phu luc D pdf 966 Notation A � area x�, y� � distances to centroid C Ix, Iy � moments of inertia with respect to the x and y axes, respectively Ixy � product of inertia with respect to the x and y axe[.]
D Properties of Plane Areas Notation: A area x苶, y苶 distances to centroid C Ix, Iy moments of inertia with respect to the x and y axes, respectively Ixy product of inertia with respect to the x and y axes IP Ix Iy polar moment of inertia with respect to the origin of the x and y axes IBB moment of inertia with respect to axis B-B y Rectangle (Origin of axes at centroid) A bh x h C x y bh3 Ix 12 b 苶x 2 h y苶 hb3 Iy 12 Ixy bh IP (h2 b2) 12 b y Rectangle (Origin of axes at corner) B bh3 Ix h O B b2h2 Ixy bh IP (h2 + b2) b h3 IBB 6(b2 h2) x b y Triangle (Origin of axes at centroid) c bh A x h C y b 966 hb3 Iy x bh3 Ix 36 bc x苶 h y苶 bh Iy (b2 bc c2) 36 bh2 Ixy (b 2c) 72 bh IP (h2 b2 bc c2) 36 APPENDIX D y Triangle (Origin of axes at vertex) c B B h O x b y bh3 Ix 12 bh Iy (3b2 3bc c2) 12 bh2 Ixy (3b 2c) 24 bh3 IBB Isosceles triangle (Origin of axes at centroid) bh A x h C y B x B b bh3 Ix 36 b x苶 h y苶 hb3 Iy 48 Ixy bh3 IBB 12 bh IP (4h2 3b2) 144 (Note: For an equilateral triangle, h 兹3苶 b/2.) y Right triangle (Origin of axes at centroid) bh A x h C y B x B bh3 Ix 36 b B bh3 Ix 12 h bh3 IBB 12 hb3 Iy 12 bh IP (h2 b2) 12 x b2h2 Ixy 24 bh3 IBB b y Trapezoid (Origin of axes at centroid) a h b2h2 Ixy 72 Right triangle (Origin of axes at vertex) B O h 苶y 3 hb3 Iy 36 bh IP (h2 b2) 36 y b 苶x 3 C B b h(a b) A y x h(2a b) 苶y 3(a b) h3(a2 4ab b2) Ix B 36(a b) h3(3a b) IBB 12 Properties of Plane Areas 967 968 APPENDIX D y Properties of Plane Areas Circle (Origin of axes at center) d = 2r pd2 A pr r x C B Ixy B 10 r pr4 pd4 IP 32 5p r 5p d IBB 64 y Semicircle (Origin of axes at centroid) C pr2 A y B x B y 11 pr pd Ix Iy 64 4r y苶 3p (9p 64)r Ix ⬇ 0.1098r4 72p p r4 Iy Ixy pr4 IBB Quarter circle (Origin of axes at center of circle) pr2 A x B B C y x O 4r x苶 y苶 3p pr4 Ix Iy 16 (9p 64)r IBB ⬇ 0.05488r4 144p r4 Ixy r y 12 Quarter-circular spandrel (Origin of axes at point of tangency) B B r 冢 x 冢 x y 13 C a a y O 冣 冢 冣 p Iy IBB r ⬇ 0.1370r4 16 Circular sector (Origin of axes at center of circle) x x (10 3p)r ⬇ 0.2234r 苶y 3(4 p) 2r ⬇ 0.7766r 苶x 3(4 p) 5p Ix r ⬇ 0.01825r 16 y C O 冣 p A r a angle in radians A ar r x (a p/2) 苶x r sin a r4 Ix (a sin a cos a) 2r sin a y苶 3a r4 Iy (a sin a cos a) Ixy ar IP APPENDIX D y 14 Circular segment (Origin of axes at center of circle) a angle in radians C a y Properties of Plane Areas a (a p/2) sin3 a 2r 苶y 3 a sin a cos a 冢 A r 2(a sin a cos a) r r4 Ix (a sin a cos a sin3 a cos a) x O 冣 Ixy r4 Iy (3a sin a cos a sin3 a cos a) 12 15 y a Circle with core removed (Origin of axes at center of circle) a angle in radians r C a a arccos r b a x a b (a p/2) 冢 冢 r4 3ab 2ab3 Ix 3a r r4 冣 冣 ab A 2r a r2 b 兹苶 r a苶2 冢 r4 ab 2ab3 Iy a r r4 2a y 16 Ellipse (Origin of axes at centroid) A pab b C x b a a Ixy pa b Ix p ba3 Iy pab IP (b2 a2) Circumference ⬇ p[1.5( a b) 兹a苶b苶 ] ⬇ 4.17b2/a 4a 17 y 冢 x2 y f (x) h 2 b x C O (0 b a/3) Parabolic semisegment (Origin of axes at corner) ertex V y = f (x) h ( a/3 b a) y b x bh A 16bh3 Ix 105 冣 3b x苶 2hb3 Iy 15 2h y苶 b2h2 Ixy 12 冣 Ixy 969 970 APPENDIX D Properties of Plane Areas y 18 Parabolic spandrel (Origin of axes at vertex) y = f (x) x ertex V hx2 y f (x) b2 h y C O bh x A b 3b x苶 bh3 Ix 21 y 19 冢 xn y f (x) h bn x C x b b2h2 Ixy 12 冣 冣 (n 0) b(n 1) 苶x 2(n 2) hn y苶 2n 2bh3n3 (n 1)(2n 1)(3n 1) hb3n Iy 3(n 3) Ix y 20 冢 n A bh n1 y O hb3 Iy Semisegment of nth degree (Origin of axes at corner) y = f (x) h 3h 苶y 10 b2h2n2 Ixy 4(n 1)(n 2) Spandrel of nth degree (Origin of axes at point of tangency) hx n y f (x) bn y = f (x) x h C y O b(n 1) x苶 n2 bh A n1 x bh3 Ix 3(3n 1) b 21 4bh A p C y B b b y hb3 Iy n3 d = 2r 冢 9p 16 冣 Thin circular ring (Origin of axes at center) Approximate formulas for case when t is small x t 冢 Ixy p d 3t Ix Iy pr 3t pd 3t IP 2p r 3t 冣 32 Iy 3 hb3 ⬇ 0.2412hb3 p p 8bh3 IBB 9p A 2prt pdt r C b2 h Ixy 4(n 1) ph y苶 x BIx 8 p bh3 ⬇ 0.08659bh3 Ixy 22 h(n 1) y苶 2(2n 1) Sine wave (Origin of axes at centroid) y h (n 0) APPENDIX D 23 B C b y b b angle in radians x O Ix r 3t(b sin b cos b) 冢 冣 Thin rectangle (Origin of axes at centroid) Approximate formulas for case when t is small A bt b b C x t B Iy r 3t(b sin b cos b) 2b sin2b cos2b IBB r 3t b Ixy y (Note: For a semicircular arc, b p/2.) r sin b 苶y b A 2brt r 24 971 Thin circular arc (Origin of axes at center of circle) Approximate formulas for case when t is small y t B Properties of Plane Areas tb3 Ix sin2 b 12 tb3 Iy cos2 b 12 tb3 IBB sin2 b B 25 Regular polygon with n sides (Origin of axes at centroid) A B b R1 R2 b C centroid (at center of polygon) n number of sides (n 3) b length of a side b central angle for a side C a 360° b n 冢 a interior angle (or vertex angle) 冣 n2 a 180° n a b 180° R1 radius of circumscribed circle (line CA) b b R1 csc 2 b b R2 cot 2 R2 radius of inscribed circle (line CB) b nb A cot 2 Ic moment of inertia about any axis through C (the centroid C is a principal point and every axis through C is a principal axis) nb4 b b Ic cot 3cot2 192 2 冢 冣冢 冣 IP 2Ic