Principles of Buckling, Finite Element Analysis and Abaqus - Dissertation Example

?Principles of Buckling, Finite Element Analysis and Abaqus 512490) Buckling Columns that are axially loaded tend to deflect laterally beyond a certain load. The column is said to have buckled in such a condition. This buckling load generates compressive stresses in the long member and it fails when the actual developed stress exceeds the allowable compressive stress. This buckling load for structural steel sections is directly dependent on the length of the member taking this load. Source: Gere James & Timoshenko Stephen, 2004 The critical forms of buckling for member columns that have different end conditions were developed by Euler in 1744. These underwent no major changes in the next 100 years. In 1845, A.H.E Lamarle proposed the theory that Eulers formula could only be effectively utilised if the slenderness ratio was beyond a certain prescribed limit. In 1889 Considere further made an addendum in the form that Eulers formula could not be used for inelastic buckling since the actual section modulus available on the concave and concave sides of a bend beam were different. This lead to the formulation of the Reduced Modulus theory for buckling and which is still undergoing lot of revisions. (Gere James & Timoshenko Stephen, 2004) Theory Euler by a series of experiments observed that the buckling stress generated in an axially loaded column is directly proportional to the Youngs Modulus, the moment of inertia of the material and inversely proportional to the effective length of the member. In other words he represented the formula by a simple equation. Pcr=л x E x I/ Le^2 Here Pcr represents the critical load, E the Youngs Modulus which is an inherent property of the material, I the moment of inertia is function of the dimensional values in terms of breadth and height of the material. Le represents the effective length of the column. A column of length L that is free at the top and fixed at the bottom has a effective length Le of 2L while a column that is hinged at one end has an effective length of L. In other words for an axially loaded beam the point at which inflection of the beam tends to take place is measured as the effective length. For a column with both ends fixed the inflection takes place at L/4 from the end. The effective length in such a case is L/2. Another combination is that of the column fixed at the base and pinned on top. Calculating this from a series of differential equation with known end conditions would provide an effective length of 0.7L. Hence the Euler’s equation for all the above commonly loaded conditions can be represented as Pcr=л^2 x E x I/ (K x L)^2 where K=2 for fixed-free column, K=1 for pinned end columns, K=1/2 for columns with fixed ends and K=0.7 for column fixed at base and pinned at top. (Gere James & Timoshenko Stephen, 2004) Source: Gere James & Timoshenko Stephen, 2004 The Euler’s formula is used to calculate the corresponding critical stress that is generated due to this critical load Pcr. Here ?cr= Pcr/ A where A is the area of cross section of the member which could further written as ?cr=л^2 x E/(L/r)^2. Here L/r can be together noted as the slenderness ratio. L as denoted earlier is the length of the column while r=v I/A is called the radius of gyration of the member. (Gere James & Timoshenko Stephen, 2004) Using Eulers Theory in Calculations For the analysis of simple beams using Eulers formula, slenderness ratios of columns should not surpass 180. For other members that absorb compression forces the L/r ratio is limited to 200. (Welded Tanks for Oil Storage, 2008) For checking whether the column provided for a section is safe, the actual compressive stress is calculated using the simple formula ? actual=P/A whether P is the external load and A is the cross section of the member. The L/r ratio of the selected member is checked and limited to 180. Thereafter maximum allowable compressive stress generated is found out by using the above formula ? allowable=л^2 x E/(L/r)^2 for columns. If the actual stress calculated is less than the maximum allowable stress then the selected member is safe. Care should be taken to provide a certain margin between the allowable stress and the actual stress to account for any kind of impact loads that may arise in future. When the generated loads are more than the allowable, then a beam of higher section modulus is taken in case of I & H sections and the schedule (thickness) chosen increased to the next higher size in case of pipes. (DeWolf John & Bicker David, 1990) It is generally noted that for long columns the allowable compressive stress reduces with increased slenderness ratios. Limitations It is however to be noted that Eulers formulas can only be effectively utilized only in the elastic stability limit. Beyond this limit the parameters do not behave elastically and hence the value obtained is not correct. Finite Element Analysis Finite Element Analysis is an advanced form of analysis of components subjected to various degrees of forces and bending moments. These analyses are usually carried out for critical components; the performance of which can affect the efficiency and the stability of the whole structure. This was first advanced by R.Courant in 1943 that used the Ritz method of numerical calculations in breaking down the component into finite elements and then applying the loads to these elements. Thereafter the structure was analysed using the variational calculus method to get the solutions to a vibration system as a whole. Further, in 1956 M.J.Turner, R.W,Clough, H.C.Martin and L.J.Topp brought out a paper that studied the stiffness and deflection of critical structures. Since this method was limited to the analysis of critical structures, FEA was used in the costly mainframe computers that were generally used by the aeronautics, defence and nuclear industry. However with the advent of new generation supercomputers and the cost of computers coming down drastically it is possible to carry out different forms of simple Finite Element Analysis using the various softwares available in the market. Theory The basic theory of the FEA method of analysis includes generating a computer model of the component to be analyzed for different moments and loading conditions. This method helps in the design of new components with specific dimensions which can affect the load bearing capacity of the structure. Designs of outdated components can further be refined and fine tuned to respond to these critical stresses. (Doyle James, 2004) This therefore can help a company redefine its product and come out with a new variant on the basis of the goodwill it already has in the market. Further, how this product would behave in real time conditions could be simulated in the computer program itself. This would help us in determining its performance prior to manufacture. An additional advantage is FEA can also be utilized in ascertaining what further changes should be done to the structure when the design conditions would undergo changes at a future point in time. (Bryan William, 1999) Studies could be conducted into the reasons that might have gone into the structural failure of a part and recommendations to alter the model could be carried out using FEM. Source: Widax Peter, 1997 The two types of analysis that are usually carried out include the 2-D modelling and the 3-D modelling. The 2-D modelling involves a simple process of analysis that can be carried out on a normal computer. However the results that are obtained are not as accurate as that obtained using 3-D modelling. The disadvantage in using 3-D modelling is that it requires an efficient computer backup to produce the results which are however more detailed and comprehensive. (Wood John, 2007) Another advantage in using the FEM analysis is that apart from the normal types of loading conditions that can be attributed to a structure, the designer can provide other algorithms in the way of loading functions that might be specific to that particular component. This ensures that a custom made design is generated for each component. Inputting these loading functions can make the component behave in a linear or non-linear manner. Linear systems usually exist in ranges of the elastic proportional limit of the stress-strain diagram while non-linear systems account for systems that undergo plastic deformation. Behaviour of such plastic systems can be ascertained till this fractures at the ultimate load. Source: Widax Peter, 1997 Using FEM The process of using the FEM involves creating a mesh of intersecting horizontal and vertical lines on the component to be analysed. The degree of intensity of this mesh or the spacing between meshing lines depend upon the degree of accuracy that is required of the analysis. For a more comprehensive analysis the mesh is more closely spaced. Mesh spacing is also dependent upon the complexity of shape of the structure to be analysed. For an irregular structure a more dense mesh spacing is required so that it covers all critical areas of the complex shape. (Wood John, 2007) The points of intersection are called nodes and after carrying out the analysis the stresses and deflection generated at each of these nodes is obtained as a tabulated report. The areas of a component that would have a higher nodal density are corners, edges, weld seams and fillet areas that are usually considered areas of high stress concentrations. (Blodgett Omer, 1991, Design of Welded Structures) After the meshing of the component has been carried out the material properties are assigned to the structure. These include the density of the material, Elastic modulus, thermal conductivity, viscosity and other mechanical properties of the material. Further, the anticipated loads are applied at locations on the structure. Analysis of the component gives us an output in the form of stresses and deflections generated at each node. (Bryan William, 1999) Comparing these with the allowable stresses and deflections give us an idea of the criticality of the structure. A number of wide ranges of variables that can affect the output of an analysis are as follows (Widax Peter, 1997) (i) Mass, temperature (ii) Volume of the component (iii) Strain energy (iv) Stress and Strain (v) Force and displacement (vi) Velocity and acceleration Loading conditions that can affect the result of a analysis include (Widax Peter, 1997) (i) Pressure applied and gravitational forces (ii) Centrifugal loads (iii) Thermal loads due to heat transfer between different media. (iv) Displacements that are effected by other sources. (v) Convection currents and the dynamics of heat flux within the analysed region. (vi) Various forms of dynamic loads. Source: Wood John, 2007 To carry out the analysis a reference library of all types of probable elements are maintained. These are used at different points of time to analyse a structure that may have a combination of all these elements. These elements include (Widax Peter, 1997) (i) Rod elements (ii) Beam Elements (iii) Mass elements (iv) Rigid Elements (v) Viscous damping elements (vi) Plates or Shell. (vii) Composite elements (viii) Shear Panels (ix) Solid Elements (x) Spring Elements Source: Wood John, 2007 FEA programs can also be used in the analysis of isotropic and anisotropic materials. Anisotropic elements are considered as those components which have different mechanical properties along each axis. Isotropic properties on the other hand have identical mechanical properties in all planes. Analysis There are a number of comprehensive engineering analyses that can be carried out using the FEM. These include the structural analysis of components that can are undergo only elastic deformation. Other areas include materials that are plastically deformed and the stresses generated during this deformation. (Doyle James, 2004) Source: Wood John, 2007 Vibration analysis of components can be carried out on structures using FEM. These vibrations can be the effect of impact loads or shock loads resulting in large vibrations. Storage tanks that are located in seismic prone areas can be analysed using FEM. Seismic studies conducted in that area could provide us the peak ground acceleration that might have generated during the past earthquakes. (API 650, 2008) This could be utilised in the study of these impulsive seismic forces generated on the tank shell and the compulsive forces in the liquid due to seismic vibrations. The study of the longitudinal and compressive forces generated due to these seismic loads on shell seams and other areas could be studied using FEM. Fatigue analysis also forms a part of Finite Element Analysis. There are components that can fail by repeated stresses due to cyclic loading. Such elements are said to have failed by fatigue. Carrying out FEM can predict the regions where cracks are likely to originate and the reinforcements or design changes that need to be made to accommodate this problem. (Wood John, 2007).Heat transfer analysis can also be carried out using FEM. These can be used extensively in the design of cryogenic or low temperature storage tanks like argon, oxygen and nitrogen tanks. The steady state heat transfer with the environment and the design factors that need to be incorporated into the tank sides in the form of insulation pearlite to limit the boil of rate of the contained liquid form an important analysis using FEM. (API 620, 2002) The Finite Element analysis therefore provides a more comprehensive form of analysis and predicts the performance of a component prior to manufacture. This is more effective than the usual procedure of manufacturing a test component and then trying to simulate conditions by subjecting it to various loads. Abaqus Abaqus is a fairly adaptable tool that is used in finite element modelling. This method provides the user to simulate different loading conditions that the product would encounter at different stages of its life cycle. This tool can be used to subject the structure to a thermal load during its initial cycle. Further down, if the process undergoes a change then the new dynamic loads that the structure would need to be bear could be simulated well in advance. Similarly fatigue loads that would invariably develop at the completion of a set number of cycles and which could cause minor cracks could be set up to act at the required stage of the life cycle. Again if a natural frequency test is done after a static analysis step then the preload stiffness will also be included. Hence the analysis provides accurate information of the life of the component when it is subjected to varying loads at different stages of its existence. This ensures an economic design which is not too conservative and has not been generated keeping the worst case scenario in mind. (Abaqus Theory manual, 2004) Abaqus provides both linear and non linear designs. In non linear designs the abaqus standard automatically selects the load increments that need to be provided at each stage of its life cycle. The user in this case only lays down the loads that are likely to be generated and the tolerances to be accounted. The system is thereafter very capable of selecting the increments that need to be provided after each stage of its life cycle. Some of the complex analyses that can be done using the Abaqus method include (Abaqus Theory manual, 2004) 1. Buckling and post buckling methods 2. Non linear dynamic loads. 3. Harmonic oscillations and the analysis of frequencies generated due to impact loads matching with the natural frequency of the structure thus leading to resonance. This resonance or high amplitude of vibration can cause severe criticalities in structure. 4. Analysis of porous components and their behaviour with respect to different loads. 5. Analysis of components that are in semi fluid solid state. 6. Analysis of Piezoelectric structures. These structures generate voltages when subjected to different ranges of loads. The analysis offers an insight into the electric conduction with the stress generated due to these loads. The Abaqus library offers a complete array of components that could be used in the geometric modelling of structures. Composite layered structures could also be modelled by specifying different materials that are present at different heights of the component. The Abaqus method also provides several kinds of models like elastomers which elicit purely elastic responses, materials that offer a large extent of ductility like mild steel or clay, materials that have the property of flow ability like sand and brittle materials like ceramics. (Abaqus Theory manual, 2004) One of the principal advantages in using the Abaqus model of analysis is that it offers the study of Hysterisis of a component. This is done by separating the analysis into two parts. One is the stress relaxation tests which are the time when the impact stress no longer ceases to be active. The other is the non linear rate-dependent deviation from its position of equilibrium which is a function of time. These two tests provide a detailed output regarding the degree of Hysteresis in a structure The other areas of interest where Abaqus can be specifically used are in determining the sliding interaction between two surfaces. This can be used in the study of interaction between two bodies that have a tendency to deform each other and also between a rigid body and a body capable of changing surface profile. In these types of loading one surface is considered as the ‘Master’ surface while the other is called ‘Slave’. These are also used in the study of the fluids penetrating contact surfaces between two bodies that might be rigid or flexible. (Abaqus Theory manual, 2004) Another very effective use of Abaqus is in the study of multi point constraints like the sliding Constraint, revolute joint like a hinge in which the axis about rotation is fixed, a universal Joint that can provide two axes of rotation about a joint. (Abaqus Theory manual, 2004) The Abaqus tool of Flexibility analysis is more robust and comprehensive tool that can be used to study the effect of different types of loads on a variety of loading scenarios. As mentioned in the FEM analysis, the Abaqus is an advanced technique of computing the complex loading scenarios using finite element analysis in the design of a component. Reference Lists 1. Wood John, 2007, Finite Element Analysis-Theory to Practice, Building a Solid Foundation, 1-18, retrieved from US Air Force Academy database. 2. Bryan William, 1999, Introduction to Finite Element Methods, ASME Career Development Series, 1-49, retrieved from ASME database. 3. Doyle James, 2004, Finite Element Methods, Modern Experimental Stress Analysis: completing the solution of partially specified problems, John Wiley & Sons Ltd 4. Widax Peter, 1997, Introduction to Finite Element Analysis, Accessed on 18th March 2011, Available at http:// www.sv.vt.edu/classes 5. DeWolf John & Bicker David, 1990, Column Base Plates, Steel Design Guide Series, Retrieved from AISC database. 6. Blodgett Omer, 1991, Design of Welded Structures, The James Lincoln Arc Welding Foundation. 7. Steel Tanks for Liquid Storage, 1992, Steel Plate Engineering Data, Retrieved from AISC data base. 8. Welded Tanks for Oil Storage, 2008, API 650, American Petroleum Institute, API Publishing Services. 9. Gere James & Timoshenko Stephen, 2004, Mechanics of Materials, CBS Publishers and Distributors. 10. Abaqus Theory manual, 2004, Accessed on 18th March 2011, Available at http://abaqusdoc.ucalgary.ca 11. Design and Construction of Large Welded, Low-Pressure Storgae Tanks, 2002, A{I 620, American Petroleum Institute, API Publishing Services. Read More