Y2 40.80707 0.000000 X3 595.0000 0.000000 Y3 33.24250 0.000000 X4 570.0000 0.000000 Y4 25.36635 0.000000 X5 540.0000 0.000000 Y5 24.69553 0.000000 X6 505.0000 0.000000 Y6 23.71802 0.000000 X7 470.0000 Y7 19.88090 X8 430.0000 Y8 19.22171 X9 390.0000 Y9 16.27885 X10 345.0000 Y10 15.44283 X11 295.0000 Y11 14.15819 X12 240.0000 Y12 12.43375 X13 180.0000 Y13 10.25516 X14 105.0000 Y14 8.441656 C1 128.0000 C2 124.0000 C3 119.0000 C4 114.0000 C5 108.0000 C6 101.0000 C7 94.00000 C8 86.00000 C9 78.00000 C10 69.00000 C11 59.00000 C12 48.00000 C13 36.00000 C14 21.00000 H1 444.0057 Row Slack or Surplus 1 320751.3 2 10.00000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -4.402540 -11.02568 -48.20120 20.01531 14.23796 -21.22961 10.54568 -15.78595 5.961521 2.137815 -1.518545 -5.145899 5.521017 1.519777 0.000000 Dual Price 1.000000 0.000000
3 20.00000 0.000000 4 25.00000 0.000000 5 25.00000 0.000000 6 30.00000 0.000000 7 35.00000 0.000000 8 35.00000 0.000000 9 40.00000 0.000000 10 40.00000 0.000000 11 45.00000 0.000000 12 50.00000 13 55.00000 14 60.00000 15 75.00000 16 85.00000 17 0.000000 18 0.000000 19 0.000000 20 0.000000 21 0.000000 22 0.000000 23 0.000000 24 0.000000 25 0.000000 26 0.000000 27 0.000000 28 0.000000 29 0.000000 30 0.000000 31 -0.6347215E-03 32 -0.7412484E-02 33 0.000000 34 0.000000 35 0.000000 36 0.000000 37 0.000000 38 0.000000 39 0.000000 40 0.000000 41 0.000000 42 0.000000 43 0.000000 44 0.000000 45 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 4.402540 11.02568 48.20120 -20.01531 -14.23796 21.22961 -10.54568 15.78595 -5.961521 -2.137815 1.518545 5.145899 -5.521017 -1.519777 0.3521803 0.2561577 0.1910750 0.1920553 0.1934765 0.1710500 0.1781742 0.1641220 0.1730769 0.1815259 0.1899165 0.1986526 0.2401614 0.3273765 0.000000
根据lingo求解可知 当级数为14时，铁心柱的有效截面积最大。 求得面积为320751.3
附录四：
以此数学建模，应用lingo工具 数学模型为
max=(x1*y1+2*x2*y2+2*x3*y3+2*x4*y4+2*x5*y5+2*x6*y6+2*x7*y7+2*x8*y8+2*x9*y9+2*x10*y10+2*x11*y11+2*x12*y12+2*x13*y13+2*x14*y14)/3.1415926/r/r; r>=325; r<=380; x1<=2*r; x1>=x2; x2>=x3; x3>=x4; x4>=x5; x5>=x6; x6>=x7; x7>=x8; x8>=x9; x9>=x10; x10>=x11; x11>=x12; x12>=x13; x13>=x14; x14>=20; x1/5=c1; x2/5=c2; x3/5=c3; x4/5=c4; x5/5=c5; x6/5=c6; x7/5=c7; x8/5=c8; x9/5=c9; x10/5=c10; x11/5=c11; x12/5=c12; x13/5=c13; x14/5=c14; @gin(c1);
@gin(c2); @gin(c3); @gin(c4); @gin(c5); @gin(c6); @gin(c7); @gin(c8); @gin(c9); @gin(c10); @gin(c11); @gin(c12); @gin(c13); @gin(c14);
(x1/2)*(x1/2)+(y1/2)*(y1/2)=r*r; (x2/2)*(x2/2)+(y1/2+y2)*(y1/2+y2)=r*r; (x3/2)*(x3/2)+(y1/2+y2+y3)*(y1/2+y2+y3)=r*r; (x4/2)*(x4/2)+(y1/2+y2+y3+y4)*(y1/2+y2+y3+y4)=r*r; (x5/2)*(x5/2)+(y1/2+y2+y3+y4+y5)*(y1/2+y2+y3+y4+y5)=r*r; (x6/2)*(x6/2)+(y1/2+y2+y3+y4+y5+y6)*(y1/2+y2+y3+y4+y5+y6)=r*r; (x7/2)*(x7/2)+(y1/2+y2+y3+y4+y5+y6+y7)*(y1/2+y2+y3+y4+y5+y6+y7)=r*r; (x8/2)*(x8/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8)*(y1/2+y2+y3+y4+y5+y6+y7+y8)=r*r; (x9/2)*(x9/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8+y9)*(y1/2+y2+y3+y4+y5+y6+y7+y8+y9)=r*r;
(x10/2)*(x10/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10)*(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10)=r*r;
(x11/2)*(x11/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11)*(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11)=r*r;
(x12/2)*(x12/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12)*(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12)=r*r;
(x13/2)*(x13/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12+y13)*(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12+y13)=r*r;
(x14/2)*(x14/2)+(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12+y13+y14)*(y1/2+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12+y13+y14)=r*r;
求得结果为：
Local optimal solution found.
Objective value: 0.9667117 Objective bound: 0.9667117 Infeasibilities: 0.5820766E-10 Extended solver steps: 1646 Total solver iterations: 159235
Variable Value X1 750.0000 Y1 122.8629 X2 730.0000 Y2 44.27486 X3 705.0000 Y3 36.21823 X4 675.0000 Y4 32.69670 X5 640.0000 Y5 30.31492 X6 600.0000 Y6 28.29941 X7 560.0000 Y7 23.66681 X8 515.0000 Y8 22.54812 X9 465.0000 Y9 21.11993 X10 410.0000 Y10 19.38868 X11 350.0000 Y11 17.34466 X12 285.0000 Y12 14.96400 X13 210.0000 Y13 12.93605 X14 120.0000 Y14 10.02788 R 379.9984 C1 150.0000 C2 146.0000 C3 141.0000 C4 135.0000 C5 128.0000 C6 120.0000 C7 112.0000 C8 103.0000 C9 93.00000 C10 82.00000