University of Nigeria Analysis of One... · 2015-08-29 · University of Nigeria Research Publications UGWUANYI, Juliet Nneka Author PG/M.Eng/05/39817 Title Dynamic Analysis of One

University of Nigeria Research Publications

UGWUANYI, Juliet Nneka

Aut

hor

PG/M.Eng/05/39817

Title

Dynamic Analysis of One Bay Industrial Building Subjected to Two Degrees of Freedom

Facu

lty

Engineering

Dep

artm

ent

Civil Engineering

Dat

e November, 2007

Sign

atur

e

DYNAMIC ANALYSIS OF ONE BAY INDUSTRIAL BUILDING SUBJECTED TO TWO DEGREES OF FREEDOM

UGWUANYI JULIET NNEKA PG/MmENG/05/39817

DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF NIGERIA, NSUKKA

NOVEMBER, 2007

DYNAMIC ANALYSIS OF ONE BAY INDUSTRIAL BUILDING SUBJECTED TO TWO DEGREES OF FREEDOM

i

UGWUANYI JULIET NNEKA PG/M,ENG/05/39817

SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE AWARD OF MASTER OF ENGINEERING IN CIVIL ENGINEERING

DEPARTMENT, AT UNIVERSITY OF NIGERIA NSUKKA.

NOVEMBER, 2007

CERTIFICATION

This thesis has been certified for the award of the degree of Master of

Engineering in Civil Engineering

Engr. Prof. N.N. Project Supervis

, / Engr. ~ k f . J.C. Agunwaniba Head of Department

Engr. Prof. 0 . M Sadiq External Examiner

Date

i i i

DEDICATION

To my parents,

sisters and brothers

for their love and understanding.

To God my one and only.

ACKNOWLEDGMENT

I am immensely grateful to my supervisor. Engr. Prof. N.N. Osadebe for not

only suggesting the project topic and creating time despite his tight schedule but for

also providing some materials that were used for this work. Great appreciation is due

to my head of department Engr. Prof. J.C. Agunwamba for his fatherly advice and

nloral support. Also special thanks are due for all the lecturers and staff of the

department of Civil Engineering, University of Nigeria, Nsukka.

I would like to thank my course mates especially Adamu Abdullahi, for their

cooperation. Also to Ubachukwu Obiekwe. who is more than a brother to me and has

assisted me in several ways, T say a very big thank you.

To my parents, Mr. and Mrs. Emmanuel Ugwuanyi that sacrificed all that they

have to make this progranme a reality, not minding the responsibilities involved,

thank you very much and may God continue to bless you for me. I also thank my

brothers and sisters for their support and cooperation.

Finally. it is with deep gratitude and affection that I pay tribute to the Almighty

God for his mercies. protection and kindness forever in my life.

May God bless you all. Amen.

v

ABSTRACT

This work considers the dynamic analysis of one bay industrial building

subjected to two degrees of freedom. Industrial buildings, being light buildings calls

for the inclusion of the dynamic wind effect in the analysis. Wind is taken as the

external source of excitation.

The excitation frequency from the wind, which can also be expressed as vortex

shedding was investigated. The wind force from the static and the dynamic effect were

also investigated and then compared to show the importance of dynainic wind analysis

because of the great force it exerts on the building. This dynamic wind force was used

as the external load distributed on the structure causing the vibration (forced

vibration).

Equations of motion governing the MDOF system under forced vibration were

formulated using the flexibility influence coefficient method at the lumped mass

points. The dynamic response of the building which include the natural frequencies as

well as the inertia forces on the system were determined and used to obtain the

bending moment of the structure.

Then the excitation frequency gotten was used to check the occurrence of

resonance with the natural frequency of the structure.

From the analysis, it was observed that the natural frequency of the structure

analyzed is more than the excitation frequency exerted on the building from the

external dynamic source, which is wind. This indicates that the structure can withstand

the forces coming from the dynamic sources and no occurrence of resonance.

Finally, this work is recommended for the analysis of industrial building

subject to two degrees of freedom with dynamic load source from wind inducing the

vibration.

TABLE OF CONTENT

Title

Certification

Dedication

Acknowledgement

Abstract

Table of Contents

List of Figures

List of Tables

CHAPTER ONE

Introduction

Industrial building

Degrees of freedom

Types of degrees of freedom

Dynamic loads

Natural frequency and factors affecting it

Resonance

0b.jectives

Scope

CHAPTER TWO

Literature Review

2.0 Introduction

2.1 Lumped parameter

2.2 Distributed parameter

2.3 Finite-element

2.4 Matrix-methods

vii

CHAPTER THREE

3.0 Introduction

3.1 Dynamic Response of industrial building

3.1.1 Force of inertia

3.2 Equation of motion of f'orced vibration of MDOF system

3.3 Solution of equation of motion

3.4 Determination of naturaI frequencies

3.5 Evaluation of Natural Frequencies

CHAPTER FOUR

4.0 Introduction

4.1 Static wind effect on industrial building

4.1.1 Numerical Example

4.2 Dynamic Effect of wind on industrial buiIding

4.2.1 Dynamic wind force on industrial building

4.3 Numerical example

4.3.1 Evaluating the forcing frequency

4.3.2 Evaluating the wind forcing on building

CHATPER FIVE

5.0 Introduction

5.1 Procedure adopted in analysis

5.2 Numerical example

5.3 Evaluation of natural frequency

5.4 Evaluating the inertia forces on the system

CHAPTER SO(

6.0 Discussion of results

6.1 Conclusions/Recomn~endation

References

viii

LIST OF FIGURES

Fig 1.1

Fig 1.2

Fig 3.1

Fig 3.2

Fig 3.3

Fig 3.4-5

Fig 4.1

Fig 4.2

Fig 4.3

Fig 5.1

Fig 5.2

Fig 5.3-6

Fig 5.7

Fig 5.8

Fig 5.9

Fig 5.10-1 1

Fig 5.12

A typical industrial building

A Rigid industrial building frame

A beam element under forced vibration

Effects of external load on the beam element

Inertia forces on the beam element

Effects of unit load at point MI and M2

The action of wind forces on the industrial building

The idealized Industrial building from fig 4.1

The Industrial building to be analyzed

Shows all the reaction on the building

The structure showing the two degrees of freedom

Base system

Bending moment from the first degree of freedom

Bending moment due to the second degree of freedom

Bending moment due to the external load

Moment diagram showing the effects of force of inertia

Final Moment diagram of the idealized structure

LIST OF TABLES

Table 4.1 Shows the static wind force as velocity varies

Tables 4.2 Shows the dynamic wind force with the excitation frequency 37

Table 5.1 Shows the dynamic responses of the building 49

CHAPTER ONE

1.0: INTRODUCTION

Most often, the behavior of structures or structural elements under static loads

only has been considered. However, there are many cases where the response of a

struchlre to moving loads or to suddenly applied loads must be investigated. These

loads have their sources fi-om earthquakes (Newmark et al, 1971, Apostolov, 1980

Dawe, 1984,, Polyakov, 1985, Smith. I988), gusty wind (Timoshenko et al., 1974,

Kolousek et al, 1984, Ambrose et al, 1987, Smith, 1988), explosions, impact loads,

moving loads, vibrating machinery, and other dynamic disturbances. The common

feature of all dynamic disturbances is that they generate vibrations in the structures

upon which they act. The consequences of the dynamic disturbances include: Over-

stressing and Collapse of structure, cracking, damage to safety related equipments,

fatigue and adverse human response (Smith, 1988) and these can be severe and

undesirable. Consequently, pre-requisite to design of such structures is a good

insight to their vibrational motions and in particular their natural frequencies.

Generally, the proneness of a structure to the dynamic excitation is assessed

by comparing the natural frequencies of the structure with the frequencies of the

dynamic excitation (Dawe, 1984). The knowledge of the natural frequencies of

structures and the associated modes of vibration enables the structural engineer not

only to evaluate some dynamic parameters necessary for the design of the structure

but also to predict the likelihood of resonance due to certain dynamic disturbances

(Gupta. 1972, Polyakov, 1985, Varbanov, 1989). In the absence of damping,

resonance implies that the dispIacements of the structure tend to infinite. Thus,

natural frequencies should be determined with some fairly good accuracy. A new

approach for the structural classification of building that makes use of the lowest

natural frequency of the building has been devised (Cook, 1985).

In the analysis of structures subjected to static forces only, the forces acting

on the structure are in general independent of the deformations of the structure

provided that the displacements are small. In a vibrating structure, the dynamic force

arises from the vibration of the mass of the elements of the structure. Since

according to Newton's second law of motion. forces acting on a mass is defined as

mass multiplied by the acceleration of the mass, following D'AIembert's principle,

the analysis of a structure which is vibrating can be treated as an equivalent structure

subjected to static external loads called inertial forces which is negative (Dawe,

1984).

Under this type of loading conditions, the elements of the structure are set in

motion. It is therefore necessary to apply the principles of dynamics rather than those

of static to determine the response. It ill be seen that the maximum deflections,

strains, stresses and various other response quantities are generally more severe

when loads of given amplitude are applied dynamically rather than statically

(Tauchert, 1974). This leads to the reasons for the dynamic analysis of structures

subjected to dynamic loads. Firstly, when a structure is subjected to dynamic load,

there is possibility of the structure entering into resonance, which is an undesirable

phenomenon. Dynamic analysis enables the structural engineer to predict the

likelihood of resonance and to take necessary steps to avoid it. Also, when a load is

dynamic, it induces vibration in the structure upon which it acts. In the cause of

vibration, some forces of inertia are generated giving extra loading on the structure,

therefore dynamic analysis dmls with the method of analyzing the structure taking

note of the forces of inertia which are completely absent when static loads are acting

on the same structure.

1.1 INDUSTRIAL BUILDING

The term industrial building according to (Linton, 1965) has come to mean a

range of structures used by industry where at least a part of the enclosed area is of

one storey height. Also according to (Macdonald, 1975), industrial building is

applied to a range af structures whose colnmon features are single-storey

construction and an open plan suitable for factory production or warehousing. The

most common type of industrial building is the simple rectangular structure, typically

single storey, which ptovides a waterproof and environmentally comfortable space

for carrying out manufacturing or for storage. Examples are building fat steel mills,

structural shops, train sheds, automotive assembly plants, spacecraft factories etc. see

fig 1.1

Fig 1.1 : A typical industrial building.

From a structural viewpoint. industrial buildings are characterized by their

comparatively low height and by their lack of interior fluors. walls and partitions. It

is this later characteristic that contributes most to their being placed in a separate

class (Lothers, 1954). The walls are frequently of light non-load bearing material,

which is capable of acting as infill bracing and stability must be provided either by

diagonal bracing or by rigid joints (Macdonald, 1975). The building structure must

be unobstructed by coIumns for a considerable length (Reedel et al, 1964,

Macdonald, 1975), and the building stabilized by malting the frames rigid

(Macdonald. 1975). Industrial building frames consist of diagonally braced side

bents or portals and severally transverse bents from each roof truss with its

supporting pair of columns. The whole structure is made rigid by the use of diagonal

bracing in the planes of the lop and bottom chords of the roof trusses. As the bracing

is largely to square the building during erection and to prevent the building from

twisting under a diagonal wind effect (Lothers, 1954, Linton E. Grinter, 1965,

Macdonald, 1975). Industrial buildings often use portal frames for their clean lines,

ease of maintenance, economy and speed of erection, thus they form an important

class of structure as in fig 1.2.

Ridge

Fig 1.2

A Rigid Industrial Building Frame

Still, another peculiarity of the industrial building (Lothers, 1954) is the

heavy, dynamic, electric loads brought to its columns by moving cranes. These

diverse structures all have the common requircmcnt of large open floor areas

frequently requiring roof trusses that provide adequate headroom for the use of an

overhead traveling crane (Lintun.E.Grinter. 1965).

1.2: DEGREES OF FREEDORI

This is the independent deflection configuration of a structure (Hurty W.C. et

al, 1964). (Clough and penzien, 1975) gave the definition as the number of

displacement components, which must be considered in order to represent the effects

of all significant inertia forces of a structure. This means the number of independent

geometric parameters necessary to determine the motion of the system (Thomson,

1993). It is equal to the number of independent types of motion possible in the

structure (Coates et.al, 1980). The number of dynamic degrees of freedom describing

its motion may define the complexity of a structure. These degrees of freedom may

be displacements, rotations or some combination of both (Boswell L.F. et al, 1993,

Smith, 1988, Biggs, 1964). Using the lumped mass procedure, the number of degrees

of freedom is equal to the number of masses,

1.2.1 TYPESOFDEGREEOFFREEODOM

Almost all the structural systems in engineering possess infinite degrees of

freedom (Osadebe, 1999). A dynamic system can either be a single degree of

freedom system (SDOF) or a many degrees of freedom system (MDOF). The

example/ilIustration below is used to explain the above types of systems. The most

basic vibrating system consists of a single lumped mass and a spring. This is said to

have one degree of freedom in that there is only one possible direction of movement

for the lumped mass. This is in contrast with the offshore platform which may sway,

twist, heave and so on, which in reality is a system with many degrees of freedom,

(.I.%[ Smith. 1988). In general, the more complicated the structure, the greater the

number of dynamic degrees of freedom, which will be required to describe its

motion (Boswell et al, 1993, Thornson, 1993, Bolton, 1994).

1.3 DYNAMIC LOADS

A dynamic load is any load of which the magnitude direction or position

varies with time (Clough and Penzien, 1975). According to "structural use of steel

work in building", dynamic b a d is part of an imposed load resulting fEom motion. It

is time dependent (Taranath, 1988) and dispIays significant dynamic effects.

Similarly, the structural response to a dynamic load i.e. the resulting deflections and

stresses are also time varying or dynamic (Clough and Penzien. 1975). The

movement of the system caused by the dynamic load in most cases has a vibrational

character. These movements are executed with acceleration which gives rise to an

additional inertia forces in the element of the system thereby causing some dynamic

reactions.

By dynamic reaction, it means the state of the system (displacements,

deformation, stresses, velocities, accelerations etc) far the period after the action of

the external dynamic loads. Dynamic reaction of a system is a function of time

because the load which gives rise to the dynamic reaction is a f~mction of these

parameters. So the dynamic reaction depends on the type of dynamic load and state

of the system. The initial stage in the solution of any vibration problem is to define

the source of vibration. Information is required about the likely levels of forces,

accelerations, amplitudes etc and the associated frequencies. There are many sources

of vibration each producing their own particular characteristics (Boswell et.al, 1993).

The different sources of dynamic loads include: earthquakes, wind, industrial

machinery, human forces, moving vehicles, blasting, pile driving, and other dynamic

disturbances (Newmark et al 1971, 'hnoshenko S. et al, 1974, Apostolov 1980,

Dawe, 1984, Kolousek et al, 1984, Polyakov, 1985: Smith, 1988). These loads are

characterized with variable intensities, frequency states, lines of actions and

directions with respect to time and under their actions, the system comes to a state of

movement around the position of static equilibrium which can be very severe and

undesirable (Smith, 1988).

1.4 NATURAL FREQUENCY AND FACTORS AFFECTING IT

The frequency of free vibration is termed the natural frequency (Hatter, 1973).

Any oscillating object has a natural frequency. which is the frequency it tends to

settle into if it is not disturbed. This is that frequency at which the system would

vibrate if deflected once and then allowed to move freely. If a system is excited by

the continued application of external forces at this fi-equency, the amplitude of the

oscillation will build up and may lead to the destruction of the system/object.

The natural frequencies are properties of the dynamical system established by

its mass and stiffness distribution (Riggs. 1964). Calculations of the stiffnesses,

where necessary are to be based on a proper evaluation of the flexural, shearing axial

and torsional deformation of the members (CTBUH, 1978). Cheng (1970) showed

that in most practical structures, the effect of shearing deformation does not

substantially affect the natural frequency of a structure. It becomes important only at

higher modes when a continuous k a m vibrates with many nodes in a span. In

general, higher frequencies are of very little interest in most common situation.

Consequently, shear deformation effect is often neglected in dynamic analysis of

structure. It is useful to know the modal frequencies of a structure as it ensures that

the frequency of any applied periodic loading which is the forcing freqr~ency will nut

coincide with a modal frequency and hence cause resonance which leads to large

oscillations that may cause damage (Hurty et. al, 1964, Timoshenku et al, 1974).

The undamped system can be used to obtain an upper bound of the natural

frequencies, if one is interested only in the maximum response of the structure

(Tauchert, 1974) and hence the calculations for the natural frequencies are generally

made on the basis of no damping (Thomson, 1993).

1.5 RESONANCE

Resonance occurs when the natural frequency, 'w' coincides with the forcing

frequency, ' 8 ' i.e. w =0 (Biggs, 1964, Clough and penzien, 1975). This is a

condition in which a system vibrates in response to a force applied at or near to the

system's natural frequency. The steady-state response of an undamped system tends

towards infinity at resonance (Clough and penzien. 1975). Resonance of a structure

is usually avoided by changing the systems stiffness or it's mass (Warbuston, 1976).

Increasing the stiffness increases the natural frequency and increasing the mass,

decreases the natural frequency. If this is impossible, (as for example, when the

forcing frequency function is unknown), then it is important to ensure that the

structure has sufficient damping to prevent the build up of excessively large

deformations (Theodore R. Tauchert, 1974). In problems of this nature, damping

plays an important role and its presence cannot be ignored.

1.6 ORJECTJVES

1. To derive the equations of motion governing the MDOF system under

forced vibration when sub.jected to two degrees of freedom using the

influence coefficients methods with the masses lumped.

2. To determine the excitation frequency from the dynamic source, which is

wind and the wind force exerted on the building and its effects on the

building when it is static and when it is dynamic.

3. Using the formulated equations to determine the dynamic response of

industrial building.

4. Comparing the excitation frequency from the dynamic source with the

natural frequency of the system to check resonance.

1.7 SCOPE OF WORK

This work involves using the flexibility influence coefficient method with the

lumped-mass method to determine the dynamic response of industrial building

subjected to two degrees of freedom applying D'Alembert's principles. The dynamic

responses of major concern herc include the natural frequency for the case of free

vibration, force of inertia and bending moment for the case of forced vibration.

The frequency of forcing from the wind effects will be evaluated and used in

the analysis to compute wind force on the building.

For the determination of natural frequency, cases of negative and complex

eigenvalue problem will be outside the scope of this work. Only real and positive (or

zero) eigenvalues will be encountered as is usuaIly the case with vibration problems

(Hatter, 1973) and which pertains to stable struct~ral systems (clough and penzien,

1975).

CHAPTER TWO

LITERATURE REVIEW

2.0 INTRODUCTION

Several methods have been adopted to aid dynamic analysis of systems subjected to

vibrations (Hurty et al, 1968).

They include:

i. Lumped parameter method

i i . Distributed parameter method

iii, Finitedement method

iv. Matrix-based method

2.1 LUMPED PARAMETER

This approach to modeling structures is the easiest of the discretised methods

of analyzing structures. The behavior of the structure is captured using a finite points

or degrees of freedom. While every structure has in reality, a continuous distribution

of mass. it is possible to approximate some structures by systems having one or more

discrete mass points (Clough and penzien, 1975). As the mass of the beam are

concentrated in discrete points or lumps. the analytical problem would be greatly

simplified because inertia forces could be developed only at these mass points. In

this case, it is necessary to define the displacements and accelerations only at these

discrete points (Clough and penzien, 1975). Such approximation reduces the number

of degrees of freedom of a structure from infinity to a finite value (Tauchert, 1974).

This method has the advantage of making the analysis as simple as possible with

regards to economy of computational efforts and still, useft11 and reasonable results

are obtained (Dawe, 1984, BosweIl et al, 1993). This is a crude method of modeling

as the amount of useful information, which can be obtained when a continuous

structure is approximated by a single degree of freedom system, is limited. Though

many times, this model can add useful insight to many problems (Tauchert, 1974)

Hence, this work will adopt the lumped mass method in its analysis of the dynamic

response. Based on this method, the following assun~ptions are made: they include,

1. The roof masses are concentrated at a single print in the structure.

2. The beam and column elements have no mass.

3. Deformations of the elements are linear. This means that when an element is

deflected, the force induced in the element is proprtional to the deflection.

2.2 DISTRIBUTED PARAMETER

Most Civil Engineering Structures are continuous and have distributed

material properties (Smith, 1988). A variety of structural elements, which are of

interest to the engineer. may be classified as distributed systems for the purpose of

dynamic analysis. These elements include: beams, plates. structures for which the

important structural properties are continuously distributed (Boswell et al, 1993).

The basis of this method resides in the assumption that the cross-sectional

dimensions of the structure are affected when the structure is vibrating. In principle,

motion of distributed sys tem is defined by an infinite number of coordinates and

hence, associated degrees of freedom.

The equation of motion for the element may be obtained by deriving an

appropriate partial differential equation, which is solved when the relevant boundary

conditions are applied. This is an exact method though extremely tedious (RosweI1 et

al, 1993). This method can be easily applied to prismatic structures. It becomes

difficult and more complex when non-prismatic structures are involved in the

analysis, \vlrich is where the finite-element mode! is advantageous. To obtain unique

solutions, the prescribed boundary conditions must be satisfied explicitly. In natusa

mode vibrations, the frequency equation stems directly from mathernatica

conditions by considering the boundary conditions.

2.3 FINITE-ELEMENT METHOD

This made its first appearance in engineering journals in a famous paper by

Turnner, Clough and Topp (1956) and it heralded a completely new era in

engineering analysis (Smith, 1988). This method was developed when the authors

had been working on a method of dynamic analysis of aircraft structures and had

introduced the concept of subdividing a structure of irregular shape into a large

number of simpler geometrical entities or elements. They discovered that if the load-

displacement equation for a single element were derived in matrix form, it was

possible to use matrix algebra to combine the interacting effects of all the elements

in a systematic and conceptually straightforward manner (Smith, 1988).

The basic concept is to divide the structure into sub-regions having simpler

geometries than the original problem. Each sub region is of finite size and has a

number of key points called nodes that control the behavior of the element. By

making the displacements or stresses at any point in an element dependent upon

those at the nodes, a finite number of differential equations of motions for such

nodes is con~puted and solved using con~puter.

This approach enables a problem with an infinite number of degrees of

freedom to be converted to one with a finite number, thereby simplifying the sdution

process. For good accuracy in the solution, the number of nodal degrees of freedom

must usually be fairly large (Weaver et al, 1990, Boswell et al, 1993). This model

has helped in the aircraft industry since aircraft structural configuration were

changing rapidly and the methods of a few years earlier were insufficiently general

to deal with the variety and complexity of the new structural shapes. From its origin

in the aircraft industry, it spread rapidly within the realm of solid mechanics into

civil and mechanical engineering.

The great strength of the finite-element model is its versatility since there is

virtually no limit to the type of structure that can be analyzed. This is an efficient

method of determining the dynamic performance of structures for these reasons: It

saves design time, it saves money in construction and also increases the safety of the

structure (Smith, 1988). It is applicable to structures of all types (Clough and

Penzien, 1975).

2.4 MATRIX-BASED METHOD

From the period around the early fifties, the developments, which were

taking, place in the realm of digital computing presented the structural analyst with

ever increasingly powerful aid in the solution of his problem. This progressively led

to a change in emphasis away from methods, which are simplified and specialized to

particular structures, so as to avoid overwhelming manual labour on the part of the

analyst towards more direct and general methods whose basic philosophy is tailored

to the digital computer.

Matrix algebra provided a basis for the efficient organization and

n~anipulation of large quantities of data. Thus, without the need to develop any

fundamentally new structural principles, the stage was set for the introduction of the

matrix methods of structural analysis. The solution of the system of equations

effectively gives the exact solution for the theoretical structures. Two kinds of

approach are commonly used in this method: the flexibility matrix and stiffness

matrix. In considering thc type of approach to select, is the type of quantities, which

are to be computed. For instance, if stiffness influence coefficients are the quantities

of prinlary interest, these can be obtained in a more direct way using the stiffness or

the displacement method. On the other hand, the flexibility coefficients are

computed more easily using the flexibility or the force approach. Hence it is

generally advisable to use that approach which involves the fewer number of

unknowns (Tauchert, 1974). When the natural frequencies and characteristic shapes

were derived from stiffness caefficients, the equations of motion are in terms of

stiffness coefficients K as:

M x + r k x = o .

In using this. the first mode obtained is the highest mode. In many cases, this

is undesirable since the fundamental or first mode is usually of greatest interest. In-

fact, when dealing with many degrees of freedom, analysis is often based on the first

few modes alones, while the higher modes are neglected.

When the equations of motions are written or derived in terms of flexibility

coefficients, the form of the equation is then

. X + C M x 6 , , = O

Where 8,, 's are flexibility coefficients. In this, the first mode obtained. rather

than being the highest mode, is the fundamental. This solves the difficulty in using

the stiffness coefficients for deriving the equations of motion.

In view of the disadvantage of obtaining the highest mode first, the only

reason for using stiffness ec~uations is that. in many cases, the stiffness coefficients

are more easily con~puted as in rigid frames by introducing unit deflections.

In other cases, the flexibility coefficients are more easily determined than are

the stiffness coeficients especially if shearing distortions are to be included.

For the purpose of this analysis, the flexibility influence coeficients will be

used to fbrmulate the equations of motion for the determination of the dynamic

response of the structure.

CHAPTER THREE

3.0 lNTRODUCTION

Dynamic analysis is similar to static analysis except that there is the extra

dimension of time to take into account (Smith, 1988). If the forces are time

dependent, then this leads to problems in dynamics in which the inertia of

accelerating masses must be taken into considcration, Given a structure with time

dependent forces applied to it (these forces may actually vanish in the special case of

natural vibration), then equations of motion whose solution yields response

information is formulated. Knowledge of certain principles of dynamics is essential

to the forn~ulation of these equations. They include: Hamilton's, Langrange,

D7Alembert's Principles (Hurty et al, 1964, Tauchert, 1974, Boswell et al, 1993).

The principal structural characteristics that affect the decision to make a

dynamic design analysis are the natural frequencies of the first few noimal modes of

free vibration and the effective size of the structure. If the structure is stiff, the first

few natural frequencies will be relatively high and there will be little energy in the

spectrum of atmospheric turbulence available to excite resonance. If the structure is

flexible, the first few natural frequencies will be relatively low and the response will

depend on the frequency of the fluctuating wind forces. At frequencies below the

first natural frequency, the structure will tend to follow closely the fluctuating forces

actions. The dynamic response will be attenuated at frequencies above the first

natural frequency but will be amplified at frequencies at or near the natural

frequency; consequently, the dynamic deflections may be appreciably greater than

the static values (Gould et.al., 1980). In this chapter, thc equations of motion

governing MDOF system under forced vibration will be formulated.

3.1 DYNAMIC RESPONSE OF INDUSTRIAL BUILDING

In most cases of dynamic loading of structures, the response is mainly in the

fundamental mode of the structure (Stafford-Smith and Coull, 1991). Consequently,

the fundamental modes are of great importance as it is used to check and compare if

the structure can withstand the excitations that comes from the dynamic loads. In

cases of industrial buildings, the possibility exists that some portions of the dynamic

response in the next few modes, say the second and third may reach appreciable

proportions. There are therefore some interests in determining thc nmde shapes and

associated frequencies not only in the fundamental mode but also in several of the

higher modes. The dynamic response can be from free vibration or forced vibration,

but this work considered forced vibration.

3.2.1 FORCE OF INERTIA

The force of inertia of the structure due to the vibrating mass 'M,' is given by:

F = - M i x i

Where:

x = Acceleration

M = mass.

The negative sign signifies that inertia force opposes motion

3.2 EQUATION OF MOTION OF FORCED VIBRATION OF MDOF

SYSTEM

Wherever a system is acted upon by a time varying external load F, (t), forced

vibration is set up (Kolousek, 1973). Thus, under forced vibration, we consider:

1. The forces of inertia.

2. Restoring forces

3. Forces due to external load.

From equilibrium. consideration of the forces of inertia and the restoring

forces should have negative signs since they oppose motion while, the force, due to

external loads should have positive signs.

For ease of the analysis, a beam element with two degrees of fieedom system

with external loading is used for the formulation of the equation of motion of MDOF

system under forced vibration.

Consider the system with two degrees of freedom as shown in fig 3.1,

assuming that it is acted upon by excitation forces and that its under forced vibration:

Fig. 3.1

F, (t), F2 (t). and F3 (t) are the amplitude of the external loads acting on the system

and they are taken as one force. The effect on the system is assumed as shown below

in fig 3.2

Fig. 3.2

X,(t) and X2(t) are the dynamic displacements of the masses MI and M2 at any

arbitrary time on the system. Since the system is in motion, inertial forces of

magnitude - M, 2, as in fig 3.3 act on it

4 1 1

-

Fig. 3.3

Applying a unit load of P = I kN at point of M,

Fig. 3.4

Also applying a unit load o l P - IKN at point of M2

Fig. 3.5

The equations of motion are expediently set up using D'AIembert's principle

and using the principle of super-position as the flexibility coefficients which are the

deflections caused to the system due to the application of the unit load p = I kN at

points of MI and MI. From the above explanations and the drawings. results in the

equations below:

x, ( t ) =Sly sin 6t - M , x , (0 *if,, - IV,X,(~)S,~ - - 3.1

x , ( 1 ) = 6 ~ ~ sin&!-M,x,(t)*S?, -M22z(~) *6 , , -- 3.2

Equations 3.1 and 3.2 can be written in the abbreviated from as:

Where;

tiij is the displacement effects on the system at the ith nodal point due to a unit load

applied at the jth nodal point.

Then equations 3. I and 3.2 can be arranged as

x,(t)+M,Y,(t)S,, +M,i , ( t )G, , =6,,sinA 3.4

x , (t) + M, X I (r)6,, + M,x2 (1)6,, = S-,, sin 64 3.5

Equations 3.4 and 3.5 are the equation of motion of MDOF system with two degrees

of freedom under the effect of external loading.

3.3 SOLUTION OF THE EQUATION OF MOTION

As the system motion is in steady state regime, the solution is sought in the

form (Kolousek et al, 1984. Paz, 1985):

x, = C, sin 6t 3.6

Where;

C, = amplitude of motion.

Sin& = frequency of forcing.

Assuming the solution of equation 3.4 and 3.5 is in the form of eqn. 3.6.

Then differentiating eqn. 3.6 twice gives:

if = -c,Q' sin Bt

Then the force of inertia is given by

-Mix, = - M I ~ , B 2 sin@

Let the amplitude of the force of inertia be 2. then:

2, = M,C,8'

From eqn. 3.9,

Substituting eqns. 3.6. 3.8. 3.10 into eqns. 3.4.3.5 that are the equations of motion of

MDOF system with two degrees of freedom under external load effects and solvillg

lead to the solution as:

SjiZ, +S ,Z , + 6 , = 0

6 ,Z , + S i z 1 +S,,, = 0

Where

M , h! ; d ,

4 1 =C I,,,, Equations 3.11 and 3.12 are the solution of the equation of motion of

subjected to two degrees of freedom.

MDOF system

In equation 3.14, Mi and Mj are bending moinents due to unit load applied at

the it" and j"' nodal points respectively.

EI is the rigidity of the beam element of the given structure.

3.4 DETERMINATION OF NATURAL FREQUENCTES

In determining the natural frequencies of the system, note that the system will

be vibrating under self-excitation. The frequency equation is obtained from equation

3.1 1 and 3.12 by:

(a) Noting that in the absence of external force, 4, which is the displacement

effect from the external load is equal to zero.

(b) Also recall that ' 8 ' is the forcing frequency therefore, the frequency of the

system will then be 'w' since the structure is vibrating on its own that is free

vibration.

Consequently, with these in mind, the equation of MDOF system under self-

excitation is given as:

Where

To solve eqns. 3.15 and 3.16, it is put in matrix form called the frequency matrix

obtained through flexibility formulation as shown below:

The formulation of eqn.3.18 is an important mathematical problem known as

eigenproblem equation.

For non-trivial solution of eqn. 3.18, that is the solution for which not all the

amplitudes are zero requires that the determinant of the matrix factor of the

amplitude vector be equal to zero, that is

Equation [3.19] is an eigen-value problem leading to the modal characteristic values

where

k p k Z--------------- -> kii.

Using eqn. 3.20. the modal frequencies are evaluated as follows:

j = 1. 2 for this work.

If the flexibility influence coefficients 'Sij' used in eqn. l3.191 are multiples of the

rigidity

'EI', then eqn. [3.21] becomes:

Where;

I,vI e 1v2 < MI, ivhich ore rhe natz~ralficqzrencies.

This formulation applies to frame structures, which wilI be used in this work

analysis.

3.5 EVALUATION OF THE NATURAL FREQUENCIES:

In a lumped mass anatysis [Ms] is a diagonal matrix and always positive

definite (that is, its inverse exists). [K] 1s also positive definite since there is no rigid

body motion.

There are several methods of determining some or all of the eigenvalues

(natural frequencies squared) and the associated eigenvectors (vibrations modes).

Some of these are the Jacobi method, determinate search, subspace iteration and the

Householder- Qr methods.

The system size, the bandwidth, and the number of required natural

frequencies and modes of vibration determine which method should be used on a

particular problem. The numerical advantages of some of these solutions techniques,

operation counts, storage requirements and algorithms have been discussed

extensively (Gupta, 1970, 1972, Wilson and Bathe, 1972, 1973).

For this work, the determinant method will be used to evaluate the natural

frequencies as in the form of section, 3.4. A basic approach to the eigenvalue

problem of an n x n matrix of snlall order is to expand the determinant and seek a

solution to the resulting characteristic polynomial equation. This approach will be

used extensively in chapter five of this work to demonstrate the manual procedure

involved. However, when the order of matrix is above three, the approach becomes

more cumbersome and difficult to manipulate. Though, because a 2 x 2 matrix is

gotten for this work, the determinant approach will be used for the analysis.

CHAPTER FOUR

4.0 INTRODUCTION

Wind is a turbulent motion of air (Reedel et.al, 1964, Ambrose et.aI, 1987),

which is characterized by highly irregular chaotic variation of velocity with time at

each point of space. The dynamic effects of wind on structures have been studied for

a much shorter time than the static effects (Kolousek et.al, 1984). This is because

dynamic effects were not so evident in the older structures but also to some extent

because the contribution of the dynamic action of wind to structural failures of the

past was not always recognized.

The action of wind 011 industrial buildings can be classified into the static and

dynamic effects. Static effect refers to the steady (time- average) forces and

pressures tending to give the structure or its component members a steady

displacement and dynamic effects refer to the tendency to set the structure oscillating

(Scruton .c.) and its effect is time varying (Norris et.al, 1959).

These static and dynamic effects of wind are as explained below.

The diagram below shows the wind forces as they are hitting the industrial building

Fig. 4.1

4.1 STATIC WIND EFFECT ON BUILDING

A basic wind speed V appropriate to the district where the structure is to be

erected is determined from the meteorological data. According to CP3. Chapter V:

Part 2, an assessment of wind load should be made as follows;

1. Determine the design wind speed from

v, = v* S, *s2*s3 ( I d s )

Where;

V = basic wind speed in mls.

S, = factor relating to topology and is generally taken as unity.

SZ = factor relating to hcight above ground and wind bracing.

S3 = a probabilistic factor relating the likelihood of the design wind speed being

exceeded to the probable life of the structure. A value of unity is recon~n~ended for

general use.

2. Convert the design wind speed to the dynamic pressure using

q = K*v,~ (Nlm2) 4.2

Where K is a constant = 0.6 13

3. The pressure exerted on the building is

P = Cp*q

Where C, = C,, - CPi

C,, = External pressure coefficient

Cpi = internal pressure coefficient

4. Then the resultant wind force on the building is calculated using

F = Cp *@A 4.4

This force is assumed to be distributed over the surface of the building. A

numerical example will be used to illustrate this.

4.1.1 NUMERICAL EXAMPLE

Using the information provided, determine the total wind force on the

structure of fig 4.2

Wind speed = 40mIs

S, = 1.0

Sz varies with height

S3 = 1.0

Ground roughness category = 3

Building size class B

Using 3m spacing

SOLUTION

From the information given, the basic design wind speed from eqn 4.1 is

v, = VSIS2S3

= 40*S2

from 4.2,the dynamic pressure 'q' = KV,

=0.613 * (40 *s212

The S2 factor varies with height using tables 13 of CP3

Therefore for 0 -5m, S2= 0.65, q = 4 1 4 ~ 1 m ~

5m - 10, S2 = 0.74, q = 537Nm2

10m- 15m, S2 = 0.83, q = 676bJ/m2

The maximum pressure among these will be used.

Consider wind load on roof

Height to eaves = 10m = h

Width of building = 18111 = w

Then Ww = 10/18 = 0.56

Slope of roof = 24'

For the pressure coeffkients, using tables 14 and 15 of CP3

Cpe wind parallel: windward = -0.4

Leeward = -0.5

Across wind: -0.8 max on both dopes

Cpi max effect is when = +0.2

Then max uplift on both slopes from eq.4.3

= (-0.8- (+0.2)) * q

= -1.oq

but q = 676~/m"

then load on each slope P = -676~irn' .This is the pressure exerted at any point on

the surface of the building and the negative sign indicates that it is a suction.

The resultant wind load on the area of the building from eq.4.4

F=676* 14 * 3

= 2S392N

= 28.39KN

So at the velocity of 40m/s, the wind force input on the building is 28.39KN. Using

the outlined procedure, the static wind forces inputs considering other velocities are

as outlined in the table below:

Table 4.1

4.2 DYNAMIC EFFECTS OF WIND ON BUILDING

WIND VELOCITY (mls)

The wind velocity V (t) can be expressed as the sum of a time dependent

STATIC WIND FORCE (KN)

- - mean con~ponent V and a fluctuating component V (t) (Vaicaitis et.al, 1975) i.e.:

Where:

V (31, t) = total wind speed

-

V ( X > = mean valve of wind speed

- Vf (X , t) = fluctuating component of wind speed

The basic wind speed (V) appropriate to the district where the structure is to be

erected is determined from meteorological data (Boswell et.aI, 1993).

Every normal structural member will be affected if it is exposed to a flow u f air.

Wind flow past a body generally results in the formation of vortices. It is really

difficult to deal with the real flow around structural shapes because the matl~ematics

needed to describe the flow is difficult and it has to be based on some assumptions

which are not exactly true to real life, so an experimentally derived, non-dimensional

strouhal number is used for each shape of cross-section (Bolton, 1994).

The vortex-shedding frequency that is also called excitation frequency is expressed

as below,

Where:

8 = Excitation frequency

V = fluid velocity normal to the member

S = strouhal number for the section

D = member diameter or width.

From equation 4.6, it implies that the forcing frequency from the wind depends upon

the dimension 'D' and velocity 'V only (Rollon, 1994). So depending on the wind

intensity at the site, the excitation frequency on the structure is calculated using

eq.4.6, which with negligible error, can be assumed to be uniformly distributed over

the fill1 height.

3 4

4.2.1 DYNAMIC WIND FORCE ON THE BUILDING:

When wind with a certain velocity depending on the site location, blows against a

building, the periodic transverse force experienced by the building may be expressed

(Norris et.al, 1959) by:

This is because the formation of vortices on either sides of the structure give rise to

an alternating force which is transverse to the flow direction (Smith, 1988: Roswel

et.al. 1993).

Equation 4.18 can also be written as:

F (t) = Fo Sin Bt

Where:

F (t) = periodic force experienced by the building and is assumed to be distributed

over the surface of the building.

Fo = Co % P v2 (x, t) D is the amplitude of the force exerted on the building in the

flow direction.

Q = Excitation frequency of wind flow

t = time

4.3 NUMERICAL EXAMPLES

1. EVALUATING THE FORCING FREQUENCY

2. EVALUATING THE WIND FORCE ON BUILDING.

Assuming wind with a velocity of 4Omls hits an industrial building of the type shown

below:

FIG 4.2 Idealized industrial building from fig 4.1

The fotlowing data are used:

Overall length of industrial building = 30111

I - section of 356 x 17 1 s 5 1 kg U.B.s

Density of air = 1 .2kg/m3

Strouhal number 'S' for the section = 0.14

Fluctuating drag coefficient = 0.4

4.3.1 Evaluating the forcing frequency '6 ' : on the bnilding.

From equation 4.6,

This means that if a wind velocity increases fiom zero to 40mls, (the excitation

frequency exerted on the building is 32.75 HZ), then the excitation frequency will

therefore vary from zero to 32.75 HZ for the section.

4.3.2: Evaluating the periodic force 'F' on the building caused by the wind

flow:

From equation 4.8:

Fo = Co 1/2t V ~ D

This is the amplitude of the exciting fiequency. For the purpose of structural

calculations, this periodic force is idealized to be uniforn~ly distributed horizontal

load that acts on the building (Macdonald, 7975) as thus:

FIG 4.3

The table below shows the excitation frequency and the dynamic force input on the

building as the wind speed varies from eqn. 4.6 and 4.8.

TABLE 4.2

WIND VELOCITY EXCITATION FREQUENCY (Hz)

The wind speed V appropriate to the district where the structure is to be

erected is determined from meteorological data (Boswell et.al, 1993). But for

this analysis, a wind speed of 4Om/s was assumed and used as in fig 4.3.

CHAPTER FIVE

5.0 INTRODUCTION

This chapter presents the manual solution procedure for the dynamic analysis

of an industrial building subjected to two degrees if freedom of fig 4.3. The

flexibility method, which is one of the methods of dynamic analysis as in chapter

two, is used. The basic procedure is presented in section 5.1 which will be used to

determine the natural frequencies, the force of inertia acting on the structure. The

wind forces on the building as were used in chapter three and chapter four will be

used in the analysis to get the bending moment diagram for the structure.

5.1 THE PROCEDURE ADOPTED USING THE CLASSICAL INFLUENCE

COEFFICTENT METHOD IN THE ANALYSIS IS AS FOLLOWS:

(a) Determine the degree of freedom of the system

(b) Determine the redundancy of the system if it is indeterminate.

(c) Remove the redundant forces to make the system determinate to get the static

bending moment diagram.

(d) Draw the unit bending moment diagrams for the necessary degrees of

freedom conditions by applying a unit load of P = I kN on the structure.

(e) Calculate the flexibility influence coefficients

(f) The linear equations are then set up and are given as:

Where:

S is the displacement o f the i"' nodal point due to a unit load applied at the j' nodal

point.

Zi is the unknown force of inertia

S is the displacement of the ith nodal point due to external excitation. 'I'

6,) and S,, are obtained from the mi and mi and mp diagrams respectively using the

local equilibrium principle.

(g) Equation 5.1 is solved and the unknown values of zj determined.

The final bending moment diagram is then obtained using the relationship;

Mr = Mp + &M, 2.j 5.2 J- I

Where:

Mf is the final bending moment diagram. Also the final shear force diagram can now

be easily obtained using the local equilibrium principle and the relationship;

5.2 NUMERICAL EXAMPLE

Consider the MDOF industrial building structure loaded as shown in flig 4.3.

SOLUTION:

From the outlined procedure,

(a) The degree of fieedom of the system = 2.

(b) The st~zrcture has three redundant force as shown in Fig 5.2

Let the redundant forces be: MA, ME and HI-:

(C) The structure is made determinate by removing the redundant forces to give the

figure below showing the degrees of freedom.

FIG: 5.2

the bending moment diagram for the first degree of freedom is then gotten as below.

Mo diagram Fig 5.3

Introducing the redundant H,

Fig 5.4

Lx,=I Fig 5.5: introducing the redundant MA

Introducing the redundant ME

Fig 5.6

From fig 5.3 - fig 5.6 solving for the flexibility coefficients gives the MI diagram

due to the vertical load appIied which gives the first degree of freedom as below:

Fig 5.7:M1 diagram

M1 diagram due to the second degree of freedom from the horizontal force:

Fig 5.8 M2 diagram

For the external force applied:

Fig 5.9: Mp diagram

(e) Writing the linear equation for this system with two degrees of freedom from

equation 3.1 1 and 3.12 gives.

S ' , , Z I + S,, 22 + 6 ,p= 0

Then the flexibility coefficients that are the deflection as cursed by the application of

the unit loads are calculated as thus using Diagram multiplication.

46

5.3 EVALUATION OF THE NATURAL FREQUENCIES (W,, W,).

From equation 3.15 and 3.16. the natural frequency is calculated from the equation

of motion for free vibration thus:

Also from equation 3.17,

S*,, = S,, - l/Mw2

Substituting equation (C) and (D) into A and B gives,

, , - M W Z + z2=0 I ?

S,, z, + (S,, - 1 ~ d ) z, = 0 --

Writing the frequency matrix of the system as from equation 3.18 and noting that:

K = I / M W ~ or "'/M\?

Then:

From equation 3.19, and substituting the values of the flexibility coefficients and

multiplying through by EI gives for non trivial solution

Expanding:

from this gave the quadratic equation:

The solution to this gives the eigenvalues as:

Where K, > K2

From which as in equation 3.22 gave the modal frequencies as:

where WI r ~2 w l k l i arc thc natural frequencies of the structtlre.

5.4 EVALUATING THE INERTIA FORCE ON THE SYSTEM (Zl Zz).

From equation 3.1 1 and 3.12,:

6*,, 21 + q, z2 + SIP= 0

6, ,z1+ 6' 22 z2+ 61p=0

From equation 3.13. evaluating 6,, gives

=(615.69/EI) - (1/1250*32.75~)

= - 5.33 lo-'

Having evaluated 6ii the equations 3.1 1 and 3.12 becomes:

(-7.28 X I O - ~ ) ZI + (1.57 XIO-') Z2 + 3.33 xlod= 0

(1.57 xloJ) Z, - (5.33 x10-') z2 + 5.56 xlod = 0

Solving simultaneously gives

z, = 7.73

Zz = 104.54

These 'Z values' are the additional forces exerted on the building causing extra

loading not present when static analysis was done (force of ineltia).

The dynamic response of the structure is summarized as in table 5.1;

TABLE 5.1

Multiplying M1 diagram and M2 diagra~n with the inertia forces

Fig 5.10: MI Z1 diagram showing the effect of the inertia force

INERTIA

FORCES (KN)

Zl = 7.73

Z2 = 104.54

NATURAL

FREQUENCY

(radlsec) -

WI = 61.09

Wz= 222.62

WTND

VELOCITY (mh)

40

EXCITATION

FREQUENCY

32.75

I-

Fig 5.11: M2 Z2 diagram showing the effect of the inertia force

Then the final bending moment diagram according to equation 5.2 gives

4.17 4.17 Fig 5.12: Final bending moment diagram:

CHAPTER SIX

6.0 DISCUSSION OF RESULTS AND CONCLUSION

1. Using the lumped mass analysis made the analysis easy because the inertia

forces were developed only at the inass points.

2. It was also observed that the greater the velocity of wind, the more the force it

exerts on the building comparing table 4.1 and table 4.5 which give the static

wind effect and dynamic wind effects at different velocities. It was observed

that the dynamic effect from the wind is even more than twice the static effect

and this should not be ignored in analysis (Gould et.al, 1980, Boswell et.al,

1993).

3. The dynamic response of the structure as in table 5.1 showed that the natural

frequency of the industrial building is greater than the excitation frequency

exerted on the building, which shows that the building cannot experience

resonance. The determination of the naturaI frequency is the foremost

fbndamental principle in the dynamic analysis and design of building

structures.

4. Also the result showed the additional force being exerted on the building in

the cause of vibration as in table 5.1. This calls for dynamic analysis, as these

forces are not prescnt when static analysis was done (Yi-Kwei wen, 1975).

This work investigated the dynamic analysis on industrial buildings subjected

to two degrees of freedom. It also showed the additional force of inertia generated

when a dynamic load induces vibration on the structure giving extra loading on the

structure as the wind changes speed and direction rapidly. These inertia forces are

completely absent when static loads are acting on the structure. For this reason, the

conventional wind force formula based on the quasi-steady assumption (inertia force

not considered) may not be applicable. The effects of inertia, therefore, need to be

examined (Y i-Kwei Wen, 1975).

Also the result of the analysis showed that the Ilatural ucque1iq IS high above

the frequency of forcing. This is used to check the possibility of the structure

entering into resonance, which is an undesirable phenomenon and avoiding it by

changing the structure's stiffi~ess or mass.

Also the dynamic wind effects on industrial building as included in the

dynamic analysis showed that wind has strong impact on industrial building being

light buildings. This is because from the analysis done, the dynamic wind effect is

greater than the static effect. So the static means of analyzing the ef'fccts of wind on

industrial building should be avoided (Yi Kwei Wen, 1975) and shou'.d only be used

in the preliminary consideration of the analysis (Boswell et.al., 1993). Therefore, to

determine the internal stresses, among which is bending momcnt, due to dynamic

effects, the basic principles enumerated in this work are recomlnended and as such,

no approximate solutions would be preferred

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