Moments and products of inertia and radii of gyration about central axes. I x ¼ I y ¼ Ix 0 ¼ a4. r x ¼ r y ¼ r 0 x ¼ 0:2887a.

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3 TBLE.1 Properties of sections NOTTION: ¼ re ðlengthþ 2 ; y ¼ distnce to extreme fiber (length); I ¼ moment of inerti ðlength Þ; r ¼ rdius of gyrtion (length); Z ¼ plstic section modulus ðlength 3 Þ;SF¼ shpe fctor. See Sec for pplictions of Z nd SF Form of section 1. Squre ¼ 2 2. Rectngle ¼ bd y c ¼ x c ¼ 2 y 0 c ¼ 0:707 cos p y c ¼ d 2 3. Hollow rectngle ¼ bd b i d i y c ¼ d 2 re nd distnces from centroid to extremities Moments nd products of inerti nd rdii of gyrtion bout centrl xes I x ¼ I y ¼ Ix 0 ¼ 1 12 r x ¼ r y ¼ r 0 x ¼ 0:2887 I x ¼ 1 12 bd3 I y ¼ 1 12 db3 I x > I y if d > b r x ¼ 0:2887d r y ¼ 0:2887b I x ¼ bd3 b i d 3 i 12 I y ¼ db3 d i b 3 i 12 r x ¼ x r y ¼ y Z x ¼ Z y ¼ 0:25 3 SF x ¼ SF y ¼ 1:5 Z x ¼ 0:25bd 2 Z y ¼ 0:25db 2 SF x ¼ SF y ¼ 1:5 Z x ¼ bd2 b i d 2 i SF x ¼ Z xd 2I x Z x ¼ db2 d i b 2 i SF y ¼ Z yb 2I y Plstic section moduli, shpe fctors, nd loctions of plstic neutrl xes 802 Formuls for Stress nd Strin [PP.

4 TBLE.1. Tee section ¼ tb þ t w d y c ¼ bt2 þ t w dð2t þ dþ 2ðtb þ t w dþ 5. Chnnel section ¼ tb þ 2t w d y c ¼ bt2 þ 2t w dð2t þ dþ 2ðtb þ 2t w dþ I x ¼ b 3 ðd þ tþ3 d3 3 ðb t wþ ðd þ t y c Þ 2 I y ¼ tb3 12 þ dt3 w 12 r x ¼ x r y ¼ y I x ¼ b 3 ðd þ tþ3 d3 3 ðb 2t wþ ðdþt y c Þ 2 ðd þ tþb3 I y ¼ dðb 2t wþ r x ¼ x r y ¼ y If t w d 5 bt, then Z x ¼ d2 t w b2 t 2 btðd þ tþ þ t w 2 Neutrl xis x is locted distnce ðbt=t w þ dþ=2 from the bottom. If t w d bt, then Z x ¼ t2 b þ t wdðt þ d t w d=2bþ 2 Neutrl xis x is locted distnce ðt w d=b þ tþ=2 from the top. SF x ¼ Z xðd þ t y c Þ I 1 Z y ¼ b2 t þ t 2 wd SF y ¼ Z yb 2I y If 2t w d 5 bt, then Z x ¼ d2 t w 2 b2 t 2 btðd þ tþ þ 8t w 2 Neutrl xis x is locted distnce ðbt=2t w þ dþ=2 from the bottom. If 2t w d bt, then Z x ¼ t2 b þ t wd tþ d t wd b Neutrl xis x is locted distnce t w d=b þ t=2 from the top. SF x ¼ Z xðd þ t y c Þ I x Z y ¼ b2 t þ t wdðb t w Þ SF y ¼ Z yb 2I y PP. ] Properties of Plne re 803

5 TBLE.1 Form of section 6. Wide-flnge bem with equl flnges ¼ 2bt þ t w d y c ¼ d 2 þ t 7. Equl-legged ngle ¼ tð2 tþ y c1 ¼ 0:7071ð2 þ t t 2 Þ 2 t y c2 ¼ 0: t x c ¼ 0: Unequl-legged ngle ¼ tðb þ d tþ x c ¼ b2 þ dt t 2 2ðb þ d tþ y c ¼ d2 þ bt t 2 2ðb þ d tþ re nd distnces from centroid to extremities Moments nd products of inerti nd rdii of gyrtion bout centrl xes bðd þ 2tÞ3 I x ¼ ðb t wþd I y ¼ b3 t 6 þ t3 wd 12 r x ¼ x r y ¼ y I x ¼ b 0:5t2 b 2 12 þ b I y ¼ b where b ¼ t 12 r x ¼ x r y ¼ y I x ¼ 1 3 ½bd3 ðb tþðd tþ 3 Š ðd y c Þ 2 I y ¼ 1 3 ½db3 ðd tþðb tþ 3 Š ðb x c Þ 2 I xy ¼ 1 ½b2 d 2 ðb tþ 2 ðd tþ 2 Š ðb x c Þðd y c Þ r x ¼ x r y ¼ y Z x ¼ t wd 2 þ btðd þ tþ SF x ¼ Z x y c I x Z y ¼ b2 t 2 þ t2 wd SF y ¼ Z yx c I y Plstic section moduli, shpe fctors, nd loctions of plstic neutrl xes Let y p be the verticl distnce from the top corner to the plstic neutrl xis. If t= 5 0:0, then # 1=2 y p ¼ t ðt=þ2 2 Z x ¼ ðy c1 0:6667y p Þ If t= 0:, then y p ¼ 0:3536ð þ 1:5tÞ Z x ¼ y c1 2:828y 2 pt þ 1:8856t 3 80 Formuls for Stress nd Strin [PP.

6 TBLE.1 9. Equilterl tringle ¼ 0: Isosceles tringle 11. Tringle y c ¼ 0:577 x c ¼ 0:5000 y 0 c ¼ 0:577 cos ¼ bd 2 y c ¼ 2 3 d ¼ bd 2 y c ¼ 2 3 d x c ¼ 2 3 b Prllelogrm ¼ bd y c ¼ d 2 x c ¼ 1 2 ðb þ Þ I x ¼ I y ¼ I x 0 ¼ 0:0180 r x ¼ r y ¼ r x 0 ¼ 0:201 I x ¼ 1 36 bd3 I y ¼ 1 8 db3 I x > I y if d > 0:866b r x ¼ 0:2357d r y ¼ 0:201b I x ¼ 1 36 bd3 I y ¼ 1 36 bdðb2 b þ 2 Þ I xy ¼ 1 72 bd2 ðb 2Þ y x ¼ 1 dðb 2Þ 2 tn 1 b 2 b þ 2 d 2 r x ¼ 0:2357d p r y ¼ 0:2357 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 b þ 2 I x ¼ 1 12 bd3 I y ¼ 1 12 bdðb2 þ 2 Þ I xy ¼ 1 12 bd2 y x ¼ 1 2d 2 tn 1 b 2 þ 2 d 2 r x ¼ 0:2887d p r y ¼ 0:2887 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 þ 2 Z x ¼ 0: ; Z y ¼ 0: SF x ¼ 2:33; SF y ¼ 2:000 Neutrl xis x is 0:2537 from the bse. Z x ¼ 0:097bd 2 ; Z y ¼ 0:0833db 2 SF x ¼ 2:33; SF y ¼ 2:000 Neutrl xis x is 0:2929d from the bse. PP. ] Properties of Plne re 805

7 TBLE.1 Form of section 13. Dimond 1. Trpezoid ¼ bd 2 y c ¼ d Solid circle ¼ pr 2 re nd distnces from centroid to extremities ¼ d ðb þ cþ 2 y c ¼ d 2b þ c 3 b þ c x c ¼ 2b2 þ 2bc b 2c c 2 3ðb þ cþ y c ¼ R Moments nd products of inerti nd rdii of gyrtion bout centrl xes I x ¼ 1 8 bd3 I y ¼ 1 8 db3 r x ¼ 0:201d r y ¼ 0:201b I x ¼ d3 b 2 þ bc þ c 2 36 b þ c d I y ¼ 36ðb þ cþ ½b þ c þ 2bcðb 2 þ c 2 Þ ðb 3 þ 3b 2 c 3bc 2 c 3 Þ þ 2 ðb 2 þ bc þ c 2 ÞŠ d 2 I xy ¼ 72ðb þ cþ ½cð3b2 3bc c 2 Þ þ b 3 ð2b 2 þ 8bc þ 2c 2 ÞŠ I x ¼ I y ¼ p R r x ¼ r y ¼ R 2 Plstic section moduli, shpe fctors, nd loctions of plstic neutrl xes Z x ¼ 0:0833bd 2 ; Z y ¼ 0:0833db 2 SF x ¼ SF y ¼ 2:000 Z x ¼ Z y ¼ 1:333R 3 SF x ¼ 1: Formuls for Stress nd Strin [PP.

8 TBLE Hollow circle ¼ pðr 2 R 2 i Þ y c ¼ R 17. Very thin nnulus ¼ 2pRt y c ¼ R 18. Sector of solid circle ¼ R 2 y c1 ¼ R 1 2 sin 3 2R sin y c2 ¼ 3 x c ¼ R sin I x ¼ I y ¼ p ðr R i Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r x ¼ r y ¼ 1 2 R 2 þ R 2 i I x ¼ I y ¼ pr 3 t r x ¼ r y ¼ 0:707R I x ¼ R! þ sin cos 16 sin2 9 I y ¼ R ð sin cos Þ ðnote: If is smll; sin cos ¼ Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r x ¼ R sin cos 1 þ 16 sin r y ¼ R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin cos 1 2 Z x ¼ Z y ¼ 1:333ðR 3 R 3 i Þ SF x ¼ 1:698 R R 3 i R R R i Z x ¼ Z y ¼ R 2 t SF x ¼ SF y ¼ p If 5:3, then Z x ¼ 0:6667R 3 3 1=2 # sin 2 tn Neutrl xis x is locted distnce Rð0:5= tn Þ 1=2 from the vertex. If 55:3, then Z x ¼ 0:6667R 3 ð2 sin 3 1 sin Þ where the expression 2 1 sin 2 1 ¼ is solved for the vlue of 1. Neutrl xis x is locted distnce R cos 1 from the vertex. If 73:09, then SF x ¼ Z xy c2 I x If 73:09 90, then SF x ¼ Z xy c1 I x Z y ¼ 0:6667R 3 ð1 cos Þ If 90, then 1 cos SF y ¼ 2:6667 sin sin cos If 590, then 1 cos SF y ¼ 2:6667 sin cos PP. ] Properties of Plne re 807

9 TBLE.1 Form of section 19. Segment of solid circle (Note: If p=, use expressions from cse 20) 20. Segment of solid circle (Note: Do not use if > p=þ re nd distnces from centroid to extremities ¼ R 2 ð sin cos Þ # 2 sin 3 y c1 ¼ R 1 3ð sin cos Þ # 2 sin 3 y c2 ¼ R cos 3ð sin cos Þ x c ¼ R sin ¼ 2 3 R2 3 ð1 0:2 2 þ 0:019 Þ y c1 ¼ 0:3R 2 ð1 0: þ 0:0028 Þ y c2 ¼ 0:2R 2 ð1 0: þ 0:0027 Þ x c ¼ Rð1 0: þ 0:0083 Þ 21. Sector of hollow circle ¼ tð2r tþ y c1 ¼ R 1 2 sin 1 t 3 R þ 1 2 t=r 2 sin y c2 ¼ R 3ð2 t=rþ þ 1 t 2 sin 3 cos R 3 (Note: If t=r is smll, cn exceed p to form n overlpped nnulus) x c ¼ R sin Moments nd products of inerti nd rdii of gyrtion bout centrl xes # I x ¼ R sin cos þ 2 sin 3 16 sin 6 cos 9ð sin cos Þ I y ¼ R 12 ð3 3 sin cos 2 sin3 cos Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r x ¼ R 1 þ 2 sin3 cos 2 sin cos 16 sin 6 9ð sin cos Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r y ¼ R 1 2 sin3 cos 2 3ð sin cos Þ I x ¼ 0:0113R 7 ð1 0:391 2 þ 0:050 Þ I y ¼ 0:1333R 5 ð1 0:762 2 þ 0:1111 Þ r x ¼ 0:1309R 2 ð1 0:075 2 Þ r y ¼ 0:72Rð1 0: þ 0:018 Þ I x ¼ R 3 t 1 3t 2R þ t2 t3 R2 R 3! þ sin cos 2 sin2 þ t2 sin 2 3R 2 1 t # ð2 t=rþ R þ t2 6R 2 I y ¼ R 3 t 1 3t 2R þ t2 t3 R2 R 3 ð sin cos Þ rffiffiffiffi rffiffiffiffi I r x ¼ x I y ; r y ¼ Plstic section moduli, shpe fctors, nd loctions of plstic neutrl xes 808 Formuls for Stress nd Strin [PP.

10 TBLE Solid semicircle ¼ p 2 R2 23. Hollow semicircle Note: b ¼ R þ R i 2 t ¼ R R i Note: If is smll: sin ¼ þ 120 ; sin cos ¼ þ 2 sin 2 ; ¼ þ 2 5 cos ¼ þ 2 ; þ sin cos 2 sin2 ¼ þ 105 y c1 ¼ 0:5756R y c2 ¼ 0:2R x c ¼ R ¼ p 2 ðr2 R 2 i Þ y c2 ¼ 3p R 3 R 2 i R 2 R 2 i or y c2 ¼ 2b p y c1 ¼ R y c2 x c ¼ R 2. Solid ellipse ¼ pb y c ¼ x c ¼ b # 1 þ ðt=bþ2 12 I x ¼ 0:1098R I y ¼ p 8 R r x ¼ 0:263R r y ¼ R 2 I x ¼ p 8 ðr R i Þ 8 ðr 3 R 3 i Þ2 9p R 2 R 2 i or I x ¼ 0:2976tb 3 þ 0:1805bt 3 0:0088t5 b I y ¼ p 8 ðr R i Þ or I y ¼ 1:5708b 3 t þ 0:3927bt 3 I x ¼ p b3 I y ¼ p b3 r x ¼ 2 r y ¼ b 2 Z x ¼ 0:350R 3 ; Z y ¼ 0:6667R 3 SF x ¼ 1:856; SF y ¼ 1:698 Plstic neutrl xis x is locted distnce 0:00R from the bse. Let y p be the verticl distnce from the bottom to the plstic neutrl xis. y p ¼ð0:7071 0:2716C 0:299C 2 þ 0:3983C 3 ÞR Z x ¼ð0:828 0:910C þ 0:725C 2 0:2850C 3 ÞR 2 t where C ¼ t=r Z y ¼ 0:6667ðR 3 R 3 i Þ Z x ¼ 1:333 2 b; SF x ¼ SF y ¼ 1:698 Z y ¼ 1:333b 2 PP. ] Properties of Plne re 809

11 TBLE.1 re nd distnces from Form of section centroid to extremities 25. Hollow ellipse ¼ pðb i b i Þ y c ¼ x c ¼ b Note: For this cse the inner nd outer perimeters re both ellipses nd the wll thickness is not constnt. For cross section with constnt wll thickness see cse Hollow ellipse with constnt wll thickness t. The midthickness perimeter is n ellipse (shown dshed). 0:2 < =b < 5 See the note on mximum wll thickness in cse 27. # 2 b ¼ ptð þ bþ 1 þ K 1 þ b where K 1 ¼ 0:26 þ 0: b þ b y c ¼ þ t 2 x c ¼ b þ t 2 Moments nd products of inerti nd rdii of gyrtion bout centrl xes I x ¼ p ðb3 b i 3 i Þ I y ¼ p ðb3 i b 3 i Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r x ¼ 1 b 3 b i 3 i 2 b i b i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r y ¼ 1 b 3 i b 3 i 2 b i b i I x ¼ p # 2 b t2 ð þ 3bÞ 1 þ K 2 þ b þ p # 2 b 16 t3 ð3 þ bþ 1 þ K 3 þ b where K 2 ¼ 0:139 þ 0:1279 b 0: b K 3 ¼ 0:139 þ 0:1279 b 0:0128 b 2 For I y interchnge nd b in the expressions for I x ; K 2,ndK 3 Plstic section moduli, shpe fctors, nd loctions of plstic neutrl xes Z x ¼ 1:333ð 2 b 2 i b i Þ Z y ¼ 1:333ðb 2 b 2 i i Þ SF x ¼ 1:698 3 b 2 i b i 3 b 3 i b i SF y ¼ 1:698 b3 b 2 i ib b 3 b 3 i i # 2 b Z x ¼ 1:3333tð þ 2bÞ 1 þ K þ t3 þ b 3 where K ¼ 0:1835 þ 0:895 b 0: b For Z y interchnge nd b in the expression for Z x nd K. 810 Formuls for Stress nd Strin [PP.

12 TBLE Hollow semiellipse with constnt wll thickness t. The midthickness perimeter is n ellipse (shown dshed). 0:2 < =b < 5 Note: There is limit on the mximum wll thickness llowed in this cse. Cusps will form in the perimeter t the ends of the mjor xis if this mximum is exceeded. If b 1; then t mx ¼ 22 b If b 5 1; then t mx ¼ 2b2 ¼ p # 2 2 tð þ bþ 1 þ K b 1 þ b where K 1 ¼ 0:26 þ 0: b þ b y c2 ¼ 2 p K 2 þ t2 6p K 3 where K 2 ¼ 1 0:331C þ 0:0136C 2 þ 0:1097C 3 K 3 ¼ 1 þ 0:9929C 0:2287C 2 0:2193C 3 Using C ¼ b þ b y c1 ¼ þ t 2 y c2 x c ¼ b þ t 2 I X ¼ p # 2 b 8 t2 ð þ 3bÞ 1 þ K þ b þ p # 2 b 32 t3 ð3 þ bþ 1 þ K 5 þ b where K ¼ 0:139 þ 0:1279 b 0: b K 5 ¼ 0:139 þ 0:1279 b 0:0128 b 2 I x ¼ I X y 2 c2 For I y use one-hlf the vlue for I y in cse 26. Let y p be the verticl distnce from the bottom to the plstic neutrl xis. y p ¼ C 1 þ C 2 =b þ C 3 ð=bþ 2 þ C ð=bþ 3 where if 0:25 < =b 1, then C 1 ¼ 0:5067 0:5588D þ 1:3820D 2 C 2 ¼ 0:3731 þ 0:1938D 1:078D 2 C 3 ¼ 0:100 þ 0:0179D þ 0:885D 2 C ¼ 0:0170 0:0079D 0:0565D 2 or if 1 =b <, then C 1 ¼ 0:829 þ 0:0725D 0:1815D 2 C 2 ¼ 0:1957 0:6608D þ 1:222D 2 C 3 ¼ 0:0203 þ 1:8999D 3:356D 2 C ¼ 0:0578 1:6666D þ 2:6012D 2 where D ¼ t=t mx nd where 0:2 < D 1 Z x ¼ C 5 þ C 6 =b þ C 7 ð=bþ 2 þ C 8 ð=bþ 3 2 t where if 0:25 < =b 1, then C 5 ¼ 0:0292 þ 0:379D 1=2 þ 0:0578D C 6 ¼ 0:367 0:8531D 1=2 þ 0:3882D C 7 ¼ 0:1218 þ 0:3563D 1=2 0:1803D C 8 ¼ 0:015 0:08D 1=2 þ 0:0233D or if 1 =b <, then C 5 ¼ 0:221 0:3922D 1=2 þ 0:2960D C 6 ¼ 0:6637 þ 2:7357D 1=2 2:082D C 7 ¼ 1:5211 5:386D 1=2 þ 3:9286D C 8 ¼ 0:898 þ 2:8763D 1=2 1:887D For Z y use one-hlf the vlue for Z y in cse 26. PP. ] Properties of Plne re 811

13 TBLE.1 Form of section 28. Regulr polygon with n sides 29. Hollow regulr polygon with n sides Properties of sections Continued) re nd distnces from centroid to extremities ¼ 2 n tn r 1 ¼ 2 sin r 2 ¼ 2 tn If n is odd y 1 ¼ y 2 ¼ r 1 cos n þ 1 p 2 2 If n=2 is odd y 1 ¼ r 1 ; y 2 ¼ r 2 If n=2 is even y 1 ¼ r 2 ; y 2 ¼ r 1 ¼ nt 1 t tn r 1 ¼ 2 sin r 2 ¼ 2 tn If n is odd y 1 ¼ y 2 ¼ r 1 cos n þ 1 p 2 2 If n=2 is odd y 1 ¼ r 1 ; y 2 ¼ r 2 If n=2 is even y 1 ¼ r 2 ; y 2 ¼ r 1 Moments nd products of inerti nd rdii of gyrtion bout centrl xes I 1 ¼ I 2 ¼ 1 2 ð6r2 1 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r 1 ¼ r 2 ¼ 2 ð6r2 1 2 Þ I 1 ¼ I 2 ¼ n3 t þ 1 tn t tn þ t tn 2 2 t tn # 3 r 1 ¼ r 2 ¼ p ffiffiffi 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 t tn 3 tn þ 2 t tn # u 2 t Plstic section moduli, shpe fctors, nd loctions of plstic neutrl xes For n ¼ 3, see cse 9. For n ¼, see cses 1 nd 13. For n ¼ 5, Z 1 ¼ Z 2 ¼ 0:8825r 3 1. For n xis perpendiculr to xis 1, Z ¼ 0:8838r 3 1. The loction of this xis is from tht side which is perpendiculr to xis 1. For n 5 6, use the following expression for neutrl xis of ny inclintion: Z ¼ r 3 1 1:333 13: þ 12:528 1 # 3 n n 812 Formuls for Stress nd Strin [PP.