Геометрични характеристики на L-сечение

L_section.png

Размери

h = 150 mm, tw = 10 mm

bf = 100 mm, tf = 10 mm

Площ

Aw = h·tw = 150·10 = 1500 mm2

Af = (bftwtf = (100 – 10)·10 = 900 mm2

A = Aw + Af = 1500 + 900 = 2400 mm2

Sz = Aw·tw/2 + Af·(bf + tw)/2 = 1500·10/2 + 900·(100 + 10)/2 = 57000 mm3

yc = Sz/A = 57000/2400 = 23.75 mm

Sy = Aw·h/2 + Af·tf/2 = 1500·150/2 + 900·10/2 = 117000 mm3

zc = Sy/A = 117000/2400 = 48.75 mm

Периметър

P = 2·(h + bf) = 2·(150 + 100) = 500 mm

Инерционни моменти

h1 = htf = 150 – 10 = 140 mm

Iy_w = Aw·(h2/12 + (zch/2)2) = 1500·(1502/12 + (48.75 – 150/2)2) = 3846094 mm4

Iy_f = Af·(tf2/12 + (zctf/2)2) = 900·(102/12 + (48.75 – 10/2)2) = 1730156 mm4

Iy = Iy_w + Iy_f = 3846094 + 1730156 = 5576250 mm4

Iz_w = h1·tw·(tw2/12 + (yctw/2)2) = 140·10·(102/12 + (23.75 – 10/2)2) = 503854 mm4

Iz_f = tf·bf·(bf2/12 + (ycbf/2)2) = 10·100·(1002/12 + (23.75 – 100/2)2) = 1522396 mm4

Iz = Iz_w + Iz_f = 503854 + 1522396 = 2026250 mm4

Iyz = Aw·(h/2 – zc)·(tw/2 – yc) + Af·(tf/2 – zc)·((bf + tw)/2 – yc) = 1500·(150/2 – 48.75)·(10/2 – 23.75) + 900·(10/2 – 48.75)·((100 + 10)/2 – 23.75) = -1968750 mm4

I1 = (Iy + Iz)/2 + √(IyIz)2/4 + Iyz2 = (5576250 + 2026250)/2 + √(5576250 – 2026250)2/4 + (-1968750)2 = 6452024 mm4

I2 = (Iy + Iz)/2 – √(IyIz)2/4 + Iyz2 = (5576250 + 2026250)/2 – √(5576250 – 2026250)2/4 + (-1968750)2 = 1150476 mm4

α1 = atan((IyI1)/Iyz) = atan((5576250 – 6452024)/(-1968750)) = 23.98

Полярен инерционен момент

Ix = I1 + I2 = 6452024 + 1150476 = 7602500 mm4

Инерционни радиуси

ry = √Iy/A = √5576250/2400 = 48.2 mm

rz = √Iz/A = √2026250/2400 = 29.06 mm

rx = √Ix/A = √7602500/2400 = 56.28 mm

r1 = √I1/A = √6452024/2400 = 51.85 mm

r2 = √I2/A = √1150476/2400 = 21.89 mm

Съпротивителни моменти

Wel_y = Iy/(hzc) = 5576250/(150 – 48.75) = 55074 mm3

Wel_z = Iz/(bfyc) = 2026250/(100 – 23.75) = 26574 mm3

Пластични съпротивителни моменти

yc_pl = bfA/(2·tf) = 100 – 2400/(2·10) = -20 mm

yc_pl < tw => yc_pl = A/(2·h) = 2400/(2·150) = 8 mm

Wpl_z = tf·(yc_pl2 + (bfyc_pl)2)/2 + (htf)·(yc_pl2 + (twyc_pl)2)/2 = 10·(82 + (100 – 8)2)/2 + (150 – 10)·(82 + (10 – 8)2)/2 = 47400 mm3

zc_pl = (A/2 – Af)/tw = (2400/2 – 900)/10 = 30 mm

Wpl_y = tw·(zc_pl2 + (hzc_pl)2)/2 + Af·(zc_pltf/2) = 10·(302 + (150 – 30)2)/2 + 900·(30 – 10/2) = 99000 mm3

Радиус на закръгление

r = 12 mm

L_section_tor.png

Инерционен момент за усукване*

Itf = bf·tf3·(1 – 0.630·tf/bf + 0.052·(tf/bf)5)/3 = 100·103·(1 – 0.630·10/100 + 0.052·(10/100)5)/3 = 31233 mm4

Itw = h1·tw3·(1 – 0.315·tw/h1 + 0.00164·(tw/h1)5)/3 = 140·103·(1 – 0.315·10/140 + 0.00164·(10/140)5)/3 = 45617 mm4

α = tw/tf·(0.07 + 0.076·r/tf) = 10/10·(0.07 + 0.076·12/10) = 0.16

D = 2·(tw + tf + 3·r – √2·(2·r + tw)·(2·r + tf)) = 2·(10 + 10 + 3·12 – √2·(2·12 + 10)·(2·12 + 10)) = 15.83

It = Itw + Itf + α·D4 = 45617 + 31233 + 0.16·15.834 = 86981 mm4

Съпротивителен момент за усукване*

C = D/(1 + π2·D4/(16·A2))·(1 + (0.118·ln(1 + D/(2·r)) + 0.238·D/(2·r))·0.7616) = 15.83/(1 + 3.142·15.834/(16·24002))·(1 + (0.118·ln(1 + 15.83/(2·12)) + 0.238·15.83/(2·12))·0.7616) = 18.32

Wt = It/C = 86981/18.32 = 4746.73 mm3

*Определени са с приблизителни формули съгласно Roark Formulas for Stress and Strain, 7th ed, W.Young, R. Budynas, McGraw-Hill, 2002

Секториален инерционен момент

h2 = htf/2 = 150 – 10/2 = 145 mm

b2 = bftw/2 = 100 – 10/2 = 95 mm

Iw = (h23·tw3 + b23·tf3)/36 = (1453·103 + 953·103)/36 = 108500000 mm6

Ефективна площ за срязване

Asy = 5/6·bf·tf = 5/6·100·10 = 833.33 mm2

Asz = 5/6·h·tw = 5/6·150·10 = 1250 mm2