The Effects of Rotational Inertia on the Response of Dynamically Loaded Structures

The Effects of Rotational Inertia on the Response of Dynamically Loaded Structures

Chapter One

Research Objectives

The main objective of this study is to review the traditional method of dynamic structural analysis which includes the effects of translational inertia only without any consideration of the effects of rotational inertia. At the end of this study, the following objectives should be achieved:

Derivation of a closed form equation of motion for an arbitrary structural member under dynamic excitation, subject to the effects of translational and rotational inertia
Determination of the response criteria of a structural member subject to dynamic loading, with consideration for the effects of translational and rotational
Comparison of the dynamic response and internal stress distributions of a dynamically loaded structural member, considering translational inertia on the one hand and translational and rotational inertia on the other
An assessment of the risk inherent in the traditional method of dynamic structural analysis, which assumes translational inertia to be the only significant inertia forces developed in structural members under dynamic

CHAPTER TWO

LITERATURE REVIEW

General Overview

A large number of techniques exist for the analysis and design of structures operating in dynamic environments. Some techniques are amenable to general applications while some may require little refinements before they can be applied to various kinds of structural systems. In carrying out any design, an accurate assessment of the dynamic character of the load and the response of the structural system will be invaluable to the engineer’s desire of designing and erecting engineering structures that will be safe, durable, economical and serviceable.

Simplifying assumptions and approximations of load influences and/or structural configurations are often made in the course of analysis and design. This is applicable in the case of both static and dynamic structural analysis. In all cases, these approximations should be kept within acceptable limits so that the true response history can be evaluated (Clough & Penzien, 1993). In the past, there have been cases of catastrophic failures, with high cost implications, as a result of inaccurate and over– ambitious design assumptions (Parker, 1993)

The structural engineer must idealize his structures before proceeding with the analysis and design. A hypothetical model is assumed which represents the structure’s true semblance in reality. It is absolutely necessary that the engineer treats the idealization of his structure with the importance it deserves. “Establishment of the analytical model is one of the most important steps of the analysis process; it requires experience and knowledge of design practices in addition to a thorough understanding of the behaviour of structures (Kassimali; 1993). The accuracy of the analysis and the prediction of the true life interactive forces depend on how closely the model approximates the real structure both in geometry, loading configuration and member interactions.

For dynamically loaded structures, and indeed all kinds of structures, the explosive release of energy that is imminent when collapse or impaired performance occurs compel the engineer to be realistic in his model formulation. He should ensure that the model employed for analysis represents, as accurately as is practically possible, the true life performance of the structure being analyzed and to consider all significant behavioural tendencies of the respective members and their connections. Accurate prediction of the natural frequencies of structures and the establishment of the various modes of vibration will lead to an optimum design of these structures. With a broad spectrum of the dynamic response criteria (displacement, moments, shear, axial, and angular distortion) known, the zones of resonance can be defined and the structural engineer will take necessary steps to prevent the occurrence of such.

Discretization Models for Dynamic Structural Analysis

The dynamic analysis of most real structures involve the application of varying degrees of complicated idealizations, depending on which method approximates, accurately, the performance of the structure. However, the structural engineer must never allow mathematical manipulations or model refinements to obscure his engineering judgment (BS 8110: 1985). Significant uncertainties, particularly loading characteristic encountered in dynamic problems are best tackled by employing simple ideal mechanisms for the analysis. Application of complex model refinements are not usually justified as they result in precision that are much greater than the input characteristic (Biggs; 1964).

Inertia forces which result from the acceleration of the elements of a structure subjected to dynamic excitation are the major distinguishing factor between dynamic and static loads. Considerations for their development and inclusion of their effects create the fundamental distinction between static and dynamic structural analysis (Clough and Penzien, 1993). Often times, some of the inertia forces will have inconsequential effects on the dynamic behaviour of the structure for a number of reasons. The accelerations producing these inertia forces may be so slow, or the displacement degree of freedom in their direction partly or fully restrained, thus, making their effects relatively inconsequential. In these cases, these inertia forces may be excluded in the dynamic analysis of the structures. Systematic formulation of the equation of motion may be employed so that the inertia forces with insignificant effect may be eliminated. Discretization method for modeling dynamic problems makes the elimination of these inertia forces easily achievable, so that only significant inertia forces will be reflected in the equations of motion.

When viewed thusly, discretization of elements in structural dynamics will not be seen as a means of limiting the number of dynamic degrees of freedom, but also as a means of considering only the significant inertia forces in dynamic systems. This agrees fully with the general principles of dynamic structural analysis which defines the number of dynamic degrees of freedom of a structure as the number of displacement components which must be considered in order to represent the effects of all significant inertia forces of the structure (Clough and penzien, 1993). This will lead to a simple, but realistic analysis procedure for dynamic problems. Some common discretization methods for dynamic structural analysis are examined below. These provide convenient and practical approaches to the dynamic response analysis of engineering structures.

Lumped Mass Idealization

Lumped mass idealization procedure for modeling dynamic problems is used creditably well for structures in which large proportion of the total mass are concentrated at a few points that are discretely defined. Examples of such structures are skeletal structures connected at their nodes and framed structures. The masses of these structures and the loads they support are lumped at these nodes and the inertia forces are assumed to develop only there. The lumped mass idealization does not give accurate results for continuous structures or some three dimensional structures like dams, domes, retaining walls etc whose main force interaction are by membrane action.

As explained in section 1.3, the principles of static equilibrium are applied to determine the masses to be lumped at the nodes. The method provides a simple means of limiting the number of dynamic degrees of freedom by lumping the mass of the structure and all supported loads only at the region where significant inertia forces will be developed. This method will be used later in this work to model the structures that will be analyzed for dynamic loads.

CHAPTER THREE

EQUATIONS OF MOTION OF MDOF SYSTEMS WITH TRANSLATIONAL AND ROTATIONAL DEGREES OF FREEDOM

Introduction

The primary objective of dynamic structural analysis is the determination of displacement-time histories and the evaluation of stress distributions in a structure subjected to time-varying perturbations. The mathematical expressions that show the relationship between the dynamic loading, time-dependent displacements and stress distributions are called equations of motion. Solutions of these equations of motion give precise displacement-time histories for deterministic dynamic analysis and probable displacement-time histories for non-deterministic dynamic analysis.

This chapter aims at deriving, in closed form, the equations of motion for the dynamic analysis of an engineering structure with translational and rotational degrees of freedom. For the purpose of this derivation, we shall arbitrarily select a simply supported beam and employ the lumped mass idealization for discretizing the beam. Various methods are currently in use for the formulation of equations of motion of dynamic models. Here, we shall use the direct equilibration method, applying D’Alembert’s principle which seeks to convert dynamic problems into their static equivalent with the addition of inertia. Even though we shall use a simply supported beam for the formulation of the equations of motion, the set of equations which will be formulated hereunder will be valid for any arbitrary engineering structure.

CHAPTER FOUR

NUMERICAL EVALUATION OF THE PERFORMANCE OF MDOF SYSTEMS

Evaluation of Flexibility Influence Coefficients

The structural response of dynamically excited structures can be conveniently evaluated and expressed in terms of the flexibility influence coefficients of the selected nodal points at which the masses of the structures are lumped. The flexibility influence coefficient (δ_ij) is defined in structural mechanics as the deflection of coordinate ‘i’ due to a unit load at coordinate ‘j’, and is taken as positive if it is in the same direction as the load p_i.

The simply supported beam model with three discrete mass points within the span will be used for the dynamic analysis. The beam system and the moment diagrams corresponding to the various unit loads in the displacement degrees of freedom are illustrated in fig. 4.1 (a-ί). It is worth mentioning that even though the masses lumped on the supports do not exhibit vertical translation, they have significant rotational inertia. This is because they do not have adequate restraint against rotation. The effects of these rotational inertia forces on the left and right supports are illustrated in fig. 4.1 (h) and 4.1 (ί).

CHAPTER FIVE

CONCLUSION AND RECOMMENDATIONS

CONCLUSION

From the results obtained in Chapter four, we can reach the following conclusions:

Structures operating in dynamic environments cannot be adequately analysed by the application of the principles of static structural analysis
The dynamic character of loads acting on structures must be assessed to determine the appropriate analysis procedure to be adopted for the structure. Wherever and whenever significant inertia forces are developed in the structure or its component parts, the effects of these inertia forces must be accounted for taken in the analysis and design of the
Inertia forces can occur in the form of translational or rotational inertia. Any of these inertia forces can have significant effect in any given structure or at any instant of time that is relevant for the analysis of the
The traditional method of analyzing dynamically loaded structures by referring to rotational inertia as ‘slave’ degree of freedom and condensing them out of the equations of motion, leaving only translational inertia as ‘master’ degrees of freedom is not appropriate and must be
The inclusion of rotational inertia in the analysis of dynamically excited structures will allow for more appropriate determination of the fundamental natural frequencies and hence the modes of vibration of structures. This will help to define the zones of resonance and hence enable dynamic structural analysts to prevent the occurrence of
The inclusion of all significant inertia forces in the analysis of dynamically loaded structures will enable more accurate determination of the magnitude and the distribution of internal stresses in the structures. This will lead to more functional designs that will be durable, economical and perform optimally throughout the design life of the

RECCOMENDATIONS

From the fore-going, the following recommendations can be made:

The dynamic character of loads acting on engineering structures should be properly assessed prior to commencement of analysis and
Dynamically loaded structures analysed with the consideration of translational and rotational inertia will give more reliable results that can simulate field conditions.
Dynamic Structural analysts are enjoined to refrain from the traditional method of analysis that ignores rotational inertia induced in dynamically loaded structures. This is because rotational inertia forces induced in structures are as significant as translational inertia
Consideration of rotational inertia forces in analysis and design will ensure that the model used for analysis truly approximates the behavior and response of structures in their design
The inclusion of rotational inertia forces in the analysis of dynamically loaded structures will lead to the determination of the critical values of shear force, bending moment and other relevant response criteria that would be required for optimized design of the

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