A Report on Buckling Load Analysis CE00449M (Applied Structural Integrity) BY UMESH KUMAR BOHRA (09004623) MSc MECHANICAL ENGINEERING Submitted To, Peter Ogrodnik Faculty of Computing Engineering and Technology Contents: S. No Title Page no. 1. Abstract 3 2. Introduction 3 3. Experimental Method 3 4. Analysis using Ansys 6 5. Theoretical Calculation 9 6. Results Comparison 9 7. Conclusion 10 7. Reference 10 List of Figures and Tables

Item Title Page No Table2. 1 Deflection values for a applied amount of Load 5 Graph2. 1 Deformation vs Load for the rod 6 Fig. 4. 1 Design of Stainless steel rod 6 Fig. 4. 2 Displacement and load applied for the rod at its end 7 Fig. 4. 3 The deformation direction for the rod after application of load 7 Fig. . 4 Face meshing of the model 8 Table 4. 1 Mesh sizing and its corresponding results of total deformation 8 Graph 4. 1 Total deformation vs No. of elements in mesh 8 Table 6. 1 Comparison of results 9 1. Abstract: In this case study we determine buckling load for a stainless steel rod by applying compressive load on the either side of the rod. The same model is designed and the buckling load for it is determined by carrying out an analysis using Ansys software.

The values obtained in Ansys are then compared with the theoretical and experimental values of the same rod. 2. Introduction: In engineering, buckling is a failure mode characterised by a sudden failure of a structural member that is subjected to high compressive stresses (Lindberg & Florence, 1987). Buckling occurs when the load applied exceeds the elastic limit of the body. In this case study we are aiming to determine the buckling load for a stainless steel using the experimental method and by also by using Ansys 12. 0 software and then they are compared with the theoretically calculated one. . Experimental Method: While carrying out the experiment we make use of the following apparatus * Deflection Transducer: It is used to determine the deflections in steel rod when the force is applied at its ends. It converts the deflection into electrical signal which is in form of voltage and thus the deflection can be read in the voltmeter. * Load cell: It is a device in which the output is proportional to the load applied. It is also a form of transducer which used to measure the amount of force which is being applied on the rod.

This Load cell converts the force into electrical signals and this signal is sent to micro strain meter. * Micro strain meter: It is used to display the values of force which is applied on the rod and the original value of the load is obtained by multiplying it with a constant 3. 5 which represents the value of each division in micrometer. While experimental Analysis the rod is being supported at its both ends and compressive load is applied on both the ends of the rod. Due to the application of this force we find some deflection at the centre of the rod.

The force is being increased in steps which can be noted using the micrometer which is connected to the micro strain gauge which is placed on the rod and due to the application of the load we find deflection in the rod, which is been measured using the deflection transducer. The deflection readings are noted down along with the load values. The values obtained in micrometer for the force is multiplied with 3. 4 which is a constant and is a conversion factor to get the original value of load for the micrometer used.

The values for the load which is calculated after multiplying it with conversion factor and deflection can be seen in the table 1. S. no| Load in N| Deflection in Volts| 1| 297. 5| 3. 591| 2| 343| 3. 596| 3| 416. 5| 3. 609| 4| 476| 3. 622| 5| 591. 5| 3. 651| 6| 633. 5| 3. 658| 7| 689. 5| 3. 672| 8| 770| 3. 7| 9| 819| 3. 718| 10| 868| 3. 738| 11| 910| 3. 756| 12| 945| 3. 772| 13| 980| 3. 791| 14| 1060. 5| 3. 835| 15| 1141| 3. 891| 16| 1225| 3. 953| 17| 1288| 4. 008| 18| 1305. 5| 4. 03| 19| 1414| 4. 148| 20| 1491| 4. 26| 21| 1533| 4. 41| 22| 1582| 4. 43| 23| 1627. 5| 4. 564| 24| 1669. 5| 4. 686| 25| 1711. 5| 5. 026| 26| 1743| 5. 384| Table2. 1. Deflection values for a applied amount of Load A graph is been plotted by taking load on X-axis and deflection on Y-axis and from this graph a sudden increase in the deflection value for a increase in load gives us the value of buckling load for the rod. Graph2. 1. Deformation vs Load for the rod As per the graph we can say that the buckling load of rod as per the experiment conducted is 1707. 5N. 4. Analysis using Ansys:

The dimension of the rod for which the buckling load is to be determined is noted and a model with same dimensions is created. For creating this model we enter into the geometry stage of the static structural analysis and in which we firstly draw the hexagonal shape of the model. The dimension of each side of this hexagonal shape is given as 4. 65mm and then it is being extruded to 185 mm. The sketch option is selected and the circular shape of 6. 9 mm diameter is drawn over this hexagonal shape and then it is extruded for a length of 67mm and again a hexagonal shape of same dimension with each side 4. 5mm length as before is drawn over this circle and is extruded to a length of 185mm. The final model obtained can be seen in fig 1 Fig. 4. 1 Design of Stainless steel rod This model is then used for the analysis. The material of which this rod is made of is selected using the engineering data. This rod while carrying out the experiment was hinged at the either ends so while carrying out the analysis using ansys we use the displacement option. For this model firstly we determine the new cylindrical co-ordinate system.

After determining the cylindrical co-ordinate system we use the displacement tool in which we fix the rod in one direction and it is free about the other two directions as shown in fig. The force is then applied on the rod on its either ends pointing inwards. Fig. 4. 2 Displacement and load applied for the rod at its end After application of the load the total deformation and the directional deformation is calculated. The static structural analysis is then integrated with the linear buckling analysis in which the total deformation and directional deformation is being calculated.

The deformation of the rod will be in downward direction which can be seen in the Fig4. 3. Fig. 4. 3 The deformation direction for the rod after application of load We then try to refine the mesh to get more accurate results for which we use the face meshing tool and the results are noted down The value obtained for the different mesh sizes can be seen in the Table 2. The mesh of the rod can be seen below after using face meshing at the centre of the rod. . Fig. 4. 4 Face meshing of the model Mesh size | No. of elements| | Total deformation| 0. 001| 8432| | 3. 981| 0. 0015| 4118| | 3. 82| 0. 002| 2537| | 3. 982| 0. 0025| 1941| | 3. 981| 0. 003| 1541| | 3. 982| 0. 004| 1488| | 3. 983| 0. 0045| 1189| | 3. 985| 0. 005| 1142| | 3. 989| 0. 006| 1174| | 4. 0015| 0. 0065| 1172| | 3. 9984| 0. 007| 1153| | 4. 0028| Table 4. 1 Mesh sizing and its corresponding results of total deformation A graph is then being plotted with these different values of mesh and its corresponding elements and the deformation values for it. Graph 4. 1 Total deformation vs No. of elements in mesh It can be seen from the graph that the result is steady from a point which is at a 1941 no. of elements.

Thus at a mesh size of 0. 0025 we attain a result for load multiplier as 3. 981 which when multiplied by the load value gives us the actual buckling load for the stainless steel rod which is 1898. Thus the buckling load of the rod by using Ansys was found to be 1898 N. This result obtained is then compared with the theoretical one and with the experimental one. 5. Theoretical Calculation: Buckling load can be calculated using the formula: Buckling load=? 2EI/l2 Where * E Modulus of elasticity * I Second moment of area * L Length of the bar For the given model: Modulus of elasticity E=200? 03 N/mm2 (since it is made of steel) Length of the bar=437 (which is the total length of the rod) I= ? /4? (7)4=1884. 75 (Since the diameter of the circular cross section is 7) After using the above data we find that the buckling load of the rod is 1918 N 6. Comparison of Results: S. no| Process| Result| 1| Experimental| 1704. 5 N| 2| Ansys| 1898 N| 3| Theoretical| 1918N| Table 6. 1 Comparison of results * Experimental and Ansys solution: The value obtained in the experimental method for buckling load of the stainless steel rod was 1704. 5N and the one obtained in the Ansys was 1898N.

The difference in these two values is a bit high and it may be because of the parallax errors and systematic errors. There may be some error done while measuring the dimensions of the stainless steel rod or improper working of the equipments used while conducting the experiment. It can be because of the change in the working conditions as well. * Theoretical and Ansys solution: The value obtained in the theoretical calculation was found to be 1918. 12N and the one obtained in Ansys was 1898N. The difference may be because of the error while measuring the dimensions of the stainless steel rod.

A small error in the dimension might lead to a difference in the Buckling load. 7. Conclusion: The Ansys Software was useful in calculating the result of the Buckling load for the stainless steel rod. It also justified with the theoretically calculated values of the buckling load. The difference in the values may be because of the small errors in the measuring the dimension. Reference 1. Lindberg, H. E & Florence, A. L 1987, ‘Buckling’, in search. com, accessed 8 May 2010, from < http://www. search. com/reference/Buckling>