Finite Element Analysis of Flat Slab

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Using numerical formulation of Bogner-Fox-Schmit (BFS) plate element

Input data

Span lengths

⃗a = hp ( [3.6; 4.2; 4.2; 3.6] )  = [3.6 4.2 4.2 3.6] m

⃗b = hp ( [3; 3.6; 3] )  = [3 3.6 3] m

Number of axes - n_sa = len ( ⃗a )  + 1 = 5 , n_sb = len ( ⃗b )  + 1 = 4

Axis coordinates - ⃗x_s = [0 3.6 7.8 12 15.6] m, ⃗y_s = [0 3 6.6 9.6] m

Slab dimensions - l_a = ⃗x_s.5 = 15.6 m, l_b = ⃗y_s.4 = 9.6 m

Thickness - t = 0.2 m

Load - q = 10 kN/m²

Modulus of elasticity - E = 35000 MPa

Poisson`s ratio - ν = 0.2

Finite element mesh

We will use Bogner-Fox-Schmit rectangular finite element with n_DOFs = 16

Element dimensions - a₁ = 0.6 m, b₁ = 0.6 m

Number of elements and joints along a and b -

⃗n_a = ceiling(⃗aa₁) = ceiling(⃗a0.6) = [6 7 7 6] , n_ea = sum ( ⃗n_a )  = 26 , n_ja = n_ea + 1 = 26 + 1 = 27

⃗n_b = ceiling(⃗bb₁) = ceiling(⃗b0.6) = [5 6 5] , n_eb = sum ( ⃗n_b )  = 16 , n_jb = n_eb + 1 = 16 + 1 = 17

Total number of elements - n_e = n_ea · n_eb = 26 · 16 = 416

Total number of joints - n_j = n_ja · n_jb = 27 · 17 = 459

Supported joints count - n_s = n_sa · n_sb = 5 · 4 = 20

Joint coordinates

⃗x_j = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.6 0.6 0.6 ... 15.6] m

⃗y_j = [0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6 6.6 7.2 7.8 8.4 9 9.6 0 0.6 1.2 ... 9.6] m

Numbers of joints at elements' corners

transp ( e_j )  = 1234567891011121314151618192021⋯441 1819202122232425262728293031323335363738⋯458 1920212223242526272829303132333436373839⋯459 23456789101112131415161719202122⋯442

Supported joints

⃗s_j = [1 6 12 17 103 108 114 119 222 227 233 238 341 346 352 357 443 448 454 459]

Joints for element e - j_e ( e )  = row ( e_j; e ) 

Finite element formulation

Shape functions

Along dimension a

Along dimension b

For vertical displacements w N1,w ( ξ; η )  = Φ1a  ( ξ )  · Φ1b  ( η )  N2,w ( ξ; η )  = Φ3a  ( ξ )  · Φ1b  ( η )  N3,w ( ξ; η )  = Φ3a  ( ξ )  · Φ3b  ( η )  N4,w ( ξ; η )  = Φ1a  ( ξ )  · Φ3b  ( η )	For rotations θₓ N1,θₓ ( ξ; η )  = Φ2a  ( ξ )  · Φ1b  ( η )  N2,θₓ ( ξ; η )  = Φ4a  ( ξ )  · Φ1b  ( η )  N3,θₓ ( ξ; η )  = Φ4a  ( ξ )  · Φ3b  ( η )  N4,θₓ ( ξ; η )  = Φ2a  ( ξ )  · Φ3b  ( η )	For rotations θᵧ N1,θᵧ ( ξ; η )  = Φ1a  ( ξ )  · Φ2b  ( η )  N2,θᵧ ( ξ; η )  = Φ3a  ( ξ )  · Φ2b  ( η )  N3,θᵧ ( ξ; η )  = Φ3a  ( ξ )  · Φ4b  ( η )  N4,θᵧ ( ξ; η )  = Φ1a  ( ξ )  · Φ4b  ( η )
For twist ψ N1,ψ ( ξ; η )  = Φ2a  ( ξ )  · Φ2b  ( η )  N2,ψ ( ξ; η )  = Φ4a  ( ξ )  · Φ2b  ( η )  N3,ψ ( ξ; η )  = Φ4a  ( ξ )  · Φ4b  ( η )  N4,ψ ( ξ; η )  = Φ2a  ( ξ )  · Φ4b  ( η )

Shape functions vector

N ( i; ξ; η )  = take ( i; N_1,w  ( ξ; η ) ; N_1,θₓ  ( ξ; η ) ; N_1,θᵧ  ( ξ; η ) ; N_1,ψ  ( ξ; η ) ; N_2,w  ( ξ; η ) ; N_2,θₓ  ( ξ; η ) ; N_2,θᵧ  ( ξ; η ) ; N_2,ψ  ( ξ; η ) ; N_3,w  ( ξ; η ) ; N_3,θₓ  ( ξ; η ) ; N_3,θᵧ  ( ξ; η ) ; N_3,ψ  ( ξ; η ) ; N_4,w  ( ξ; η ) ; N_4,θₓ  ( ξ; η ) ; N_4,θᵧ  ( ξ; η ) ; N_4,ψ  ( ξ; η )  ) 

Constitutive matrix (stress - strain relationship)

D = E · t³12 ·  ( 1 − ν² )  · hp([1; ν; 0 | ν; 1; 0 | 0; 0; 1 − ν2]) = 35000 · 0.2³12 ·  ( 1 − 0.2² )  · hp([1; 0.2; 0 | 0.2; 1; 0 | 0; 0; 1 − 0.22]) = 24.3055564.8611110 4.86111124.3055560 009.722222 kNm

Strain-displacement matrix

B₁ ( j; ξ; η )  = take ( j; Φ″_1a  ( ξ )  · Φ_1b  ( η ) ; Φ″_2a  ( ξ )  · Φ_1b  ( η ) ; Φ″_1a  ( ξ )  · Φ_2b  ( η ) ; Φ″_2a  ( ξ )  · Φ_2b  ( η ) ; Φ″_3a  ( ξ )  · Φ_1b  ( η ) ; Φ″_4a  ( ξ )  · Φ_1b  ( η ) ; Φ″_3a  ( ξ )  · Φ_2b  ( η ) ; Φ″_4a  ( ξ )  · Φ_2b  ( η ) ; Φ″_3a  ( ξ )  · Φ_3b  ( η ) ; Φ″_4a  ( ξ )  · Φ_3b  ( η ) ; Φ″_3a  ( ξ )  · Φ_4b  ( η ) ; Φ″_4a  ( ξ )  · Φ_4b  ( η ) ; Φ″_1a  ( ξ )  · Φ_3b  ( η ) ; Φ″_2a  ( ξ )  · Φ_3b  ( η ) ; Φ″_1a  ( ξ )  · Φ_4b  ( η ) ; Φ″_2a  ( ξ )  · Φ_4b  ( η )  ) 

B₂ ( j; ξ; η )  = take ( j; Φ_1a  ( ξ )  · Φ″_1b  ( η ) ; Φ_2a  ( ξ )  · Φ″_1b  ( η ) ; Φ_1a  ( ξ )  · Φ″_2b  ( η ) ; Φ_2a  ( ξ )  · Φ″_2b  ( η ) ; Φ_3a  ( ξ )  · Φ″_1b  ( η ) ; Φ_4a  ( ξ )  · Φ″_1b  ( η ) ; Φ_3a  ( ξ )  · Φ″_2b  ( η ) ; Φ_4a  ( ξ )  · Φ″_2b  ( η ) ; Φ_3a  ( ξ )  · Φ″_3b  ( η ) ; Φ_4a  ( ξ )  · Φ″_3b  ( η ) ; Φ_3a  ( ξ )  · Φ″_4b  ( η ) ; Φ_4a  ( ξ )  · Φ″_4b  ( η ) ; Φ_1a  ( ξ )  · Φ″_3b  ( η ) ; Φ_2a  ( ξ )  · Φ″_3b  ( η ) ; Φ_1a  ( ξ )  · Φ″_4b  ( η ) ; Φ_2a  ( ξ )  · Φ″_4b  ( η )  ) 

B₃ ( j; ξ; η )  = 2 · take ( j; Φ′_1a  ( ξ )  · Φ′_1b  ( η ) ; Φ′_2a  ( ξ )  · Φ′_1b  ( η ) ; Φ′_1a  ( ξ )  · Φ′_2b  ( η ) ; Φ′_2a  ( ξ )  · Φ′_2b  ( η ) ; Φ′_3a  ( ξ )  · Φ′_1b  ( η ) ; Φ′_4a  ( ξ )  · Φ′_1b  ( η ) ; Φ′_3a  ( ξ )  · Φ′_2b  ( η ) ; Φ′_4a  ( ξ )  · Φ′_2b  ( η ) ; Φ′_3a  ( ξ )  · Φ′_3b  ( η ) ; Φ′_4a  ( ξ )  · Φ′_3b  ( η ) ; Φ′_3a  ( ξ )  · Φ′_4b  ( η ) ; Φ′_4a  ( ξ )  · Φ′_4b  ( η ) ; Φ′_1a  ( ξ )  · Φ′_3b  ( η ) ; Φ′_2a  ( ξ )  · Φ′_3b  ( η ) ; Φ′_1a  ( ξ )  · Φ′_4b  ( η ) ; Φ′_2a  ( ξ )  · Φ′_4b  ( η )  ) 

B ( j; ξ; η )  = hp ( [B₁  ( j; ξ; η ) ; B₂  ( j; ξ; η ) ; B₃  ( j; ξ; η ) ] ) 

The coefficients of the stiffness matrix will be calculated by using the equation

K_e,ij = a₁ · b₁ ·  1∫0  1∫0 B_i  ( ξ; η ) ^T · D · B_j  ( ξ; η )  dξ dη

Element stiffness matrix

BTDB_e ( i; j; ξ; η )  = transp ( B  ( i; ξ; η )  )  · D · B  ( j; ξ; η ) 

K_e ( i; j )  = a₁ · b₁ ·  1∫0  1∫0 BTDB_e  ( i; j; ξ; η )  dη dξ

$Repeat{$Repeat{K_{e.i, j} = K_e  ( i; j )  for j = i...n} for i = 1...n} = 0.9777823

K_e = 796.296296135.185185135.18518516.736111-391.20370484.953704-13.6574074.097222-13.88888936.57407436.574074-8.541667-391.203704-13.65740784.9537044.097222 046.66666721.5972224.666667-84.95370414.027778-4.0948240.7083333-36.57407410.2777788.541667-1.624997-13.6574071.9444444.097222-0.5833333 0046.6666674.666667-13.6574074.0948241.944444-0.5833333-36.5740748.54166710.277778-1.625-84.953704-4.09722214.0277780.7083333 0000.9777823-4.0972220.70833330.5833333-0.1611111-8.5416671.6249971.625-0.2305541-4.0972220.58333330.7083333-0.1611111 0000796.296296-135.185185135.185185-16.736111-391.20370413.65740784.953704-4.097222-13.888889-36.57407436.5740748.541667 0000046.666667-21.5972224.66666713.6574071.944444-4.097222-0.583333336.57407410.277778-8.541667-1.624997 00000046.666667-4.666667-84.9537044.09722214.027778-0.7083333-36.574074-8.54166710.2777781.625 00000000.97778234.0972220.5833333-0.7083333-0.16111118.5416671.624997-1.625-0.2305541 00000000796.296296-135.185185-135.18518516.736111-391.203704-84.95370413.6574074.097222 00000000046.66666721.597222-4.66666784.95370414.027778-4.094824-0.7083333 000000000046.666667-4.66666713.6574074.0948241.9444440.5833333 000000000000.9777823-4.097222-0.7083333-0.5833333-0.1611111 000000000000796.296296135.185185-135.185185-16.736111 000000000000046.666667-21.597222-4.666667 0000000000000046.6666674.666667 0000000000000000.9777823

Element load vector

F_e,i = a₁ · b₁ ·  1∫0  1∫0 N_i  ( ξ; η ) ^T · q dξ dη

⃗F_e = [0.9 0.09 0.09 0.009 0.9 -0.09 0.09 -0.009 0.9 -0.09 -0.09 0.009 0.9 0.09 -0.09 -0.009] kN

Solution

Global stiffness matrix

K = 10²⁰135.185185135.18518516.736111-391.203704-13.65740784.9537044.097222000000000000⋯0 135.18518546.66666721.5972224.666667-13.6574071.9444444.097222-0.5833333000000000000⋯0 135.18518521.59722246.6666674.666667-84.953704-4.09722214.0277780.7083333000000000000⋯0 16.7361114.6666674.6666670.9777823-4.0972220.58333330.7083333-0.1611111000000000000⋯0 -391.203704-13.657407-84.953704-4.0972221592.592593270.3703700-391.203704-13.65740784.9537044.09722200000000⋯0 -13.6574071.944444-4.0972220.5833333270.3703793.33333300-13.6574071.9444444.097222-0.583333300000000⋯0 84.9537044.09722214.0277780.70833330093.3333339.333333-84.953704-4.09722214.0277780.708333300000000⋯0 4.097222-0.58333330.7083333-0.1611111009.3333331.955565-4.0972220.58333330.7083333-0.161111100000000⋯0 0000-391.203704-13.657407-84.953704-4.0972221592.592593270.3703700-391.203704-13.65740784.9537044.0972220000⋯0 0000-13.6574071.944444-4.0972220.5833333270.3703793.33333300-13.6574071.9444444.097222-0.58333330000⋯0 000084.9537044.09722214.0277780.70833330093.3333339.333333-84.953704-4.09722214.0277780.70833330000⋯0 00004.097222-0.58333330.7083333-0.1611111009.3333331.955565-4.0972220.58333330.7083333-0.16111110000⋯0 00000000-391.203704-13.657407-84.953704-4.0972221592.592593270.3703700-391.203704-13.65740784.9537044.097222⋯0 00000000-13.6574071.944444-4.0972220.5833333270.3703793.33333300-13.6574071.9444444.097222-0.5833333⋯0 0000000084.9537044.09722214.0277780.70833330093.3333339.333333-84.953704-4.09722214.0277780.7083333⋯0 000000004.097222-0.58333330.7083333-0.1611111009.3333331.955565-4.0972220.58333330.7083333-0.1611111⋯0 000000000000-391.203704-13.657407-84.953704-4.0972221592.592593270.3703700⋯0 000000000000-13.6574071.944444-4.0972220.5833333270.3703793.33333300⋯0 00000000000084.9537044.09722214.0277780.70833330093.3333339.333333⋯0 0000000000004.097222-0.58333330.7083333-0.1611111009.3333331.955565⋯0 ⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋮00000000000000000000⋯0.9777823

Global load vector

⃗F = [0.9 0.09 0.09 0.009 1.8 0.18 0 0 1.8 0.18 0 0 1.8 0.18 0 0 1.8 0.18 0 0 ... 0.009] kN

Solution of the system of equations

⃗Z = slsolve ( K; ⃗F )  = [0 0.5523613 0.3827392 -0.4161287 0.2028064 0.3732832 0.2648463 -0.1937261 0.2989047 0.3091182 0.04830843 -0.02501454 0.261207 0.3426386 -0.1651493 0.1275073 0.1211489 0.4681287 -0.2671179 0.2293302 ... -0.4161286] mm

Results

Joint displacements

transp ( W_z )  = 00.3030.4880.5120.3830.16500.1390.340.4690.4720.3470.14600.1460.3470.4720.4690.340.139⋯0 0.2030.4190.5620.5810.4850.3370.250.310.4380.5310.5330.4430.3110.2420.3110.4430.5330.5310.4380.31⋯0.203 0.2990.4820.6050.620.5370.4170.350.3870.4850.5620.5640.4890.3870.3380.3870.4890.5640.5620.4850.387⋯0.299 0.2610.4610.5930.6080.5130.3750.2990.3420.4550.5420.5440.460.3450.2890.3450.460.5440.5420.4550.342⋯0.261 0.1210.3860.550.5650.4430.2530.1380.2170.3810.4930.4970.3890.2250.1340.2250.3890.4970.4930.3810.217⋯0.121 00.340.5270.5420.4040.17300.1350.3360.4630.4670.3450.14500.1450.3450.4670.4630.3360.135⋯0 0.1390.3980.5560.5660.440.2470.1290.2060.3670.4780.4810.3740.2110.1210.2110.3740.4810.4780.3670.206⋯0.139 0.2990.4870.6080.6120.5110.3690.2870.3250.4330.5160.5190.4370.3240.270.3240.4370.5190.5160.4330.325⋯0.299 0.3630.5260.6320.6350.5440.420.350.3760.4640.5360.5380.4670.3730.3290.3730.4670.5380.5360.4640.376⋯0.363 0.2990.4870.6080.6120.5110.3690.2870.3250.4330.5160.5190.4370.3240.270.3240.4370.5190.5160.4330.325⋯0.299 0.1390.3980.5560.5660.440.2470.1290.2060.3670.4780.4810.3740.2110.1210.2110.3740.4810.4780.3670.206⋯0.139 00.340.5270.5420.4040.17300.1350.3360.4630.4670.3450.14500.1450.3450.4670.4630.3360.135⋯0 0.1210.3860.550.5650.4430.2530.1380.2170.3810.4930.4970.3890.2250.1340.2250.3890.4970.4930.3810.217⋯0.121 0.2610.4610.5930.6080.5130.3750.2990.3420.4550.5420.5440.460.3450.2890.3450.460.5440.5420.4550.342⋯0.261 0.2990.4820.6050.620.5370.4170.350.3870.4850.5620.5640.4890.3870.3380.3870.4890.5640.5620.4850.387⋯0.299 0.2030.4190.5620.5810.4850.3370.250.310.4380.5310.5330.4430.3110.2420.3110.4430.5330.5310.4380.31⋯0.203 00.3030.4880.5120.3830.16500.1390.340.4690.4720.3470.14600.1460.3470.4720.4690.340.139⋯0 mm

Bending moments

Z_j ( j )  = slice ( ⃗Z; k₁ ·  ( j − 1 )  + 1; k₁ · j ) 

Z_e ( e )  = hp ( [Z_j  ( e_{j.e, 1} ) ; Z_j  ( e_{j.e, 2} ) ; Z_j  ( e_{j.e, 3} ) ; Z_j  ( e_{j.e, 4} ) ] ) 

Average bending moments at joints, kNm/m

M_j = 1.4984650.30973720.21973220.15634330.15705190.99833020.15641930.15197410.194220.15197420.15641970.99833030.15704750.15633970.21974860.30988081.498588.5029986.4791995.777684⋯1.498434 1.5669147.8051019.337787.6680973.333-28.3617353.146397.3238888.8950197.3238883.14639-28.3617343.3329977.6680939.3377917.805181.5668130.32059725.3822017.126518⋯1.56695 8.0913913.7668960.4863938-2.479308-4.4591990.15508644.7803592.8427231.18852×10^-8-2.842723-4.780359-0.15508664.4591992.479312-0.4863957-3.766967-8.091394.1108572.5729420.367139⋯8.09139

Bending moments for the plate

Bending moments - M_x

transp ( Mx )  = 1.4984658.50299811.0665210.477416.7974770.8838727-31.2352710.30150175.6444428.7799958.8776125.9200110.6854447-30.1330410.68544515.920018.8776138.7799955.6444430.3015014⋯1.498543 0.30973726.4791999.43228.8772754.922637-2.695317-10.242892-3.2602943.8087977.2504257.3613934.127752-2.764349-9.733242-2.7643494.1277527.3613937.2504263.808797-3.260293⋯0.3098324 0.21973225.7776848.7162938.1350554.039483-2.257879-6.060711-2.8192912.9321446.5167586.6405433.291674-2.294253-5.615624-2.2942533.2916746.6405436.5167582.932144-2.819291⋯0.2197447 0.15634335.9931769.1259938.4927894.05777-3.257751-7.98846-3.8326772.9136746.7974696.9268223.304597-3.205198-7.381383-3.2051983.3045976.9268226.7974692.913674-3.832677⋯0.1563395 0.15705197.26964410.3722499.5566044.972553-5.268409-16.118864-5.8654913.7668327.7132287.8382434.157472-5.096284-15.086013-5.0962844.1574727.8382437.7132283.766832-5.865491⋯0.1570491 0.99833029.03852511.08041810.1227775.76385-2.135317-38.650059-2.7455574.521798.1908048.3150414.904088-2.088413-36.536458-2.0884134.9040888.3150418.1908044.52179-2.745557⋯0.9983301 0.15641937.22104710.2820019.4364654.848242-5.358009-16.18287-5.963023.6242627.5601117.6988554.05327-5.138805-15.091525-5.1388054.053277.6988557.5601113.624262-5.96302⋯0.1564295 0.15197415.8731868.927958.2297553.773453-3.419119-8.032747-4.0090422.5942596.4697616.6264533.062913-3.269947-7.306587-3.2699473.0629136.6264536.4697612.594259-4.009042⋯0.151956 0.194225.5141478.3802137.6929313.468337-2.491105-5.742638-3.0733842.3104685.9792226.142652.786902-2.376825-5.118402-2.3768252.7869026.142655.9792222.310468-3.073384⋯0.1942406 0.15197425.8731868.927958.2297553.773453-3.419119-8.032747-4.0090422.5942596.4697616.6264533.062913-3.269947-7.306587-3.2699473.0629136.6264536.4697612.594259-4.009042⋯0.1519563 0.15641977.22104710.2820019.4364654.848242-5.358009-16.18287-5.963023.6242627.5601117.6988554.05327-5.138805-15.091525-5.1388054.053277.6988557.5601113.624262-5.96302⋯0.1564287 0.99833039.03852411.08041810.1227775.76385-2.135317-38.650059-2.7455574.521798.1908048.3150414.904088-2.088413-36.536458-2.0884134.9040888.3150418.1908044.52179-2.745557⋯0.99833 0.15704757.26964610.3722499.5566044.972554-5.268409-16.118864-5.8654913.7668327.7132287.8382434.157472-5.096284-15.086013-5.0962844.1574727.8382437.7132283.766832-5.865491⋯0.1570551 0.15633975.9931779.1259948.4927894.05777-3.257751-7.98846-3.8326772.9136746.7974696.9268223.304597-3.205198-7.381383-3.2051983.3045976.9268226.7974692.913674-3.832677⋯0.1563387 0.21974865.7776858.7162928.1350524.039485-2.25788-6.060711-2.8192912.9321446.5167586.6405433.291674-2.294252-5.615624-2.2942533.2916746.6405436.5167582.932144-2.819291⋯0.2197395 0.30988086.4791779.4321788.8772824.922634-2.695314-10.242892-3.2602933.8087977.2504267.3613934.127752-2.764349-9.733242-2.7643494.1277527.3613937.2504263.808797-3.260293⋯0.309677 1.498588.50289311.06653710.4774016.7974870.8838683-31.235270.3015015.6444438.7799948.8776145.9200090.6854455-30.1330410.68544495.9200118.8776138.7799955.6444420.3015017⋯1.498434 kNm/m

Bending moments M_y

transp ( My )  = 1.5669140.32059720.23335870.21478520.16828160.17980351.0157890.17943010.16698940.20981910.20919060.16634670.17761190.98914820.17761230.16634590.20919150.20981810.16699030.1794296⋯1.566846 7.8051015.3822014.3084124.1349524.7423316.3403958.2307146.283164.6119813.892283.8674994.5305346.122587.9858966.122584.5305343.8674993.892284.6119816.28316⋯7.805153 9.337787.1265185.8000745.5530326.4148988.115779.0466668.0147046.1888365.144855.1167776.0905897.8212248.7772257.8212256.0905895.1167775.144856.1888368.014704⋯9.337789 7.6680975.417564.2624933.9204134.2336925.4198226.3445915.3011043.9794783.4779983.4809043.9685825.2228376.1814595.2228373.9685823.4809043.4779983.9794785.301104⋯7.668093 3.333-0.32807790.53963460.405616-1.17438-3.412974-0.3301829-3.531204-1.3995740.099649960.1609008-1.210571-3.234664-0.2184219-3.234664-1.2105710.16090080.09964995-1.399574-3.531204⋯3.332998 -28.361735-7.038366-1.936162-1.646652-4.993974-13.644678-36.317523-13.771126-5.215881-1.858171-1.764027-4.886867-13.033174-34.571125-13.033174-4.886867-1.764027-1.858171-5.215881-13.771126⋯-28.361736 3.14639-0.50666960.34851460.1799353-1.456247-3.762435-0.7233495-3.922981-1.765819-0.2520091-0.1968807-1.596341-3.661657-0.6661947-3.661657-1.596341-0.1968807-0.2520091-1.765819-3.922981⋯3.146398 7.3238885.0945483.9089783.4891393.6754684.6858655.498734.4839693.2541992.7943762.7885493.2144954.3499645.2436514.3499643.2144952.7885492.7943763.2541994.483969⋯7.323874 8.8950196.7401225.3508634.961265.6089376.9130637.5725296.6931585.1417844.1805254.1496375.0277916.4568627.2129736.4568625.0277914.1496374.1805255.1417846.693158⋯8.895035 7.3238885.0945483.9089783.4891393.6754684.6858655.498734.4839693.2541992.7943762.7885493.2144954.3499645.2436514.3499643.2144952.7885492.7943763.2541994.483969⋯7.323874 3.14639-0.50666950.34851450.1799354-1.456247-3.762435-0.7233495-3.922981-1.765819-0.2520091-0.1968807-1.596341-3.661657-0.6661947-3.661657-1.596341-0.1968807-0.2520091-1.765819-3.922981⋯3.146397 -28.361734-7.038366-1.936162-1.646652-4.993974-13.644678-36.317523-13.771126-5.215881-1.858171-1.764027-4.886867-13.033174-34.571125-13.033174-4.886867-1.764027-1.858171-5.215881-13.771126⋯-28.361736 3.332997-0.32807810.53963450.405616-1.174379-3.412974-0.3301829-3.531204-1.3995740.099649940.1609008-1.210571-3.234664-0.2184219-3.234664-1.2105710.16090080.09964996-1.399574-3.531204⋯3.333003 7.6680935.417564.2624933.9204144.2336915.4198226.3445915.3011043.9794783.4779983.4809043.9685825.2228376.1814595.2228373.9685823.4809043.4779983.9794785.301104⋯7.668093 9.3377917.1265295.8000795.5530296.4148998.1157699.0466678.0147046.1888365.144855.1167776.0905897.8212258.7772257.8212256.0905895.1167775.144856.1888368.014704⋯9.337788 7.805185.3822474.3083974.1349584.7423266.3403988.2307136.2831614.611983.8922813.8674984.5305346.122587.9858966.122584.5305343.8674993.892284.6119816.28316⋯7.805067 1.5668130.32041280.23337480.21477460.16829440.17979831.0157890.17942920.16699140.20981690.20919270.1663450.17761280.98914820.1776120.16634630.2091910.20981870.16698980.1794301⋯1.56695 kNm/m

Bending moments M_xy

transp ( Mxy )  = 8.0913914.1108571.417137-0.9458016-3.233923-4.7834720.021753974.8406963.3363691.142106-1.025808-3.195694-4.6752918.203632×10^-94.6752913.1956941.025808-1.142106-3.336369-4.840696⋯-8.09139 3.7668962.5729420.9831798-0.5157529-2.056961-3.2290760.058233953.3549532.2134050.7311327-0.6697977-2.124616-3.226088-1.924384×10^-83.2260882.1246150.6697977-0.7311327-2.213405-3.354953⋯-3.766944 0.48639380.3671390.21047150.08573235-0.1195606-0.31383010.073481280.46129660.26670830.05683469-0.07988453-0.2739108-0.43182734.135681×10^-90.43182730.27391090.07988453-0.05683469-0.2667083-0.4612966⋯-0.4863951 -2.479308-1.795977-0.60125750.61674581.7468932.0926540.07628287-1.948193-1.634066-0.58147370.49193451.5362541.879329-9.399297×10^-10-1.879329-1.536254-0.49193450.58147371.6340661.948193⋯2.479311 -4.459199-3.239942-0.89774270.68266692.3709224.5249970.07578146-4.382303-2.264662-0.68176990.60076632.1521054.2108782.347034×10^-10-4.210878-2.152105-0.60076630.68176992.2646624.382303⋯4.459199 0.15508640.16117180.14924740.12459870.096391450.077977030.073881980.069886510.051250560.02105121-0.008413693-0.02659669-0.02344966-6.046931×10^-110.023449660.026596690.008413693-0.02105121-0.05125056-0.06988651⋯-0.1550867 4.7803593.5659681.189369-0.4505428-2.206755-4.3992850.063484924.5354542.379370.7252095-0.6263983-2.225358-4.2792771.578955×10^-114.2792772.2253580.6263983-0.7252095-2.37937-4.535454⋯-4.780358 2.8427232.134380.8682924-0.4377229-1.679594-2.0840690.037787042.168351.7961290.6296557-0.5417072-1.677932-2.037682-4.744667×10^-122.0376821.6779320.5417072-0.6296557-1.796129-2.16835⋯-2.842723 1.18852×10^-8-6.576042×10^-9-7.016847×10^-101.244056×10^-102.191792×10^-10-7.501514×10^-111.8954×10^-11-1.177769×10^-133.047327×10^-121.092988×10^-12-4.777445×10^-125.435255×10^-12-3.554444×10^-123.228026×10^-12-2.990397×10^-121.375859×10^-12-1.298123×10^-127.488428×10^-132.937394×10^-12-1.450797×10^-11⋯-3.835288×10^-9 -2.842723-2.13438-0.86829240.43772291.6795942.084069-0.03778704-2.16835-1.796129-0.62965570.54170721.6779322.037682-7.622464×10^-12-2.037682-1.677932-0.54170720.62965571.7961292.16835⋯2.842723 -4.780359-3.565968-1.1893690.45054282.2067554.399285-0.06348492-4.535454-2.37937-0.72520950.62639832.2253584.2792772.756043×10^-11-4.279277-2.225358-0.62639830.72520952.379374.535454⋯4.780358 -0.1550866-0.1611716-0.1492474-0.1245987-0.09639145-0.07797703-0.07388197-0.06988651-0.05125056-0.021051210.0084136930.026596690.02344966-1.059221×10^-10-0.02344966-0.02659669-0.0084136930.021051210.051250560.06988651⋯0.1550864 4.4591993.2399410.8977427-0.6826669-2.370922-4.524997-0.075781474.3823032.2646620.6817699-0.6007663-2.152105-4.2108784.094768×10^-104.2108782.1521050.6007663-0.6817699-2.264662-4.382303⋯-4.459198 2.4793121.7959760.6012571-0.6167458-1.746893-2.092654-0.076282861.9481931.6340660.5814737-0.4919345-1.536254-1.879329-1.611647×10^-91.8793291.5362540.4919345-0.5814737-1.634066-1.948193⋯-2.479307 -0.4863957-0.3671418-0.2104716-0.085731850.11956050.3138302-0.07348132-0.4612966-0.2667083-0.056834690.079884520.27391090.43182736.850159×10^-9-0.4318273-0.2739109-0.079884530.056834690.26670830.4612966⋯0.4863938 -3.766967-2.57294-0.98317550.51575362.0569623.229076-0.05823381-3.354953-2.213405-0.73113270.66979772.1246153.226088-3.125512×10^-8-3.226088-2.124615-0.66979770.73113272.2134053.354953⋯3.766876 -8.09139-4.110792-1.4171370.94579833.2339224.783472-0.02175395-4.840696-3.336369-1.1421061.0258083.1956944.6752911.254052×10^-8-4.675291-3.195694-1.0258081.1421063.3363694.840696⋯8.09139 kNm/m

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