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QUESTON No 5

2‐NODE BEAM ELEMENT

The beam problem given in the question statement was modeled using a single 2‐Node Beam Element. Followings are the details of different parameters used during modeling and analysis in Ansys. 2 3 1 2 10 2

12 23 10000 0.3

Figure‐1 shows the resultant model in Ansys with loads and boundary conditions. The deflection of the beam is shown in Figure‐2. Resultant vertical deflection (uy) and Stress (σx) values calculated at each node are tabulated in Table‐1. 1.00 0.1 10 0.1 10

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Figure 1 2‐Node Beam Element with Loads and Boundary Conditions

Figure 2 Deflection in 2‐Node Beam Element after Application of Load

Table 1

Node No Vertical Displacement (uy) Stress (σx) 1 0 0.1 E32 2 1 0.1 E32

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4‐NODE QUADRILATERAL ELEMENT

The same problem was solved in Ansys using 4‐Node Quadrilateral Element. For meshing purpose, the whole beam was divided into 5 elements of similar size and shape. In Ansys, this element is named as Plane42 Element. Unit thickness was assumed. The boundary conditions were applied. The geometry, meshing and boundary conditions are shown in Figure‐3. After running the solution, the resultant deflection profile is shown in Figure‐4. Resultant vertical deflection (uy) and Stress (σx) values calculated at each node are tabulated in Table‐2. (For Node Numbers Refer Figure 5). In the given theoretical calculations, Vc is the vertical displacement at center of the right end of the beam. In case of 4‐Node Beam, we cannot have a node at this point so the results exactly at this location are not calculated. However, we can find the vertical displacement at this point by averaging out the results at Node No 2 and 7. So 2 1.016 270 2 10 102 10

Figure 3

Beam Analysis with 5 4‐Node Quad Elements

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Figure 4

Deflection Profile after Application of Load

Figure 5

Node Numbering for 4‐Node Quad Elements

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Table 2

Node No Vertical Displacement (uy) Stress (σx) 1 0 270 2 1.0177 30 3 5.92E‐02 240 4 0.21437 180 5 0.44172 120 6 0.71634 60 7 1.0143 ‐30 8 0 ‐270 9 0.71726 ‐60

10 0.44148 ‐120 11 0.21443 ‐180 12 5.92E‐02 ‐240

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4‐NODE QUADRILATERAL ELEMENT (POOR ASPECT RATIO)

The above problem was repeated with same element type but with badly shaped elements. For this, during meshing, the elements were designed to have poor aspect ratio. The resultant meshed beam with deflection profile is shown in Figure‐6. Table‐3 shows the resultant vertical deflection (uy) and Stress (σx) values calculated at each node.

In this case also, we cannot have a node at this point so the results exactly at this location are not calculated. So, we find the vertical displacement at this point by averaging out the results at Node No 11 and 12. So 2 0.99823 278.09 2 19.047 13.3332 16.19

Figure 6 Meshing with Poor Aspect Ratio

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Table 3

Node No Vertical Displacement (uy) Stress (σx) 1 0 278.09 2 1.53E‐02 254.7 3 3.55E‐02 ‐254.31 4 0 ‐275 5 0.11855 214.97 6 6.07E‐02 ‐217.54 7 0.30874 157.25 8 0.16092 ‐172.81 9 0.70453 82.459

10 0.4292 ‐95.674 11 1.0041 39.048 12 0.99236 ‐52.381

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8‐NODE QUADRILATERAL ELEMENT

In Ansys, this element is named as Plane82 Element. The same problem was solved again using 8‐Node Quadrilateral Element. For meshing purpose, the whole beam was divided into 5 elements of similar size and shape. The geometry, meshing and boundary conditions are shown in Figure‐7. After running the solution, the resultant deflection profile is shown in Figure‐8. Resultant vertical deflection (uy) and Stress (σx) values calculated at each node are tabulated in Table‐4. (For Node Numbers Refer Figure‐9). Now finding out the resultant parameters from the analysis: 1.0105 300 10

Figure 7

8‐Node Quadrilateral Element

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Figure 8 Deflection Profile of 8‐Node Quadrilateral Element

Figure 9

Node Numbering for 8‐Node Quad Element

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Table 4

Node No Vertical Displacement (uy) Stress (σx) 1 0 3002 1.0101 5.57E‐123 1.76E‐02 4 6.13E‐02 2405 0.12689 6 0.21375 1807 0.31916 8 0.43948 1209 0.57171

10 0.71299 6011 0.86011 12 1.0101 ‐2.98E‐1213 1.0105 14 0 ‐30015 0.86011 16 0.71299 ‐6017 0.57171 18 0.43948 ‐12019 0.31916 20 0.21375 ‐18021 0.12689 22 6.13E‐02 ‐24023 1.76E‐02 25 5.65E‐02 26 0.21137 27 0.43763 28 0.712

MAXIMUM SHEAR STRESS FROM BEAM THEORY

From beam theory, the expression for maximum shear stress at point ‘c’ is as follows:

10

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COMPARISON OF RESULTS

Element Type

σxx‐B ζxy‐c Vc

Beam Theory Results 300 10 1.031 2‐Node Beam

Result 0.1X1032 0.1X1032 1.00 Erroneous values of stress Deviation from Theory Erroneous 3%

Q‐4 Result 270 10 1.016 Good results Deviation from Theory 10% 0% 1.45%

Q‐4 (Poor A/R)

Result 278.09 16.19 0.99823 Great diversity in results Deviation from Theory 7.3% 62% 3.18%

Q‐8 Result 300 10 1.0105 Extremely good results Deviation from Theory 0% 0% 1.99%

Discussion on Results: A comparison of results obtained by using different types of elements and mesh shapes, reveals following general trends: ‐

(a) Accuracy of the results increases with increasing the number of nodes in each element.

(b) A poorly shaped mesh may give unpredictably wrong results.

With 2‐Node Beam element, we got extremely erroneous results for the stresses. The main reason for that was the use of only one element for the meshing. As FEM approach uses piecewise approximation to find the overall effect, using only one element and that too with very simple assumptions enhances the probability of getting the results far removed from the actual ones. In this case, although we got acceptable deflection results; these results are less accurate than other two types (Q‐4 and Q‐8 elements). The error in results shows a decreasing trend once we shift from Quad‐4 to Quad‐8 element. This is due to the increase in the number of nodes per element. More number of nodes provides more number of calculation points. This makes the distance between two calculation points smaller and thus chances of error reduce. Therefore, the overall best results were obtained from 8‐Node Quadrilateral Element Mesh. Another point to note here is that once the mesh shape was distorted while keeping all other parameters same, the meshing resulted in poorly shaped elements with bad aspect ratios. Using such type of mesh gives highly inaccurate results. The reason for this is that once we use poor aspect ratios for element, we violate the basic assumptions of that element which are based on the peculiar shape of that very element. So the error is incorporated due to violation of shapes of element.