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

Finite element mesh

⃗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

⃗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

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

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

Finite element formulation

D_1,1 = E · t³12 · ( 1 − ν² ) = 35000 · 0.2³12 · ( 1 − 0.2² ) = 24.305556 kNm

D = D_1,1 · hp([1; ν; 0 | ν; 1; 0 | 0; 0; 1 − ν2]) = 24.305556 · hp([1; 0.2; 0 | 0.2; 1; 0 | 0; 0; 1 − 0.22]) = 24.3055564.8611110 4.86111124.3055560 009.722222 kNm

Element stiffness matrix calculation

α = b₁a₁ = 0.60.6 = 1

K_e_{.1, 1} = 15635 · (αa₁² + 1α · b₁²) + 7225 · A₁ = 15635 · (10.6² + 11 · 0.6²) + 7225 · 0.36 = 32.761905

K_e_{.5, 5} = K_e_{.1, 1} = 32.761905 , K_e_{.9, 9} = K_e_{.1, 1} = 32.761905 , K_e_{.13, 13} = K_e_{.1, 1} = 32.761905

K_e_{.1, 2} = 235 · (39 · αa₁ + 11α² · b₁) + 30 · ν + 625 · b₁ = 235 · (39 · 10.6 + 111 · 0.6) + 30 · 0.2 + 625 · 0.6 = 5.561905

K_e_{.5, 6} = -K_e_{.1, 2} = -5.561905 , K_e_{.9, 10} = -K_e_{.1, 2} = -5.561905 , K_e_{.13, 14} = K_e_{.1, 2} = 5.561905

K_e_{.1, 3} = 235 · (11 · α²a₁ + 39α · b₁) + 30 · ν + 625 · a₁ = 235 · (11 · 10.6 + 391 · 0.6) + 30 · 0.2 + 625 · 0.6 = 5.561905

K_e_{.5, 7} = K_e_{.1, 3} = 5.561905 , K_e_{.9, 11} = -K_e_{.1, 3} = -5.561905 , K_e_{.13, 15} = -K_e_{.1, 3} = -5.561905

K_e_{.1, 4} = 1135 · (α² + 1α²) + 10 · ν + 150 = 1135 · (1 + 11) + 10 · 0.2 + 150 = 0.6885714

K_e_{.5, 8} = -K_e_{.1, 4} = -0.6885714 , K_e_{.9, 12} = K_e_{.1, 4} = 0.6885714 , K_e_{.13, 16} = -K_e_{.1, 4} = -0.6885714

K_e_{.1, 5} = 235 · (27α · b₁² − 78 · αa₁²) − 7225 · A₁ = 235 · (271 · 0.6² − 78 · 10.6²) − 7225 · 0.36 = -16.095238

K_e_{.9, 13} = K_e_{.1, 5} = -16.095238

K_e_{.1, 6} = 1335 · (6 · αa₁ − 1α² · b₁) + 625 · b₁ = 1335 · (6 · 10.6 − 11 · 0.6) + 625 · 0.6 = 3.495238

K_e_{.2, 5} = -K_e_{.1, 6} = -3.495238 , K_e_{.9, 14} = -K_e_{.1, 6} = -3.495238 , K_e_{.10, 13} = K_e_{.1, 6} = 3.495238

K_e_{.1, 7} = 135 · (27α · b₁ − 22 · α²a₁) − 30 · ν + 625 · a₁ = 135 · (271 · 0.6 − 22 · 10.6) − 30 · 0.2 + 625 · 0.6 = -0.5619048

K_e_{.3, 5} = K_e_{.1, 7} = -0.5619048 , K_e_{.9, 15} = -K_e_{.1, 7} = 0.5619048 , K_e_{.11, 13} = -K_e_{.1, 7} = 0.5619048

K_e_{.2, 4} = 235 · (11 · α · b₁3 + a₁α²) + 2 · a₁ · ( 5 · ν + 1 ) 75 = 235 · (11 · 1 · 0.63 + 0.61) + 2 · 0.6 · ( 5 · 0.2 + 1 ) 75 = 0.192

K_e_{.6, 8} = K_e_{.2, 4} = 0.192 , K_e_{.10, 12} = -K_e_{.2, 4} = -0.192 , K_e_{.14, 16} = -K_e_{.2, 4} = -0.192

K_e_{.2, 6} = 135 · (26 · α − 3α³) − 225 · α = 135 · (26 · 1 − 31³) − 225 · 1 = 0.5771429

K_e_{.10, 14} = K_e_{.2, 6} = 0.5771429

K_e_{.2, 8} = 135 · (11 · α · b₁3 − 3 · a₁2 · α²) − a₁ · ( 5 · ν + 1 ) 150 = 135 · (11 · 1 · 0.63 − 3 · 0.62 · 1) − 0.6 · ( 5 · 0.2 + 1 ) 150 = 0.02914286

K_e_{.4, 6} = K_e_{.2, 8} = 0.02914286 , K_e_{.10, 16} = -K_e_{.2, 8} = -0.02914286 , K_e_{.12, 14} = -K_e_{.2, 8} = -0.02914286

K_e_{.2, 10} = 335 · (1α³ + 3 · α) + 225 · α = 335 · (11³ + 3 · 1) + 225 · 1 = 0.4228571

K_e_{.6, 14} = K_e_{.2, 10} = 0.4228571

K_e_{.2, 12} = - 170 · (3 · a₁α² + 13 · α · b₁3) − a₁150 = - 170 · (3 · 0.61 + 13 · 1 · 0.63) − 0.6150 = -0.06685714

K_e_{.4, 10} = -K_e_{.2, 12} = 0.06685714 , K_e_{.6, 16} = K_e_{.2, 12} = -0.06685714 , K_e_{.8, 14} = -K_e_{.2, 12} = 0.06685714

K_e_{.2, 14} = 235 · (9 · α − 2α³) − 825 · α = 235 · (9 · 1 − 21³) − 825 · 1 = 0.08

K_e_{.6, 10} = K_e_{.2, 14} = 0.08

K_e_{.2, 15} = 135 · (11α² − 13 · α²2) + 5 · ν + 150 = 135 · (111 − 13 · 12) + 5 · 0.2 + 150 = 0.1685714

K_e_{.3, 14} = -K_e_{.2, 15} = -0.1685714 , K_e_{.6, 11} = -K_e_{.2, 15} = -0.1685714 , K_e_{.7, 10} = K_e_{.2, 15} = 0.1685714

K_e_{.1, 14} = 135 · (27 · αa₁ − 22α² · b₁) − 30 · ν + 625 · b₁ = 135 · (27 · 10.6 − 221 · 0.6) − 30 · 0.2 + 625 · 0.6 = -0.5619048

K_e_{.2, 13} = K_e_{.1, 14} = -0.5619048 , K_e_{.5, 10} = -K_e_{.1, 14} = 0.5619048 , K_e_{.6, 9} = -K_e_{.1, 14} = 0.5619048

K_e_{.1, 15} = 1335 · (6α · b₁ − α²a₁) + 625 · a₁ = 1335 · (61 · 0.6 − 10.6) + 625 · 0.6 = 3.495238

K_e_{.3, 13} = -K_e_{.1, 15} = -3.495238 , K_e_{.5, 11} = K_e_{.1, 15} = 3.495238 , K_e_{.7, 9} = -K_e_{.1, 15} = -3.495238

K_e_{.1, 16} = 135 · (11α² − 13 · α²2) + 5 · ν + 150 = 135 · (111 − 13 · 12) + 5 · 0.2 + 150 = 0.1685714

K_e_{.4, 13} = -K_e_{.1, 16} = -0.1685714 , K_e_{.8, 9} = K_e_{.1, 16} = 0.1685714 , K_e_{.5, 12} = -K_e_{.1, 16} = -0.1685714

K_e_{.2, 2} = 435 · (13 · α + 1α³) + 825 · α = 435 · (13 · 1 + 11³) + 825 · 1 = 1.92

K_e_{.6, 6} = K_e_{.2, 2} = 1.92 , K_e_{.10, 10} = K_e_{.2, 2} = 1.92 , K_e_{.14, 14} = K_e_{.2, 2} = 1.92

K_e_{.2, 3} = 1135 · (α² + 1α²) + 60 · ν + 150 = 1135 · (1 + 11) + 60 · 0.2 + 150 = 0.8885714

K_e_{.6, 7} = -K_e_{.2, 3} = -0.8885714 , K_e_{.10, 11} = K_e_{.2, 3} = 0.8885714 , K_e_{.14, 15} = -K_e_{.2, 3} = -0.8885714

K_e_{.3, 16} = - 135 · (3 · α² · b₁2 − 11 · a₁3 · α) − b₁ · ( 5 · ν + 1 ) 150 = - 135 · (3 · 1 · 0.62 − 11 · 0.63 · 1) − 0.6 · ( 5 · 0.2 + 1 ) 150 = 0.02914286

K_e_{.4, 15} = K_e_{.3, 16} = 0.02914286 , K_e_{.7, 12} = -K_e_{.3, 16} = -0.02914286 , K_e_{.8, 11} = -K_e_{.3, 16} = -0.02914286

K_e_{.4, 4} = 4105 · (α · b₁² + a₁²α) + 8 · A₁225 = 4105 · (1 · 0.6² + 0.6²1) + 8 · 0.36225 = 0.04022857

K_e_{.8, 8} = K_e_{.4, 4} = 0.04022857 , K_e_{.12, 12} = K_e_{.4, 4} = 0.04022857 , K_e_{.16, 16} = K_e_{.4, 4} = 0.04022857

K_e_{.4, 8} = 135 · (2 · α · b₁²3 − a₁²α) − 2 · A₁225 = 135 · (2 · 1 · 0.6²3 − 0.6²1) − 2 · 0.36225 = -0.006628571

K_e_{.12, 16} = K_e_{.4, 8} = -0.006628571

K_e_{.4, 12} = - 170 · (a₁²α + α · b₁²) + A₁450 = - 170 · (0.6²1 + 1 · 0.6²) + 0.36450 = -0.009485714

K_e_{.8, 16} = K_e_{.4, 12} = -0.009485714

K_e_{.4, 16} = 135 · (2 · a₁²3 · α − α · b₁²) − 2 · A₁225 = 135 · (2 · 0.6²3 · 1 − 1 · 0.6²) − 2 · 0.36225 = -0.006628571

K_e_{.8, 12} = K_e_{.4, 16} = -0.006628571

K_e_{.1, 8} = 135 · (11 · α² − 132 · α²) + 5 · ν + 150 = 135 · (11 · 1 − 132 · 1) + 5 · 0.2 + 150 = 0.1685714

K_e_{.2, 7} = -K_e_{.1, 8} = -0.1685714 , K_e_{.3, 6} = K_e_{.1, 8} = 0.1685714 , K_e_{.4, 5} = -K_e_{.1, 8} = -0.1685714 , K_e_{.9, 16} = K_e_{.1, 8} = 0.1685714

K_e_{.10, 15} = -K_e_{.1, 8} = -0.1685714 , K_e_{.11, 14} = K_e_{.1, 8} = 0.1685714 , K_e_{.12, 13} = -K_e_{.1, 8} = -0.1685714

K_e_{.1, 9} = - 5435 · (αa₁² + 1α · b₁²) + 7225 · A₁ = - 5435 · (10.6² + 11 · 0.6²) + 7225 · 0.36 = -0.5714286

K_e_{.5, 13} = K_e_{.1, 9} = -0.5714286

K_e_{.1, 10} = 135 · (27 · αa₁ + 13α² · b₁) − 625 · b₁ = 135 · (27 · 10.6 + 131 · 0.6) − 625 · 0.6 = 1.504762

K_e_{.2, 9} = -K_e_{.1, 10} = -1.504762 , K_e_{.5, 14} = -K_e_{.1, 10} = -1.504762 , K_e_{.6, 13} = K_e_{.1, 10} = 1.504762

K_e_{.1, 11} = 135 · (13 · α²a₁ + 27α · b₁) − 625 · a₁ = 135 · (13 · 10.6 + 271 · 0.6) − 625 · 0.6 = 1.504762

K_e_{.3, 9} = -K_e_{.1, 11} = -1.504762 , K_e_{.5, 15} = K_e_{.1, 11} = 1.504762 , K_e_{.7, 13} = -K_e_{.1, 11} = -1.504762

K_e_{.1, 12} = - 1370 · (α² + 1α²) + 150 = - 1370 · (1 + 11) + 150 = -0.3514286

K_e_{.2, 11} = -K_e_{.1, 12} = 0.3514286 , K_e_{.3, 10} = -K_e_{.1, 12} = 0.3514286 , K_e_{.4, 9} = K_e_{.1, 12} = -0.3514286 , K_e_{.5, 16} = -K_e_{.1, 12} = 0.3514286

K_e_{.6, 15} = K_e_{.1, 12} = -0.3514286 , K_e_{.7, 14} = K_e_{.1, 12} = -0.3514286 , K_e_{.8, 13} = -K_e_{.1, 12} = 0.3514286

K_e_{.1, 13} = 235 · (27 · αa₁² − 78α · b₁²) − 7225 · A₁ = 235 · (27 · 10.6² − 781 · 0.6²) − 7225 · 0.36 = -16.095238

K_e_{.5, 9} = K_e_{.1, 13} = -16.095238

K_e_{.2, 16} = 135 · (2 · a₁α² − 13 · α · b₁3) + 2 · a₁75 = 135 · (2 · 0.61 − 13 · 1 · 0.63) + 2 · 0.675 = -0.024

K_e_{.4, 14} = -K_e_{.2, 16} = 0.024 , K_e_{.6, 12} = K_e_{.2, 16} = -0.024 , K_e_{.8, 10} = -K_e_{.2, 16} = 0.024

K_e_{.3, 3} = 435 · (α³ + 13α) + 8 · α25 = 435 · (1³ + 131) + 8 · 125 = 1.92

K_e_{.7, 7} = K_e_{.3, 3} = 1.92 , K_e_{.11, 11} = K_e_{.3, 3} = 1.92 , K_e_{.15, 15} = K_e_{.3, 3} = 1.92

K_e_{.3, 4} = 235 · (α² · b₁ + 11 · a₁3 · α) + 2 · b₁ · ( 5 · ν + 1 ) 75 = 235 · (1 · 0.6 + 11 · 0.63 · 1) + 2 · 0.6 · ( 5 · 0.2 + 1 ) 75 = 0.192

K_e_{.7, 8} = -K_e_{.3, 4} = -0.192 , K_e_{.11, 12} = -K_e_{.3, 4} = -0.192 , K_e_{.15, 16} = K_e_{.3, 4} = 0.192

K_e_{.3, 7} = 235 · (-2 · α³ + 9α) − 8 · α25 = 235 · (-2 · 1³ + 91) − 8 · 125 = 0.08

K_e_{.11, 15} = K_e_{.3, 7} = 0.08

K_e_{.3, 8} = 135 · (2 · α² · b₁ − 13 · a₁3 · α) + 2 · b₁75 = 135 · (2 · 1 · 0.6 − 13 · 0.63 · 1) + 2 · 0.675 = -0.024

K_e_{.4, 7} = -K_e_{.3, 8} = 0.024 , K_e_{.11, 16} = -K_e_{.3, 8} = 0.024 , K_e_{.12, 15} = K_e_{.3, 8} = -0.024

K_e_{.3, 11} = 335 · (α³ + 3α) + 2 · α25 = 335 · (1³ + 31) + 2 · 125 = 0.4228571

K_e_{.7, 15} = K_e_{.3, 11} = 0.4228571

K_e_{.3, 12} = - 170 · (3 · α² · b₁ + 13 · a₁3 · α) − b₁150 = - 170 · (3 · 1 · 0.6 + 13 · 0.63 · 1) − 0.6150 = -0.06685714

K_e_{.4, 11} = -K_e_{.3, 12} = 0.06685714 , K_e_{.7, 16} = -K_e_{.3, 12} = 0.06685714 , K_e_{.8, 15} = K_e_{.3, 12} = -0.06685714

K_e_{.3, 15} = 135 · (26α − 3 · α³) − 2 · α25 = 135 · (261 − 3 · 1³) − 2 · 125 = 0.5771429

K_e_{.7, 11} = K_e_{.3, 15} = 0.5771429

K_e = D_1,1 · K_e = 24.305556 · 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 046.66666721.5972224.666667-84.95370414.027778-4.0972220.7083333-36.57407410.2777788.541667-1.625-13.6574071.9444444.097222-0.5833333 0046.6666674.666667-13.6574074.0972221.944444-0.5833333-36.5740748.54166710.277778-1.625-84.953704-4.09722214.0277780.7083333 0000.9777778-4.0972220.70833330.5833333-0.1611111-8.5416671.6251.625-0.2305556-4.0972220.58333330.7083333-0.1611111 0000796.296296-135.185185135.185185-16.736111-391.20370413.65740784.953704-4.097222-13.888889-36.57407436.5740748.541667 0000046.666667-21.5972224.66666713.6574071.944444-4.097222-0.583333336.57407410.277778-8.541667-1.625 00000046.666667-4.666667-84.9537044.09722214.027778-0.7083333-36.574074-8.54166710.2777781.625 00000000.97777784.0972220.5833333-0.7083333-0.16111118.5416671.625-1.625-0.2305556 00000000796.296296-135.185185-135.18518516.736111-391.203704-84.95370413.6574074.097222 00000000046.66666721.597222-4.66666784.95370414.027778-4.097222-0.7083333 000000000046.666667-4.66666713.6574074.0972221.9444440.5833333 000000000000.9777778-4.097222-0.7083333-0.5833333-0.1611111 000000000000796.296296135.185185-135.185185-16.736111 000000000000046.666667-21.597222-4.666667 0000000000000046.6666674.666667 0000000000000000.9777778

⃗F_e = q · A₁24 · [6; a₁; b₁; A₁6; 6; -a₁; b₁; -A₁6; 6; -a₁; -b₁; A₁6; 6; a₁; -b₁; -A₁6] = 10 · 0.3624 · [6; 0.6; 0.6; 0.366; 6; -0.6; 0.6; -0.366; 6; -0.6; -0.6; 0.366; 6; 0.6; -0.6; -0.366] = [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

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 16.7361114.6666674.6666670.9777778-4.0972220.58333330.7083333-0.1611111000000000000⋯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 4.097222-0.58333330.7083333-0.1611111009.3333331.955556-4.0972220.58333330.7083333-0.161111100000000⋯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 00004.097222-0.58333330.7083333-0.1611111009.3333331.955556-4.0972220.58333330.7083333-0.16111110000⋯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 000000004.097222-0.58333330.7083333-0.1611111009.3333331.955556-4.0972220.58333330.7083333-0.1611111⋯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 0000000000004.097222-0.58333330.7083333-0.1611111009.3333331.955556⋯0 ⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋱⋮00000000000000000000⋯0.9777778

⃗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

⃗Z = slsolve ( K; ⃗F ) = [0 0.5523352 0.382696 -0.4159901 0.2028057 0.3732616 0.2648531 -0.1936926 0.2989084 0.3091046 0.04831304 -0.02500182 0.2612117 0.3426307 -0.1651507 0.1275183 0.1211517 0.4681261 -0.2671228 0.2293407 ... -0.4159902] mm

Results

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

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} ) ] )

M_j = 1.5002750.30977470.21973790.15634340.15704780.99832250.15642330.15196360.19424650.1519630.15642380.99832240.15704870.15634320.21973730.30975011.5002588.502216.479335.777678⋯1.500075 1.5613237.8059929.3381267.66833.333101-28.3619893.1463977.3238838.8950477.3238823.146398-28.3619893.3331027.66839.3381267.8059781.561340.32204035.3824847.126847⋯1.561519 8.0886973.7662450.4861465-2.479522-4.4594030.15500784.7804022.842763-2.410353×10^-8-2.842763-4.780402-0.15500784.4594032.479522-0.4861463-3.766235-8.0886974.1090252.5724750.3669336⋯8.088698

transp ( Mx ) = 1.5002758.5022111.06578310.4766316.7969380.8844119-31.2340010.30206295.6439378.7792398.8768585.919520.6860137-30.1316490.68601425.9195198.8768598.7792395.6439380.3020627⋯1.500259 0.30977476.479339.4318288.8768314.922389-2.695046-10.242271-3.2600123.8085627.2499787.3609524.127543-2.764018-9.732553-2.7640184.1275437.3609527.2499783.808562-3.260011⋯0.3097374 0.21973795.7776788.7161968.1349194.039416-2.257802-6.060553-2.8192132.932076.5165976.6403923.291629-2.294127-5.615407-2.2941273.2916296.6403926.5165962.93207-2.819213⋯0.2197445 0.15634345.9931719.1259748.4927624.057756-3.257726-7.988419-3.8326612.9136446.7974156.9267753.304594-3.205142-7.381295-3.2051423.3045946.9267756.7974152.913644-3.832661⋯0.1563406 0.15704787.2696610.372279.5566174.972566-5.268395-16.118845-5.8654883.7668247.7132127.8382354.157483-5.096247-15.085958-5.0962474.1574837.8382357.7132123.766824-5.865488⋯0.1570494 0.99832259.03856711.08045310.1228085.763875-2.1353-38.650028-2.7455524.5217928.1908078.3150474.904107-2.088386-36.536389-2.0883864.9041078.3150478.1908074.521792-2.745552⋯0.9983223 0.15642337.22106610.2820369.4364984.848272-5.357982-16.182847-5.9630023.624277.5601197.6988674.053292-5.138773-15.091479-5.1387734.0532927.6988677.5601193.62427-5.963002⋯0.1564268 0.15196365.8732038.927978.2297863.773483-3.41909-8.032715-4.0090252.5942716.4697716.6264673.062933-3.269913-7.306549-3.2699133.0629336.6264676.4697712.594271-4.009025⋯0.1519571 0.19424655.5141468.3802347.6929583.468366-2.491073-5.742609-3.073362.3104825.9792326.1426642.786926-2.376796-5.118358-2.3767962.7869266.1426645.9792322.310482-3.07336⋯0.194254 0.1519635.8732048.927978.2297863.773483-3.41909-8.032715-4.0090252.5942716.4697716.6264673.062933-3.269913-7.306549-3.2699133.0629336.6264676.4697712.594271-4.009025⋯0.1519564 0.15642387.22106610.2820369.4364984.848272-5.357982-16.182847-5.9630023.624277.5601197.6988674.053292-5.138773-15.091479-5.1387734.0532927.6988677.5601193.62427-5.963002⋯0.1564266 0.99832249.03856711.08045310.1228085.763875-2.1353-38.650028-2.7455524.5217928.1908078.3150474.904107-2.088386-36.536389-2.0883864.9041078.3150478.1908074.521792-2.745552⋯0.9983221 0.15704877.26965910.372279.5566174.972566-5.268395-16.118845-5.8654883.7668247.7132127.8382354.157483-5.096247-15.085958-5.0962474.1574837.8382357.7132123.766824-5.865488⋯0.1570582 0.15634325.9931719.1259748.4927624.057756-3.257726-7.988419-3.8326612.9136446.7974156.9267753.304594-3.205142-7.381295-3.2051423.3045946.9267756.7974152.913644-3.832661⋯0.156342 0.21973735.7776778.7161968.1349194.039416-2.257802-6.060553-2.8192132.932076.5165976.6403923.291629-2.294127-5.615407-2.2941273.2916296.6403926.5165962.93207-2.819213⋯0.2197303 0.30975016.4793359.431838.8768314.922388-2.695046-10.242271-3.2600123.8085627.2499787.3609534.127543-2.764018-9.732553-2.7640194.1275437.3609527.2499783.808562-3.260012⋯0.3094814 1.5002588.50222411.06578310.476636.7969390.8844112-31.2340010.3020635.6439378.7792398.8768585.919520.6860137-30.1316490.68601435.9195198.8768598.7792395.6439380.3020627⋯1.500075 kNm/m

transp ( My ) = 1.5613230.32204030.23543350.21672910.16938360.17865381.0129710.17816160.16785760.21141710.21080950.16727620.17645270.98644480.17645330.16727530.21081050.21141620.16785820.1781613⋯1.561345 7.8059925.3824844.3092534.1357694.7426476.340088.2301146.282764.6121333.8928683.8680944.5307066.12227.9852796.12224.5307063.8680933.8928694.6121326.28276⋯7.80597 9.3381267.1268475.8004925.5534226.4150978.115739.0465048.0146136.1889345.1451025.117036.0906867.821128.777037.821126.0906865.117035.1451016.1889348.014613⋯9.338132 7.66835.4177814.2627213.9206234.2338295.4198786.3445995.3011323.9795653.478143.4810433.9686645.2228496.181445.2228493.9686643.4810433.478143.9795655.301132⋯7.668298 3.333101-0.32799450.53974810.4057356-1.174273-3.412893-0.3301173-3.53113-1.3994770.099760860.1610125-1.210479-3.234594-0.2183684-3.234594-1.2104790.16101250.09976085-1.399477-3.53113⋯3.333102 -28.361989-7.038398-1.936124-1.64658-4.99389-13.644591-36.317421-13.771031-5.215778-1.858062-1.763919-4.886758-13.033067-34.570998-13.033067-4.886758-1.763919-1.858062-5.215778-13.771031⋯-28.361989 3.146397-0.50668750.34853730.1799829-1.45618-3.762354-0.7232665-3.922882-1.765717-0.2519029-0.1967723-1.596231-3.661551-0.6660923-3.661551-1.596231-0.1967723-0.2519029-1.765717-3.922882⋯3.1464 7.3238835.0945673.9093.4891813.6755264.6859375.4988174.4840593.2542962.7944772.7886533.2145984.350075.2437544.350073.2145982.7886532.7944773.2542964.484059⋯7.323878 8.8950476.7401375.3508914.9612985.6089926.9131347.5726116.6932525.1418814.1806234.1497385.0278956.4569637.2130836.4569635.0278954.1497384.1806235.1418816.693252⋯8.895053 7.3238825.0945673.9089993.4891813.6755264.6859375.4988174.4840593.2542962.7944772.7886533.2145984.350075.2437544.350073.2145982.7886532.7944773.2542964.484059⋯7.323877 3.146398-0.50668770.34853730.1799829-1.45618-3.762354-0.7232665-3.922882-1.765717-0.2519029-0.1967723-1.596231-3.661551-0.6660923-3.661551-1.596231-0.1967723-0.2519029-1.765717-3.922882⋯3.1464 -28.361989-7.038398-1.936124-1.64658-4.99389-13.644591-36.317421-13.771031-5.215778-1.858062-1.763919-4.886758-13.033067-34.570998-13.033067-4.886758-1.763919-1.858062-5.215778-13.771031⋯-28.36199 3.333102-0.32799460.53974820.4057355-1.174273-3.412893-0.3301173-3.53113-1.3994770.099760860.1610125-1.210479-3.234594-0.2183684-3.234594-1.2104790.16101250.09976085-1.399477-3.53113⋯3.333109 7.66835.4177814.2627213.9206234.2338295.4198786.3445995.3011323.9795653.478143.4810433.9686645.2228496.181445.2228493.9686643.4810433.478143.9795655.301132⋯7.6683 9.3381267.1268455.8004915.5534226.4150978.115739.0465048.0146136.1889345.1451025.117036.0906867.821128.777037.821126.0906865.117035.1451016.1889348.014613⋯9.338126 7.8059785.3824794.3092544.135774.7426476.3400818.2301146.282764.6121333.8928683.8680944.5307066.12227.9852796.12224.5307063.8680933.8928694.6121326.28276⋯7.805828 1.561340.32206640.23543390.21672770.16938450.17865291.0129710.17816180.16785750.21141720.21080940.16727630.17645270.98644480.17645340.1672750.21081080.2114160.16785810.1781614⋯1.561519 kNm/m

transp ( Mxy ) = 8.0886974.1090251.416336-0.9453079-3.232331-4.7813530.021763664.8385873.3347481.141499-1.025238-3.194104-4.6732159.790543×10^-94.6732153.1941041.025238-1.141499-3.334748-4.838587⋯-8.088697 3.7662452.5724750.9829256-0.515595-2.056503-3.2286130.058255113.3545242.2129550.7309473-0.6696087-2.124156-3.225638-2.241521×10^-83.2256382.1241560.6696087-0.7309473-2.212955-3.354524⋯-3.766235 0.48614650.36693360.210340.08576266-0.1194038-0.31368240.07347950.46113970.26652590.05675809-0.07980332-0.2737226-0.4316644.761034×10^-90.4316640.27372260.07980332-0.05675809-0.2665259-0.4611397⋯-0.4861468 -2.479522-1.796141-0.6013640.61671451.7469222.092690.07626484-1.948268-1.634146-0.58150580.49196681.5363311.879395-1.027996×10^-9-1.879395-1.536331-0.49196680.58150581.6341461.948268⋯2.479522 -4.459403-3.240091-0.89783760.68261572.3709064.5250.07575709-4.382355-2.264704-0.68178660.60077812.1521384.2109132.395547×10^-10-4.210913-2.152138-0.60077810.68178662.2647044.382355⋯4.459403 0.15500780.16110120.14918770.12455210.096356070.077948450.073859910.069867520.051235710.0210446-0.008412859-0.02659053-0.02344541-5.804372×10^-110.023445410.026590530.008412859-0.0210446-0.05123571-0.06986752⋯-0.1550078 4.7804023.5659761.18935-0.4505739-2.206794-4.3993290.063467544.5354632.3793750.7252099-0.6264028-2.225368-4.2792951.444385×10^-114.2792952.2253680.6264028-0.7252099-2.379375-4.535463⋯-4.780401 2.8427632.1343930.8682899-0.4377384-1.679619-2.0840940.037778392.1683581.7961380.6296579-0.5417111-1.677943-2.037693-3.548115×10^-122.0376931.6779430.5417111-0.6296579-1.796138-2.168358⋯-2.842763 -2.410353×10^-83.062399×10^-8-4.397001×10^-93.066824×10^-10-8.12632×10^-113.636829×10^-11-1.325189×10^-114.764349×10^-12-7.14584×10^-131.790365×10^-13-3.8029×10^-132.910935×10^-1301.375317×10^-13-9.949168×10^-132.393×10^-12-3.712996×10^-122.380218×10^-123.027025×10^-12-1.394228×10^-11⋯3.938371×10^-8 -2.842763-2.134393-0.86828990.43773841.6796192.084094-0.03777839-2.168358-1.796138-0.62965790.54171111.6779432.0376934.127238×10^-12-2.037693-1.677943-0.54171110.62965791.7961382.168358⋯2.842763 -4.780402-3.565976-1.189350.45057392.2067944.399329-0.06346754-4.535463-2.379375-0.72520990.62640282.2253684.279295-1.911611×10^-11-4.279295-2.225368-0.62640280.72520992.3793754.535463⋯4.780401 -0.1550078-0.1611013-0.1491877-0.1245521-0.09635607-0.07794845-0.07385991-0.06986752-0.05123571-0.02104460.0084128590.026590530.023445417.954439×10^-11-0.02344541-0.02659053-0.0084128590.02104460.051235710.06986752⋯0.1550075 4.4594033.2400910.8978376-0.6826157-2.370906-4.525-0.075757094.3823552.2647040.6817866-0.6007781-2.152138-4.210913-3.335117×10^-104.2109132.1521380.6007781-0.6817866-2.264704-4.382355⋯-4.459401 2.4795221.7961410.601364-0.6167145-1.746922-2.09269-0.076264841.9482681.6341460.5815058-0.4919668-1.536331-1.8793951.414276×10^-91.8793951.5363310.4919668-0.5815058-1.634146-1.948268⋯-2.479516 -0.4861463-0.3669331-0.21034-0.085762740.11940380.3136824-0.07347951-0.4611397-0.2665259-0.056758090.079803320.27372260.431664-6.33653×10^-9-0.431664-0.2737226-0.079803320.056758090.26652590.4611397⋯0.4861443 -3.766235-2.572475-0.98292630.51559492.0565033.228613-0.05825506-3.354524-2.212955-0.73094730.66960872.1241563.2256382.942834×10^-8-3.225638-2.124156-0.66960870.73094732.2129553.354524⋯3.766119 -8.088697-4.109035-1.4163360.94530843.2323314.781353-0.0217637-4.838587-3.334748-1.1414991.0252383.1941044.673215-1.247931×10^-8-4.673215-3.194104-1.0252381.1414993.3347484.838587⋯8.088698 kNm/m

Finite Element Analysis of Flat Slab

Input data

Finite element mesh

Finite element formulation

Element stiffness matrix calculation

Solution

Results

`N`_1,w shape function plot	`N`_1,θₓ shape function plot