Ravi Patel - 1101066
FEA Assignment
University Universi ty of Birmingham
Introduction The aim of this task was to t o use FEA (Finite Element Analysis) software (ABAQUS) to determine the maximum stress and maximum vertical deflection of a cant ilever beam subject to a load of 20kg the end and compare the effect of mesh density and element size on these values. An additional task was to change the load to 600kg for the largest mesh density range and examine the effects. The material properties for the aluminium beam are given below:
ABAQUS Methodology Firstly the rectangular beam was sketched and t he correct dimensions were applied, 550mm by 25mm and the rectangle was extruded outwards by 2mm creating the 3D beam. Next, the material properties parameters had to be inputted. Since the beam will undergo elastic-plastic behaviour and deformation, certain values needed to be calculated before analysis can be carried out. The Young ’s Modulus and Poisson’ Poisson ’s ratio were already supplied with the data and can be put in straight away, but as the material also behaves plastically beyond the yield stress, it had to be assigned a plastic strain value and the respective stress value. This can easily be calculated analytically: analytically:
Where:
Two sets of stress and plastic strain values must be entered and these were 503 MPa @ 0 value of plastic strain and 634.92 MPa @ 0.0955 plastic strain. As we only have two sets of
Ravi Patel - 1101066
FEA Assignment
University Universi ty of Birmingham
values there will be some inaccuracy as the best guess for the plastic behaviour of the material on a stress-strain curve will be linear, however the values should be sufficient enough to produce a close enough value. Next, the part was defined as a homogenous solid section and the section was assigned to the part. The model was then assembled by creating an Instance and creating a step in addition to the initial step already created by the software. Then a boundary condition was applied to the fixed end of the cantilever beam and it was set to “ENCASTRE” ENCASTRE ” which constrains all degrees of freedom at the fixed end. Then the load was applied as a concentrated load at the opposite end of the beam where the 20kg weight would hang. The load had to be split between the 2 vertices of the beam at that end as you can only apply a load to a node. These loads were 98.1N each in the –Y –Y direction. Now the model is ready to be meshed for analysis. It must be seeded first and the global seeds must be input. This T his value can be increased or decreased until the number of elements in your mesh falls within the range you require. Now the part is meshed and is ready for analysis.
Beam showing Boundary condition at fixed end, Load on other end (left) and meshed beam (right)
Creating a job (deform) and submitting it for a full analysis would give the results for this task. The results were then viewed and also note the CPU time taken from the monitor. Changing the output variables to Von Mises stress and Displacement in the Y direction will give you the results required. But first, let us calculate the analytical/true values for deflection for this cantilever beam with an end load of 20kg so it can be compared to the results obtained from FEA analysis:
Ravi Patel - 1101066
University Universi ty of Birmingham
2.604x 10-9 m4
FEA Assignment
0.05828m = 58.28mm
Where: b = breath (2mm), h= height (25mm), F= force (196.2N), E = Young’s Young ’s Modulus (71.7 x 109), I = Second moment of inertia, = maximum deflection
Results
Element Range Required
No of Elements
A
30-60
31
B
160-300
207
C
600-1200
828
D
2500-6000
4396
~ 9000
7790
E (Table 1)
Max Vertical Deflection
Max Von Mises Stress
CPU Time
A
58.21
528.21
0.2
B
58.13
555.44
0.4
C
58.23
567.38
0.8
D
58.30
527.58
6.0
58.32
535.07
10.6
E (Table 2)
Ravi Patel - 1101066
FEA Assignment
University Universi ty of Birmingham
Graph Plots and Diagrams 570.00 565.00 ) 560.00 a P M555.00 ( s s 550.00 e r t S 545.00 m u540.00 m i x 535.00 a M530.00
(Graph 1)
525.00 520.00 0
2000
4000
6000
8000
10000
Number of Elements (Mesh Density 58.35 ) m58.30 m ( n o i t c 58.25 e l f e D m58.20 u m i x a M58.15
(Graph 2)
58.10 0
2000
4000
6000
8000
10000
Number of Elements (Mesh Density
12.0 10.0 e 8.0 m i t
U P 6.0 C l a t o 4.0 T
2.0
(Graph 3) 0.0 0
2000
4000
6000
8000
Number of Elements (Mesh Density)
10000
Ravi Patel - 1101066
FEA Assignment
University Universi ty of Birmingham
Deformed Beam representation with contours from ABAQUS:
Deformed Beam with 31 elements, showing both Von Mises Stress and Vertical Displacement Displacement
Deformed Beam with 7790 elements, showing both Von Mises Stress and Vertical Displacement Discussion The general trend from the results obtained is that as the number of elements of the part (mesh density) increases, the stress and deflection values begin to roughly converge. For maximum stress (Graph 1), the f luctuation begins to lessen as you increase the number of elements and it is possible to see that especially when using over 6000 elements, the results become much more accurate and convergence is evident. The graphs suggest that they would converge to a value of 535 MPa for maximum stress. With the stress-no of elements curve, there is a large fluctuation between 0-1000 elements and decreases dramatically dramatically after 4000 elements, however however to converge to a true value for maximum stress, further FEA analysis must be carried out to verify the predicted convergence preferably using 8000+ elements.
Ravi Patel - 1101066
FEA Assignment
University Universi ty of Birmingham
The contour plots of the deformed shape show that as you increase mesh density, it becomes more accurate as for the Von Mises stress analysis, it is much easier to locate the areas of high stress and low stress just by looking at the contour plots. The deflection- no of elements curve (Graph 2) seems to hover around the true value of 58.28mm and excluding what seems to an anomaly for the 207 element result, there is definite convergence to a value of 58.32mm for deflection. The 207 element result at first glance seems to be a large anomaly however the FEA output values were never more than 0.26% out, the 207 element result having the largest error from the true value. This could be because deflection can be calculated analytically and so just by k nowing the load and its location, beam dimensions and material data, the program could get a rough value for deflection with simple calculations whereas Von Mises Stress is a lot more difficult to calculate and requires much more complex calculations and analysis. There is definitely a linear relationship between CPU analysis time and the number of elements. The graph (Graph 3) seems to suggest that the analysis time increases by one second for around every 1700 elements you add to your mesh. If a more complex part was being simulated in ABAQUS, the CPU time would dramatically increase and would be much more difficult to use a large number of elements in analysis due to a large memory being occupied and the CPU taking a much longer time to compute the results. When a 600kg load (5886N) was applied in the program in place of the original 20kg load, the program presented an error message suggesting that the strain increment has exceeded fifty times the strain to cause first yield and that the program will not attempt plasticity calculations due to this large strain increment. increment. To verify this, the stress and deflection for the 600kg load will be calculated analytically:
1.75m
This theoretical value of vertical deflection is over 3 times the beam own length and when the plastic strain value was only 0.0955 the beam had to have been undergoing plastic strain and without a doubt would have passed its ultimate tensile stress of 572MPa resulting in failure of the beam which is why ABAQUS was unable to run the calculation due to the strain increment being too high i.e. over 50 times the strain at the yield strength, which was 0.0125 so once it reached a strain value of more than 0.625, the program would not have completed the job analysis. The beam has most definitely surpassed its rupture strength.
Ravi Patel - 1101066
FEA Assignment
University Universi ty of Birmingham
Conclusion Overall it is fair to say that as you increase the number of elements (mesh density), your results become much more accurate and tend to converge to the true value. When the element size is small, the FEA software cannot perform precise complex calculations as these can only be performed at the nodes. Therefore by increasing the number of elements (decreasing their size), the software will produce more accurate results due t o the larger number of finite elements. However, this will only be true up till convergence occurs because after this, increasing the number of elements will have little or no effect on the precision of results but instead be counter-productive as it will just be increasing CPU time and memory usage. For other complex part analysis, a reasonable compromise must be made between the two.
