Buckling analysis of cylindrical composite shells under lateral pressure with numerical and analytical methods

Nowadays, carbon fiber materials are increasingly important. These materials are used not only for strength applications but also for low density, which means that low mass applications. Carbon fiber materials have some advantages and some disadvantages. The advantages can be listed as; low mass, high strength, which means that high strength/weight ratio, high stiffness, good surface finish and so on. The disadvantages can be listed as; high cost, difficult to manufacture, the dimensional tolerances are dependent to the manufacturing method. Buckling strength of a material is different when compared with other kinds of strength properties suck as axial loading, bending and torsion. As known, every material has a stress strain curve. For axial loading, bending and torsion strength, the applied force is known and the problem is to find whether or not the material resist that force. To do that, the yield strength of the material is known from the stress strain curve of that specific material. First of all, the stress occurred because of the force exerted on the material, is calculated from the strength equations. Then, the calculated stress value is compared with the yield strength of the material to see if the material is strong enough. The buckling strength of the material is independent from the stress occurred on the material. The yield strength of the material is compared with the stress occurred on the material whereas in buckling point of view the critical buckling load is different from one geometry to the other geometry and it cannot be generalized. The important thing in buckling is the limit value (the critical buckling load) is calculated by using the buckling equations but for axial loading, bending and torsion; the limit value (yield strength) is known and it is a material property. Another difficult and critical thing for buckling is the buckling failure is seen in the elastic region of the material's stress strain curve. As it is known, for the axial loading, bending and torsional strength problems, the material fractures after the permanent deformation occurs. For that reason, the material gives signals for fracture. In the first stage, by increasing the applied force, the material elongates elastically which means that if the force is reduced, the material saves its original shape. After that, again by increasing the force, the material reaches its yield point. Beyond that point, the original dimensions of the material has changed permanently even if the applied force is reduced to zero. By increasing the force after the yield point, the material deforms more and the necking situation is seen. This situation gives a brief signal for the fracture region because the necking situations occurs where the fracture occurs. If the force is increased more the material fractures from the necking point. For the compression, bending and torsional strength problems, as it is seen the fracture region is seen before the fracture. However; for buckling problems, since the buckling occurs in the elastic region, there is no sign when or where the fracture occurs. This makes the buckling problem is very important. There are also some types of buckling such as, buckling under compression load, buckling due to the hydrostatic pressure, buckling due to bending and buckling due to torsion. This means that buckling not only occurs because of compression but also occurs because of torsion or bending. Fraction due to buckling is an immediate fracture that is because it occurs in the elastic region. This shows the importance of buckling problem. The other thing that this thesis studies is shell theories. Many structures are made of shells. Therefore, the strength analysis of shells is important. The shell theories are much more complex when they are compared to the beam theories. The differential equations become more complex. This study is about the buckling of composite shells. The purpose of this study is to find the optimum fiber angles that carry the maximum buckling load in different loading conditions for the same geometrical propertied by analytically and Finite Element Method. The fiber angle orientation plays a very crucial role for strength of the carbon fiber structures in every loading condition. If the loading direction and sequence is known, the fiber angle orientation can be calculated for plies. For buckling point of view, the loading conditions are important because, the fiber angle orientation differs from the load direction. This also makes the buckling modes different. The shape that occurs from buckling deformation differs due to loading condition and the fiber angle orientation. Another crucial thing in this study is to see if the part is fractured due to yield or buckling. To see that first, the optimum fiber angles that carries the maximum load is calculated iteratively. Then, Tsai – Hill failure theory is applied to the structure to see if the structure is yielded or buckled. If the structure is yielded, the maximum load that makes the structure was not to yield but buckled is found out. In this study, MATLAB is used for the analytical solution procedure. The buckling equations for composite shell structures are used for calculating the critical buckling load. As mentioned before, the shell theories are harder than other strength equations. In addition, the composite equations are complex because the carbon fiber material is not isotropic material. A carbon fiber lamina is an orthotropic material, which means that, it must have 9 independent material constants and the stiffness matrix is 6x6. A simplification can be performed by using plane stress assumption. This reduces the material's stiffness matrix from 6x6 to 3x3. This assumption can be performed because the stress distribution along the layer thickness axis is negligible when compared the longitudinal and lateral axes. Sine a shell type of material is very thin this assumption does not make a big change. Moreover, as mentioned before, the crucial thing in this thesis is to make angle optimization. This is also performed by MATLAB. To do that, an angle matrix is created randomly depending on the precision of the program, and the layer amount. The precision of the MATLAB program can be defined in the MATLAB code. This is the angle increments from 0° to 90° such as, 0-30-60-90 or 0-15-30-45-..-90 or 0-1-2-3-…-90. This angle matrix consists of all angle configurations. The column number of this matrix gives the angle configurations. The stiffness matrix must be computed for all angle configurations and at the end of the program, the buckling load must be calculated for all angle configurations. The angle configuration that gives the maximum buckling load for the specified geometry (diameter, length, and layer thicknesses) is the optimum angle configuration for that geometry. In addition, one more thing must be checked. That is the yielding or buckling. To check that Tsai – Hill failure theory is used. If the structure is yielded before it is buckled, the load that makes the structure is buckled but not yielded is found. After all, a finite element analysis is performed to verify the analytical solution. To do that, ABAQUS CAE is used. The structure is defined as shell and all of the properties are written from the results of MATLAB analytical solution. At the end, the results are compared for MATLAB analytical solution and ABAQUS finite element solution.