Demeter G. Fertis

Elastic and Inelastic Analysis of Variable Thickness Plates, Using Equivalent Systems*

ABSTRACT This paper develops a closed-form solution that can be used for both elastic and inelastic analysis of thin rectangular and circular plates with variable thickness in one dimension. The solution is given in the form of an equivalent system of flat plates that replaces the original variable thickness plate. At a given point, the sum of the deflections, rotations, bending moments, or bending stresses of the flat plates is equal to that of the corresponding elements of the original variable thickness plate. A simplified equivalent system of constant rigidity is also obtained, in order to reduce the mathematical complexity of the variable thickness problem. It should also be noted that any known method of analysis, closed-form or numerical, may be used to solve the equivalent system of flat plates. The method is general, and it can be applied equally well for both elastic and inelastic analysis. Various cases of loading and thickness variation are solved using both exact and simplified equivalent sys...

Nonlinear transient finite element analysis of rotor-bearing-stator systems☆

Abstract This paper extends the finite element scheme to handle the highly nonlinear interfacial fields generated in the fluid filled annulli of squeeze film and journal bearings so as to model the transient response of rotor-bearingstator systems. Since such simulations are highly nonlinear, direct numerical integration schemes are employed to generate the overall response. In this context, the paper gives consideration to such items as (i) numerical efficiency/stability, (ii) comparison of implicit and explicit schemes, (iii) determines extent of response nonlinearity as well as (iv) extensively benchmarks the overall concept/methodologies.

Inelastic analysis of flexible bars using simplified nonlinear equivalent systems

Abstract The research here deals with the inelastic analysis of flexible bars of any arbitrary variation in their moment of inertial along their length, that are subjected to complicated loading conditions. Since the material of such flexible members is permitted to be stressed well beyond its elastic limit, practically all the way to failure, their modulus of elasticity along their length will also vary. Therefore, the already complicated nonlinear analysis of such members must also take into consideration the actual variation of their modulus of elasticity along their length when large deflections and rotations are calculated. The analysis and mathematical formulation of this problem is based on the method of the equivalent systems which was developed by the first author of this research. This method permits the replacement of the initial nonlinear problem with complicated moment of inertia variations and loading conditions with an equivalent simpler nonlinear system of constant stiffness E1I1, that has the same length, elastic line, and boundary conditions, as the initial nonlinear system. The constant stiffness nonlinear equivalent system will be loaded with a few concentrated equivalent loads. The solution of the constant stiffness equivalent nonlinear system may be obtained by using either one of the following two ways: (a) By using simpler nonlinear analysis if such analysis is readily obtainable, or (b) by deriving a pseudolinear equivalent system of constant stiffness and applying linear analysis. The choice of method would largely depend upon individual preference, but utilization of equivalent pseudolinear systems coupled with pseudolinear analysis, was proven to be the most convenient. The method is considered to be general and it can be applied to many types of flexible beam problems. Exact, as well as accurate approximate solutions are obtained, and the results are compared.

Inelastic analysis of prismatic and nonprismatic members with axial restraints

Abstract This paper deals with the analysis of prismatic and nonprismatic members with axial restraints, where their material is permitted to be stressed well beyond its elastic limit. This loading on the members causes the modulus of elasticity of the material to vary along their length. The mathematical formulation of this problem, as well as its analysis, is based on the method of equivalent systems that was developed by the first author. This method permits replacement of the original nonlinear member of variable stiffness ExIx, with one of uniform stiffness E1Il, that has the same length, boundary conditions, and elastic line as the original variable stiffness member. It is proven mathematically that this equivalency exists and that the solution of the equivalent system using linear analysis yields the same results as the solution of the original nonlinear variable stiffness member. Deflections and rotations may be easily obtained using equivalent systems, and the member can be analyzed in both elast...

A new pre-loaded beam geometric stiffness matrix with full rigid body capabilities

Abstract Space structures, due to economic considerations, must be light-weight. Accurate prediction of the natural frequencies and mode shapes is critical for determining the structural adequacy of components, and designing a control system. The total stiffness of a member, in many cases, includes both the elastic stiffness of the material as well as additional geometric stiffness due to pre-load (initial stress stiffness). The pre-load causes serious reservations on the use of standard finite element techniques of solution. In particular, a phenomenon known as “grounding”, or false stiffening, of the stiffness matrix occurs during rigid body rotation. The author has previously shown that that the grounding of a beam element is caused by a lack of rigid body rotational capability, and is typical of beam geometric stiffness matrices formulated by others, including those with higher-order effects. Having identified the source of the problem as the force imbalance inherent in the formulations, the author developed a beam stiffness matrix from a directed force perspective, and showed that the resultant global stiffness matrix contained complete rigid body mode capability, and performed well in the diagonalization methodology customarily used in dynamic analysis. In this paper, the authors investigate the “grounding” of membrane elements, and develop a new membrane element with rigid body rotational capabilities.

Equivalent systems for inelastic analysis of non-prismatic members

Abstract The research in this paper involves the analysis of non-prismatic members where the material is permitted to be stressed well beyond its elastic limit, thus causing the modulus of elasticity to vary along the length. The deflection characteristics of such members are determined by using the first authors method of the equivalent systems (see for example Fertis [D. G. Fertis, Dynamics and Vibration of Structures . John Wiley, New York (1973); D. G. Fertis, Dynamics and Vibration of Structures (revised edition). Robert E. Krieger, Malabar, FL (1984)], Fertis and Keene [D. G. Fertis and M. E. Keene, J. struct. Engng, Proc. ASCE 16 (1990)] and Fertis and Pallaki [D. G. Fertis and S. Pallaki, J. Engng Mech., Proc. ASCE 115 (1989)]), which permits replacement of the original member of variable stiffness E x I x with one of uniform stiffness E 1 I 1 , whose elastic line is identical to the one of the original variable stiffness member. It is proven mathematically that the inelastic analysis of members with variable moment of inertia I x and variable modulus of elasticity E x can be carried out by using equivalent linear systems of constant stiffness E 1 I 1 and applying known methods of linear elementary mechanics. The member can be analyzed for both elastic and inelastic ranges, all the way to failure, thus permitting observation of progressive deterioration of the members ability to resist stress and deformation, and establish useful practical critical limits regarding these properties.

Inelastic analysis of prismatic and nonprismatic aluminum members

Abstract Today, many structures are subjected to stresses beyond the elastic limit of the material, creating a loading condition of great significance to the design engineer. This is particularly important for aircraft and aerospace structures and for any structure subject to earthquake and blast loads. Since both prismatic and nonprismatic members may be used in these structures, the analysis can be very complicated, especially when the moment of inertia Ix and the modulus of elasticity Ex vary along the length of the member. The method of equivalent systems development by Fertis [Dynamics and Vibration of Structures. John Wiley (1973); revised Edn., Krieger (1984)], Fertis and Keene [J. Struct. Engng116(2) (1990)] and Fertis and Taneja [J. Struct. Engng. 117(2) (1991)] provides an accurate and simplified method of analysis for the solution of such problems in ferrous alloys. This paper deals with the utilization of equivalent systems of constant stiffness E1I1 where both Ex and Ix of the original member are permitted to vary in any arbitrary manner, in order to determine the rotations and deflections of prismatic and nonprismatic aluminum alloy members. In the analysis, the material of the member is permitted to be stressed well beyond its elastic limit, and all the way to failure.

Grounding of space structures

Abstract Space structures, such as the Space Station solar arrays, must be extremely light-weight, flexible structures. Accurate prediction of the natural frequencies and mode shapes is essential for determining the structural adequacy of components, and designing a controls system. The tension pre-load in the ‘blanket’ of photovoltaic solar collectors, and the free/free boundary conditions of a structure in space, causes serious reservations on the use of standard finite element techniques of solution. In particular, a phenomenon known as ‘grounding’, or false stiffening, of the stiffness matrix occurs during rigid body rotation. This paper examines the grounding phenomenon in detail. Numerous stiffness matrices developed by others are examined for rigid body rotation capability, and found lacking. A force imbalance inherent in the formulations examined is the likely cause of the grounding problem, suggesting the need for a directed force formulation.

Dynamic analysis of space-related linear and non-linear structures

Abstract In order to be cost-effective, space structures must be extremely light-weight, and subsequently, very flexible structures. The power system for Space Station ‘Freedom’ is such a structure. Each array consists of a deployable truss mast and a split ‘blanket’ of photo-voltaic solar collectors. The solar arrays are deployed in orbit, and the blanket is stretched into position as the mast is extended. Geometric stiffness due to the preload make this an interesting non-linear problem. The space station will be subjected to various dynamic loads, during shuttle docking, solar tracking, attitude adjustment, etc. Accurate prediction of the natural frequencies and mode shapes of the space station components, including the solar arrays, is critical for determining the structural adequacy of the components, and for designing a dynamic controls system. This paper chronicles the process used in developing and verifying the finite element dynamic model of the photo-voltaic arrays. Various problems were identified in the investigation, such as grounding effects due to geometric stiffness, large displacement effects, and pseudo-stiffness (grounding) due to lack of required rigid body modes. Various analysis techniques, such as development of rigorous solutions using continuum mechanics, finite element solution sequence altering, equivalent systems using a curvature basis, Craig-Bampton superelement approach, and modal ordering schemes were utilized. This paper emphasizes the grounding problems associated with the geometric stiffness.

Free vibration of variable stiffness flexible bars

Abstract The research work here deals with the free undamped vibration analysis of flexible nonprismatic members. During vibration, the beam carries its own weight, as well as other weights that are attached to the member and participate in its vibrational motion. The variation at the mass can be of any arbitrary nature, and its moment of inertia, or stiffness EI, where I is the cross-sectional moment of inertia and E is the Youngs modulus, may also vary in an arbitrary manner. Since vibrations of flexible members are taking place with respect to a large static configuration position, it becomes important to locate this position as accurately as possible. The method of the equivalent systems as developed by Fertis and co-workers can be used for this purpose. Therefore, the free vibration analysis here is done in two steps. The first step involves the solution of the Euler-Bernoulli nonlinear differential equation, in conjunction with the method of the equivalent systems, in order to establish the static equilibrium configuration and position. From this position, the small vibrations of the member can be determined by using appropriate differential equations that incorporate the effect of the large static deformation. The results are compared by using more than one method to calculate the frequencies.