Assignment 4 Beam Element AE5030 Advanced Finite Element Method [PDF]

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ASSIGNMENT 4 BEAM ELEMENT AE5030 Advanced Finite Element Method By: Budi Aji Warsiyanto 23620014



Lecturer: Dr. Ir. Muhamad Giri Suada



AEROSPACE ENGINEERING STUDY PROGRAM FACULTY OF MECHANICAL AND AEROSPACE ENGINEERING BANDUNG INSTITUTE OF TECHNOLOGY 2020



1.



PROBLEM



The taper beam element illustration is given as in Figure 1-1 and its properties are shown in Table 1-1. From this information, the deformation value of the beam element can be calculated using finite element methods (Abaqus software). To support the solution of the finite element method, setup or modeling views are required. The setup or modeling consists of the geometric (coordinates), material property, load, and mesh. Table 1-1. Beam structure properties and load Parameter



Value



Modulus of elasticity, E



100000 N/mm2



Length, L



1080 mm



Root height, H0



270 mm



Tip height, HL



90 mm



Thickness, t



10 mm



Load, P



10 N



𝐇𝟎



𝐇𝐱



𝐇𝐋



𝐏 𝐱 𝐋 Figure 1-1. A beam element structure 2.



SOLUTION



To solve this problem, the structure must be divided into several elements consisting of a length (𝑑𝑥 ) and height (𝐻𝑥 ). Therefore, an equation is needed to determine the dimensions of the height based on continuous length. Figure 2-1 shows an element used to determine a continuous length equation (Eq. (1)) to calculate the value of 𝐻𝑥 .



𝐇𝐱



𝐝𝐱 Figure 2-1. Beam element discretization



1



𝑥 2 .𝐻 𝐿 3 0 2𝑥 𝐻𝑥 = 𝐻0 (1 − ) 3𝐿



𝐻𝑥 = 𝐻0 −



(1)



The Abaqus application is used as a finite element (numeric) solution to the problem in the beam element above. Abaqus was chosen because it is one of the advanced applications for analyzing a structure. The process using the finite element method application is divided into three, namely pre-processing, solution, and post-processing. In addition, several modules are used to set up numerical modeling of beam elements. The following shows the module used to perform numerical modeling: a. Part (geometry) In this module, the structural model is created based on the dimensions of length 𝐿 shown in Table 1-1, namely 1080 mm. Figures 2-2 (a), (b), (c), and (d) show the different elements in the beam structure, namely 2, 5, 10, and 20 elements are created in millimeters (mm) unit. For more elements (40, 80, and 160) it is not displayed because the details of the dimension of the elements are close to each other so that they appear stacked. The difference of elements is intended to approximate the beam element which has a tapered shape. This aims to determine the effect of the number of elements on the deformation value obtained from the finite element method computation.



(a)



(b)



(c)



(d) Figure 2-2. Difference in dimensions per element based on the number of elements: (a) 2, (b) 5, (c) 10, and (d) 20 elements b. Property The modulus of elasticity, 𝐻𝑥 and thickness 𝑡 is inputted on this module. Figures 2-3 (a), (b), (c), and (d) show the beam element that has inputted its properties so that it has an area that extends along 𝑥-axis. The thickness visualization extends along 𝑧-axis so that it is not visible on the figure. This figure is only an illustration of the profile (rendering) of the beam element to show that the model matches the property intended by the user. The non-rendered beam structure model is shown in Figure 2-3 (e). 2



Hx



x L (a)



(b)



(c)



(d) (e) Figure 2-3. Rendering of (a) 2, (b) 5, (c) 10, (b) 20 elements and (e) non-rendering beam element that have inputted by the property



3



c. Step The step is used to determine the type of simulation, whether static or dynamic. In this assignment, the static simulation type linear perturbation is used because the force 𝑃 given has a constant value by time and the problem is still in the scope of linear analysis. Figure 2-4 shows the setup used in this module. "Nlgeom: Off" is chosen because the structure does not experience large displacement during loading.



Figure 2-4. Setup in the module step d. Load Figure 2-5 (a) shows the beam element that has applied boundary conditions and loads in the form of concentrated force (CF). Figures 2-5 (b) and (c) show the boundary condition setup which constrained from movement in all directions (ENCASTRE).



BC



P



(a)



(b)



(c)



Figure 2-5. Setup on "Module Load": (a) illustration of the beam with 160 elements that have applied boundary conditions, (b) boundary conditions, and (c) load 4



e. Mesh The mesh module is used to define the mesh setup, such as the size or number and type (structured, unstructured) of the elements. In this assignment, because the beam elements are modeled with a onedimensional line, only the size or number of elements can be determined. Figures 2-6 (a), (b), (c), and (d) show the difference of element and node numbers on an element beam. More elements are not displayed because the node and element labels are too many so that appears stacked. The coordinates of each node from Figure 2-6 are shown in Table 2-1 based on the difference of elements is created in the Part module (see Figure 2-1 and attachment for details). Figure 2-7 shows the element type of the beam structure model, namely B21. node



element



(a)



(b)



(c)



(d) Figure 2-6. The number of elements and nodes in the beam structure: (a) 2, (b) 5, (c) 10, and (b) 20 elements Table 2-1. The coordinates for each node in the 10 and 20 element beam structures



Node 1 2 3 4 5 6



10 elements Coordinate Node (x, y) (0, 0) 7 (108, 0) 8 (216, 0) 9 (324, 0) 10 (432, 0) 11 (540, 0)



Coordinate (x, y) (648, 0) (756, 0) (864, 0) (972, 0) (1080, 0)



Node 1 2 3 4 5 6 7 8 9 10 11



5



20 elements Coordinate Node (x, y) (0, 0) 12 (54, 0) 13 (108, 0) 14 (162, 0) 15 (216, 0) 16 (270, 0) 17 (324, 0) 18 (378, 0) 19 (432, 0) 20 (486, 0) 21 (540, 0)



Coordinate (x, y) (594, 0) (648, 0) (702, 0) (756, 0) (810, 0) (864, 0) (918, 0) (972, 0) (1026, 0) (1080, 0)



Figure 2-7. The element type of beam structure f. Job The job module is used to perform the computation process (running) modeling that has been created. Figure 2-8 shows the job manager in the Abaqus application.



Figure 2-8. Job manager 2.1.1 Results The simulation results are maximum deformation in 𝑦-direction (U2) which is displayed in the form of visualization and curves (graphs) are plotted based on the difference in the number of elements. Figure 2-9 (a), (b), (c), (d), (e), (f), and (g) shows the deformation value of the beam structure based on the color gradient and the difference in the number of elements, namely 2, 5, 10, 20, 40, 80, and 160 elements, respectively. The color gradient is described by blue which indicates the minimum deformation value and progressively increases to red which indicates the maximum deformation value. The negative sign indicates the direction of the deformed structure, which is the deformation moves in the negative 𝑦-direction due to the force which is also in the negative 𝑦-direction. The “Min” notation in the figure refers to the legend which shows the red color as the minimum value. Figure 2-10 shows the comparison of the maximum deformation values for different elements. It can be seen that more elements provide a maximum deformation value that is increasingly converging (values tend to be constant).



6



(a)



(b)



(c)



7



(d)



(e)



(f)



(g) Figure 2-9. Visualization of the deformation of the beam structure for (a) 2, (b) 5, (c) 10, (d) 20, (e) 40, (f) 80, and (g) 160 elements



8



x 10-3



Deformation (mm)



16



Deformation



14 12 10 8 6 4 0



40



80 120 Element numbers



160



Figure 2-10. The maximum deformation values based on the difference of element numbers ATTACHMENT The mesh coordinates for the varying number of elements will be different. The following is the data for the creating of the beam structure geometry with the different number of elements: • Two (2) elements Node



Coordinate (x, y)



1 2 3



0, 0 540, 0 1080, 0



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



180



1800



90



900



• Five (5) elements Node



Coordinate (x, y)



1 2 3



0, 0 216, 0 432, 0



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



234



2340



198



1980



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



265.5



2655



261 256.5 252



2610 2565 2520



Node



Coordinate (x, y)



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



4 5 6



648, 0 864, 0 1080, 0



162 126 90



1620 1260 900



Node



Coordinate (x, y)



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



22 23 24 25 26



567, 0 594, 0 621, 0 648, 0 675, 0



175.5 171 166.5 162 157.5



1755 1710 1665 1620 1575



• Forty (40) elements Node



Coordinate (x, y)



1 2 3 4 5



0, 0 27, 0 54, 0 81, 0 108, 0



9



6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21



135, 0 162, 0 189, 0 216, 0 243, 0 270, 0 297, 0 324, 0 351, 0 378, 0 405, 0 432, 0 459, 0 486, 0 513, 0 540, 0



247.5 243 238.5 234 229.5 225 220.5 216 211.5 207 202.5 198 193.5 189 184.5 180



2475 2430 2385 2340 2295 2250 2205 2160 2115 2070 2025 1980 1935 1890 1845 1800



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



267.75



2677.5



265.5 263.25 261 258.75 256.5 254.25 252 249.75 247.5 245.25 243 240.75 238.5 236.25 234 231.75 229.5 227.25 225 222.75 220.5



2655 2632.5 2610 2587.5 2565 2542.5 2520 2497.5 2475 2452.5 2430 2407.5 2385 2362.5 2340 2317.5 2295 2272.5 2250 2227.5 2205



27 28 29 30 31 32 33 34 35 36 37 38 39 40 41



702, 0 729, 0 756, 0 783, 0 810, 0 837, 0 864, 0 891, 0 918, 0 945, 0 972, 0 999, 0 1026, 0 1053, 0 1080, 0



153 148.5 144 139.5 135 130.5 126 121.5 117 112.5 108 103.5 99 94.5 90



1530 1485 1440 1395 1350 1305 1260 1215 1170 1125 1080 1035 990 945 900



• Eighty (80) elements Node



Coordinate (x, y)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23



0, 0 13.5, 0 27, 0 40.5, 0 54, 0 67.5, 0 81, 0 94.5, 0 108, 0 121.5, 0 135, 0 148.5, 0 162, 0 175.5, 0 189, 0 202.5, 0 216, 0 229.5, 0 243, 0 256.5, 0 270, 0 283.5, 0 297, 0



10



Node



Coordinate (x, y)



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64



553.5, 0 567, 0 580.5, 0 594, 0 607.5, 0 621, 0 634.5, 0 648, 0 661.5, 0 675, 0 688.5, 0 702, 0 715.5, 0 729, 0 742.5, 0 756, 0 769.5, 0 783, 0 796.5, 0 810, 0 823.5, 0 837, 0 850.5, 0



177.75 175.5 173.25 171 168.75 166.5 164.25 162 159.75 157.5 155.25 153 150.75 148.5 146.25 144 141.75 139.5 137.25 135 132.75 130.5 128.25



1777.5 1755 1732.5 1710 1687.5 1665 1642.5 1620 1597.5 1575 1552.5 1530 1507.5 1485 1462.5 1440 1417.5 1395 1372.5 1350 1327.5 1305 1282.5



24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41



310.5, 0 324, 0 337.5, 0 351, 0 364.5, 0 378, 0 391.5, 0 405, 0 418.5, 0 432, 0 445.5, 0 459, 0 472.5, 0 486, 0 499.5, 0 513, 0 526.5, 0 540, 0



218.25 216 213.75 211.5 209.25 207 204.75 202.5 200.25 198 195.75 193.5 191.25 189 186.75 184.5 182.25 180



2182.5 2160 2137.5 2115 2092.5 2070 2047.5 2025 2002.5 1980 1957.5 1935 1912.5 1890 1867.5 1845 1822.5 1800



65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81



864, 0 877.5, 0 891, 0 904.5, 0 918, 0 931.5, 0 945, 0 958.5, 0 972, 0 985.5, 0 999, 0 1012.5, 0 1026, 0 1039.5, 0 1053, 0 1066.5, 0 1080, 0



126 123.75 121.5 119.25 117 114.75 112.5 110.25 108 105.75 103.5 101.25 99 96.75 94.5 92.25 90



1260 1237.5 1215 1192.5 1170 1147.5 1125 1102.5 1080 1057.5 1035 1012.5 990 967.5 945 922.5 900



• One hundred and sixty (160) elements Node



Coordinate (x, y)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21



0, 0 6.75, 0 13.5, 0 20.25, 0 27, 0 33.75, 0 40.5, 0 47.25, 0 54, 0 60.75, 0 67.5, 0 74.25, 0 81, 0 87.75, 0 94.5, 0 101.25, 0 108, 0 114.75, 0 121.5, 0 128.25, 0 135, 0



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



268.875 2688.75 267.75 266.625 265.5 264.375 263.25 262.125 261 259.875 258.75 257.625 256.5 255.375 254.25 253.125 252 250.875 249.75 248.625 247.5



2677.5 2666.25 2655 2643.75 2632.5 2621.25 2610 2598.75 2587.5 2576.25 2565 2553.75 2542.5 2531.25 2520 2508.75 2497.5 2486.25 2475 11



Node



Coordinate (x, y)



𝐇𝐱 (mm)



𝐀𝐱 (mm2)



82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102



546.75, 0 553.5, 0 560.25, 0 567, 0 573.75, 0 580.5, 0 587.25, 0 594, 0 600.75, 0 607.5, 0 614.25, 0 621, 0 627.75, 0 634.5, 0 641.25, 0 648, 0 654.75, 0 661.5, 0 668.25, 0 675, 0 681.75, 0



178.875 177.75 176.625 175.5 174.375 173.25 172.125 171 169.875 168.75 167.625 166.5 165.375 164.25 163.125 162 160.875 159.75 158.625 157.5 156.375



1788.75 1777.5 1766.25 1755 1743.75 1732.5 1721.25 1710 1698.75 1687.5 1676.25 1665 1653.75 1642.5 1631.25 1620 1608.75 1597.5 1586.25 1575 1563.75



22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66



141.75, 0 148.5, 0 155.25, 0 162, 0 168.75, 0 175.5, 0 182.25, 0 189, 0 195.75, 0 202.5, 0 209.25, 0 216, 0 222.75, 0 229.5, 0 236.25, 0 243, 0 249.75, 0 256.5, 0 263.25, 0 270, 0 276.75, 0 283.5, 0 290.25, 0 297, 0 303.75, 0 310.5, 0 317.25, 0 324, 0 330.75, 0 337.5, 0 344.25, 0 351, 0 357.75, 0 364.5, 0 371.25, 0 378, 0 384.75, 0 391.5, 0 398.25, 0 405, 0 411.75, 0 418.5, 0 425.25, 0 432, 0 438.75, 0



246.375 245.25 244.125 243 241.875 240.75 239.625 238.5 237.375 236.25 235.125 234 232.875 231.75 230.625 229.5 228.375 227.25 226.125 225 223.875 222.75 221.625 220.5 219.375 218.25 217.125 216 214.875 213.75 212.625 211.5 210.375 209.25 208.125 207 205.875 204.75 203.625 202.5 201.375 200.25 199.125 198 196.875



2463.75 2452.5 2441.25 2430 2418.75 2407.5 2396.25 2385 2373.75 2362.5 2351.25 2340 2328.75 2317.5 2306.25 2295 2283.75 2272.5 2261.25 2250 2238.75 2227.5 2216.25 2205 2193.75 2182.5 2171.25 2160 2148.75 2137.5 2126.25 2115 2103.75 2092.5 2081.25 2070 2058.75 2047.5 2036.25 2025 2013.75 2002.5 1991.25 1980 1968.75 12



103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147



688.5, 0 695.25, 0 702, 0 708.75, 0 715.5, 0 722.25, 0 729, 0 735.75, 0 742.5, 0 749.25, 0 756, 0 762.75, 0 769.5, 0 776.25, 0 783, 0 789.75, 0 796.5, 0 803.25, 0 810, 0 816.75, 0 823.5, 0 830.25, 0 837, 0 843.75, 0 850.5, 0 857.25, 0 864, 0 870.75, 0 877.5, 0 884.25, 0 891, 0 897.75, 0 904.5, 0 911.25, 0 918, 0 924.75, 0 931.5, 0 938.25, 0 945, 0 951.75, 0 958.5, 0 965.25, 0 972, 0 978.75, 0 985.5, 0



155.25 154.125 153 151.875 150.75 149.625 148.5 147.375 146.25 145.125 144 142.875 141.75 140.625 139.5 138.375 137.25 136.125 135 133.875 132.75 131.625 130.5 129.375 128.25 127.125 126 124.875 123.75 122.625 121.5 120.375 119.25 118.125 117 115.875 114.75 113.625 112.5 111.375 110.25 109.125 108 106.875 105.75



1552.5 1541.25 1530 1518.75 1507.5 1496.25 1485 1473.75 1462.5 1451.25 1440 1428.75 1417.5 1406.25 1395 1383.75 1372.5 1361.25 1350 1338.75 1327.5 1316.25 1305 1293.75 1282.5 1271.25 1260 1248.75 1237.5 1226.25 1215 1203.75 1192.5 1181.25 1170 1158.75 1147.5 1136.25 1125 1113.75 1102.5 1091.25 1080 1068.75 1057.5



67 68 69 70 71 72 73 74 75 76 77 78 79 80 81



445.5, 0 452.25, 0 459, 0 465.75, 0 472.5, 0 479.25, 0 486, 0 492.75, 0 499.5, 0 506.25, 0 513, 0 519.75, 0 526.5, 0 533.25, 0 540, 0



195.75 194.625 193.5 192.375 191.25 190.125 189 187.875 186.75 185.625 184.5 183.375 182.25 181.125 180



1957.5 1946.25 1935 1923.75 1912.5 1901.25 1890 1878.75 1867.5 1856.25 1845 1833.75 1822.5 1811.25 1800



13



148 149 150 151 152 153 154 155 156 157 158 159 160 161



992.25, 0 999, 0 1005.75, 0 1012.5, 0 1019.25, 0 1026, 0 1032.75, 0 1039.5, 0 1046.25, 0 1053, 0 1059.75, 0 1066.5, 0 1073.25, 0 1080, 0



104.625 103.5 102.375 101.25 100.125 99 97.875 96.75 95.625 94.5 93.375 92.25 91.125 90



1046.25 1035 1023.75 1012.5 1001.25 990 978.75 967.5 956.25 945 933.75 922.5 911.25 900