Plane Stress Elements Research Articles

Numerical stability of complex finite elements

The use of complex finite elements has been shown to give considerable economies in computing time but there is an inherent danger that numerical instability can arise during the formulation of the stiffness matrix for such elements. This problem is studied with particular reference to rectangular plane stress elements for which any number of nodes along a side may be prescribed. It is shown that the occurrence of numerical instability is directly related to the degree of the polynomial assumed for the displacement function and to the number of significant figures used in the calculation. The particular matrix operations which are susceptible to error are indicated.

Structural analysis of aircraft impact on a nuclear powered ship

As part of a general safety analysis, the reliability against structural damage due to an aircraft crash on a nuclear powered ship is evaluated. This structural analysis is an aid in safety design. It is assumed that a Phantom military jet-fighter hits a nuclear powered ship. The total reaction force due to such an aircraft impact on a rigid barrier is specified in the guidelines of the Reaktor-Sicherheitskommission (German Safety Advisory Committee) for pressurized water reactors. This paper investigates the aircraft impact on the collision barrier at the side of the ship. The aircraft impact on top of the reactor hatchway is investigated by another analysis. It appears that the most unfavorable angle of impact is always normal to the surface of the collision barrier. Consequently, only normal impact will be considered here. For the specific case of an aircraft striking a nuclear powered ship, the following two effects are considered: Local penetration and dynamic response of the structure. The local penetration occurs at points where the engines or other rigid objects hit the structure. It is assumed that the aircraft engine is a rigid body projectile and the side wall of the ship is the target. The applied steel penetration formulae for projectiles were empirically derived for military applications, where both the projectile and the target are unlike those of an impact of an aircraft engine. For this reason it is expedient to calculate the upper and the lower limit values of the penetration depths. The results show that the highest penetration depth is less than the sum of all wall thicknesses of the collision barrier. The solution of the dynamic analysis is obtained by using the finite element method. The results are the eigenmodes, the eigenfrequencies, the displacements of the nodes, and the stresses in the applied plane stress elements. It is shown that the maximum stress which only appears in one element is on the same level as the yield stress of the St. 42 steel. The structural analysis shows that the collision barrier is a sufficient safeguard against the perforation of the engine and against the cracking of the structure as a result of the dynamic response to an aircraft impact.

Analysis of Shear Cores Using Superelements

A number of multistory buildings, containing concrete shear cores to house vertical services, have sustained major damage in recent earthquakes. Simplified analytical models for the shear cores cannot account for the relative stiffness of intersecting walls and for the influence of openings on the wall stiffness. Wall stress calculations are also beyond the capabilities of many of these continuum models. To remedy this, a three-dimensional shear core analytical model composed of two-dimensional rectangular plane stress and plate bending finite elements was developed. A subassemblage of finite elements forms a superelement and a stack of superelements is assembled in series elimination fashion to model the core tower. Determination of vibration frequencies, mode shapes, and nodal displacements for several isolated shear cores demonstrates the capabilities of the model. This general purpose analytical model can be readily incorporated into existing multistory-building frame models, and can serve as a basis for evaluation of simpler continuum models.

Coupled Shear Wall Analysis by Lagrange Multipliers

A special finite element technique is described which permits interconnection of line elements in flexure and first-order plane stress elements. Continuity of displacements and rotations at the junction of the two types of elements is satisfied by imposing certain constraint conditions through use of Lagrange multipliers. The method can be employed to analyze coupled shear walls as well as frame-shear wall systems. Accuracy is checked by analyzing a coupled shear wall system with three degrees of mesh refinement. Results are further compared with those from a conventional finite element scheme, which idealizes the coupling beams by plane stress elements. The proposed method is shown to be more accurate and more efficient than the conventional method.

The rectangular plane stress element by Turner, Pian and Wilson

AbstractOne of the first finite elements presented was that by Turner et al.1 Pian2 presented a hybrid plane stress element and Wilson et al.3 presented a non‐conforming displacement element. It is shown that the three resulting stiffness matrices are identical.

Non‐linear finite element dynamic analysis of axisymmetric solids

AbstractThe subject of this paper is the finite element analysis of axisymmetric solids subjected to axisymmetric static and dynamic loading, and taking into account material as well as geometric non‐linearities. A general Lagrangian formulation forms the basis for the incremental equations of motion which are solved using direct integration methods. Solution accuracy is improved by applying equilibrium correction loads at each step. Finite element discretization is achieved through the use of quadrilateral plane stress and axisymmetric elements with incompatible modes added for improvement of the element flexural characteristics. Several numerical examples are presented to demonstrate the effectiveness of the developed computer program.

Anisotropic tapered elements using displacment models

AbstractTwo FORTRAN IV subroutines that generate the stiffness, mass and geometric matrices for plate bending and plane stress elements using displacement models are described. The elements can be of either rectangular or triangular shape with anisotropic properties together with a linear variation in thickness. Although such a formulation is straightforward theoretically, emphasis has been placed on the programming technique where the amount of algebra involved in the matrix manipulations is avoided. Exact integration is automatically performed since a closed‐form solution con be obtained.

Elastic soil

This paper presents a refined finite element analysis for the analysis of two-dimensional plane stress and plane strain structures with particular emphasis being placed on the ability to solve problems of soil-structure interaction under earthquake loadings. The structure and the soil are idealized as an assemblage of quadrilateral plane stress and plane strain elements having a cubic variation in displacement enabling a more accurate representation of the stiffness properties of the system than that previously available. The response of the system to the earthquake acceleration history is achieved by a superposition of normal mode responses and the methods of obtaining the mode shapes and frequencies are outlined. Examples are presented to illustrate the capability of this approach.

Elastic and Equilibrium Matrices of General Semi-Monocoque Membrane Elements

Many structures are made up of skin material which is stiffened by reinforcing members. Configurations of this type are predominant in the aircraft industry where they are referred to as semi-monocoque structures. When analysing such structures, using finite element techniques, the reinforcing members (stringers) are lumped together to form pure axial elements and the skin is then represented by isotropic plane stress elements.

A Derivation Procedure for the Dynamic Flexibility Matrix of a Triangular Bending Element

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

Finite-element stress analysis of a non-linear problem: A connecting-road eye loaded by means of a pin

The stress analysis of a connecting rod under a tensile load represents a non-linear problem, since the distribution of the forces, acting between the eye of the rod and the pin, changes as the loading proceeds. This is due to the tendency of the eye to deform, thus wrapping itself around the pin as the clearance between the parts is reduced. In consequence, the stress pattern in the eye also changes. Such a problem has been solved with a linear elastic finite-element computer programme. A connecting-rod eye was idealized as a plane-stress problem and special elements were introduced to connect the pin and rod together. These elements were assigned stiffnesses which were changed in a prescribed manner after each iteration until compatibility of forces and displacements, with due allowance for initial clearances, was obtained at each connecting point. Although eight iterations were required to achieve the correct displacements, it was found that the circumferential stresses did not change appreciably after the first few iterations. Good agreement was found between the calculated stresses and those measured in a laboratory test with electrical-resistance strain gauges.

Thermo-elastic analysis of rectangular plates in plane stress by the finite-element displacement method

The finite-element displacement method is used to solve a number of thermo-elastic plate problems involving different boundary conditions and variations of thickness. A plane-stress element stiffness matrix and temperature-load column matrix are derived on the assumption that thickness and temperature representations within the element are based on corner-point values. Results obtained from different element models are compared with other analytical solutions in order to establish the best element representation.