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TITLE PAGE

ANALYSIS OF CONTINUOUS THIN-WALLED BOX BEAM IN

PURE AND WARPING TORSION

A MASTER DEGREE RESEARCH PROJECT SUBMITTED TO

THE DEPARTMENT OF CIVIL ENGINEERING, UNIVERSITY OF

NIGERIA, NSUKKA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE AWARD OF THE DEGREE OF MASTER OF ENGINEERING

IN CIVIL ENGINEERING (STRUCTURAL ENGINEERING)

BY

MBACHU, VITALIS CHIDOZIE

(PG/M.ENG/07/42768)

ENGR. PROF. OSADEBE, N.N [SUPERVISOR]

MARCH, 2010

APPROVAL /CERTIFICATION

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This Thesis has been approved / certified for the award of the Degree of Master of Engineering

(M.ENG) in Civil Engineering

BY

REV. ENGR. PROF. OSADEBE, N.N. ENGR. PROF.ONYEYILI, I.O.

SUPERVISOR EXTERNALEXAMINER

ENGR. EZEOKONKWO.J.C

HEAD, CIVIL ENGINEERING

DEPARTMENT

ENGR. PROF. AGUNWAMBA, J.C

DEAN, FACULTY OF ENGINEERING.

DEAN, SCHOOL OF POST GRADUATE STUDIES.

DEDICATION

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TO

GOD Almighty,

The Giver of every thing that is good.

And to my parents,

Sir and Lady S.O MBACHU (KSM)

For the Love and Care shown to me.

ACKNOWLEDGEMENT

My sincere appreciation goes to God almighty, the giver of life and all good gifts.

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Also to my parents Sir and Lady S.O Mbachu, for their perseverance and belief in me. Anthony

Mbachu, Sylvia Mbachu, Miriam Mbachu Ayogu and Esther Mbachu and not forgetting the baby

Chimdi-uto Michael Emeka Ayogu (a light that brought a beacon of hope to the family)Ify Stella

Onubogu, Ngozi, Obiorah Onubogu and Pastor Sam Onubogu. May God bless you all.

I would sincerely like to express my gratitude to my Supervisor, Dean of Engineering faculty and

academic mentor, Rev .Engr. Prof. Osadebe, N.N, who provided me with the materials to go about

this thesis coupled with his incisive lectures on Dynamics of Structures, Soil-structure interaction

and Thin-walled structures and his consistent guidance, Thanks for being there.

Engr. Prof. Agunwamba, J.C. for his persistent encouragement. Engr. Dr. Nwoji who

recommended me to a job while studying. Engr. Dr. Okafor, F.O, Engr Amodou Adamou, Engr.

Nnaji Charles and staffs of Civil Engineering dept. UNN and my under graduate supervisor, Engr

Ezeokpube, G.C. Thanks for all your contributions .

All Catholic post graduate students of Nkrumah Catholic community CAPS executives of 2007/08

executives and members who loved and prayed for me when I served as President. Members of my

executives worthy to mention are Okeke Ogochukwu, Eze Ifeoma, Eboh Francis, Osuala Uzor,

Desmond Nnamani, Ndukaihe Izuchukwu and Engr. Mojekwu Emmanuel (Electronic engineering)

who was more than a friend to me,

Others are Anthony Ajah, Raphael Emeka, Emmanuel Ibuot and Amoke Louis the president and my

immediate successor and all the executives of 2008/2009 Catholic post graduate students. Thank

you for all your love and prayers.

My colleagues(class mates) and close friends worthy to mention are Engr. Oguaghamba

Onyedikachi, Engr. Mokwugwo Blessing, Engr Okereke Anyaora, Engr. Okonkwo Chidozie, Engr.

Gabriel Oluka Nke, Engr. Ojianya Dennis Engr. Duru Samuel, Engr. Idachaba and Engr. Onodugo

for the great times we shared together.

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May God in his mercy bless you all.

ABSTRACT

This work considers the response of thin-walled box structures in pure and warping torsion. In

practice, thin-walled members subjected to torsion have a distinguishing feature in that plane

sections do not remain plane. Eccentricity of load on the beams cross section generates torsion in

bridges and other structural members. The methodology of the study commenced with the

derivation of the torsional equilibrium equation on the basis of Vlasov’s stress displacement

relations by the calculus of variations. The existing equation due to St. Venant’s classical torsion

theory which was based on the independent summation of the combined torsional stresses was

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compared with the derived equation by Osadebe which took into consideration the interactive effect

of pure and warping torsion. Values obtained from the derivation were used to obtain closed form

expressions that explicitly described the torsional stresses along the continuous beam under various

loading and support conditions using the initial value approach to obtain deflections, rotations and

bi-moments. The results of the comparative analysis based on the numerical study further revealed

that the present equation based on the interactive effect of the combined torsional state shows a

significant effect on torsional rigidity, giving rise to higher stresses, thereby offering safe design.

TABLE OF CONTENT

Title page Page

Certification I

Dedication II

Acknowledgement III

Table of content IV

List of tables V

List of figures VI

List of symbols VII

Abstract VIII

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CHAPTER ONE 1

1.1 Introduction 1

1.2 Statement of problem 5

1.3 Assumptions in torsion theory 5

1.4 Significance of torsion 6

1.5 Objective of study 8

1.6 Limitations of study 8

1.7 Significance of study 9

CHAPTER TWO 10

LITERATURE REVIEW 10

2.1 Historical background of torsional analysis 10

2.2 Equilibrium torsion and compatibility torsion 11

2.3 Torsional theory 12

2.4 Pure torsion 13

2.5 Warping torsion 14

2.6 Behaviour of box section under general loading 14

2.7 Vlasov’s stress displacement relations 19

2.8 General torsional boundary conditions 20

2.9. Membrane analogy in torsion 21

2.10. Differential representation of st. Venant torsion constant 22

CHAPTER THREE

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3.1 Derivation and formulation of related equations and terms 27

3.2 Alternative formulation of torsional equilibrium equation 28

3.3 Vlasov’s displacement functions 29

3.31 Derivation of torsional equilibrium equation 30

3.4 Solution of the combined torsional resistance 34

3.5 Beam theory analysis 37

3.6 Concentrated torque 38

3.7 Initial value approach 40

3.8 Particular cases of support 43

CHAPTER FOUR

4.1 Torsional stress analysis 50

4.2 Continuous thin walled box beam subjected to an internal concentrated torque and uniformly

distributed torque 52

4.3 Box girder interface 54

4.4 Discussion of results 66

CHAPTER FIVE

5.1 Analysis of results and conclusion. 67

REFERENCES 69

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LIST OF FIGURES

Figure 2.1 Warping of a closed cross section

Figure2.2 Generalized loads separated into pure bending and a force couple

Figure 2.3 Force couple separated into pure torsion and distortion

Figure 2.4 Representation of torques on box girders

Figure 2.5a. Effective loading from eccentric gravity loading

Figure 2.5b. Example of torsion on spandrel beams

Figure 3.1. Free body diagram showing the relationship between distributed torque and

concentrated torque.

Figure 3.2. Simply supported thin walled box beam subjected to external concentrated

Torque.

Figure 3.3 Simply supported thin walled box beam showing reaction torques

Figure 3.4 Single cell box girder subjected to torque per unit length with arbitrary supports

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Figure 3.5. Single span with Fixed supports

Figure 3.6. Single span with Pinned supports

Figure 3.7. Box beam with Fixed-Pinned supports

Figure 4.1. Continuous thin walled box beam showing distributed torque

Figure 4.2. Continuous thin walled box beam showing applied eccentric loads.

Figure 4.3. Discretization of continuous span showing internal concentrated torque

Figure 4.41. Maximum torsional deflection values against ratio with table of values

Figure 4.42 Maximum first order twist values against ratio with table of values

Figure 4.43 Maximum second order twist values against ratio with table of value

Figure 4.44 Predicted values of torsional deflection with Fixed supports(distributed torque

Figure 4.45 Predicted values of first order twist (table of values) with Fixed supports

Figure 4.46 Predicted values of second order twist (table of values) with Fixed supports Figure

4.47 Predicted values of torsional deflection (table of values) with Fixed-Pinned

supports

Figure 4.48 Predicted values of second order twist (table of values) with Fixed-Pinned supports

Figure 4.49 Predicted values of first order twist (table of values) with Fixed-pinned supports

Figure 4.50 Predicted values of torsional deflection (table of values) with Pinned supports

Figure 4.51 Predicted values of first order twist (table of values) with Pinned supports

Figure 4.52 Predicted values of second order twist (table of values) with Pinned supports

Figure 4.53 Predicted values of torsional deflection (table of values) with Pinned supports

Figure 4.54 Predicted values of second order twist of the continuous span subjected to an internal

concentrated torque and a distributed torque. Figure 4.55

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Predicted values of second order twist of the continuous span subjected to an internal concentrated

torque and a distributed torque.

LIST OF SYMBOLS

Pure torsion

Modulus of elasticity in shear

Warping torsion

Torque about shear centre

Flexural rigidity

Torsional deflection (twist)

' First derivative of torsional deflection (slope)

Second derivative of torsional deflection (bi-moments)

Uniformly distributed torque

External concentrated torque

Improved St. Venant’s pure torsion constant

Existing St. Venant’s pure torsion constant

Internal indeterminate concentrated torque

Warping constant.

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Torsional parameter.

Eccentricity.

Uniform shear flow.

Moment generated in the cross section.

Potential energy.

Strain energy.

Work done.

change in rotation per unit length

LIST OF TABLES

Table 2.1- General torsional boundary conditions

Table 2.2- Analogy of membrane theory and torsion theory

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CHAPTER ONE

1.1

INTRODUCTION

Thin-walled structures comprise an important and growing proportion of engineering construction

with areas of application becoming increasingly diverse, ranging from aircraft, bridges, ships and

oil rigs to storage vessels, industrial buildings and warehouses. According to Schafer many

factors, including cost and weight economy, new materials and processes and the growth of

powerful methods of analysis have contributed to the growth in the use of thin walled elements [1-

3]. Thin walled structures are structures made from thin plates joined along at their edges. A

distinguishing characteristic of thin-walled structural elements is that two dimensions are much

larger than the third one, that is, the thickness. In general, thin-walled structures are structures in

which the ratio of the thickness (t) to the two other linear dimensions (h, l) ranges within the limits:

In a thin walled beam, the shear stresses and strains are much larger relative to those in solid

rectangular beams. When thin wall structures are twisted, there is a phenomenon called warping of

the cross section. The cross section warps and Bernoulli’s hypothesis is violated. Warping is

defined as the out-of-plane distortion of the beam’s cross section in the direction of its longitudinal

axis due to the twisting of its cross section [5]. This has led to the development of this research

work which concentrates specifically on single cell and continuous single cell thin wall structures in

which problems arise due to the torsion response of thin walled box beams. Structural codes during

the first half of the century were silent regarding torsion design. Thin wall open sections usually

have a very low torsional rigidity, combined bending and twisting occurs in axially loaded members

such as angles and channels, whose shear center axis and centroidal axis does not coincide.

According to Holgate , thin wall structures are generally assumed to include: structures made

of thin steel plates, such as box columns and beams, cranes, pressure vessels, pipelines, air- craft

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wings and steel silos, Also a limited range of concrete structures such as shells, folded plate, cooling

towers and some bridge elements. In complex structures such as curved beams, helical stairways

and eccentrically loaded box beams, torsional response dominates the structural behavior. Torsional

moments tend to twist the structural member around its longitudinal axis, inducing shear stresses.

However, structural members are subjected to pure torsion moment. In most cases, torsion moment

act concurrently with bending moment and shear forces.

In basic engineering structures such as bridges and buildings are subjected to dynamical loads, these

structural elements set up vibrations and interacting forces (bending moment stresses and torsion)

hinders its stability under load. Analytical formulations in structural engineering helps to

understand the behavior of complex structural systems .The subject of torsion is widely avoided

in most analysis due to the shear complication of solving a plethora of differential equations and

then correctly combining them with any additional planar shear and bending forces on the

structure[6-9]. According to Osadebe (2005), the recent multitude of research efforts invested in

thin-walled structures are justified by:

The optimization of profile geometries to fit aesthetic, strength and connectivity requirements, the

complexity of thin-walled structure behavior and the need for brief but accurate and reliable design

method in order to remain time competitive.

Stress is defined as the intensity of force per unit area, the components of stress includes normal

stress which is perpendicular to the section and shearing stress which is tangential to the system, a

stress resultant may either be a normal or shearing force, similarly, a stress couple may be a bending

moment resulting from the normal stress components, or twisting moment resulting from shearing

stress components. A twisting moment is also called a torque. Broadly speaking, the term “force” is

used to imply either force or couple. Hence the term stress resultant may be used to imply either

internal force or internal couple.

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Torsional stress may also be transmitted through diagonal members to the lateral bracing and the

girders resistance to such stress is a very important factor in bridge design because the main girder

would have to function as a torsion bar [10]. Torsion is an internal stress resultant, a distortion

caused by applying torque in opposite direction to the end of an object. The torsion response of

structural elements of structural elements can be classified into two categories namely:

i. Pure torsion

ii. Warping torsion.

Engineering materials employed in rotating machine parts, such as engine crankshafts and ship-

propeller shafts, must resist torsion stress. Torsion (twisting) is created when moment or turning

force is applied to a structural member making it to deflect at an angle(twist)- a moment which

causes twisting is called twisting moment or torsion moment. Torsion actually produces shear

stresses inside the material- a beam in torsion will fail in shear (the twisting action causes the

molecules to slide apart). Torsion loading occurs when the line of action does not pass through the

shear centre of the beam. The shear center is an imaginary point on a section, where a shear force

can be applied without inducing any torsion. In general, the shear center is not the centroid. For

cross-sectional areas having one axis of symmetry, the shear center is located on the axis of

symmetry. For those having two axes of symmetry, the shear center lies on the centroid of the cross-

section. The shear centre for I-beam as well as box-tubular-, and circular-shaped beams, coincides

with the centre of gravity. However, it does not coincide with the centre of gravity of channel-

shaped and other non-symmetrical open sections beams. The twisting moment, or torque induced by

torsion loading is similar to bending moment in that its value is the product of the load and its

perpendicular distance to the shear centre.

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Shearing, a term frequently used in continuum mechanics refers to the occurrence of shear strain- a

deformation of a material in which parallel internal surfaces slide past one another. Shearing is

induced by a shear stress in the material.

Torsion on open sections causes three distinct types of stresses namely

i. Torsion shear stresses.

ii. Warping shear stresses

iii. Warping normal stresses

All these stresses must be algebraically added to the stresses caused by the normal bending of the

beam. This summation must be ensured that it is within the prescribed limits of AISC.

Straight girders in bridges develop such forces where torsion and bending are simultaneously

instituted. The problem of torsion arises when the load applied on the structure is eccentric (off

centre) with respect to the girder axis (neutral axis) or shear centre for a beam that has a centroidal

axis. Engineers are thus compelled to determine shear centre, resolve forces into bending and

torsion forces and then determine stress resultants and deformations.

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1.2 THE STATEMENT OF THE PROBLEM

This work seeks to investigate the torsional stresses emanating from statically applied torsional

loads on structural thin walled continuous box beams and single span box beams over varying

support and loading conditions. It also compared the conventional equation by St. Venant’s which

was based on the independent summation of the combined torsional states and the improved

equation by Osadebe (1993) was based on the modal interaction of pure and warping torsion. In the

numerical analysis, the method of initial value approach was used.

1.3 ASSUMPTIONS IN TORSION THEORY

Classical theory due to St. Venant assumes that

(i) Members are prismatic and straight.

(ii) Behavior is elastic(Hooke’s law is valid)

= ' (1.1)

Equation (1.1) assumes further that,

(iii) Cross section is thin walled and displacements are small compared to the length of the

member.

(iv) Navier’s hypothesis holds- plane section remains plane during the loading of the section.

This assumption allows the strains to be proportional to the neutral axis (except for

warping torsion).

(v) Thickness of the beam is much smaller than a representative dimension of the cross section.

(vi) Out of plane distortion (warping) of the beam section vary with the rate of twist.

(vii) Beam has a constant flexural rigidity.

(viii) Warping function remains constant along wall thickness.

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(ix) The effect of the displacement of the axis of the beam and change of geometry is not taken

into account in the equilibrium equation.

1.4 SIGNIFICANCE OF TORSION

Torsion loading occurs when the line of action does not pass through the shear centre of the beam.

The shear centre is an imaginary point on a section, where a shear force can be applied without

inducing any torsion. In general, the shear centre is not the centroid. For the cross sectional areas

having one axis of symmetry, the shear centre is located on the axis of symmetry. For those having

two axis of symmetry, the shear center lies on the centroid of the cross section. The shear centre for

I-beams as box, tubular and circular shaped beams coincides with the centre of gravity. However, it

does not coincide with the centre of gravity of channel-shaped and other non- symmetrical open

section beams. The twisting moment, or torque induced by torsion loading is similar to bending

moment in that its value is the product of the load and its perpendicular distance to the shear centre.

The effect of torsion in a box girder is influenced by the radius of curvature, span length, and out-

to-out width of box structure, depth of structure and thickness of deck, soffit, and exterior girder

webs. Increasing is the use of voided and box beams with no diaphragms or other internal stiffeners,

the resultant problem of what torsional stiffness to allow and how to know the effect of warping or

shear deformation becomes important. The effect of warping cannot be neglected in obtaining the

steel strength properties since warping constant plays an important role in finding the value of the

equivalent slenderness of the beam. Torsion is seldom relied upon as a significant load path in the

design of steel buildings. There is no simple rule of thumb to determine whether torsion is a

significant factor to be considered in the design of any particular curved structure. Structures

generally do not carry torsional moments, but torsion can be an integral part of stability problem

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[11] as stability of a flexural member is often a function of its torsional stiffness. Some of the

significant effect of torsion are:

i. Imparts angular displacement of one end cross section with respect to the other.

ii. Sets up differential girder rotation causing super elevation misalignment due to

shearing stresses.

Frequently, torsion is a secondary, though not necessarily a minor effect that must be considered in

combination with other types of behavior, although circular shafts such as tubes or pipes are ideal in

torsion loading situations. These shapes do not carry or transfer other type of loading in an efficient

manner. A beam has its materials distributed very far from its centroid. Thus, their resistance to

bending at the extreme fibers is increased since most of its materials are positioned very far from

the neutral axis.

The fundamental concept of the theory of elasticity which is primarily concerned with the study of

the response of elastic bodies under the action of forces will be used to obtain the torsional

equilibrium equation. When the relative positions of points in a continuous body are altered, the

body is strained. The change in the relative positions of points is called deformation.

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1.5 OBJECTIVES OF THE STUDY.

This study is carried out with the intent on the following objectives:

1) To review earlier work done on the analysis of box girders in pure and warping torsion.

2) To analyze the behavior of box girders, taking into account the dimensional properties of its

cross section.

3) To establish particular closed form expressions for the analysis of continuous box girder

structures using the initial value approach

4) To compare the results of the conventional equation of St. Venant which was based on the

independent summation of the torsional states and the present equation based on the

interactive effect of the combined torsional states generated by Osadebe that took into

consideration the effect of modal interaction.

5) To determine the torsional deflection, first and second order twist (rotation and bi-moment)

of a thin walled box beam.

1.6 LIMITATION OF THE STUDY

1) This study treats the torsional stress analysis of only static loads on box beams neglecting

dynamic loads.

2) Torsion is an integral part of stability as stability analysis is not carried out.

3) The study also does not treat or consider torsional analysis of thin-walled box girder

subjected to local buckling and overall buckling.

4) Beam of variable cross section was not considered.

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1.7 SIGNIFICANCE OF STUDY

This study will hopefully be relevant in the following respects:

1) The analysis and design of thin wall structures in torsion.

2) This work would be of immense appeal to post graduate student beginners of thin wall

structures and it’s analysis in box beams.

3) To the members of Academia.

4) The regulatory body for the structural designs.

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CHAPTER TWO

LITERATURE REVIEW

2.1 HISTORICAL BACKGROUND OF TORSIONAL ANALYSIS.

Pure torsion or St. Venant torsion generally occurs when the cross section remains planar during

twisting. Warping is the out of plane effect which causes the cross section lateral displacement. The

stresses acting on a cross section when torsional moment is applied to a member consists of pure

torsional stresses, shear stresses due to warping and normal stresses due to warping. Pure torsional

shear stresses are the in- plane shear stresses that vary linearly through the thickness of the cross

section and act in a direction parallel to the edge of the cross section. They are maximum and equal,

but in opposite directions along the edges of the cross section, but vary in magnitude along the

width of the cross section. Torsion is usually associated with bending moments and shearing forces,

and the interaction among these forces is important [12]. Due to the limitations of Bernoulli’s

theory as regards thin wall structures under generalized loading, the search for a more acceptable

method of analysis was necessitated. The theory proposed by V.Z Vlasov, a Russian scientist

amongst others was accepted and was adopted [13-14].

Scientists and a few structural engineers have over the years immensely contributed to the research

in torsion analysis. Below are some of the key contributors on the historical development of

torsional analysis: French Engineer, Adhemar Jean Barre de Saint Venant (1853) presented the

classical torsion theory to the French academy of science. A. Michell and L. Prandtl’s (1899)

presented results on flexural torsional buckling. S.P Timoshenko (1905) presented a paper on the

effect of warping torsion in I-beams. C. Bach (1909) noted the existence of warping stresses not

predicted by classical torsion theory when shear centre and centroid do not coincide. V. Wagner

(1929) began to develop a general theory of flexural torsional buckling. Von Karman and

Christensen (1944) developed a theory for closed sections. Benscoter (1954) developed a more

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accurate theory for closed sections. Urban (1955) developed a theory for closed section isotropic

beam with uniform cross section.

V.Z Vlasov (1906-1958) the pioneer of thin walled beam theory presented a solution for isotropic ,

closed section beams containing of flat walls, his solution in pure torsion, he assumed independent

warping functions for each wall segment and hence, no cross sectional properties were presented.

He also developed the theory of general bending and twisting of thin walled beams. Osadebe (1993)

in his work on modal interaction revealed the deficiency of St. Venant’s classical torsion theory on

thin walled box girder structures using Vlasov’s variational approach. Osadebe and Mbajiorgu

(2002) carried out advanced finite element formulation of the torsional behavior of thin walled

beams with arbitrary open and closed cross section.

2.2 EQUILIBRIUM TORSION AND COMPATIBILITY TORSION.

Whatever the cause of the internal forces and deformations in a structure, three basic conditions

must be observed. These are

(i) the equilibrium of forces.

(ii) (ii) The compatibility of displacements and

(iii)(iii) the laws of material behavior [15].

The first condition requires that the internal forces balance the externally applied loads/forces. The

second condition requires that the deformed structure fits together, i.e. that the deformations of the

members are compatible. The third condition shows the relation in linear elasticity in which

Hooke’s law is valid.

Equilibrium torsion occurs when the torsional resistance is required to maintain static equilibrium.

In this case, if sufficient torsional resistance is not provided, the structure will become unstable and

collapse. External loads have no alternative load path and must be resisted by torsion. Compatibility

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torsion develops where redistribution of torsional moments to adjacent members can occur. The

term compatibility refers to the compatibility of deformations between adjacent parts of the

structure. As an example, a spandrel beam supporting an external slab. As load on the slab

increases, so does the negative slab end moment, which induces torsion in the spandrel beam. The

negative slab moment will be proportional to the torsional stiffness of the spandrel beam. In cases

where equilibrium torsion exists or torsional behavior dominates the structural response, engineers

must design for maximum torsional moments [16].

2.3 TORSIONAL THEORY

Torsional moments in thin wall sections such as box girder are resisted by the shear stresses on the

components that make up the girder cross section. Although both pure and warping torsion are

developed in box girder section, pure torsion often dominates the stiffness of the closed cross

section. Box girders are made of two flange plates and two web plates imparting to them high

torsional strength and rigidity. Flanges in a box girder and webs in plates and box girders are often

reinforced with stiffeners to allow for efficient use of thin plates.

Saint Venant’s torsion is the result of pure shear deformations in the planes of the plates that

comprises the thin wall member, while warping torsion is associated with the bending deformation

in the planes of the individual plate. Although, St. Venant’s pure and warping torsion are developed

in box girder sections, St. Venant’s torsion often dominates the stiffness of the closed cross section.

The longitudinal normal stresses due to warping torsion are usually negligible (kollbrunner and

Basler 1969). The torsion of a thin – walled box section is a function of the shear modulus of the

material and the torsional constant, which is related to the geometric profile of the cross section.

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2.4 PURE TORSION

The basic governing equation for an elastic member subjected to torsion is given as

(2.1)

St. Venant torsion is a function of the rotation of the beam and its inherent torsional stiffness.

The shear stress due to St. Venant torsion can be solved using Prandtl’s membrane analogy. For

girders with a single cell cross section, a uniform shear flow, , develops along the perimeter of

the box and can be determined using the Bredt’s equation[17].

For closed sections, the pure torsion constant or torsional moment of inertia is given by equation

(2.3)

For a Thin wall box beam, St. Venant pure torsion constant is given by

Fig 2.1 Warping of a closed cross section

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2.5 WARPING TORSION

The basic governing equation for an elastic member subjected to warping torsion is given by (Basler

and Kollbrunner, 1969).

Where is the elastic modulus of elasticity and is the warping constant. Warping moment is an

innate characteristic

c of every beam. It abets to resist any externally applied torque in combination with moments

caused by pure torsion. As a result of the uneven axial deformation caused by warping stresses,

plane section no longer remains plane. Shearing stresses cause out of plane deformation of the

beams cross section resulting in warping. For a thin walled box beam used in this research;

2.6 BEHAVIOUR OF BOX SECTION UNDER GENERAL LOADING

Study has shown that a general load applied to a box section is a combination of three components –

namely bending, torsion and distortion.

In the first step, the force can be separated into two components: a pair of symmetric vertical loads

as shown in figure 2.2

In the second step, the force couple (torsion) is separated into pure torsion and distortion

components as shown in figure 2.3

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Fig 2.2. General loads separated into pure bending and a force couple

Fig 2.3. Force couple separated into pure torsion (rotation) and distortion

The magnitude of the torsional shear stresses component as shown in Figure 2.3 is evaluated.

From torsional components )

From pure torsional state

From pure distortional state

From equation 2.6c

P/222222

P/2

P/2

P/2

=

+

a

b

P

Y

Torsional component

+

=

Y

X

Y

Pure torsional state

Pure distortional state

Y

a

b

X

X

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From equation 2.6a,

Stresses on the box beam obtained by fig. 2.3 evaluated is given by equations 2.7a and 2.7b

Members subjected to torsion have a distinguishing feature in that plane sections do not remain

plane when loaded. This phenomenon causes the cross section of members to warp. Pure torsion

resists the applied loads through shear stresses in the plane of the cross section. During loading,

displacements occur in and out of plane. Torsional stiffness of a cross section consists of both

warping and St. Venant component; however, the high torsional stiffness of boxes is primarily due

to the large St Venant’s component that results from a closed cross section. Since the St. Venant’s

term dominates the torsional stiffness, torsional warping stresses in boxes are usually relatively

small (Kollbrunner and Basler 1969). Depending on the distribution of torsional loads, the cross

section of a box girder may distort from its original shape. This distortion of the cross section can

lead to significant transverse bending and longitudinal warping stresses, which are in addition to

torsional warping stresses. Warping stresses that develop as a result of distortion of cross section are

appropriately referred to as distortional warping stresses.

A box girder is a special case of a folded plate structure in which the plates are arranged to form a

closed section [18]. Segmental box girder structures are common nowadays for the construction of

long elevated superstructures over busy road or obstacles. This construction minimizes the amount

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of formwork during construction and reduces the assembly time. Torsion in box girders is usually

the result of horizontal curvature of the girder or unbalanced gravity loading that results in an

eccentricity of the load on the cross section.

From the figure, e is eccentricity, M is moment and V is shear force.

Fig 2.5(a).effective loading from eccentric gravity loading.

Fig 2.5(b) example of torsion in a spandrel beam.

Vlasov ) first studied the distortion of box girders while investigating the torsional behavior of

thin walled beams with a closed cross section. Dabrowski ( ) established a more rigorous

theory when he developed the governing equation for box girder distortion.

Distortional analysis requires the separation of the distortional components from the applied

torsional loads as all practical loading cases causes some cross sectional change in geometry.

Fig2.5(b)

e M

V

30

CLOSED THIN WALL SECTIONS

Fig 2.6 Box beam Cross-Section

Not withstanding the availability of multitude of profiles, thin-walled structural elements are

generally grouped into; open and closed profiles. When in use as columns or beams, thin-walled

box structures can have three different types of arbitrary cross-sections.

Mono symmetric sections

Asymmetric sections

Doubly symmetric sections

31

2.7 VLASOV STRESS DISPLACEMENT RELATIONS

On the basis of Vlasov theory, the longitudinal displacement field U(x, y) of the thin walled box

beam structure is given by

Where are the unknown functions which express the laws governing the variation of

the displacement along the X-axis of the box structure.

are the strain modes of the single cell box cell respectively. Out of plane (m-

displacement) and in plane displacement (n-displacement).

From the theory of elasticity, the normal and shear strain are given by

The bending moment due to the distortion of the cross section

Where is the moment generated in the cross section by the application of a unit lateral

displacement.

32

2.8 GENERAL TORSION BOUNDARY CONDITIONS

The torsional properties for calculating warping normal stresses and pure stresses depend on the

boundary conditions that prevail at the ends of the member. These boundary conditions are defined

as free, pinned, or fixed.

Free: represents the boundary condition such as that which exists at the free end of a cantilever

beam. No other member is connected at that point

Pinned: Pinned represents the condition that corresponds to a pinned support defined at the joint

through the support

Fixed: Fixed represent the condition where a fixed support exists at a joint conforming to the

assumptions of equation 1.1,

The generalized torsional boundary and interface conditions are valid for any beam arrangement.

The boundary conditions are common for end restraints and are listed with the various end

conditions in table 2.1.

Table GENERAL TORSIONAL BOUNDARY CONDITIONS

33

location Geometric

boundary

conditions

Equilibrium boundary

conditions

Free end --- ″

n

Pinned end ″

Fixed end

Free interface

(j total)

Loaded

unsupported

interface(n total)

Interior supports

(I supports)

2.9 MEMBRANE ANALOGY IN TORSION

The solution of the basic torsion equation was first proposed by Prandtl’s in 1903. He identified this

analogy for the stress distribution of non circular shafts under tension. The technique permits the

visualization of the ψ function by considering the inflation of a hypothetical thin membrane over the

34

exact cross section, which is subjected to a torsional moment . A membrane is a structure whose

thickness is small in comparison to the surface area. It has a negligible bending rigidity, a

membrane is the two dimensional equivalent of a string under tension just as a plate is a two

dimensional equivalent of a beam, an example of a membrane is a soap film. This analogy is

applied only to isotropic materials. The applicability of the membrane theory is used in determining

torsional constant.

2.10. DIFFERENTIAL REPRESENTATION OF ST. VENANT’S TORSION CONSTANT.

In general, deformations consist of rotations of cross sections together with displacements (out of

plane of the cross section) is termed warping.

From the fundamental concept of elasticity theory, the expression for torque about the shear centre

from an element is given by

(2.15)

The element must be in equilibrium

This equation boils down to

The relationship between rotation of the section and displacement will permit an interrelationship

between geometry, material properties and stresses.

In plane distortions are related to the rotation of the cross section as follows

35

Where U and V are horizontal and vertical displacements respectively

Shearing stresses results in z and x axis

Differentiating equations 2.18 and 2.19 to obtain the rate of change of displacement

(2.22)

(2.23)

Change in rotation per unit length

For an elastic body,

=

(2.27)

Also, the 3D form of Hooke’s law indicates the stress field

For displacements parallel to the x- and y-axes. Ѳ is the rotation of the cross section per unit length.

Assuming the warping is represented by the function.

W

36

Longitudinal normal stresses do not exist. Substituting the above relations into the differential

equation of equilibrium

Yields

Since

Two equations exist for three equations. Introducing Airy’s stress functions

(2.36)

Substituting the variable is eliminated

obtaining

Equation Equation 2.15 and 2.38 are the required equations to solve

for stresses and distortions of a section. Equation is now modified

37

+

Using integration by parts, Equation becomes

Using membrane analogy as developed, shearing stresses is given by

Summation of moment about shear centre gives

(Area enclosed by the wall elements)

(2.44)

From the analogy of the membrane,

G (2.47)

Or,

38

The above pure torsion constant for a thin walled box beam is

Equation 2.47b as developed from the membrane theory and stated by Prandtl’s is the torsional

constant analogous to equation 2.48, which is used in the present formulation.

Table 2.2. ANALOGY OF THE MEMBRANE THEORY AND TORSION

Membrane problem Torsion problem

Main variables Displacement normal to membrane,

U1,pressure under membrane, tension along

membrane, assumed constant

Stress function,

Shear modulus G. twist rate

assumed constant

Governing PDE

Volume Volume under membrane

Resisting moment

Boundary

conditions

on contour on boundary

The displacement function in the partial differential equation of the membrane is mathematically

analogous to the stress function

39

CHAPTER THREE

METHODOLOGY

3.1 DERIVATION AND FORMULATION OF THE RELATED EQUATIONS AND

TERMS.

INTRODUCTION

This chapter examines the action of a box section subjected to a torque as resisted by pure torsion

stresses (St. Venant torsion) and warping torsion stresses. Basic derivations of torsional equilibrium

equations that will lead to values as obtained by Osadebe to evaluate torsional stresses (deflection

(twist), rotations (slope) and bi-moments) in continuous thin walled box beams will be developed in

this chapter. The solution to the governing differential equation for torsional equilibrium will be

derived. The complementary and particular solutions will be obtained. Formulations of closed form

40

expressions will be developed with respect to various support conditions using the initial value

approach for various spans will be developed.

Fig 3.1 Free body diagram showing the relationship between uniformly distributed torque and

concentrated torque.

The equilibrium condition for a discrete element length acted upon by a distributed torsional

load is given by equation 3.2

or

3.2 ALTERNATIVE FORMULATION OF TORSIONAL EQUILIBRIUM EQUATION.

Osadebe (1993) used Vlasov’s stress- displacement functions and related stress –strain expressions

to derive the equation of torsional equilibrium of a single cell box structure subjected to pure and

warping torsion. The obtained equation is given by

(3.3)

Where E and G are the modulli’s of elasticity and rigidity.

The conventional equation by St. Venant’s classical torsion theory is

(3.4)

41

From the relation,

Where,

St. Venant’s pure torsion constant and warping torsion constant is given by

These values as obtained by Osadebe will be used in this work as a continuous single cell box

structure will be analyzed using both equations (3.3 and 3.4)

3.3 VLASOV’S V.Z DISPLACEMENT FUNCTIONS.

Osadebe (1993) on the basis of Vlasov’s theory derived expressions of Torsional equilibrium by the

calculus of variations.

From Lagrange’s principle, Vlasov expressed displacement in the longitudinal and transverse

direction in a series form for a thin-walled closed structure subjected to an external torque.

42

Where are the unknown functions which express the laws governing the variation

of the displacement along the X-axis of the box structure.

Components are the strain modes of the single cell box cell respectively.

Out of plane (m-displacement) and in plane displacement (n-displacement).

3.31 DERIVATION OF TORSIONAL EQUILIBRIUM EQUATION.

From Vlasov’s concept, the position of a point on the middle cross section of a thin walled closed

box structure is determined by the coordinates “x” in the longitudinal X-direction and “y” in the

transverse S-direction. Vlasov assumed that the plates are in a plane state stress, the strain can

therefore be completely described knowing , the displacement along the tangent to the

contour of the transverse section produced by the applied torsional load.

From the theory of elasticity [19-23], the normal and shear strain are given as

43

Consequently, the normal and shear stresses are given by equation 3.16 and 3.17,

Substituting equation 3.13 into 3.15 gives

Bending moment due to the distortion of the cross section is given by,

The strain energy of the box structure is given by

Work done by the externally applied torque is given by

44

Potential energy of the torsionally loaded thin wall box beam is given by

Using the constitutive relations

Similarly,

Substituting 3.27 and 3.30 into 3.25.

45

Expanding the above and integrating the series

The functional of the above is minimized using the Euler- Lagrange equation by the calculus of

variations [24-25].

Substituting 3.34 into 3.351 and 3.352

46

Dividing equations 3.38 by G

Differentiating equation 3.40 with respect to x,

Adding equation 3.40 to 3.39 eliminates the warping function and its derivative

The derived equation using Vlasov’s variational approach derived by Osadebe is given by

Equation 3.43 is also reduced by equation 3.2 for particular concentrated loading cases

3.5 SOLUTION OF COMBINED TORSIONAL RESISTANCE

The combined torsional resistance of a box beam is comprised of the pure resisting torque and the

warping resisting torque. Solution of equation (3.43) is broken down into two parts since it is a non

homogenous differential equation that entails a homogenous solution and a particular solution.

For a concentrated torque, the governing differential equation is given by equation 3.44

Applying the equilibrium condition for a discrete element length acted upon by a distributed

torsional load as given by equation 3.2 yields equation 3.46

47

Let and

In order to solve for the homogenous part, equation (3.45) is written as

Assume the solution is of the form

.

Total homogenous equation can be expressed as

The particular solution of 3.48 is obtained since the right hand side of the non homogenous equation

is a constant, and the lowest order derivative of in the O.D.E is one. Assuming where

A is a constant determined by substituting the assumed particular equation into 4.4. Thus, the

particular solution.

Let

48

Total solution is the sum of given by equation 3.522 and 3.51

The exponential function in hyperbolic equivalent is

=

Equation 3.46 is solved similarly,

Equation 3.55 has the complementary solution,

Roots of the equation are:

The complementary function for equation 3.55 is given by,

+ (3.58)

The particular solution is given by,

Let

49

Substituting 3.61 into 3.59:

Adding equations 3.58 and 3.62 to obtain equation 3.63:

= + )

3.6 BEAM THEORY ANALOGY IN TORSION ANALYSIS

In Bernoulli-beam theory, the moment curvature relationship is written as

(3.64)

Where is the displacement function and x) is the variation of bending moment of the beam

having a constant flexural rigidity. A similar expression exist in torsion analysis

(3.65)

Where is referred to as the bimoment of the beam. In ordinary flexural analysis, the bending

moment is a function of second derivative of displacement. Similarly, in torsion analysis, the

bimoment is a function of the derivative of the rotation or the out of plane displacements. If the

50

derivative of the governing bending moment equation is taken, the shearing force acting on the

beam can be determined since the shear force is the first order

change of moment. Analogously, the derivative of the bimoment yields warping moment; hence, the

warping moment corresponds to the shear force in a beam subjected to pure flexure.

Substituting the equation for bimoment into the equation for normal warping stresses gives the

derivative of the warping moment gives the equation for the loading on the beam.

(3.66)

Equation 3.66.an expression in torsion analysis is similar to the plain bending stress equation given

in strength of materials.

(3.67)

INITIAL VALUE.

3.7 CONCENTRATED TORQUE

Simply supported thin walled box beam subjected to an external concentrated torque. Concentrated

torque may be internal or external.

Fig 3.2 Simply supported box beam subjected to an external concentrated torque.

Fig 3.3 simply supported box beam subjected to an external concentrated torque applied at mid span

showing reaction torques

51

The Homogenous equation is

(3.70)

Has the complementary function given in equation 3.71

(3.71)

The last component of equation 3.72 and 3.73 is the particular integral for the beginning and end of

the section respectively.

The two solutions are identical at mid-span and by symmetry

52

Considering the left half of the beam first, the boundary conditions at the origin are

:

Hence, for the left half of the beam,

For the right hand side of the beam, placing the origin at the right end running to the left

3.8 INITIAL VALUE APPROACH

In engineering practice and applications we are not only interested in the general solution

(complimentary solution and particular integral) of a given differential equations, but also in

the particular solution satisfying the given initial conditions.

The initial value approach is adopted such that all the conditions necessary for obtaining the desired

particular solution are all given at the origin.

Initial conditions

53

At

Consider shown in figure 3.4

Fig 3.4 Single cell box girder subjected to a torque per unit length with arbitrary support conditions

Taking equation 3.63 and its successive derivatives and obtaining the following equations

INITIAL VALUES (at x=0)

(a)

(b)

(c)

(d)

From equation 3.79

54

From equation 3.80

From equation 3.81

Thus,

=

Substituting the constants into equations (3.79-3.82)

= + +

55

Equation 3.83 through 3.86 is the closed form expressions to be used depending on the support

conditions and torsional boundary conditions.

3.8 PARTICULAR CASES OF SUPPORT

Fig 3.5 Single span box beam subjected to distributed torque with fixed supports

At x=0; 0, At x=L; 0,

Equation 3.83 reduces as,

= +

(Equation 3.84 reduces as)

Invoking the torsional boundary condition at L

x

LENGTH L

56

Re arranging above,

Using 0

Solving 3.90 and 3.92 simultaneously to obtain

Expressions for can be obtained by substituting into the general expressions

will permit a solution for uniformly distributed torque over a thin walled box beam fixed at both

ends.

x

57

Fig 3.6 Box beam under a distributed torque with pinned supports.

At x=0; 0, At x=L; 0,

From the torsional boundary conditions, equation 3.57 reduces as

= +

Invoking the condition at L. 0 on equation 3.57 gives

0= +

The boundary condition

Substituting the expressions of and into equation 3.83 to 3.86 will permit a solution of a

torque per unit length over a thin walled box beam torsionally pinned at both ends.

58

x=0; 0, x=L; 0,

Fig 3.7. Fixed ended thin walled box beam under a concentrated torque.

Boundary conditions

At ,:

)

At ,:

)

)

From a ;

(3.102)

From (b);

From (d);

Substituting the constants into the equations gives the closed form expressions

59

x=0; 0, x=l; 0,

Fig 3.8. Fixed-Hinged box beam under the effect of an externally applied torque.

Boundary conditions app

At ,:

At ,:

60

From (a)

From (b)

From (d)

LENGTH L

x

61

x=0; 0, x=l; 0,

Fig 3.9 Box beam under a distributed torque with fixed-pinned supports.

At x=0; 0, At x=L; 0,

Conditions applied at x=0.equation (3.83-3.86) becomes

= + (3.112)

(3.113)

62

CHAPTER FOUR

4.1 TORSIONAL STRESS ANALYSIS

In an indeterminate continuous box beam, it is required to determine the torsional deflection and its

successive derivatives. In essence, the method of the initial value will be used as it gives all the

solution of the beam at the origin. The continuous span in figure 4.1 is a life continuous span

subjected to eccentric loading creating torsion. End conditions are assumed simple which translates

torsionally to

The statically indeterminate internal torque will be obtained on the separation of the spans as

obtained in figure 4.3. Continuous beams are indeterminate between simple spans and fixed –ended

beams. Interior spans resemble the fixed case more closely as observed from the boundary

conditions and the free end of the exterior span may be closer to the simple span case. Continuity in

spans over supports can significantly reduce torsional moment. The unknown constants inherent in

the each of the equations 4.1 and 4.4 will be appropriately determined using the table of torsional

boundary conditions as given in table 2.1.

The torsional response of a continuous doubly symmetric thin wall beam will be solved by applying

the basic equation.

63

Fig 4.1. A continuous box beam showing distributed torque.

Fig 4.2. Hollow Box cross section showing Eccentric load

x= L x=+L X=0

Girder 1 Girder 2

64

Fig 4.3. Discretization of the continuous span showing the internal concentrated torque that exist.

4.2 CONTINUOUS THIN WALL BOX BEAM SUBJECTED TO AN INTERNAL

CONCENTRATED TORQUE AND A UNIFORMLY DISTRIBUTED TORQUE.

The general response of girder 1 is given from (40)

=

Equation 4.1 gives

Torsional response of the second span earlier obtained in equation 3.63 is given by

Successive derivatives of equation 4.4 gives

From equations (4.1 to 4.6) eight unknown constants exists as shown in the matrix form in equation

4.7

65

These boundary conditions will be obtained from table 2.1

Solution of girder 1 is given by

=

At

(4.8a)

(4.8b)

At

From 4.3,

At

From 4.8b,

Substituting equation 4.8b into 4.9

written as

Solving 4.1a and 4.12 simultaneously, gives

Recalling that,

66

Substituting as given in 4.12a into 4.12,

Torsional response of Girder one in deflection and its successive derivatives are obtained in

equations 4.14, 4.15 and 4.16.

4.3 BOX GIRDER INTERFACE

The conditions are given in equation 4.17 and 4.18.

At

(4.17)

(4.18)

And reducing the statically internal indeterminate internal concentrated torque as obtained in

equation (4.21)

67

Torsional response of girder 2 is given by

The values obtained on the basis of the difference of the torsional constants will be used in the

graphs plotted.

The constants are given by equations 4.24 through 4.27

68

The calculations yielding the graphs (values) were obtained using the Microsoft excel calculation

spread sheet. The results obtained from the given example leads to the plotting of the following

graphs.

(i) Torsional deflection(twist) against beam length

(ii) First order twist(rotation) against beam length

(iii) Second order twist(bimoment) against beam length

(iv) Torsional deflection maximum values against and ratios.

These values are plotted for the following support conditions for uniformly distributed

torque and concentrated torque.

(i) Fixed –Fixed support.

(ii) Hinged –Hinged support.

(iii) Fixed- Hinged support.

These constants will permit the evaluation of torsional stresses long the continuous span

69

; ,

FIG 4.41 Plot of Maximum Torsional Deflection against (H1/ H2 Ratio) At The Mid span (x =

2.5m)

FIG 4.42 Plot of first order twist against (h1/ h2 ratio)

; ;

70

FIG 4.43 Torsional Deflection Values(Fixed-Fixed Support) For Both Equations

; ,

FIG 4.44 Torsional Rotational Values (Fixed-Fixed Support) For Both Equations.

71

FIG 4.45. Second Derivative of Torsional Deflection (Fixed-Fixed) For Both Equations.

FIG 4.46. Torsional Deflection Values (Fixed-Pinned Support) For Both Equations

; ;

72

FIG 4.47 Comparison Of Second Derivative Of Torsional Deflection Values (Fixed-Pinned

Support) Subjected To A Uniformly Distributed Torque.

FIG 4.48 First derivatives of torsional deflection values (Fixed-Pinned Support).

73

FIG 4.49 Torsional Deflection Values (Pinned Supports) For Both Equations.

FIG 4.50. Torsional Rotational Values (Pinned Supports) For Both Equations.

74

FIG 4.51. Comparison of Second Derivative of Torsional Deflection Values (Pinned-Pinned

Support).

FIG 4.52. Predicted Values Of Second Derivative For Torsional Deflection For The Two Span

Continuous Box Girder.

75

Fig 4.53 Values Of First Order Derivative Of Torsional Deflection For The Continuous Span.

76

FIG 4.54 Values Of Torsional Deflection For The Continuous Span

FIG 4.55 Combined Torsional Stresses For Fixed-Pinned Box Beam Subjected To A Uniformly

Distributed Torque

FIG 4.56 Combined Torsional Stresses For A Thin-Wall Box Beam (Pinned Supports) Subjected

To A Uniformly Distributed Torque.

77

Fig 4.57 Simpy supported thin walled box beam subjected to an external concentrated torque

78

4.4 DISCUSSION OF RESULTS

Figure 4.41 shows the relationship between the equations as the torsional rigidities become equal

when the box cross section and . At values less than 1 and greater than 1, both

equations exhibit inverse relationships for the maximun stresses obtained at the midspan.the same

also goes for figure 4.42.

Figure 4.43 also reveals that the torsional stress obtained on the basis of the interactive effect of the

combined torsional states generates a significant higher stresses than the conventional equation. The

fixed ended box beam possesing boundary conditions for torsional deflection generates a relatively

significant stress in the comparative analysis. This same result is extended to figure 4.44 and figure

4.45 which shows the rotation and second order derivative for torsional deflection(bi-moments).

Continuous spans generally behave as fixed ended beams, this is due to their similarities in

boundary conditions as torsional moments are significantly reduced and the statically indeterminate

internal torque explicitly contributes to the stresses generated in addition to the torsional stress

induced by the external eccentric unifomly distributed torque. This is shown in figure 4.52, figure

4.53 and figure 4.54 for the second order derivative of torsional deflection, rotation and deflection

respectively. From the graphs, the internal torsional boundary conditions are equal to satisfy

equilibrium and compatibility conditions. The results generally show that higher stresses are

generated in the analysis of the conventional and present equations.

79

CHAPTER FIVE

5.1 SUMMARY OF RESULTS AND CONCLUSION

The solution to torsional problems in structural mechanics is very complicated as regards thin wall

structures. The doubly symmetric closed cross section used in this study whose torsional

equilibrium equation was derived by the variational approach using Vlasov’s stress displacement

relation generated a combined interactive effect of pure and warping torsion which created a fourth

order differential equation whose closed form expression was shown on various loading and support

conditions. The concept of St. Venant’s torsion was examined and shown on the basis of the elastic

theory and the membrane analogy was shown to obtain St. Venant’s pure torsion. The phenomena

of warping torsion was also examined. Warping was shown to be an out-of -plane effect arising

when the out of plane displacements are constrained. Osadebe in his work on modal interaction

revealed the deficiency of the conventional torsion equation was obtained on the independent

summation of St. Venant’s pure torsion and the warping torsion effect, he however showed that the

conventional torsional equation is safe to use only in the absence of modal interaction and derived a

modified pure torsion constant was obtained on the basis of the interactive effect of pure and

warping torsion. Study has also shown that the torsional states don’t occur separately. The values

obtained were used in this work. The derived pure torsion constant which took into consideration

the combined torsional states was extended to a continuous thin wall box beam. It was also shown

that both equations become equal when both torsional rigidities becomes equal at

i.e. both torsional rigidities become equal for square box cross sections.

This work has proved the inadequacy of St. Venant’s equation as it gives false torsional constant

which is higher than the actual value as given by the present equation which shows that the

combined torsional states occur simultaneously. A closed comparison of the numerical results

80

obtained however show from the graphs plotted that higher stresses are obtained with the present

formulation. This interactive torsional effect as shown relative to stresses is paramount in the

analysis and design of thin-walled box girders both for economy and safe design.

81

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