K3 þ K4 r r r where for 0:2 4 a=b 4 1:4; K1 ¼ 0:6366 a2 a K2 ¼ 0:0023 þ 0:0168 þ 0:0093 b b a2 a K3 ¼ 0:0052 þ 0:0225 0:3300 b b a2 a K4 ¼ 0:0984 0:4936 þ 0:2179 b b (Data from Refs. 12 and 13) [CHAP. 10 a2 a 0:0264 0:1187 þ 0:0868 b b a2 a K3 ¼ 0:2017 þ 0:9019 0:4947 b b a2 a K4 ¼ 0:2911 1:4875 þ 2:0651 b b K2 ¼ TB b b where B varies with as follows. For 0:2 4 4 0:4; r3 r r K1 ¼ K1 ¼ K ¼ 2Cr4 where C varies with At M ; t ¼ B ¼ K1 þ K2 where for 0:3 4 a=b 4 1:2; 31. Shaft with one spline Formula for shear stress Formulas for Stress and Strain other quantities involved, and case no. 410 TABLE 10.1 SEC. TABLE 10.1 Formulas for torsional deformation and stress (Continued) K ¼ 2Cr4 where C varies with For 0 4 b as follows. r b 4 0:5: r C ¼ K1 þ K2 At M ; t ¼ TB b b as follows. For 0 4 4 0:5; where B varies with r3 r r B ¼ K1 þ K2 2 3 b b b þ K4 þ K3 r r r 2 3 b b b þ K4 þ K3 r r r where for 0:2 4 a=b 4 1:4; K1 ¼ where for 0:2 4 a=b 4 1:4; 10.7] 32. Shaft with two splines 0:6366 a2 a 0:0069 0:0229 þ 0:0637 b b a a2 K3 ¼ 0:0675 þ 0:3996 1:0514 b b a2 a K4 ¼ 0:3582 1:8324 þ 1:5393 b b K2 ¼ K1 ¼ 0:7854 a2 a 0:0204 0:1307 þ 0:1157 b b a a2 K3 ¼ 0:2075 þ 1:1544 0:5937 b b a2 a K4 ¼ 0:3608 2:2582 þ 3:7336 b b K2 ¼ 33. Shaft with four splines K ¼ 2Cr4 where C varies with For 0 4 b as follows. r b 4 0:5: r C ¼ K1 þ K2 2 3 b b b þ K3 þ K4 r r r where for 0:2 4 a=b 4 1:0; (Data from Refs. 12 and 13) At M ; t ¼ TB b b where B varies with as follows. For 0 4 4 0:5; r3 r r B ¼ K1 þ K2 2 3 b b b þ K4 þ K3 r r r where for 0:2 4 a=b 4 1:0; K1 ¼ 0:6366 a2 a 0:0114 0:0789 þ 0:1767 b b a2 a K3 ¼ 0:1207 þ 1:0291 2:3589 b b a2 a K4 ¼ 0:5132 3:4300 þ 4:0226 b b K2 ¼ K1 ¼ 0:7854 (Data from Refs. 12 and 13) Torsion a2 a 0:0595 0:3397 þ 0:3239 b b a2 a K3 ¼ 0:6008 þ 3:1396 2:0693 b b a2 a K4 ¼ 1:0869 6:2451 þ 9:4190 b b K2 ¼ 411 412 TABLE 10.1 Formulas for torsional deformation and stress (Continued) other quantities involved, and case no. 34. Pinned shaft with one, two, or four grooves Formula for K in y ¼ K ¼ 2Cr4 where C varies with 04 TL KG a over the range r a 4 0:5 as follows. For one groove: r Formula for shear stress At M ; t ¼ TB a over the where B varies with r3 r range 0:1 4 a 4 0:5 as follows. For one groove: r a2 a3 a þ 0:9167 C ¼ 0:7854 0:0225 1:4154 r r r a2 a3 a B ¼ 1:0259 þ 1:1802 2:7897 þ 3:7092 r r r For two grooves: For two grooves: a2 a3 a C ¼ 0:7854 0:0147 3:0649 þ 2:5453 r r r a2 a3 a þ 7:0534 B ¼ 1:0055 þ 1:5427 2:9501 r r r For four grooves: For four grooves: a2 a3 a C ¼ 0:7854 0:0409 6:2371 þ 7:2538 r r r a2 a3 a4 a B ¼ 1:2135 2:9697 þ 33:713 99:506 þ 130:49 r r r r Formulas for Stress and Strain Form and dimensions of cross sections, (Data from Refs. 12 and 13) 35. Cross shaft r K ¼ 2Cs where C varies with over the s 4 range 0 4 r 4 0:9 as follows: s r2 r3 r C ¼ 1:1266 0:3210 þ 3:1519 14:347 s s s r4 r5 þ 15:223 4:7767 s s At M ; t ¼ BM T r r where BM varies with over the range 0 4 4 0:5 as follows: s3 s s r2 r3 r4 r þ 3:7335 2:8686 BM ¼ 0:6010 þ 0:1059 0:9180 s s s s At N; t ¼ BN T r r where BN varies with over the range 0:3 4 4 0:9 as follows: s3 s s r2 r3 r4 r5 r þ 109:04 133:95 þ 66:054 BN ¼ 0:3281 þ 9:1405 42:520 s s s s s (Data from Refs. 12 and 13) [CHAP. 10 (Note: BN > BM for r=s > 0:32Þ TABLE 10.2 Formulas for torsional properties and stresses in thin-walled open cross sections sixth power); t1 ¼ shear stress due to torsional rigidity of the cross section (force per unit area); t2 ¼ shear stress due to warping rigidity of the cross section (force per unit area); sx ¼ bending stress unit area) The appropriate values of y0 ; y00 , and y000 are found in Table 10.3 for the loading and boundary restraints desired Cross section, reference no. 1. Channel Constants 3b h þ 6b K¼ t3 ðh þ 2bÞ 3 Cw ¼ 2. C-section e¼b h2 b3 t 2h þ 3b 12 h þ 6b h3 3h2 b þ 6h2 b1 8b31 þ 6h2 b þ 6h2 b1 þ 8b31 12hb21 t3 ðh þ 2b þ 2b1 Þ 3 2 2 h b b 2eb1 2b21 Cw ¼ t b1 þ e þ 2 b h 3 K¼ þ 3. Hat section Selected maximum values 2 e¼ e¼b ðsx Þmax ¼ hb h þ 3b 00 Ey throughout the thickness at corners A and D 2 h þ 6b ðt2 Þmax ¼ 2 hb2 h þ 3b h þ 3b from corners A and D Ey000 throughout the thickness at a distance b h þ 6b 4 h þ 6b ðt1 Þmax ¼ tGy0 at the surface everywhere ðsx Þmax ¼ h ðb eÞ þ b1 ðb þ eÞ Ey00 throughout the thickness at corners A and F 2 ðt2 Þmax ¼ h b2 ðb eÞð2b1 þ b eÞ þ 1 ðb þ eÞ Ey000 throughout the thickness on the top and bottom flanges at a 4 2 distance e from corners C and D ðt1 Þmax ¼ tGy0 at the surface everywhere h2 e2 h 2b2 2b3 b þ b1 þ 1 þ 1 ðb þ eÞ2 2 h 3 6 3h2 b þ 6h2 b1 8b31 h3 þ 6h2 b þ 6h2 b1 þ 8b31 þ 12hb21 h ðb eÞ b1 ðb þ eÞ Ey00 throughout the thickness at corners A and F 2 h ðb eÞEy00 throughout the thickness at corners B and E 2 " # h2 ðb eÞ2 b21 hb hðb eÞ þ ðb þ eÞ 1 ðb eÞ Ey000 throughout the thickness at a distance t2 ¼ 2ðb þ eÞ 8ðb þ eÞ 2 2 sx ¼ from corner B toward corner A 2 b hb h t2 ¼ 1 ðb þ eÞ 1 ðb eÞ ðb eÞ2 Ey000 throughout the thickness at a distance e 4 2 2 from corner C toward corner B 413 h2 e2 h 2b2 2b3 b þ b1 þ þ 1 þ 1 ðb þ eÞ2 6 2 h 3 sx ¼ Torsion t3 ðh þ 2b þ 2b1 Þ 3 2 2 h b b 2eb1 2b21 b1 þ e Cw ¼ t 3 2 b h K¼ þ 10.7] due to warping rigidity of the cross section (force per unit area); E ¼ modulus of elasticity of the material (force per unit area); and G ¼ modulus of rigidity (shear modulus) of the material (force per SEC. NOTATION: Point 0 indicates the shear center. e ¼ distance from a reference to the shear center; K ¼ torsional stiffness constant (length to the fourth power); Cw ¼ warping constant (length to the t1 ¼ tGy0 at the surface everywhere TABLE 10.2 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued ) 4. Twin channel with flanges inward Constants K¼ t3 ð2b þ 4b1 Þ 3 tb2 ð8b31 þ 6h2 b1 þ h2 b þ 12b21 hÞ 24 ðsx Þmax b h b þ Ey00 throughout the thickness at points A and D ¼ 2 1 2 ðt2 Þmax ¼ b ð4b21 þ 4b1 h þ hbÞEy000 throughout the thickness midway between corners B and C 16 ðt1 Þmax ¼ tGy0 at the surface everywhere 5. Twin channel with flanges outward K¼ t3 ð2b þ 4b1 Þ 3 Cw ¼ tb2 ð8b31 þ 6h2 b1 þ h2 b 12b21 hÞ 24 hb 00 Ey throughout the thickness at points B and C if h > b1 4 hb bb1 ¼ Ey00 throughout the thickness at points A and D if h < b1 2 4 ðsx Þmax ¼ ðsx Þmax ðt2 Þmax ¼ ðt2 Þmax ¼ 2 b h h b1 Ey000 throughout the thickness at a distance from corner B toward point A if 4 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! h 1 b b1 > 1þ þ 2 2 2h Formulas for Stress and Strain Cw ¼ Selected maximum values 414 Cross section, reference no. b 2 hb b hb1 Ey000 throughout the thickness at a point midway between corners B and C if 4 1 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! h 1 b b1 < 1þ þ 2 2 2h ðt1 Þmax ¼ tGy0 at the surface everywhere 6. Wide flanged beam with equal flanges K ¼ 13 ð2t3 b þ t3w hÞ Cw ¼ h2 tb3 24 ðsx Þmax ¼ hb 00 Ey throughout the thickness at points A and B 4 hb2 Ey000 throughout the thickness at a point midway between A and B 16 ðt1 Þmax ¼ tGy0 at the surface everywhere [CHAP. 10 ðt2 Þmax ¼ TABLE 10.2 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued t1 b31 h t1 b31 þ t2 b32 K ¼ 13 ðt31 b1 þ t32 b2 þ t3w hÞ Cw ¼ h2 t1 t2 b31 b32 12ðt1 b31 þ t2 b32 Þ ðsx Þmax ¼ hb1 t2 b32 Ey00 throughout the thickness at points A and B if t2 b22 > t1 b21 2 t1 b31 þ t2 b32 ðsx Þmax ¼ hb2 t1 b31 Ey00 throughout the thickness at points C and D if t2 b22 < t1 b21 2 t1 b31 þ t2 b32 ðt2 Þmax ¼ 1 ht2 b32 b21 Ey000 throughout the thickness at a point midway between A and B if t2 b2 > t1 b1 8 t1 b31 þ t2 b32 ðt2 Þmax ¼ 1 ht1 b31 b22 Ey000 throughout the thickness at a point midway between C and D if t2 b2 < t1 b1 8 t1 b31 þ t2 b32 10.7] e¼ SEC. 7. Wide flanged beam with unequal flanges ) ðt1 Þmax ¼ tmax Gy0 at the surface on the thickest portion 8. Z-section t3 ð2b þ hÞ 3 th2 b3 b þ 2h Cw ¼ 2b þ h 12 K¼ ðsx Þmax ¼ hb b þ h Ey00 throughout the thickness at points A and D 2 2b þ h ðt2 Þmax ¼ 2 hb2 b þ h bðb þ hÞ from point A Ey000 throughout the thickness at a distance 4 2b þ h 2b þ h ðt1 Þmax ¼ tGy0 at the surface everywhere 9. Segment of a circular tube e ¼ 2r sin a a cos a a sin a cos a K ¼ 23 t3 ra Cw ¼ ra2 Ey000 throughout the thickness at midlength ðt2 Þmax ¼ r2 eð1 cos aÞ 2 ðt1 Þmax ¼ tGy0 at the surface everywhere Torsion 415 (Note: If t=r is small, a can be larger than p to evaluate constants for the case when the walls overlap) " # 2tr5 3 ðsin a a cos aÞ2 a 6 3 a sin a cos a ðsx Þmax ¼ ðr2 a re sin aÞEy00 throughout the thickness at points A and B 416 TABLE 10.2 Formulas for torsional properties and stresses in thin-walled open cross sections (Continued ) 10. Constants e ¼ 0:707ab2 3a 2b 2a3 ða bÞ3 K ¼ 23 t3 ða þ bÞ Selected maximum values ðsx Þmax ¼ t2 ¼ 4 3 Cw ¼ 11. ta b 4a þ 3b 6 2a3 ða bÞ3 K ¼ 13 ð4t3 b þ t3w aÞ a2 b 2a2 þ 3ab b2 00 Ey throughout the thickness at points A and E 2 2a3 ða bÞ3 a2 b2 a2 2ab b2 Ey000 throughout the thickness at point C 4 2a3 ða bÞ3 ðt1 Þmax ¼ tGy0 at the surface everywhere ðsx Þmax ¼ ab cos aEy00 throughout the thickness at points A and C 2 ðt2 Þmax ¼ ab2 cos aEy000 throughout the thickness at point B 4 Formulas for Stress and Strain Cross section, reference no. 2 3 Cw ¼ a b t cos2 a 3 (Note: Expressions are equally valid for þ and a) ðt1 Þmax ¼ tGy0 at the surface everywhere [CHAP. 10 d~ secoOl\ G. ...~ S em 'I.. lO. \ b c...VV'l s<::: sor('.e.:\-~ Q uY\ e<; FLX:f 2.0 ton; "OY\C1 n 1-- e T -::: 12 fS i«o f - m. . \)c:.--tc-r m \ '!\t:: c: \ C~f'.x:.flD Lor -to n 1: e en c.a.dct una c\e. \o.~ c.uo+ro pare..d~ d<:- dlc.ha tv bo , ~upot'\icl')do ~ ~) UY\ esre~or Uf\lfF0rfY\C d~ OdYo6UY\ ~ h) dos. poredes dc- Q.SO'SCr'rl hJe«tplo. 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