Buckling Load Analysis of Sigmoid Functionally Graded Timoshenko Beam on Pasternak Elastic Foundation

The development of many technologies that make our existence so comfortable has been intimately associated with the accessibility of suitable materials. In fact, early civilizations have been designated by the level of their materials development (Stone Age, Bronze Age, Iron Age). Thus, tens of thousands of different materials have evolved with rather specialized characteristics of materials. Many of our modern technologies require materials with unusual combinations of properties that cannot be met by the conventional metal alloys, Ceramics & Polymeric materials. In this project, you will get to know about a Functionally Graded Composite material which kind of solve the above problems. You will analyze this material using the Finite Element Method by modelling it using MATLAB.

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The FGM is the one which can solve problems arising from the production of a new type of composite material. These materials are used in all engineering areas especially in civil & aerospace engineering. Composite materials will fail under extreme working conditions through a process called Delamination (separation of fibres from the matrix). This can happen for example, in high temperature application where two metals with a different coefficient of expansion are used. To solve this problem, researchers in Japan in the mid 1980s, confronted with this challenge in a hypersonic space plane project requiring a thermal barrier (with outside temperature of 2000K and inside temperature of 1000K across less than 10 mm thickness), came up with a novel material called Functionally Graded Material (FGM).

Problem Description:

You need to determine the critical buckling loads of a Simple Supported Sigmoid functionally graded Timoshenko Beam by using Finite Element Method and MATLAB. The beam is made up of Steel-Alumina which is enriched by steel at the bottom part of the beam. The properties of steel and alumina are as Steel: E=2.1x1011 Pa, G=0.8x1011 Pa, =7.85X103 Kg/m3, Alumina: E=3.9x1011 Pa, G=1.37x1011 Pa, =3.9X103 Kg/m3. The dimensions of the beam are, Length L=0.5m, breadth b=0.1m and variable thickness h. At the end, you have to very different parameters of the beam such as Power law index (n), Sinusoidal foundation (?), Winkler Modulus (K1), Pasternak (K2), L/h Ratio and you need to find out the corresponding Critical Buckling load of the beam. Use result data and plot Graphs and observe the variation.

Project Description:

1. Composite Material: Materials like Iron, Steel are better in tension but poor in compression, similarly Wood, Cast Iron are better in compression but poor in tension. To gain the benefit of having more such properties in one material we combine two or more materials in some arrangement. These types of materials are called composite materials.
2. Functionally Graded Material(FGM): Functionally graded materials (FGM) are a particular kind of composite materials, characterized by the continuous variation of the properties of their components, in order to accomplish speci?c functions. This material is a type of material whose composition designed to change continuously within the solid. The gradient compositional variation of the constituents from one surface to the other provides an elegant solution to the problem of high transverse shear stresses that are induced when two dissimilar materials with a large difference in material properties are bonded. Their mechanical and thermal properties vary continuously on the space directions according to a specific gradation law, in order to accomplish certain functions. Functionally Graded Materials are orthographic materials usually made from a mixture of ceramics and metal. The overall properties of FMG are unique and different from any of the individual material that forms it. In FGM, the volume fraction of fundamental materials is gradually varying layer by layer, so that their properties of materials demonstrate a smooth and continuous change from one layer to the immediate interference layer.
3. Finite Element Method: The main rule that involves in the finite element method is “DIVIDE and ANALYZE”. The greatest unique feature which separates the finite element method from other methods is “it divides the given domain into a set of sub domains” called “finite elements”.
1. General Steps in the Finite Element Method:
1. Discretize and Select the Element Types
2. Select a Primary variable function
3. Define relations
4. Derive the element stiffness Matrix and Equations
5. Assembling the Element equations and introduce boundary conditions
6. Solve for the primary unknowns
7. Solve for secondary unknowns
2. Assembling the Element Equation & Introduce Boundary Conditions: In this, the individual element nodal equilibrium equations generated in the previous step are assembled into the global nodal equilibrium equations. One more direct method of superposition, whose basis is nodal force equilibrium, can be used to obtain the global equations.
4. Timoshenko Beam: The Timoshenko beam theory was developed by Stephen Timoshenko early in the 20th century. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behavior of short beams, sandwich composite beams, or beams subjected to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order but, unlike ordinary Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted Eigen frequencies for a given set of boundary conditions. The latter effects more noticeable for higher frequencies as the wavelength becomes shorter, and thus the distance between opposing shear forces decreases.

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Project Implementation:

1. At first do some research on this project. Because the nature of this project is a bit complex.
2. The next thing is to derive the required differential equation which describes the physics of the given problem. Follow the below process to get a generalized idea on the derivation process.
3. First, find out generalized displacement vector for the element which is shown in the figure.
4. Then formulate the equation of motion for the element subjected to axial force as nodal degrees of freedom.
5. Substitute the equation of axial force in the equation of motion.
6. Then convert the element motion matrix to global motion matrix. This global motion matrix will be of second order.
7. Solution to this 2nd order differential equation can be found out using floquet theory.
8. Use the above solution and substitute it in the global motion matrix.
9. For further simplification and making your equation rime independent, consider the critical buckling load only.
10. Then formulate the stress-strain relation matrix for the element and convert it to the global stress-strain relation matrix.
11. Then assume that the material properties of the FGM vary along the thickness of the beam, which follows a Sigmoid distribution.
12. Then find out the equation of Kinetic energy and Elastic strain energy.
13. Derive the governing differential equation by Hamilton’s principal, which will express Kinetic energy and strain energy for an element of FGM.
14. Solving these differential equations will result in shape function.
15. Now, express the displacement field in terms of nodal degree of freedom.
16. Then by using shape functions, formulate element elastic stiffness matrix and geometric stiffness matrix. In this step, you completed the mathematical modelling of a given problem.
17. Now, assemble all the matrix and solve those by writing MAT lab code.
18. In the end, you have to very different parameters of the beam such as Power law index (n), Sinusoidal foundation (?), Winkler Modulus (K1), Pasternak (K2), L/h Ratio and you need to find out the corresponding Critical Buckling load of the beam. Use result data and plot Graphs and observe the variation.

Project Brief: From the various comparison you can observe the following results,

1. The critical buckling load remains constant with an increase in the value of the power law index.
2. The critical buckling load linearly decreases with an increase in the value of the sinusoidal foundation.
3. The critical buckling load linearly increases with increase in the value of Winkler modulus.
4. The critical buckling load increases with increase in the value of L/h Ratio.
5. The critical buckling load linearly increases with increase in the value of Pasternak modulus.

Software requirements:

MATLAB: MATLAB is a multi-paradigm numerical computing environment and proprietary programming language developed by Math-Works. This software is handy in solving complex Matrices.

Programming language: MATLAB programming language

• MATLAB

• FEM
• Buckling load analysis
• Mechanical
• simulation
• MATLAB

Buckling Load Analysis of Sigmoid Functionally Graded Timoshenko Beam on Pasternak Elastic Foundation