## ABSTRACT

*Engineering structures subjected to dynamic loading exhibit vibration motions. These vibrations induce accelerations on the structures and their component parts. The accelerations in turn generate inertia forces that propagate the vibrations and significantly affect the response of the structures and their components parts to external loading. This cyclic cause and effect situation result in randomly oriented and time-dependent displacements as the basic response criteria of structures that are subjected to dynamic perturbations. Researches have shown that translational and rotational inertia are exhibited in the cause of these vibrations. Consequently, it was concluded that translational and rotational inertia are generated in structures under dynamic loading. However, translational inertia has, over time, been the subject of research in dynamic analysis of engineering structures. It is often the only inertia force considered in analysis, design and determination of the important response criteria of structures operating in dynamic environments. Indeed the effects of all significant inertia forces should be considered in the analysis and design of dynamically loaded structures. This is to ensure that the results to be obtained will truly simulate real conditions. This work seeks to investigate the effects of rotational inertia on the response criteria of structures subjected to dynamic perturbations. The results obtained from the numerical solutions of the equations of motion developed show that some important fundamental natural frequencies are obtained with the consideration of rotational inertia. The solution of the equations of motion also show that there are significant increases in the internal stress distributions evaluated when rotational inertia is taken into consideration. It is evident; therefore, that rotational inertia can no longer be ignored in the analysis of dynamically loaded structures because the internal stress distributions and other response criteria are significantly affected by rotational inertia forces.*

## TABLE OF CONTENTS

Title page – – – – – – – – – – i

Abstract – – – – – – – – – – v

Table of content – – – – – – – – – vi

List of figures – – – – – – – – – – viii

List of tables – – – – – – – – – – x

List of symbols – – – – – – – – – – xi

CHAPTER 1

INTRODUCTION

1.1 Background – – – – – – – – – 1

1.2 Concept of Dynamic loading – – – – – – 4

1.3 Overview of Dynamic Structural Analysis – – – – 8

1.4 Discretization of Structural Elements – – – – – 11

1.5 Dynamic Degrees of Freedom – – – – – – 14

1.6 Statement of the Problem – – – – – – 15

1.7 Research Objectives – – – – – – – 16

1.8 Significance of the Study – – – – – – – 17

1.9 Scope of the Study – – – – – – – – 18

CHAPTER 2

LITERATURE REVIEW

2.1 General Overview – – – – – – – – 19

2.2 Discretization Models for Dynamic Structural Analysis – – 20

2.2.1 Lumped Mass Idealization – – – – – – 22

2.2.2 Generalized Displacement Procedure – – – – – 22

2.2.3 Finite Element Idealization- – – – – – – 23

2.2.4 Merits and Demerits of Discretization – – – – – 25

2.3 Distributed mass model for Dynamic Analysis – – – 26

2.4 Rotational Inertia in Dynamic Structural Analysis – – – 28

2.5 Natural Frequency of Dynamically loaded Structures – – 31

2.6 Important Deductions from Literature Review – – – 32

CHAPTER 3

EQUATIONS OF MOTION OF MDOF SYSTEMS WITH

TRANSLATIONAL AND ROTATIONAL DEGREES OF FREEDOM

3.1 Introduction – – – – – – – – – 33

3.2 Selection of Dynamic Degree of Freedom – – – – 34

3.3 Formulation of Equations of Motion – – – – – 36

3.4 Evaluation of Internal Stress Distributions for Moment (M),

Shear Force (Q) and Normal Force (N) – – – – – 51

3.5 Evaluation of Natural Frequencies of MDOF System – – 52

3.5.1 Evaluation of Natural Frequencies with Translational and

Rotational Degrees of Freedom – – – – – – 57

3.5.2 Evaluation of Natural Frequencies with Translational

Degree of Freedom Only – – – – – – – 59

CHAPTER 4

NUMERICAL EVALUATION OF PERFORMANCE OF MDOF SYSTEMS

4.1 Evaluation of Flexibility Influence Coefficients – – – 62

4.2 Beam Model with Translational and Rotational Inertia – – 74

4.2.1 Evaluation of Natural Frequencies – – – – – 81

4.2.2 Evaluation of Amplitude Values of Force of Inertia, Z_{i }– –_{ }85

4.2.3 Evaluation of Internal Stress Distributions: M_{i}, Q_{i}, N_{i }– – 86

4.3 Beam Model with Translational Inertia Only – – – – 89

4.3.1 Evaluation of Natural Frequencies – – – – – 93

4.3.2 Evaluation of Amplitude Values of Force of Inertia, Z_{i }– – 95

4.3.3 Evaluation of Internal Stress Distributions: M_{i}, Q_{i}, N_{i }– – 95

4.4 Results – – – – – – – – – 97

4.5 Discussion of Results – – – – – – – 109

_{ }

CHAPTER 5

CONCLUSION AND RECOMMENDATIONS

5.1 Conclusion – – – – – – – – – 112

5.2 Recommendations – – – – – – – – 113

References – – – – – – – – – – 114

Appendixes – – – – – – – – – – 117

** **

**LIST OF FIGURES**

** Pages**

Fig. 1.1 Variation of Periodic Dynamic load with time 7

Fig. 1.2 Variation of Random periodic load with time 7

Fig. 1.3 Variation of stationary impact load with time 8

Fig. 1.4 Variation of seismic load with time 8

Fig. 1.5 Lumped mass idealization of a simply supported beam 13

Fig. 3.1 Lumped masses at discrete points on a simply supported beam 34

Fig. 3.2 Lumped masses at discrete points on a continuous beam 35

Fig. 3.3 Displacement components of Lumped masses on a simply supported

Beam with translational and rotational degrees of freedom 36

Fig. 3.4 Inertia Forces due to Lumped masses on a simply supported beam

With translational and rotational degrees of freedom 37

Fig. 3.5 Flexibility Influence Coefficients for translational displacement

Of lumped masses on a simply supported beam 39

Fig. 3.6 Flexibility Influence Coefficients for rotational displacement of

Lumped masses on a simply supported beam 40

Fig. 3.7 Displacement components of the nodal points due to External loads 41

Fig. 3.8 Inertia Forces acting on a simply supported beam with translational

And rotational degrees of freedom 48

Fig. 3.9 Inertia Forces acting on a simply supported beam with translational

Degrees of freedom only 59

Fig. 4.1 Unit moment diagrams for simply supported beams with lumped masses 63

## CHAPTER ONE

** INTRODUCTION**

**1.1 Background**

Engineering structures are designed with the basic aim of achieving acceptable probabilities that structures being designed will perform satisfactorily during their intended life (BS 8110: 1985). Design processes encompass considerations which should make the intended structures buildable, durable and serviceable. These design objectives demand compliance with clearly defined standards for workmanship, production and installation of component parts of the structures, as well as maintenance and use of the structure during the design life.

Well designed and properly erected structures whose designs are based on good engineering judgments perform satisfactorily, and with optimum economic values. The judgment of the designer and the level of his objectivity grow with experience and exposure in engineering. Of critical importance are the designer’s considerations for materials strengths and design loads. Materials strengths, like other physical and elastic properties of construction materials are easily determinable through standardized laboratory and field tests. Predictions and extrapolations can also be used with the aid of statistical data, to estimate design strengths. These often come to great advantage when material strengths cannot be measures directly. Design loads however, cannot be so treated.

The assessment of load magnitudes and their actual distribution on an engineering structure is one of the most critical aspects of structural analysis and design. Even when a fairly good value of the loading on a structure is known or is predictable, load increases due to miss-use, overloading, stress re-distributions, settlement, shrinkage and creep can pose serious dangers and can alter, greatly, the results of structural analysis and design. Assessments of loading by statistical methods, as is applicable to material strengths, are not yet possible due to inadequate data. Consequently, care and precision are employed in the assessment of loads for structural analysis, so that the model used for analysis can be truly simulative of service conditions.

The loads that act on structures can be classified into two broad categories: static loads and dynamic loads. Static loads are those loads that maintain constant value and position while acting on the structure. Conversely, dynamic loads are loads whose magnitude, direction and/or position vary with time. Dynamic loads manifest in different forms and affect all kinds of structural members in various ways. Consequently, the structural responses to dynamic loads (i.e. distribution of internal stresses, deflections, settlements, etc) are also time dependent, or dynamic (Clough and Penzien, 1993).

Standard methods of structural analysis are applied to structures subjected to static loads, for which magnitude, position and/or direction do not vary with time. Thus, the structure will be in a state of static equilibrium with the applied loads only developing internal stresses and deformations necessary to counteract the applied external stresses. This principle of analysis cannot be used to predict the behavior of structures operating in dynamic environment because their responses are bound to be time dependent.

Earliest considerations for dynamic loading in structural analysis provided for allowance to be made for dynamic effect, including impact by increasing the dead weight values of static loads by adequate percentage (BS 449, Part 2: 1969). This has been cited as the simplest way of obtaining the dimensions of structural members for preliminary member sizing and for trial designs. Dynamic calculations may follow, to check and perhaps modify the design (Smith; 1988). Indeed this resulted from lack of information on design criteria for dynamic loads. Generalized approach of this kind to treatment of dynamic loads in design, irrespective of type of load, mode of generation, method of application, or type of structure can lead to structures that, at best, are safe and prohibitively un-economical, or at worst, risky, unsafe and disastrous.

Dynamic loads induce acceleration on structures and all their components parts. These accelerations in turn generate inertia forces that significantly affect the response of structures to external loading. The inertial forces and resultant vibrations are randomly oriented, and so are the time-dependent displacements which are the basic response criteria of structures that are subjected to dynamic perturbations. Researchers have shown that translational and rotational displacements are exhibited in the course of these perturbations. Consequently, the inertia forces generated under dynamic loadings can be classified as translational and rotational inertia.

Translational inertia has for long been the subject of research in dynamic analysis and its effects on structures are very well taken care of in the design of dynamically loaded structures. It is occasionally the only inertia force considered in the calculation of important response parameters of dynamically loaded structures. Not much has been done about rotational inertia and its influence on the structural response of structures under dynamic excitations. Good design ethics require that the effects of all significant inertia forces must be considered in the analysis and design of structures operating in dynamic environments.

This work aims at investigating the influence of rotational inertia on the response of structures subjected to dynamic loading. Effort will be made to evaluate the risk or otherwise, inherent in the traditional methods of analysis of dynamic loads which consider only translational inertia forces, without evaluating the effects rotational inertia forces. The famous d’Alembert’s principle which seeks to convert dynamic problems into their static equivalent, with the addition of inertia, will be used in the analysis. Engineering structures, even those under dynamic excitations are known to maintain a state of equilibrium with the applied loads. Thus, the utilization of d’Alembert’s principle, a veritable tool in the hands of structural dynamic analyst, essentially converts the applied loads, the inertia forces developed in the members and the elastic resistance of the structural members to a continually changing state of dynamic equilibrium (Smith; 1988)

Structural systems are essentially members with continuous mass connected together. Inertia forces develop at various parts of the structure, at which these masses are defined. With the random orientations of these inertia forces, the structures are in effect composed of members with infinite degrees of freedom. Discrete idealization of the masses shall be employed in the analysis to convert them to members with Multi-degree of Freedom (MDOF) systems that are more amenable to analysis and design than the infinite degree of freedom systems.

**1.2 Concept of Dynamic Loading**

Dynamics is literally defined as the scientific study of the forces that produce motion. As explained earlier, dynamic loads in structural engineering context are loads whose magnitudes, positions and/or directions vary with respect to time. Thus, in effect dynamic loads may have differing mathematical values, act at different locations on the structures, and/or be oriented towards different directions at various times of interest in the analysis. The concept of dynamic loading and dynamic structural analysis arose from the understanding that loads whose parameters vary with respect to time affect structural behaviours and responses differently when compared to static loads. Static loads have constant magnitude and act at definite locations on the structures. Establishment of force equilibrium mechanisms compatible with the geometry of the structure leads directly to a definite solution of structural behavior under static loading. But by virtue of their constantly changing parameters (magnitude, position, direction), dynamic loads require a different approach for analysis, and they seldom have singular solutions.

Various forms of dynamic loads exist in practice. They arise from different influences and almost any type of structural system may be subjected to one form of dynamic loading or another during its design life. In practical engineering design, the first requirement is to identify the sources of dynamic excitation and to assess their magnitude and significance in comparison to the static loads (Smith; 1988). In identifying dynamic loads, they can be divided into two different categories; prescribed dynamic loads and random dynamic loads.

When the time histories (i.e. time-based variations) of dynamic loadings are fully known, the loading is referred to as prescribed dynamic loading. If, however, the variations in loading are not completely known but statistical data exist that give its variation or method of evaluation, the loading is referred to as random dynamic loading. All dynamic loads have a lot in common, but different methods are used for evaluating their effects on structures, depending on the characteristics of the loads. Analysis of prescribed dynamic loading leads directly to definite displacement-time histories that correspond to the defined loading. This is known as deterministic dynamic analysis. When the dynamic loading is randomly defined, with highly probabilistic tendencies, the displacement-time relationships obtained from the analysis are stochastic in nature and the analysis is referred to as non-deterministic dynamic analysis.

Dynamic loads are characterized by vibrations of the structures and all their components parts. Structures under dynamic excitation still maintain a state of equilibrium, though dynamic equilibrium, resulting in to-and-fro oscillations of the fixed structures. These resulting vibrations and oscillation activate the elasticity properties of the structures which then act as restoring forces that damp the oscillations. The vibrations and oscillations have serious adverse consequences on the performance of dynamically loaded structures especially when they result from loadings moving at ultra-high speeds.

Examples of dynamic loads encountered in structural engineering abound. Un-balanced mass effects due to rotating machinery installed in a building or the regular pulsating forces applied to the foundations of a reciprocating engine readily come to mind. These are the simplest forms of periodic loading. These kinds of periodic loading exhibit the same time variation successively for a large number of cycles and are typically represented by the sinusoidal wave function shown in fig 1.1. Their occurrence may indeed not be completely periodic, since in practice the fluctuation may not be perfectly sinusoidal, but it will be of known magnitude and at constant frequency.

Another form of periodic dynamic loading, though more complex, result from hydro-dynamic pressure generated by propellers of submerged structures, inertia forces of reciprocating machinery or dynamic pressures acting on tall buildings due to fluctuating wind velocities. This second group of dynamic load is illustrated in fig 1.2. The random nature of the loading is evident and necessitates the use of statistical methods for establishing an appropriate design loading (Smith; 1988).

Other kinds of dynamic loading are stationary impact load that occur, for example, during pile driving or during the use of Schmidt hammer for non-destructive testing of concrete. Of interest also, are short duration or instantaneous developing loads that occur during bombs blast explosion, rock blasting, open cast mining, sonic boom from ultra-high speed machines, etc. These short-duration impulsive loadings usually generate similar loading curves typified by an initial peak, followed by an almost linear decay often followed by some suction. This is illustrated in fig 1.3. The loading parameters like length of pulses, amplitude values, rate of decay, etc depend on many factors such as distance from source of impulse, shape and size of impulse originator and presence or absence of damping mechanisms.

Long-duration general forms of impulsive loading such as seismic loads are indeed a special class of dynamic load. These forms of loading result from earthquakes, landslides and subsidence due to ground motions around built up areas. The dynamic character of seismic loading is often hard to predict or allow for explicitly in design (Polyakov; 1985). They are about the most important of all forms of dynamic loads considered in the design of fixed engineering structures, when compared in terms of their potential for disastrous consequences. The pattern of variation is highly oscillatory, as depicted in fig 1.4. Advanced level probability theory is employed to determine the loading parameters to adopt in structural analysis and design of structures which they perturb.

Force (P)

Static weight (Ṗ)

Time t

Fig. 1.1 Periodic Dynamic Loading

Force (P)

Mean Velocity (v)

Wind Velocity Fig. 1.2 Random Dynamic Loading

Force (P)

t

Time (t)

Fig. 1.3 Impact Load

Force (P)

t

Time

Fig. 1.4 Seismic Load

**1.3 Overview of Dynamic Structural Analysis**

Dynamic structural analysis may be viewed as a special form of (static) structural analysis, with consideration for loading with dynamic parameters. In most respects, this viewpoint is correct because dynamic analysis seeks to establish dynamic equilibrium between the structure and its elastic resistance forces on the one hand and the applied external loading and induced inertia forces on the other hand. This is indeed an extension of standard methods of structural analysis which generally considers static loading acting on a structure. When linear elastic structures are to be analyzed, it is usual to separate the applied loading into static and dynamic components, determine the response of the structure to the components separately and superimpose the results to get the total response parameters. The approaches to each of the analysis modes vary, and this amplifies the difference between static and dynamic structural analysis.

There are two fundamental distinctions between static and dynamic analysis of engineering structures. Firstly, from definition, dynamic structural analysis involves the determination of stresses and deflections of structures subjected to time-varying loads. These loads are exemplified in fluctuating wind pressures and blast forces from explosion, loads from pulsating reciprocating machinery, impact loads from suspensions of moving vehicles and locomotives, ground movements and foundation settlements of structures due to earthquakes, subsidence and earth tremors. Consequently, the corresponding structural response to a dynamic load will also be time-dependent, or dynamic. Dynamic analysis does not have a single definite solution as static analysis does; rather it generates a set of solutions which satisfy the various times of interest which the structural dynamic analyst may wish to investigate.

The second and indeed major distinction between static and dynamic structural analysis is noted in the response criteria resulting from the two methods of analysis. For static analysis, the internal stress distributions (moment, shear, deflections etc) depend only on the externally applied loading and the geometry of the structural members. These can always be evaluated by the applications of principles of static force equilibrium. In the case of dynamic analysis, the resulting internal stresses depend not only on the applied loading and the geometry of the structure, but also on the inertia forces developed due to the acceleration of the structure under the dynamic influence of the external loads. The acceleration of the structure implies the presence of momentum in the component parts, which tend to cause the structure to overshoot its natural position of maximum static deflection (rest), decelerate and come to rest momentarily before returning in an oscillatory manner (Smith, 1988). Inertia forces correspond to these changing momentums and are distributed along the structure in proportion of its mass.

The loading acting on a structure at a given instant of time or in a given section may be static, dynamic or combinations of these. The magnitude of inertia forces developed will determine the analysis procedure to be adopted. If the inertia forces developed in the structure are significant in magnitude, then the dynamic character of the problem must be reflected in the analysis. However, if the inertia forces developed are negligibly small as a result of slow motions of the loads and consequent low accelerations produced by them on the structure, the analysis of the structure at any given time may be made by applying the principles of static structural analysis. This analysis will be acceptably correct even though the loading and resulting internal stress distributions may be time dependent.

The engineer, therefore, ought to consider the effects of all forces that may have significant dynamic character, while carrying out the analysis of all engineering structures. Suffice it to state that wherever and whenever significant magnitudes of inertia forces are developed in the structure as a result of external loading, the effects of these inertia forces must be considered in the analysis of the structural response. The civil engineer, in trying to fulfill his task of designing and building structures that would neither collapse nor sustain detrimental damages when exposed to loading, must strive to protect his structures against the deadly powers of nature, dynamic excitations inclusive (Polyakov; 1985). Establishment of the analytical model is also one of the most critical steps of the structural analysis process. It requires experience and knowledge of design practices, in addition to a thorough understanding of the behaviour of structures under loading.

**1.4 Discretization of Structural Elements**

Newton’s first law of motion states that, “A body will continue in its state of rest or of uniform motion, unless it is compelled otherwise”. This universal law recognizes that mass is a measure of the “quantity of matter” in an object. These matter build up to produce reluctance against a motion while at rest or a deceleration while in motion. This reluctance was named inertia by Newton, and it must be overcome before the object can be compelled against its state of rest or uniform motion.

Inertia develops wherever the mass of the object are defined. For most engineering structures, the masses, and hence inertia are distributed over their entire span. When vibrations occur under the influence of dynamically applied loading, the structural time-varying displacements cause inertia forces to be developed along the entire span of the structure. To account for the effects of all these inertia forces, displacement components must be considered for this infinite number of regions. When analyzed thusly for dynamic response, the structures are referred to as members with distributed parameters. Analyses of structures with distributed parameters with the consideration of infinite number of displacement coordinates are painstakingly tedious and rarely embarked upon. However, by an appropriate choice of the number of displacement coordinates, refined solutions with some measure of precision can be obtained for such structures (Clough and Penzien, 1993).

Structural systems with distributed parameters are often transformed, for the purpose of analysis, to systems with discrete parameters. This provides an acceptable alternative to infinite displacement coordinates when analyzing systems with distributed parameters. The transformation is an idealization procedure which aims at lumping the masses of the structure at a finite number of discrete points, thus allowing the structure itself to be “weightless”. With this approach, displacements and accelerations are defined only at these discrete points. Also, inertia forces will subsequently develop only at these discrete points because the masses of the system are fully concentrated there.

The lumped mass idealization in dynamic structural analysis is most effective when a large proportion of the total mass of the structure under study are actually concentrated at a few discrete points. In this way, it will simulate, very closely, the dynamic response of the actual structure. It is absolutely necessary that a structure be modeled in such a manner that its dynamic behaviour can be evaluated approximately (Smith, 1988).

The conventional way of transforming a system with distributed mass to a system with lumped masses is by simple statics, as illustrated in fig 1.5 for a simple beam. Nodal points 1, 2, 3 … r are selected arbitrarily. The beam is treated as series of simply supported structures with points of support occurring at nodes. The mass of the beam and the external loads are converted into lumped masses m_{1}, m_{2}, m_{3} ……. m_{r }at the discrete points. The supporting beam is considered weightless and acts merely as an elastic system supporting the lumped masses

The lumped mass idealization procedure will be employed in this work to transform members with distributed parameter to members with discrete parameters. From fig 1.5, it follows that the mass concentrated at nodal points ‘n’ is given by (Clough and Penzien, 1993):

y

Where = mass at node n

= cross sectional Area

= density of structural member

= dimension between node (n-1) and node n

= dimension between node n and node (n + 1)

- Distributed mass

L

(b) Selection of

Arbitrary nodal points

1 2 3 (r – 1) r

(c) Equivalent simply

Supported beams

(d) Reactions

(e)Lumped masses

Fig 1.5 Lumped mass idealization of a simply supported beam model

**1.5 Dynamic Degrees of Freedom**

Structural members subject to perturbations from dynamic loadings have the tendency to vibrate and deform in directions for which they are not restrained. Inertia forces that develop in the course of these vibrations are oriented along these unrestrained directions, thus providing the time- dependent displacement characteristics of the dynamic loading. For distributed parameter systems these displacements are infinitely numbered and randomly oriented with respect to the structure. For discrete parameter systems like the lumped mass system described earlier, the time – dependent displacements are defined only at the discrete points where the masses are lumped. The displacements also occur only in the directions in which the lumped masses are not restrained, or free to move.

All the displacement components of a vibrating system must be accounted for in the analysis of the dynamic response of the system. The number of independent displacement coordinates (both linear and rotational ) necessary to represent the effects of the significant inertia forces of a vibrating system at any instant of time is termed the number of dynamic degrees of freedom (Polyakov, 1985). Any real structure composed of members with distributed mass is essentially a system within finite number of degrees of freedom. This explains why distributed parameter systems are notoriously difficult to solve, thus necessitating their conversion to systems with finite displacement coordinates and solved as multi-degrees of freedom (MDOF) Systems.

Lumped mass idealization is essentially a way of reducing the number of dynamic degrees of vibrating systems to make analysis easier. Exact solutions are rarely possible because the number of degrees of freedom of the original system has been truncated (Smith, 1988). Satisfactory results, with good degree of accuracy, are required after analysis and design so that the performance of the structure after construction will closely approximate the design assumptions. This can be achieved by an appropriate selection of the number of degrees of freedom for analysis and the formulation of the mathematical model of the structure because the validity of the evaluated results depends largely on the accuracy of the models (Clough and Penzien, 1993).

Linear (translational) and rotational degrees of freedom are frequently encountered in the analysis of the structural systems under dynamic excitations. Exclusion and/or inclusion of any of these degrees of freedom in any given analysis or at any instant of time must be objectively assessed by the structural dynamic analyst. Increasing the number of degrees of freedom assigned to a design model will lead to a more accurate result, though it may result in more rigorous mathematical calculations. The cost implications of analyzing systems with exceedingly large number of degrees of freedom should also be considered, especially when computer modeling and analysis are involved. The dynamic structural engineer is therefore confronted with the difficult task of choosing the minimum number of degrees of freedom at which the error would be acceptably low (Polyakov, 1985). His ability or inability to overcome the risk inherent in the choice of number of degrees of freedom will indeed make worthy or mar all his efforts aimed at controlling the powerful forces of nature.

**1.6 Statement of the Problem**

The major distinction between static and dynamic structural analysis is in the response criteria resulting from the two methods of analysis. For static analysis, the internal stress distributions depend on the externally applied loading and the geometry of the structure. In the case of dynamic analysis, the resulting internal stresses depend not only on the applied loading and the geometry of the structure, but also on the inertia forces developed in the structure.

In structural analysis, it is difficult to establish the boundary where static analysis should stop and where dynamic analysis could start. The approach to be adopted for any particular problem or for any instant of time depends on the inherent characteristics of the applied loading. The magnitude of inertia forces developed will determine the classification of the loading, the method of analysis to be adopted and hence the inertia forces to be included in the analysis. If the inertia forces developed in the structure are significant in magnitude, then the dynamic character of the problem must be reflected in the analysis. However, if the inertia forces developed are negligibly small as a result of slow motions of the loads and consequent low accelerations produced by them on the structure, the analysis of the structure at any given time may be made by applying the principles of static structural analysis. The structural response evaluated from the analysis of the chosen model is valid only to the extent that the model represents and simulates the actual structure and its loading configurations (Kassimali; 1993). The major problem in dynamic structural analysis therefore, is in choosing the appropriate model for analysis and the inclusion of all significant inertia forces acting on the structure.

**1.7 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 forces.
- Determination of the response criteria of a structural member subject to dynamic loading, with consideration for the effects of translational and rotational inertia.
- 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 hand.
- 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 excitation.

**1.8 Significance of the Study**

The characteristics of dynamic loadings on present day structures are highly erratic. Load increases due to unforeseen effects, stress re-distributions, impulsive or sudden impact loads can make the assessment of load effects very critical in analysis and design of dynamically loaded structures. The present day method of analysis of dynamically loaded structures gives considerations to the effects of translational inertia only.

Good design ethics require the inclusion of the effects of all loadings capable of influencing the response of a structural system. A model for the analysis of structures operating in dynamic environments should be truly simulative of the field conditions, to ensure an acceptably accurate result. This study aims to highlight the need to consider the effects of all significant inertia forces in the analysis of structures under dynamic excitations; these include translational and rotational inertia. This will ensure that the results obtained from the analysis approximate the field conditions of the structures.

**1.9 Scope of Study**

A closed form solution of the equations of motion for dynamically loaded structures subject to translational and rotational inertia has been formulated in this work. The equations of motion were based on a consideration of the inertia forces developed in the structures and their displacement degrees of freedom. The equations were used to investigate the response of a beam subjected to an arbitrary dynamic excitation. The equations of motion are of a general form and can be applied to other structures, provided the necessary boundary conditions are applied correctly.

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