DEVELOPMENT AND DEMONSTRATION OF HIGHLY PRECISE RECTANGULAR CYLINDRICAL SHELL ELEMENTS

 1. Many authors have been occupied with the development of elements for rectangular cylindrical shells. Sabir⁴⁾ proposed a new procedure using strain compatibility equations to make displacement fields of cylindrical shells. His element enables get considerably precise solutions of both static and dynamic problems of cylindrical shells. Here we adopt the strain stress function^10,11) of a cylindrical shell to get the displacement fields of the shell. The strain stress function is a single partial differential equation reduced the three equilibrium equations for the shell. Applying algebraic terms to the function enables us to get not only displacement vectors of the shell that satisfy the strain compatibility equations but also rigid motions and large displacement modes with small strains. An approach using the function is employed to develop new rectangular cylindrical shell elements that apply to the analyses of frequencies and a typical benchmark example of cylindrical shells.<br><br> 2. Basic Equations of Cylindrical Shell and the Strain Stress Equation<br> Three homogeneous equations of equilibrium is expressed by Eq.(3), of which coefficients are constant values with the coordinate parameter ξ and η. The strain stress equation of the shell is expressed by Eq.(5). Applying algebraic terms on Pascal’s triangle to Eq.(5) let obtain displacement vectors , where e2 = 0. For example, applying G = ξⁿη^m to Eq.(5) yield a displacement vector u_nm. When introducing the vectors in Eqs.(A9), we get expressions where rigid motions are defined by power series of displacement vectors u_nm.<br><br> 3. Construction of Rectangular Element of the Cylindrical Shell<br> 3.1 Small Strain and Large Displacement Vectors<br> Four rigid motions expressed by power series of displacement vectors unm, make no strains but some curvature and torsion, which are displacement vectors for large deformations with small strain energy.<br> 3.2 In-plane Displacement Vectors, 3.4 Approximate Expressions of Rigid Body Motions<br> Instead of strict rigid body motion vectors, we set vectors Eq.(19) that approximate them. The displacement vectors with underlines are adopted in precedent sections 3.1, and 3.2. Eventually, this approximation gives only slight different results compared to the strict rigid motion.<br> 4. Numerical Analyses<br> 4.1, 4.2 Frequencies Analyses of Cylindrical Tank with Strict Rigid Motion and with Approximate Rigid Motion<br> 4.3 A Typical Pinched Cylinder, 4.4 Shape Functions derived by Basing upon the Approximate Rigid Motion<br><br> 5. Conclusion<br> 1. We utilize displacement mode vectors of the cylindrical shell element in which applying algebraic terms of two-dimensional coordinates to the strain stress function. The method is the first attempt to the knowledge of the authors.<br> 2. The rigid body motion of the element was expressed with a series of the displacement mode vectors.<br> 3. Some examples of cylindrical shell tank and pinched cylinder were analyzed. Obtained solutions are compared with the solutions obtained from the representative solution method of the past.<br> 4. Displacement vectors of u_Re5 and u_Im5 were introduced to the second order terms of the in-plane displacements. The main reason for realizing high-precision analyses is that the second-order terms of in-plane displacement are improved.<br> 5. By adopting software for mathematical expression processing, it is possible to explicitly express the shape function of the method that adopts approximate rigid body motions. This shape function for the cylindrical shell elements enables to apply not only to static linear analyses but also geometrical nonlinear problems and elastic-plastic analyses.