# Grid on Elastic Sub-soil

FOUNDATION ON ELASTIC SUBSOIL ZAHRA SHARIF KHODAEI INTRODUCTION u=0 v=0 w(x,y,z)=w(x,y) (z) From the geometrical eq : x 0, y 0, xy 0, z w Extended Hook's law: d w z Eoed w , xz G , dz x d , dz yz G xz w , y w w , yz x y Eoed E (1 ) (1 )(1 2 )

INTRODUCTION Virtual work of internal forces: Ei ( z z xz xz yz yz )d = { wC1w C2 [ h C1 Eoed ( 0 w w ( w) ( w) ]}dxdy x x y y d 2 ) dz N/m 3 dz h C2 G 2 dz 0 N/m PROCEDURE FOR FEM Divide the structure into nodes and elements Find the joint forces and initial displacements Calculate the end forces for each element Assemble the global joint forces (localize) Calculate the local stiffness matrix for each element Transform the local stiffness matrix to global stiffness matrix Localize to obtain the overall stiffness matrix Solve: Ku = f for the displacements u. Compute the local displacements for each member Calculate the inner forces from fl = Kl ul

EXAMPLE MATERIAL PROPERTIES: TYPE OF SOIL: S5 Edef E0 b C1 C2 h 12 MPa 19.35 MPa 0.35 50 MPa/m 1.68 MPam 5.6 m E G 30500 MPa 13260MPa b H 2.8m 1.1m A I Ip J 3.08 m2 0.31056 m4 2.32283 m4 0.96855 m4 CONCRETE:25/30 FOUNDATION DESIGN: CROSS SECTION PROPERTIES

INTERNAL COLUMN EDGE COLUMN WALL STRIP 7623.2 KN 3891.8 KN 665.1 KN/m MESH Y X SIGN CONVENTION Something about CAL The command to solve a grid on elastic subsoil is: GRID km E=? G=? I=? J=? L=? CW=? CP=? B=? K=? Where: km element stiffness matrix E modulus of elasticity G shear modulus I moment of inertia J torsional moment of inertia L length of the element CW Winklers constant C1 CP Pasternaks constant C2 B width of the cross section K = 1 takes into account the neighborhood of the foundation RESULTS: INTERNAL FORCES IN THE MEMBERS F1 -348.44 796.41 -26.783 196.12 149.49 -1167 F2 -196.12 -149.49 1167 121.19

1523.8 2954.2 F3 121.19 1523.8 -2954.2 -196.12 -149.49 -1167 F4 196.12 149.49 1167 -348.44 796.41 26.783 F5 26.783 -796.41 348.44 180.29 -48.349 689.87 F6 2.38E-15 844.29 -242.37 1.68E-14 260.08 -673.76 F7 -26.783 -796.41 348.44 -180.29 -48.349 689.87 F8 F9 F10 -180.29 -4.79E-14 180.29

48.349 -260.08 48.349 -689.87 673.76 -689.87 404.32 7.01E-14 -404.32 -698.17 1639.6 -698.17 -449.42 2085.5 -449.42 F12 56.353 -175.09 1671.3 -119.76 2224.9 4259.5 F13 -119.76 2224.9 -4259.5 56.353 -175.09 -1671.3 F14 F15 F16 -56.353 232.12 1.04E-13 175.09 -582.02 1533.8 1671.3 434.24 -1846 15.182 -1.47E-13 -8.38E-14 1280.2 1.14E-13 1.42E-12 636.43 420.72 -446.5

F17 F18 F19 -232.12 -3.08E-13 9.53E-14 -582.02 2.27E-13 3.07E-12 434.24 -420.72 446.5 1.39E-12 232.12 -8.31E-14 3.41E-13 -582.02 1533.8 420.72 -434.24 1846 F23 119.76 2224.9 -4259.5 -56.353 -175.09 -1671.3 F24 56.353 175.09 1671.3 -15.182 1280.2 636.43 F25 F26 F27 404.32 1.32E-13 -404.32 -698.17 1639.6 -698.17 449.42 -2085.5 449.42 -180.29 -1.29E-13 180.29 48.349 -260.08 48.349 689.87 -673.76

689.87 F28 F29 F30 180.29 7.77E-14 -180.29 -48.349 260.08 -48.349 -689.87 673.76 -689.87 26.783 -8.53E-14 -26.783 -796.41 844.29 -796.41 -348.44 242.37 -348.44 F20 F21 -1.97E-12 -15.182 1.02E-12 1280.2 -420.72 -636.43 -232.12 56.353 -582.02 175.09 -434.24 -1671.3 F31 348.44 796.41 -26.783 -196.12 149.49 -1167 F32 196.12 -149.49 1167 -121.19 1523.8 2954.2 F11

15.182 1280.2 -636.43 -56.353 175.09 -1671.3 F22 -56.353 -175.09 1671.3 119.76 2224.9 4259.5 F33 -121.19 1523.8 -2954.2 196.12 -149.49 -1167 F34 -196.12 149.49 1167 348.44 796.41 26.783 DEFLECTIONS: BENDING MOMENTS: COMPARISON b/h 0.5 1 1.5 2 2.5 Max Deflection Max Moment 10.047 5327.5 6.129 5003.7 4.274

4621.7 3.22 4259.5 2.759 4047.9 C1 9.3 20 33.6 50 61.5 C2 5.32 3.22 2.31 1.68 1.54 C1 C2 70 6 61.5 60 5.32 5 50 50 MPam MPa/m 4 40 33.6 30

3.22 3 2.31 20 2 20 10 1.68 1.54 1 9.3 0 0 0.5 1 1.5 b/h 2 2.5 0.5 1 1.5 b/h 2 2.5 MAX DEFLECTION

MAX MOMENTS 12 5500 5300 10.047 10 5327.5 5100 5003.7 4900 8 KN/m mm 4700 6.129 6 4300 4.274 4 4621.7 4500 4259.5 4100 3.22 2.759 2

4047.9 3900 3700 0 3500 0.5 1 1.5 2 2.5 b/h wmax b 0.81737 5.85616( ) h 0.5 1 1.5 2 2.5 b/h M max 5727.92042e b .14208( ) h

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