Beam on Elastic Foundation

Closed-form Solutions Solutions of Beam on Elastic Foundation (Winkler & Hentenyi Method) fc' =

25.00 MPa

E = 4730000√fc'   = 2.365E+07 2.365E+07 kPa kPa Beam width B = 1.000 m Beam thickness T = 0.400 m Beam length L = 11.900 m Beam inertia I = 5.333E-03  m Modulus of subgrade reaction: ks = 65,000 kN/m³ ks' = B·ks = 65,000 kN/m² l = [ks'/(4EI)]   = 0.5991090  m 0.5991090 m l·L = 7.1294 (2.2694·π)

Concentrated load: P, kN a, m 155.00 0.175 210.00 4.025 210.00 7.875 155.00 11.725

 

Semi-infinite Beam Infinite Beam

   

b, m 11.725 7.875 4.025 0.175 11.900 11.900 11.900 11.900 11.900 11.900 11.900 11.900

W , m Points 0.350 19 0.350 19 0.350 19 0.350

11.900 11.900 11.900 Applied moment: M, kN-m a, m 40.00 0.175 -25.00 4.025 -25.00 7.875 40.00 11.725 Bearing Pressure Diagram qmax = 63.42 0.0

2.0

4.0

6.0

qmin = 43.25 8.0

10.0

12.0

14.0

10.0

12.0

14.0

0 10 20

   ²   m30    /    N    k 40  ,   q

50 60 70 Bending Moment Diagram Mmax = 74.01

Mmin = -35.12

-60 -40 -20

0.0

2.0

4.0

6.0

8.0

  m 0      N    k 20  ,    M

40 60 80 Deflection 0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.00E+00 2.00E-04 4.00E-04   m 6.00E-04  ,   y

8.00E-04 1.00E-03 1.20E-03

208827661.xls.ms_office:Analyse

Issued Date: June 21, 2007

Printed :2/4/2014 8:51 AM

Verification Table (Winkler) lx  Alx Blx 0.0 0.1 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.5000

1.0000 0.0000 0.9907 0.0903 0.8231 0.2908 0.5083 0.3096 0.2384 0.2226 0.0667 0.1231 -0.0423 0.0070 -0.0258 -0.0139 -0.0045 -0.0065 0.0017 -0.0007 0.0013 0.0006 0.0003 0.0003 -0.0001 0.0001 0.8231

0.2908

Clx 1.0000 0.8100 0.2415 -0.1108 -0.2068 -0.1794 -0.0563 0.0019 0.0084 0.0031 0.0001 -0.0004 -0.0002

Dlx 1.0000 0.9003 0.5323 0.1988 0.0158 -0.0563 -0.0493 -0.0120 0.0019 0.0024 0.0007 0.0000 -0.0001

0.2415

0.5323

Verification Table : Semi-Infinite Beam (P) Distance y lL = 1.0000 Load at 0.0·L 0.00 4.0000 1.00 0.7951 Load at 0.4·L 0.00 2.4696 0.20 1.9817 0.50 1.3145 0.80 0.7731 1.00 0.4909 0.2500 P= L= x 0.00 2.38 5.95 9.52 11.90

Q'

la

la = 0.0000 2.0000 -0.6191 -0.2216 la = 0.0000 1.2348 -0.1159 0.7900 -0.2073 0.2982 -0.2297 -0.0115 -0.2206 -0.1368

  0.00 0.00 0.00   0.40 0.40 0.40 0.40 0.40 0.40

M

0.0000 -1.2382 0.0000 -0.4017 -0.7181 -0.7960 -0.7645

Q

1.8636 -0.4758 -0.1373 1.1293 ks = 65,000 l =   0.59911 200.00 11.900 a= 2.380 y M q 1.2794E-04 0.00 8.32 4.4396E-06 -2.75 0.29 -3.3020E-06 0.13 -0.21 3.5689E-07 0.02 0.02 6.7942E-08 -0.01 0.00

Dla

a

b

1.0000 1.0000

1.0000 1.0000

3.0000 3.0000

2 .0000 2.0000

0.00 1.00

0.3564 0.3564 0.3564 0.3564 0.3564

0.6174 0.6174 0.6174 0.6174 0.6174

1.5912 1.5912 1.5912 1.5912 1.5912

0 .9738 0 .9738 0 .9738 0.9738 0.9738

0.00 0.20 0.50 0.80 1.00

0 M 0.00 0.00 0.00 0.00 0.00

q 239.64 8.32 -6.19 0.67 0.13

2.26936·π

lL =

a = 0.000 y 3.6868E-03 1.2794E-04 -9.5155E-05 1.0285E-05 1.9579E-06

Deflection 1.4000E-04 1.2000E-04 1.0000E-04 8.0000E-05 6.0000E-05 4.0000E-05 2.0000E-05 0.0000E+00 0.00 -2.0000E-05

2.00

4.00

6.00

8.00

Bearing Pressure 9.00 8.00 7.00

lx

Cla

10.00

12.00

14.00

6.00 5.00 4.00 3.00 2.00 1.00 0.00 -1.000.00

2.00

4.00

6.00

8.00

Verification Table : Semi-Infinite Beam (P) - Hetenyi la Distance y M Q l L = 1.0000 la =   0.00 0.0000 Load at 0.0·L 0.00 4.0378 0.0000 0.0000 0.00 0.20 2.8005 -0.2550 0 .3164 1.00 0.50 0.9814 -0.2478 -0.2496 1.00 1.00 -1.9717 0.0000 0.0000 1.00 la =   0.50 0.5000 Load at 0.5·L 0.00 0.9814 0.0000 0.0000 0.50 0.50 1.0124 0.2486 0.5000 0.50 l L = 2.0000 la =   0.80 0.4000 Load at 0.4·L 0.00 0.6708 0.0000 0.0000 0.80 0.20 0.6716 0.1075 0.2688 0.80 0.50 0.5734 0.2664 0.3468 1.20 0.80 0.3015 0.0267 0.0802 1.20 1.00 0.0991 0.0000 0.0000 1.20 l L = 2.0000 0.80 0.3015

P= L= x 0.00 2.38 5.95 9.52 11.90

   

200.00 11.900 A' -0.1107 0.1416 0.3478 -0.0206 -0.0116

10.00

12.00

14.00

term_1

y term_2

0.0000 0.8000 0.5000 0.0000

0.6730 2.7176 0.6730 -1.2365 0.6730 -1.3132 0.6730 -1.3270

0 .0000 1 .5782 0.9979 0.0000

0.5000 0.5000

0.0000 0.5000

0.6730 0.6730

0.6605 0.6536

0 .0000 0 .9979

1.2000 1.2000 0.8000 0.8000 0.8000

0.0000 0.4000 1.0000 0.4000 0.0000

12.3273 12.3273 12.3273 12.3273 12.3273

8.2692 8.2339 1.0182 1.2160 1.2212

0.0000 0.7993 1.9334 0.7993 0.0000

0.4000 Load at 0.4·L 0.0267 0.0802 0.80 1.6000 ks = 65,000 l =   0.59911 lL = 2.26936·π a= 4.760   0.4000  Load at 0.4·L Excel Formula B' C' y M q 0.0000 0.0000 -2.0402E-04 0.00 -13.26 -0.0786 0.0032 2.6105E-04 -13.12 16.97 0.0232 0.1853 6.4105E-04 3.87 41.67 -0.0355 -0.0276 -3.8042E-05 -5.93 -2.47 0.0000 0.0000 -2.1373E-05 0.00 -1.39

Q 0.00 0.64 37.06 -5.53 0.00

lb

lx

1.0000 0.0000 0.0000 0.0000

Div.

Bearing Pressure Semi-Infinite

M Inifinite

Se

50.00

10.00

40.00

5.00

30.00

0.00 0.00 -5.00

20.00

5.00

10.00 0.00 0.00 -10.00

-10.00 5.00

10.00

15.00

-20.00

-15.00 -20.00

Deflection Semi-Infinite

Infinite

Se

7.00E-04 6.00E-04 5.00E-04 4.00E-04 3.00E-04 2.00E-04 1.00E-04 0.00E+00 -1.00E-040.00 -2.00E-04 -3.00E-04

5.00

10.00

15.00

40.00 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 -5.000.00 -10.00

5.00

 Alx

Blx

Clx

Dlx

l(a+x)

Al(a+x)

l(a-x)

Cl(a-x)

Dl(a-x)

1.0000 0.5083

0.0000 0.3096

1.0000 -0.1108

1.0000 0.1988

0.00 1.00

1.0000 0.5083

0.00 1.00

1.0000 -0.1108

1.0000 0.1988

1.0000 0.9651 0.8231 0.6354 0.5083

0.0000 0.1627 0.2908 0.3223 0.3096

1.0000 0.6398 0.2415 -0.0093 -0.1108

1.0000 0.8024 0.5323 0.3131 0.1988

0.40 0.60 0.90 1.20 1.40

0.8784 0.7628 0.5712 0.3899 0.2849

0.40 0.60 0.90 1.20 1.40

0.3564 0.1431 -0.0657 -0.1716 -0.2011

0.6174 0.4530 0.2527 0.1091 0.0419

Moment -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.00 0.50

2.00

4.00

6.00

8.00

10.00

12.00

14.00

term_3

term_4

term_1

-1.3811 -0.7081 0.0000 0.9889 0.9889 -0.8454 0.9889 0.9889 -0.3315 0.9889 0.9889 0.0000

M term_2

term_3

term_4

term_1

0.0000 -1.3811 0.3407 0.9889 0.0833 0.9889 0.0000 0.9889

-0.7081 0.9889 0.9889 0.9889

0.0000 -1.0471 -0.6621 0.0000

Q term_2

term_3

term_4

0.0000 -1.3811 -0.7081 0.6371 0.9889 0.9889 0.2498 0.9889 0.9889 0.0000 0.9889 0.9889

0.0979 -0.0701 0.0979 -0.0701

0.0000 0.1650

0.0000 0.0833

0.0979 0.0979

-0.0701 -0.0701

0.0000 0.3296

0.0000 0.2498

0.0979 -0.0701 0.0979 -0.0701

0.8967 0.8967 3.3539 3.3539 3.3539

0.0000 1.3227 1.2077 0.1953 0.0000

0.0000 0.0427 0.6635 0.0427 0.0000

0.8967 0.8967 3.3539 3.3539 3.3539

-0.8409 -0.8409 -0.2248 -0.2248 -0.2248

0.0000 3.3049 1.1806 0.4881 0.0000

0.0000 0.1600 0.9889 0.1600 0.0000

0.8967 0.8967 3.3539 3.3539 3.3539

VBA Function M q 0.00 -13.26 - 13.12 16.97 3.87 41.67 -5.93 -2.47 0.00 -1.39

Q 0.00 0.64 37.06 -5.53 0.00

VBA Function : Infinite Beam y M q -3.5792E-05 -6.00 -2.33 2.5115E-04 - 16.95 16.32 6.3730E-04 4.19 41.42 -3.5792E-05 -6.00 -2.33 -1.6986E-05 0.56 -1.10

Q -5.53 3.47 37.08 -5.53 -0.58

y -2.0402E-04 2.6105E-04 6.4105E-04 -3.8042E-05 -2.1373E-05

-0.8409 -0.8409 -0.2248 -0.2248 -0.2248

3.1416E+00 λL > π 0.78539816 λL < π/4 π/4 ≤ λL ≤ π 7.12939758

oment i -Infi nite

Inf inite

10.00

Shear mi-Infinite

Infinite

15.00

-0.8409 -0.8409 -0.2248 -0.2248 -0.2248

10.00

15.00