Finite Element Method for Solids and Structures 4 Paul Dryburgh S Assignment: Modelling of shallow and deep beams Due: 15/10/2014

Transcription

1 Finite Element Method for Solids and Structures 4 Paul Dryburgh S Assignment: Modelling of shallow and deep beams Due: 15/10/2014 1

2 2 Table of Contents Theory... 3 Question Question 2 Theoretical Deflection... 4 Beam Elements... 6 Question 3 Input File... 6 Input file... 6 Question 4 Euler Bernoulli Element... 7 Results... 7 Comparison... 8 Question 5 Timoshenko Elements Results Question 6 Element Suitability Plane Elements Question 7 Exaggerated Deformed Shape Input file Deformed Shapes Question 8 Bending Stress Distribution Question 9 Shear Stress Distribution Question 10 Deformed Shape and Displacements Question 11 Element Comparison Question 12 Symmetry References... 42

3 3 Theory Figure 1 Beam Image Figure 1 shows the visual representation of the simply supported beam for analysis in this assignment. The depth of the beam shown is either 0.2 for the shallow case or 1m for the deep case, with a constant width of 0.2m. Figure 2 Deflected beam shape [1] Figure 2 shows the deflected shape of a simply supported beam under point load at centre, the uniform load of the assignment beam can be resolved into a single point load at centre, thus giving the same deflected shape. This figure is used to show the expected deformation of the beam, and will be referred back to throughout this report. Question 1 Shown below are the shear force and bending moment diagrams for both the shallow and deep beam.

4 4 Figure 3 Shallow beam shear force diagram Figure 4 Deep beam shear force diagram Figure 5 Shallow beam bending moment diagram Figure 6 Deep beam bending moment diagram The maximum absolute shear force occurs at the location of the supports, 0 and 5m along the length of the beam. As shown by Figure 3 and Figure 4, the maximum shear force for the shallow case is 25kN and 250kN for the deep case. The maximum bending moment occurs at the mid-span, 2.5m, in both beam cases, with a maximum bending moment of knm for the shallow beam and knm for the deep beam as shown by Figure 5 and Figure 6 respectively. Question 2 Theoretical Deflection Basic beam deflection for a determinate simply supported beam can be found using Equation 1. δ = 5ωL4 384EI Equation 1 Basic beam theory equation for the maximum deflection at the centre point of a simply supported beam

5 5 This is generally considered sufficient for shallow beams however for deep beams in order to more accurately calculate the displacement, the displacement caused by shear must also be included in the calculation. This can be found using Equation 2. δ = 5ωL4 384EI (1 + 48f sei 5GAL 2 ) Equation 2 Maximum deflection including delection caused by shear The values of Young s modulus and Poisson s ratio are taken from literature [2] and are shown in Table 1. Based upon these values the shear modulus, G, can be calculated using Equation 3. Young s modulus, E (GPa) 200 Poisson s ratio, υ 0.28 Table 1 Mechanical properties of the steel used for simulation G = E 2(1 + υ) Equation 3 Shear modlus equation for poisson s ratio and Young s modulus Shallow beam I (m 4 ) 1.3E Ѡ (kn/m) L (m) 5 5 f s A (m 2 ) 0.2 Deep beam G (GPa) Table 2 Properties of the shallow and deep beam respectively Using the predicted deflections of the shallow beam and deep beam are found to be 3.059mm and mm respectively. Using the predicted deflection of the deep beam when shear displacements are considered is found to be mm. This represents a difference of 9.83% when shear is considered.

6 6 Beam Elements Question 3 Input File Input file *HEADING SIMPLY SUPPORTED SLENDER BEAM *NODE 1,0,0 2,1.25,0 3,2.5,0 4,3.75 5,5,0 *ELEMENT, TYPE=B21 1,1,3 *ELGEN, ELSET =EALL 1,2,2,1,1,1,1 *MATERIAL,NAME=STEEL *ELASTIC 200.E9,0.28 *BEAM SECTION, ELSET=EALL,MATERIAL=STEEL, SECTION=RECT 0.2,0.2 *STEP,PERTURBATION *STATIC *BOUNDARY 1,1,2 5,2 *DLOAD EALL,PY,-10E3 *EL PRINT COORD S SF E *EL PRINT, POSITION=INTEGRATION POINTS, FREQ=1 SF S *EL PRINT, POSITION=NODES, FREQ=1 SF S *EL PRINT, POSITION=AVERAGED AT NODES, FREQ=1 SF S *NODE PRINT U RF *END STEP

7 7 Figure 7 Meshed beam Figure 7 shows a 2 Element B23 beam meshed and simulated using Abaqus. Following the computation the results can be visualised using Abaqus CAE. Question 4 Euler Bernoulli Element Results Shallow Beam Theory B23 Euler-Bernoulli 2 Element 10 Element Max Deflection (mm) Max Bending Moment (knm) Rotation at end points E E-03 (radians) Bending stress at midpoints (MPa) Table 3 Results of simulation with B23 Euler-Bernoulli element for shallow beam case Deep Beam Theory B23 Euler Bernoulli 2 Element 10 Element Max Deflection (mm) Max Bending Moment (knm) Rotation at end point E E E-04 (radians) Bending stress at midpoint (MPa) Table 4 Results of simulation with B23 Euler-Bernoulli element for deep beam case

8 8 Comparison Figure 8 Deflection B23 2 Element Figure 9 Deflection B23 10 Element As shown by Figure 8, the 2 element simulation does not accurately model the curved shape of the deflected beam. This is caused by there only being 1 node which is free to displace as the end nodes are fixed. The elements do not display a perfectly linear shape as the B23 element is a cubic element with 3 integration points which allow for some curvature. Theoretically increasing the number of elements should result in the beam being able to curve to a greater extent and more accurately represent the true deflected shape. This is confirmed by increasing the number of elements to 10, as shown in Figure 9, the deformed shape displays a more curvature which better reflects reality.

9 Bending moment (knm) Bending moment (knm) 9 Shallow Theoretical B23 2 Elements B23 10 Elements Distance along beam (m) Figure 10 Bending moment diagram along shallow beam Deep Distance along beam (m) Theortetical B23 2 Elements B23 10 Elements Figure 11 Bending Moment along Deep beam As shown by Figure 10 the 2 element simulation over predicts the bending moment along the length of the shallow beam, the 10 element simulation however produces a result that accurately matches the theoretical result. The 2 element result does not start or end with a bending moment of 0kNm at the supports as would be expected with theory. This starting value roughly represents the overshoot of the maximum value, thus removing this value from all of the results in the data set, would make the 2 element case fall in line with theory. The bending moment not having a value of 0kNm at the supports is likely caused by bending stress being interpolated outwards from the Gauss points to the nodes. Were a greater number of elements used

10 10 the bending moment at the support would converge to zero as the interpolation would be carried out with a greater degree of accuracy. These findings are repeated for the bending moment along the deep beam, as shown in Figure 11. The 10 element case again shows little discrepancy with the theoretical results, whilst the 2 element case over predicts the value of the bending moment along the beam. It can be seen that the values of rotation and deflection for the shallow beam agree fairly well, but the values for bending show some variance when compared to theory. A possible reason for this is that all the values for the assignment that have been recorded from Abaqus have been taken from nodal values. Force, rotation and displacement are all calculated at the nodes themselves, however stresses and strains are calculated at the integration points. To read the values of the stress at the node the calculated value at the integration point, found via a suitable method of integration e.g. Gauss-Legendre, then needs to be extrapolated outwards using interpolation, via shape functions, to find go find the value at the node [3]. Most nodes are also shared by more than one element, all of the nodes except the 2 end nodes when using the beam elements. The two elements adjacent to the node will have differing values of stress, thus in order to produce smooth contour plots without discontinuities the stress is averaged at the node, however a purely linear interpolation is not an accurate reflection of the stress relationship from node to node [4]. Whilst it has been discussed that the values of the stress and strain at the nodes are inherently inaccurate, and the values calculated at the interpolation points are much closer to the true value, it is generally not possible to record the value of the interpolation points stress as the maximum stress tends to occur on the outer edges. It would be possible to use the stress value at the integration point only for the beam elements as the integration points lie along the same line as the nodes. Were the values of stress taken from the integration points, it is expected that they would have followed the theoretical results with a smaller error. By increasing the resolution of the study (using more elements), the distance between the integration points and the nodes will decrease thus there is less error associated with the interpolation, as such the value at the node converges towards the true value. It is possible to compare the average stress values at the node against the elemental stress values that that node connects to, should there be a wide disparity in results it is clear the mesh needs re-fined.

11 11 Figure 12 Maximum bending stress along shallow beam Figure 13 Maximum Bending stress along deep beam Values of bending stress are taken at the outer fibre, top and bottom nodes, in order to find the maximum value of bending stress. The maximum bending stress along the length of the beam should follow a similar pattern to the bending moment along the beam, as according to Equation 4, the bending stress is only dependant on the moment variation when the beam is of constant cross-section and mechanical properties. That is where a model over predicts the bending moment along the beam, it would be expected that it would also over predict the value

12 12 of the bending stress. Figure 12 shows that as expected the 2 element model also over predicts the maximum bending stress along the length of the beam, whilst the 10 element case has converged well with the theoretical results. This results show that the physical properties of the beam have not had an effect on causing errors in the results of the simulation, instead the inaccuracies can be linked to errors interpolating stress out from the integration points. Similar findings are shown for the deep beam case in Figure 13. σ = My I xx Equation 4 Bending stress It is possible to calculate the theoretical rotation of the beam at any point along the length based upon basic static equilibrium using discontinuity functions, staring with Equation 5. Equation 5 Moment balance for beam EI d2 y dx 2 = M Where the moment with respect to x for the shallow beam is equal to M = 25,000x 10,000 x 2 2 The integral of the negative moment is equal to the rotation of the beam, as shown by M = EI d2 y dy = EI dx2 dx + a Equation 6 Equating the moment equation to the slope of the beam Where dy/dx is equal to the rotation, θ. This can then be integrated again and boundary conditions applied in order to calculate the constants of integration. From this the rotation at any point along the beams length can be found.

13 13 Figure 14 Shallow Beam Rotation Figure 15 Deep Beam Rotation Figure 14 and Figure 15 show the rotation of nodes along the beams length in comparison to the theoretical rotation. For both the deep and shallow beam the rotation of the nodes fits very well with the theoretical values.

14 14 Shear force diagrams cannot be obtained for Euler-Bernoulli beam elements as this method disregards shear force in the calculation, therefore comparison cannot be made between the theoretical shear force diagram and the Abaqus generated graph. It is clear from the figures and tables presented above that the 2 beam element analysis provides insufficient resolution to provide a suitable answer, in comparison the 10 element model provides values much closer to those predicted by theory. There is a notably large jump in resolution between 2 and 10 elements but this increase in accuracy would not continue at the same rate for an increased resolution. For example were 1000 beam elements used instead the differences between that and the 10 beam element model would be small as the 10 beam model is already well converged. It is undesirable to use a resolution finer than required as this causes unnecessary time to be put into both the modelling by the user and the time taken for the computer to run the simulation. As can be seen from Table 3 and Table 4, the B23 element under predicts the deflection of the beam by a much larger amount for the deep beam than for the shallow beam, which is relatively well converged on the solution. This inaccuracy comes from Euler-Bernoulli theory, when the ratio of beam length to depth ratio is less than 20 [5], E-B beam theory inaccurately predicts the beam theory as it does not take into account the deflection caused by transverse shear strain. The length to depth ratio of the deep beam is 5 much lower than 20, whereas the shallow beam has a ratio of 25. This theory is backed up by inspection of the theoretical deep beam deflection when shear is not considered. This result matches much better with results from Abaqus. The inaccuracies in deflection when shear is not considered is given by H [6]. L 2

15 15 Question 5 Timoshenko Elements Results Shallow Max Deflection (mm) Max Bending Moment (knm) Max Shear Force (kn) Rotation at end point (radians) Bending stress at mid-point (MPa) Theory B21 Linear Timoshenko B22 Quadratic Timoshenko 2 Element 10 Element 2 Element 10 Element E E E E Table 5 Results of Timoshenko beam elements for shallow beam case Deep Theory B21 Linear Timoshenko B22 Quadratic Timoshenko 2 Element 10 Element 2 Element 10 Element Max Deflection (mm) Max Bending Moment (knm) Max Shear Force (kn) Rotation at end point (radians) Bending stress at mid-point (MPa) E E E E E Table 6 Results of Timoshenko beam elements for deep beam case

16 16 Figure 16 Deflection B21 2 Elements Figure 17 Deflection B21 10 Elements Figure 18 Deflection B22 2 Elements Figure 19 Deflection B22 10 Elements As Figure 16 shows the B21 2 element model does not deform with a shape that matches the theoretical shape that would be exhibited in the deflection of a real life simply supported beam. Rather than forming a curve the model forms 2 straight lines which form a point at the centre. This incorrect deformation shape is caused by the elements being linear in nature, meaning the elements cannot bend but must remain straight. As there

17 17 are only 2 elements used for this simulation this allows for only 1 node which can move as 2 are fixed at the supports, hence the deformed shape. When the model is increased to use 10 elements, as shown in Figure 17, the shape produce matches much better with theory. The improvement between 1 and 10 elements is caused by increasing the number elements allows for a greater number of nodes that can deflect thus the shape of the deformation can more accurately resemble a curve. As the number of elements continues to increase the closer the simulation will be to convergence. Figure 18 shows that the B22 2 element case produces a deformed shape that begins to resemble the true curvature. B22 produces the most accurate 2 element model, this happens as the B22 element contains 3 nodes per element which allows for an extra node which can deflect in every element, this allows for the beam elements to more accurately deform in-line with the theory calculations. Figure 19 shows that the 10 element 21 case represents the expected deformed shape with a small divergence. Figure 20 Bending moment diagram Shallow

18 18 Figure 21 Bending moment diagram Deep Figure 20 shows the bending moment calculated along the length of the shallow beam. The calculated theoretical bending moment follows a 2 nd order polynomial shape. The B21 2 node case does not fit this pattern and shows a bending moment constant along its length, however the 10 element model has a more accurate distribution and fits the theoretical results reasonably well. The B22 2 element case follows the correct pattern but consistently over predicts the bending moment along the beams length. Using 10 elements for the B22 model produces a range of bending moments that fit very well with theory, as can be seen from Figure 20. Similar findings are shown for the deep beam case in Figure 21. The reason the B21 2 element case displays a constant average is due to the number of integration points in the model. B21 elements have only 1 integration point, where the bending stress is calculated at. As there will be only 2 integration points to calculate the bending moment at in the model and as the bending moment is symmetrical about the mid-point both of these integration points will yield the same value. As such when the values are averaged out to the nodes it is taking the average of the same value, hence Abaqus interprets this as being a constant bending moment along the length of the beam. Instead the actual bending moment will be a curve similar to the others in Figure 21, which passes through the points (1.25, 155) and (3.75, 155), where the x-values are taken from the location of the integration points. Whilst the 10 element case shows values much closer to theory this is caused by the increased number of elements allowing for better accommodation of the beam bending. The overshoot from the B22 2 element model can again be attributed to errors in interpolating outwards from gauss points, the more elements the less distance the interpolation must occur over hence the values at the nodes become more accurate.

19 19 Figure 22 Maximum bending stress along shallow beam Figure 23 Maximum bending stress along deep beam As would be expected the bending stress distribution follows the pattern shown by the bending moment diagram. This again confirms at that the inaccuracies results not from the stress calculations at the integration points but the interpolation outwards from these points to the nodes where the values are recorded at. As can be seen from Figure 22 and Figure 23 the B22 10 element model produces the results that most accurately follow theory.

20 20 Figure 24 Shear force diagram Shallow Figure 25 Shear force diagram Deep

21 21 Figure 24 shows the shear force along the length of the shallow beam. It can be seen that even with 2 elements the B22 element simulation is already well converged and produces results comparable with theory. Comparably the results produced by the 21 beam elements do not fit well with theory. The 2 Element model is particularly badly matched with theory, whilst the 10 element model shows some improvement. It is likely that if a much larger number of elements were used in a B21 simulation the results would converge to the true result. It is likely what is happening in the B21 element is a phenomena known as shear locking, which can occur in all first order linear elements [7]. In the real world a beam under bending experiences a shape change in order to take on a curved shape. If the dotted lines on the example beam shown in Figure 26, are taken as an individual element, it is clear that under real bending the elements remain perpendicular to each other and the angle at each element corner will remain 90 degrees, as stated by classic beam theory [8]. In first order elements however they cannot bend to take on a curved shape, instead they will take on the shape of a quadrilateral shown in Figure 27. This no longer gives elements with right-angle corners. This leads to an artificial and incorrect shear stress being applied to the element. This results in the beam becoming overly stiff, or locked, thus the results of deflection are under-predicted. This causes the computation to produce fallacious results. This would also explain why increasing the number of elements produces a result closer to basic theory, as the misalignment of the element will decrease as the number of elements used increase. Shear locking does not affect second or higher order elements, hence B22 elements are unaffected by this phenomena. There are many other locking effects that cause fallacious results, for example Poisson s ratio locking, which occurs when υ approaches 0.5, causing poor convergence of displacements [9]. This does not effect this particular problem set at Poisson s ratio is not close to 0.28 but it is an example of the other problems that can arise in FEA that the user must be wary of. Figure 26 Ideal element deformation under bending [10]

22 22 Figure 27 First order element deformation under bending, known as shear-locking [10] This shear locking phenomena is the reason for the shear force in the beam being under-predicted. The beam acts too stiff and thus the shear force acting on it is incorrectly calculated. Figure 28 Shallow Rotation

23 23 Figure 29 Deep Rotation It is immediately clear that the 2 element results from the B21 beam do produce reasonable results compared to the theoretical results. The difference in deflection between theory and the B21 element improves dramatically when 10 elements are used. This suggest that the simulation is not resolved at 2 elements and a finer resolution is required to converge on the solution, this again is caused by shear locking. Timoshenko beam theory states that rotation of the cross-section is the rotation of the neutral axis minus shear strain, given by Equation 7 Timoshenko rotation θ = μ γ As the elements used are fixed at the end points there is only 1 node and 2 integration points which are free to deflect in the model. As the beam element is linear the deflection from the end support outwards must be at a constant rate. The problem is not from Timoshenko beam theory itself as the deflection of a Timoshenko beam element can be found using Equation 8. Equation 8 Timoshenko rotation equation dθ dx = M EI This gives the deflection per metre as mm, meaning at 2.5m, and the half span length where deflection would be at a maximum, the theoretical Timoshenko deflection is mm, which closely matches the result from simple beam theory.

24 24 Question 6 Element Suitability Max Deflection (mm) Max Bending Moment (knm) Rotation at end point (radians) Bending stress at mid-point (MPa) Theory B23 Euler Bernoulli B22 Quadratic Timoshenko 2 Element 10 Element 2 Element 10 Element E E E E Table 7 B23 and B22 beam element result comparison Euler-Bernoulli beam theory assumes that plane sections always remain plane and neglects the effects of shear force. In contrast Timoshenko beam theory does not assume plane sections remain plane and takes into account the effects of shear loading and rotational inertia. Based upon the results shown in Question 5, produced using the beam elements it is clear that the Timoshenko element B21 is not the optimal choice for beam modelling as it requires a large number of nodes to approach convergence. This leaves B22 and B23 elements. Whilst it is clear that B22 elements out-perform B23 for deep beams, the question asks for the most suitable beam element for modelling slender beams, those with a length to depth ratio greater than 20. For the slender case the B22 elements were 0.011mm different from theory, in comparison B23 beam elements gave a difference of 0.001mm. The B22 element proved to be more accurate for both maximum bending moment and bending stress however, whilst the results of the rotation study were exactly the same. As the differences in deflection are small the variance in bending moments are large. Further to this the B22 element allows shear force diagrams to be generated whilst the B23 element does not. Based upon all of this the B22 beam element is the optimal choice for modelling shallow beams. This greater accuracy for bending calculations can be attributed to there being 3 nodes from which the stresses are interpolated to rather than just 2, and 1 of these is between the integration points thus should have a high degree of accuracy. The more accurate deflection value from the B23 element can be attributed to the B23 being of a higher order polynomial form, order 3 vs order 2 for B22, thus it can better model the curvature of the real beam.

25 25 Further to the points discussed, whilst computation time does not have an effect on this particular project as the simulations modelled are relatively small and converge with a small number of elements, for real life projects on larger scales computation time must be considered. Upon inspection of the output file for both the B22 and B23 beam elements it can be seen that the computation of the B22 model was performed faster than the B23 model. This further confirms B22 as the optimal beam element model for simulating deep beams similar to that in this project.

26 26 Plane Elements Question 7 Exaggerated Deformed Shape Input file *HEADING SIMPLY SUPPORTED SLENDER BEAM *NODE 1,0,0 21,5,0 801,0,1 821,5,1 *NGEN, NSET=PINNED 1,801,40 *NGEN, NSET=ROLLER 21,821,40 *NFILL PINNED,ROLLER, 20,1 *ELEMENT, TYPE=CPS4I 1,1,2,42,41 *ELGEN, ELSET=EALL 1,20,1,1,20,40,20 *MATERIAL,NAME=STEEL *ELASTIC 200.E9,0.28 *SOLID SECTION, MATERIAL=STEEL, ELSET=EALL 0.2 *STEP,PERTURBATION *STATIC *BOUNDARY 1,1,2 21,2 *DLOAD 200,BY,-1E7 199,BY,-1E7 198,BY,-1E7 197,BY,-1E7 196,BY,-1E7 195,BY,-1E7 194,BY,-1E7 193,BY,-1E7 192,BY,-1E7 191,BY,-1E7 190,BY,-1E7 189,BY,-1E7 188,BY,-1E7 187,BY,-1E7 186,BY,-1E7 185,BY,-1E7

27 27 184,BY,-1E7 183,BY,-1E7 182,BY,-1E7 181,BY,-1E7 *EL PRINT COORD S E *EL PRINT,POSITION =AVERAGED AT NODES S *NODE PRINT U RF *END STEP Deformed Shapes Figure 30 CPS4 Figure 31 CPS4I Figure 32 CPS4R Figure 33 CPS8 The above Figures show the exaggerated deflections for the 2D plane element models.

28 28 Figure 34 Hour glassing Figure 35 Hour glassing Figure 36 Example of hour glassing showing zero stress at centre of element [10] As can be seen from the CPS4R plane element type produces an hourglass style deformation pattern. This is caused by the fact that the CPS4R has a reduced number of integration points, 1 integration point (at the centre) compared to the 4 of CPS4. Reduced integration point elements are used in modelling when computation time is more crucial than an exact answer. The lower number of Gauss points means the computation runs much faster but accuracy is sacrificed. This can be used in areas where only an estimate of stress is needed or where the values are unimportant to the overall design. This single integration point makes it possible for the element to deform but the centre point and angle of it to remain unchanged, as shown in Figure 36. This results in the strain calculated at each integration point to be zero, this is known as a zero energy mode as no strain energy is produced. As the element has no resistance to zero strain energy deformation it cannot resist this, causing unrealistic and strange deformation shapes to form. When a coarse mesh is used this phenomena will propagate through the simulation [11]. In order to correct this a finer mesh must be used, this is achieved by increasing the number of elements in the mesh.

29 29 However this is not always desirable or possible as the resolution many need to be greatly increased to eliminate the presence of hour glassing, causing a large increase in the computation time. As such many FEA packages offer tools to control hour glassing. These methods include, artificial viscosity, artificial stiffness, complete element integration using Taylor series expansion and boundary smoothing [11]. Hour glassing would not occur in CPS8R even though it is reduced integration it still has 4 integration points present in the element thus the zero strain energy mode deformation is not possible. Question 8 Bending Stress Distribution The bending stress at any point in the cross section in the beam at a given length can be found using Equation 9. σ = My I xx Equation 9 Bending stress Where M is the applied moment, y the distance from the neutral axis of the point of interest and I xx the second moment of area about the x-axis of the cross-section of the beam. The second moment of area has already been calculated in Question 1 and found to be m 4. The value of y will alter from 0, the location of the maximum, outwards to the extreme fibre of the cross-section at y=0.5. As the cross-section is symmetrical about the neutral axis it is expected that the absolute values will also be symmetrical with one half positive and one negative. The applied bending moment can be found via the bending moment diagram, at a distance of 1m in from the end support the bending moment is equal to 200kNm. Using Equation 9, the maximum bending stress was found to be ±6MPa, with a zero point at neutral axis as would be expected. The bending stress distribution is shown by Figure 38 to Figure 41. Theory (MPa) CPS4 (MPa) CPS4I (MPa) CPS4R (MPa) CPS8 (MPa) Table 8 Maximum bending stress in plane elements

30 30 Figure 37 Theoretical bending stress distribution Figure 38 CPS4 Figure 39 CPS4I Figure 40 CPS4R Figure 41 CPS8

31 31 As can be seen from Figure 38 to Figure 41 the values for bending stress distribution over the beam crosssection agrees well with the theoretical results. The CPS8 and CPS4I fit the theoretical data set particularly well, as would be expected as these elements have improved qualities for replicating bending moments. It is likely that the pattern of the bending stress distorting towards the outer fibres of the beam in the CP4R is caused by the presence of hour glassing. Errors will again rise from the value of stress being read from the nodes, whilst the actual calculation is made at the integration points inside the element. The improved accuracy of CPS4I in comparison to CPS4 is due to CPS4I element making use of a feature known as incompatible modes. This is designed as a work-around for shear locking in the Abaqus FEA package. In planar elements the 4 node quadrilateral cannot accurately model the strain distribution caused by bending. The incompatibles modes prevents shear locking by using more precise interpolation functions which better model bending, the name incompatibles modes comes from the fact the strain does not have to be compatible with the interpolation functions used for displacement. Theoretically this prevents the model becoming overly stiff and should produce a larger value of deflection and bending, this is true for this study and can be seen from Table 8 and Table 10. These values should more accurate results for the study. Question 9 Shear Stress Distribution The shear stress, τ, at any distance from the neutral axis in a beam cross section can be found using the general formula shown by Equation 10. Equation 10 Shear stress τ = VAy I xx b Where V is the applied shear force, A the cross-sectional area of the beam, y the distance from the neutral axis at the location of interest, b the breadth of the cross-sectional area and I xx the second moment of area about the x-axis. This equation is for a general beam cross-section however the formula can be simplified for a specific rectangular beam. This simplified form is show by Equation 11. τ = 6V bd 3 [d2 4 y2 ] Equation 11 Simplified shear stress formula for a rectangular cross section The only quantity not given in the brief for this equation is the applied shear force at 1m from the end support. This can be found by using the shear force diagram. The equation of the shear force between 0 and 2.5m is found to be

32 32 y = 100x For 1m from the right hand side the shear force is found to be 150kN. Using the maximum shear stress, at the neutral axis, is found to be 1.125MPa, with an average of 0.75MPa. The shear stress distribution is shown by Figure 42. Theory CPS4 CPS4I CPS4R CPS8 Maximum Shear Stress (MPa) Average Shear Stress (MPa) Max Shear Stress calculated from 1.5 x Average Table 9 Shear stress in plane elements As Table 9 shows there discrepancies between the maximum shear stress recorded from Abaqus and the calculated values based on the average value of shear stress. There are several contributing factors associated with these discrepancies such as, the shear stress distribution does follow a perfect parabolic profile as it generated by the theoretical data set, this is caused by the fixities problem as previously discussed causing the neutral axis to shirt and errors associated with interpolation out from Gauss points when reading stress values at the nodes. Furthermore the method used to calculate the average shear stress can be seen as very rough. The total value of shear stress was summed and divided by the number of nodes used to give the average value. This is a rough approximation, but a more accurate method would require the function of the curve to be known and that would be difficult given the distributions do not follow the parabolic profile exactly as stated.

33 33 Figure 42 Theoretical shear stress distribution Figure 43 CPS4 Figure 44 CPS4I

34 34 Figure 45 CPS4R Figure 46 CPS8 The reason the values of actual maximum shear stress and the shear stress calculated from the average stress does not match is caused by the stress distribution across the cross-section does not fit the expected parabolic profile exactly. As would be expected the theoretical data shows the turning point of the curve, where the stress is at a maximum falls on the neutral axis. However the data from the four 2D elements shows the point of maximum stress not on the neutral axis as defined by the graph. From looking at the formula and the set-up of the model the only probable explanation for this is not that the maximum shear stress occurs away from the neutral axis but that the neutral axis itself has shifted. From inspection of the model the likely cause of this in comparison to theoretical model from beam theory is the location of the fixities. Beam theory assumes that the neutral axis is in line with the fixities, where as in the model used the fixities are placed at the bottom of the beam. Should the fixities be moved to the midpoint of the beam in line with the neutral axis the results should match better with beam theory. In order to confirm this theory the simulation is repeated for the CPS4I case with the fixities in line with the neutral axis. Figure 47 shows the adjusted shear stress distribution when the fixities are set in line with the neutral axis. As the graph shows, the distribution now matches theory very well, actutley predicting the value and location of maximum shear stress. This confirms that the results from the previous charts are likely caused by the fixities issue.

35 35 Figure 47 Shear stress distribution for CPS4I with fixities set in line with Neutral Axis The shear stress at the outer nodes is still not at zero as would be expected. This may be an inaccuracy as caused by the interpolation outwards from the Gauss points, however another discrepancy between beam theory and the model is the location of the loading. Beam theory assumes that the loading occurs on the neutral axis, where as the model used sets the load on the top face of the beam. Again this can be adjusted and the loading set to the midpoint for the vertical to replicate the beam theory case. Doing so however produces results with little variance to those shown in Figure 48, this suggests that the location of the loading with respect to the neutral axis does not have a large effect on the results and the results not returning to zero at the fixities is caused by interpolation errors. It is useful to ensure that the simulation can be modelled to match the theoretical values as this ensures that the simulation is running correctly and there are no errors and bugs that could be over-looked if there was no a data line to compare the FEA results with. However it can be argued that beam theory does not accurately model the real life scenario as it is unlikely the fixities will be placed exactly in line with the neutral axis of the beam. The likelihood is that the real scenario would fall somewhere between the bounds set by the FEA modelling and the theoretical values.

36 36 Question 10 Deformed Shape and Displacements Theory CPS4 CPS4I CPS4R CPS8 Max Displacement (mm) Table 10 Deflection of plane elements Similar to the rotation calculations carried out in section 5 the theoretical deflections of the beam can be calculated using discontinuity functions. EI d2 y 100 = M = 250x dx2 2 x2 θ = EI dy dx = x x3 + a δ = EIy = x x4 + ax + b Setting the boundary conditions as y(0) =0, y(5) =0 the constants of integration can be found to give the equation for the deflection along the beams length. This does not include the deflection caused by the shear force however this can be included by multiplying the deflection by the shear deflection factor from Equation 2. Both sets of deflection values are plotted along with the deflection values from each 2D element case for comparison in.

37 37 Figure 48 Deflected shape comparison As can be seen from Figure 48, the CPS4R element has a deflected shape which resembles a saw tooth and does not accurately match theoretical deflected shape of the beam. This shape is caused by the hour glassing effect discussed in section 7. All of the other plane elements produce a shape similar to that of theory, with a maximum deflection at the centre and zero deflection at the two ends. The shape of the deflection resembles a power 2 polynomial. It can be seen from Figure 48 that all of the models over predict the values of deflection along the length of the beam. Whilst as stated the results of the CPS4R can be disregarded they are incorrect, the other 3 models are assumed to be valid. The probable reason that they over predict the results in comparison to beam theory is related to the location of the fixity. In beam theory the fixity is defined just based upon an x coordinate, a 1 dimensional system, similar to the beam elements used in the first section. However with the two dimensional planar element the y-coordinate of the fixity must also be defined. Beam theory assumes the fixity is in line with the neutral axis, and this is replicated by the beam elements, hence the small discrepancy between results. However for this study using the 2D planar elements the fixities were set at the bottom node on either side. As this is not in line with the neutral axis the results will not match beam theory exactly. Were the fixities

38 38 moved to the midpoint on the vertical on either side the results would produce deflections more reflective of theory. On investigation of the model it can be said that the theoretical line plotted on the graph is not a true reflection of the model used in the simulation, thus they should not be compared. There is the issue with fixities as discussed previously, further to the data plotted from the simulations show the deflected shape, thus the points of deflection are taken from along the bottom axis. However the theoretical deflections values are taken from beam theory thus the deflections refer to the neutral axis. In order to confirm the model is set correctly and compare to theory the simulations are repeated with the fixities in line with the neutral axis and the deflections read along the N.A. The results of this new set of studies plotted in Figure 49. Figure 49 Deflections along the Neutral axis As the above figure shows the results compare well with the theoretical deflection, when shear is included. As would be expected the CPS8 and CPS4I elements perform particularly well showing little variance between the expected values. This is caused by CPS8 having more nodes to more accurately model the deflection and the curvature shape. Whilst the CPS4I element uses complex interpolation functions to calculate the bending more. CPS8R still shows signs of hour glassing however the results are not as dramatic, it can be seen then that the effects of hour glassing increase outwards from the neutral axis towards the extreme fibres.

39 39 Question 11 Element Comparison Theory (mm) B22 (mm) B23 (mm) CPS4 (mm) CPS4I (mm) CPS4R (mm) CPS8 (mm) 2 elements 10 elements 2 elements 10 elements Table 11 Deflection results for both plane and beam elements Based upon the values of maximum deflection for the deep beam model, as shown in Table 11, the model which produces the best accurate deflection result is the B22 Timoshenko beam element. Whilst the 10 Element model produces more accurate values for bending in the beam, considering only deflection the 2 element model can be considered sufficient. In comparison to the results produced by the B22 beam element the 2D planar elements produce values that match less well with the predicted results based on theory. Furthermore the 2D planar models are more complex to input and take a longer time to compute. Based upon this it is clear the most suitable element type for modelling a deep beam similar to that modelled in this report, would be the B22 Timoshenko beam element. Similar to the issues encountered in Question 9 and Question 10, the values of deflection of beam theory and the beam elements refer to the deflections in the neutral axis, with the fixities in line with the neutral axis, thus they cannot be properly compared with the values in Table 11 as these represent the maximum deflection taken from the extreme fibre on the bottom. As simulations have been ran for the various models for Question 9 and 10, these neutral axis deflections can be used to compare the elements types, as shown in Table 12. Theory B22 (mm) B23 (mm) CPS4 CPS4I CPS4R CPS8 (mm) (mm) (mm) (mm) (mm) elements elements elements elements Table 12 Deflection results for both plane and beam elements taken at neutral axis The results shown above are much more comparable to the results of the beam elements and the theoretical data set. Whilst the 2D sections now out-perform the B23 element, this is caused by the Euler-Bernoulli element not including deflection caused by shear whilst the 2D elements do, they still prove to be less accurate than the B22 element.

40 40 Another factor to consider when deciding the most efficient element to use for modelling is the computation time. The simpler beam elements take a significantly less time compared to the 2D elements. For example B22 10 element case takes 0.27 seconds to compute, whereas the CPS4I takes 0.51 seconds to compute. Considering more than just the accuracy of the deflections, the definition of a suitable element mesh is that it is robust, simple, locking-free response, good convergence with a relatively course mesh, computationally efficient and stable [12]. The B22 element satisfies all of these conditions in comparison to the B23 element and the 2D planar Elements. Looking at the just the values of deflection and at all of the factors combined in both cases it is clear that the optimal element of choice is the B22 Timoshenko element, when modelling basic simply supported deep beams. In addition it can be said that it would always be preferable to use the simplest possible element type for the given problem. The like hood is that as the element types increase in complexity, i.e. from 1D to 2D or 2D to 3D, the potentially issues causing incorrect results also increase thus the simpler the element the more accurate the results relatively. Question 12 Symmetry Symmetry can be used to simplify the analysis of the beam by cutting the beam in half directly through the middle and only considering one of the ends. This is possible as the reaction forces and bending moment on the beam are exactly symmetric about the mid-point. First the beam must be cut in halve reducing length from 5m to 2.5m. The right hand end of the beam must be fitted with a vertical roller support. This support prevents the beam from moving horizontally but allows vertical deflection. This leaves the system with 3-degrees of freedom ensuring the system is still determinate. As the beam has halved in size the number of beam elements required to reach solution convergence is also halved. The three degrees of freedom as indicated by Figure 51, are rotation at the left hand-end support, rotation at the right hand end support and vertical deflection at the right hand end support. For the beam elements this would mean that either 1 or 5 elements would need to be used compared to either 2 or 10 beam elements. For the planer 2D meshing both the depth and the length of the beam must be considered. The depth of the beam remains unchanged thus to maintain a resolution vertically the same number of elements, 20 for the CPS4 cases, must be used, however as the elements are rectangular 10 elements horizontally, whilst maintaining the same resolution, this gives a 10x20 mesh. This will improve computation and set-up time.

41 41 Figure 50 Original beam Figure 51 Simplified beam using symmetry

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