Friction in Sliding Bearings and Its Influence on Column ...pelleth/MahirMScThesis2006.pdf ·...

M A H I R Ü L K E R

Friction in Sliding Bearingsand Its Influence on

Column Buckling

Master of Science ThesisStockholm, Sweden 2006

KTH Engineering Sciences

Friction in Sliding Bearings and Its

Influence on Column Buckling

by

Mahir Ulker

February 2006Master of Science Thesis from

The Royal Institute of TechnologyDepartment of Mechanics

SE-100 44 Stockholm, Sweden

Preface

This thesis is the result of a question raised by structural engineering consultantsin the field of bridges in Sweden. The initiative to start working with this problemwas taken by the structural engineering company Tyrens AB and was carried outat the Department of Mechanics at The Royal Institute of Technology in Stockholmbetween August 2005 and February 2006.

First and foremost I would like to thank my supervisor, Adjunct Prof. Dr. Per-OlofThomasson, for handing me this problem and supporting me throughout the processof solution.

At the Department of Mechanics, I have only met helpful and hard working peopleto whom I owe the deepest gratitude for helping and inspiring me in my work withthis thesis. For aiding me in my theoretical queries, I would like to thank Prof.Anders Eriksson and Dr. Gunnar Tibert. I also must give a special grant to ParEkstrand, who is responsible for the computers and the network at the department,for his patience and help.

I would also like to thank Tommy Lindstrand at Spannteknik AB, for generouslysharing information about the bearings they manufacture.

For my family and friends, who have supported me at all times, I have nothing butlove.

Stockholm, February 2006

Mahir Ulker

iii

Abstract

This thesis is intended to initiate a more thorough study of the friction developedin pot sliding bearings for bridges. Today, the Swedish design rules for columnswith pot sliding bearings are based on simplifying assumptions which lead to highlyunfavourable support conditions at the top of the column. The Swedish bridge code,BRO 2004, states that the friction coefficient should be taken as 5% or zero, if it isless beneficial [1].

The support conditions of a column, subjected to axial forces, have great influenceon the buckling load of the column. It also influences the moment distributionalong the column. For a clamped–free column, the maximum bending moment ismore than three times greater than that for a clamped–guided column and thedifference becomes even greater when comparing with the clamped–hinged and theclamped–clamped columns. The buckling load for a clamped–guided column is fourtimes greater than that for the clamped–free column and is increased as the supportconditions become more favorable. Therefore, a more detailed design rule for bridgecolumns with pot sliding bearings, can make way for more cost-efficient designs.

Simple reasoning around the small deformation (Euler-Bernoulli) beam theory andthe definition of Coulomb friction, lead to more beneficial support conditions. Potbearings have a small rotational resistance, due mainly to the deformation of therubber plate which enables the rotational capacity of the bearing. A fixed potbearing can be regarded as a hinge with a weak rotational spring. The slidingcapacity of the bearing depends on the friction developed in a greased stainlesssteel-polytetrafluoroethylene contact. The available friction force increases with theload applied to the superstructure, in accordance with the Coulomb friction law.If it can be shown that the friction force is large enough to sustain the horizontalreaction force, which develops at the column top as the column is subjected to axialforces, one can justify the use of more favorable support conditions. The result ofthis thesis suggests that the support conditions which should be used are hingeswith rotational springs. However, it also indicates that the combination of a smallcoefficient of friction and a too flexible superstructure, may enable the free slidingof the column top.

v

Contents

Preface iii

Abstract v

1 Introduction 1

1.1 Aims of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Pot Bearings 5

3 The Limiting Cases 7

3.1 Column Buckling and Equilibrium Paths . . . . . . . . . . . . . . . . 7

3.2 The Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 A Continuation Method . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 The Limiting Cases — Result and Discussion . . . . . . . . . . . . . 10

4 The System - A Parametric Study 13

4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.1 Material properties for the beam and the column . . . . . . . 14

4.2.2 Material properties for the bearing . . . . . . . . . . . . . . . 15

4.3 The Coefficient of Friction . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3.1 Friction within the bearing . . . . . . . . . . . . . . . . . . . . 16

4.3.2 Temperature and contact pressure dependence . . . . . . . . . 17

4.3.3 Static and kinematic coefficient of friction . . . . . . . . . . . 18

vii

4.4 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Modelling the System in ABAQUS/CAE 19

5.1 Surface Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2 Hyper Elastic Material Models . . . . . . . . . . . . . . . . . . . . . . 22

5.2.1 Hybrid elements . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.4 Modelling the Beam and the Column . . . . . . . . . . . . . . . . . . 23

5.5 Modelling the Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.6 The Rotational Resistance of the Bearing Model . . . . . . . . . . . . 24

6 Results 27

6.1 Limit Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.2 Varying the Bending Stiffness of the Beam . . . . . . . . . . . . . . . 29

7 Discussion 31

7.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.2 FE simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.3 Suggested Further Research . . . . . . . . . . . . . . . . . . . . . . . 33

7.4 Modelling Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 37

A Limit Loads 39

A.1 µs = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.2 µs = 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.3 µs = 0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A.4 µs = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

B Varying the Bending Stiffness of the Beam 53

B.1 µs = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

B.2 µs = 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

viii

B.3 µs = 0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

C The intermediate Euler cases 67

ix

Chapter 1

Introduction

The objective of this thesis is to study how the friction force, which appears in potsliding bearings for bridges, influence the buckling length of bridge columns. Thefriction developed between the sliding surfaces of a pot sliding bearing introduceslateral support conditions at the column top which vary depending on the loadsacting on the deck. Bridge design in Sweden is regulated by the Swedish bridgecode, which is governed by the Swedish Road Administration (VV). The normativeapproach is to use simplifications which yield conservative design rules. The Swedishbridge code, BRO 2004, states that the friction coefficient should be taken as 5% orzero, if it is less beneficial [1]. Swedish structural engineers have been discussing theformulation of this rule and its implications. The implementation of this rule whendetermining the critical load of a column leads to the following system, where theelastic properties of the foundation have been omitted for the sake of simplicity:

Figure 1.1: The system given by following the Swedish bridge code, BRO 2004.

• The first criteria of the rule states that a horizontal force, FN = 0.05N , whereN is the axial force, should be added at the column top.

1

CHAPTER 1. INTRODUCTION

• The second criteria states that the column top should be regarded as free.

One subject of discussion amongst engineers is the fact that the friction force actsso as to oppose movement. The classic law of friction, can be stated as follows [11]:

1. The friction force (at the onset of sliding and during sliding) is proportionalto the normal contact force,

|F | = µN (1.1)

The coefficient of proportionality, µ, is known as the coefficient of friction.Often two values of µ are quoted: the coefficient of static friction, µs whichapplies to the onset of sliding and the coefficient of kinetic friction, µk, whichapplies during sliding motion.

2. The coefficient of friction is independent of the apparent area of contact.

3. The static coefficient is greater than the kinetic coefficient.

4. When tangential motion occurs, the friction force acts in the direction of therelative velocity, but in opposite sense:

F = −µNuT /|uT |. (1.2)

If the available friction force is large enough, it will hold the column top in place,so that it follows the translations of the beam. The connection between deck andcolumn can then be regarded as a hinge with a rotational spring. This is an inter-mediate case between the clamped–hinged and the clamped–clamped columns. Thespring represents the rotational resistance, or the restoring moment, of the bear-ing. The boundary conditions of a bridge column are always non-homogeneous anddependent on the deformation of the whole system.

The critical load of a clamped–hinged column is about eight times greater thanthe critical load of a clamped–free column (Figure 3.2). The maximum momentof a clamped–free column subjected to an axial force is approximately four timesgreater than that of a clamped–hinged column (Figure 1.2). The column used todetermine the second-order moment diagrams of Figure 1.2 was the same as thatused throughout this thesis. The parameters I, L and E are given in chapter 4. Theimperfection used was the deflection of a clamped column with a horizontal pointload acting at the top. The initial displacement at the column top was 50mm andthe load applied to the column top was 33kN. Considering the great difference inload carrying capacity for columns with different boundary conditions, the economicbenefit of a better understanding of this phenomenon cannot be neglected.

1.1 Aims of the Thesis

The assumption which has formed the basis of this study is, that the friction forcedeveloped between the sliding surfaces of the bearing is sufficient to keep the rela-tive motion between these surfaces very small during the application of a collapse

2

1.1. AIMS OF THE THESIS

load. To study the behavior of the column, the Finite Element Method (FEM)has been used to model a simple, two-dimensional, beam-column system with athree-dimensional pot bearing. The analysis was performed using the commercialFE-software ABAQUS. In order to grasp the problem, some simplifying assumptionswere made:

• The structural members, i.e. the beam and the column (Figure 4.1), areassumed to be linearly elastic.

• The column is assumed to be clamped to the foundation.

Bridge columns are often made of reinforced concrete. The effect of cracks, creep,relaxation etc. have been omitted. The foundation of a bridge column is never fullyrigid but this condition has been assumed.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

−1

0

1

2

3

4

5Second order moment for boundary conditions corresponding to the Euler cases I, II, III and IV

x [m], zero at column base

Ben

ding

mom

ent [

kNm

]

Clamped−FreeClamped−GuidedClamped−HingedClamped−Clamped

Figure 1.2: Bending moment diagrams for a column subjected to an axial force.The column has an imperfection due to a horizontal force acting at thecolumn top.

3

CHAPTER 1. INTRODUCTION

1.2 Structure of the Thesis

The thesis is outlined as follows:

• Chapter 2 describes the essential details of a pot bearing.

• Chapter 3 treats the cases of column buckling relevant to the thesis.

• Chapter 4 deals with all the necessary details of the system which is studied.

• Chapter 5 describes the relevant aspects of using a general FE-software toanalyze the problem at hand.

• Chapter 6 holds the results of the analysis performed.

• Chapter 7 is a discussion of the results.

4

Chapter 2

Pot Bearings

The most important aspect of this problem is the function of the pot bearing. Therotational capacity of a pot bearing is due to a rubber plate, which is enclosed by apot and a piston. A pot sliding bearing has a sliding plate mounted on the piston.Pot sliding bearings can be unidirectional or multidirectional. The unidirectionalpot sliding bearing has a guide which forces the sliding plate to move in a givendirection. The design of pot bearings for use in Sweden is regulated by Europeancode (EN 1337) together with, national rules stated in the Swedish bridge code(BRO 2004).

Figure 2.1: The parts of a pot sliding bearing (redrawn from [13]).

In a sliding pot bearing, friction develops in three different interfaces:

• Between the piston and the sliding plate.

• Between the piston and the pot.

• Between the rubber plate and its enclosing parts.

The most commonly used combination of materials for the sliding surfaces are stain-less steel and polytetrafluoroethylene (PTFE) with a silicon grease. This tribologicalsystem has a static coefficient of friction ranging between 0.02 and 0.10 [7]. Thisrelatively low friction, enables the superstructure to translate freely over the sub-structure.

The most important factors affecting the coefficient of friction of the PTFE–steelcontact are temperature, contact pressure and wear. Other factors are sliding veloc-ity and pre-load time. It is the temperature dependency which gives rise to the large

5

CHAPTER 2. POT BEARINGS

span of the static coefficient of friction. Measurements of the coefficient of frictionfor the greased steel-PTFE contact have been performed at the Staatliche Materi-alprufungsanstalt (MPA), situated at the University of Stuttgart in Germany [7].

Figure 2.2: Typical influences of important parameters on the friction of PTFE slid-ing bearings, redrawn from [7].

Within the bearing, low friction is needed to allow the superstructure to rotate overits supports. The steel–steel contact between the piston and the pot is greased.The maximum coefficient of friction allowed by EN 1337-5 is 0.2. As can be seenin Figure 2.1, there is a seal between the piston and the pot. The purpose of thisseal is to keep dirt and other particles out. If particles with greater hardness thanthe steel surfaces mix into the grease, they may have an abrasive effect on the steelsurfaces. This would generate more particles and alter the surface structure, makingit rougher. The inevitable effect of abrasive wear of the steel–steel contact surfacesis a higher coefficient of friction.

The inner seal has a quite different purpose. It is not as important to keep parti-cles out of the rubber–steel contact since here, the particles would be pressed intothe very soft rubber surface. Rubber is an elastomer; it is a hyperelastic, almostincompressible material. The effect of the incompressibility of rubber is that whenrubber is concealed and pressurized, it behaves much like a fluid. Without the innerseal, the rubber would flow out through the small gap between the pot and thepiston [7]. However, the fluid-like behavior of the rubber plate is highly dependenton the frictional properties of the rubber–steel interface. The friction between therubber plate and the concealing surfaces must be low enough to let the rubber slidefreely within the compartment. For this reason, the rubber–steel interface is greased.The normative rule of EN-1337 states that ”The elastomeric pad shall be assumedto have hydrostatic characteristics under pressure”. Specific demands regarding thecoefficient of friction are not given. Qualitative measures of the rotational resistanceof a pot bearing are too complicated to calculate with the accuracy needed. For thispurpose, tests are performed.

6

Chapter 3

The Limiting Cases

This chapter describes the determination of the limiting cases, or, the boundarieswithin which the results of this investigation were expected to lie. It also describesthe comparison of results for different boundary conditions.

3.1 Column Buckling and Equilibrium Paths

The stability of an equilibrium configuration of a structural system is, in generalterms, a dynamical problem [6, 8]. However, the critical state at which the systembecomes unstable can often be determined using static methods. Knowledge aboutthe critical state and the load at which the critical state is reached, are almostalways sufficient for practical engineering situations, at least within the field of civilengineering.

A classical example of how static methods can be used to determine the criticalload of a structural system is the Euler column. The static moment equilibriumequation, assuming small deformations, for a geometrically perfect column in com-pression leads to a Sturm-Liouville problem. This problem is readily solved for ahomogeneous column with a constant cross-section. The solution of the Euler col-umn consists of a critical load and a buckling mode but no knowledge regarding theamplitude of the deformation. At the critical load (also referred to as a bifurcationpoint since the solution has more than one branch emerging from that point), twopossible solutions are available: either the load can be increased infinitely with nodeformation at all or, if some small disturbance in the direction of collapse occurs,the system begins to move.

In order to determine a realistic critical load, imperfections in geometry and mate-rials must also be taken into account. The load-displacement relation can still beobtained using a static approach, but the critical load is no longer defined as theload value at a bifurcation point. Instead, the limit load is introduced. A distinctionbetween these two cases is made by associating a critical load and its bifurcationpoint to buckling and a limit load and a limit point to snapping, or snap-throughbuckling.

7

CHAPTER 3. THE LIMITING CASES

Figure 3.1: Characteristic equilibrium paths for buckling and snapping.

3.2 The Limiting Cases

The pot bearing is a hinge with a rotational spring. This would imply that thebuckling load of a column which is clamped at the base and loaded via a pot bearingat the top, is a mix of the Euler cases III and IV (see Figure 3.2). Displacementsand rotations of the column top, caused by the deformation of the beam, constitutegeometrical imperfections which should make the critical load slightly smaller.

The equilibrium of the system constituted by Euler case III requires a horizontalreaction force. If there was no friction in the bearing sliding surfaces, no reactionforce would develop and the column top would translate freely in the horizontaldirection, giving the Euler case I, or possibly the Euler case II buckling mode. Onthe other hand, if the friction force was greater than the reaction force, it wouldhold the column top in place with the Euler III buckling mode as the result. Themix between Euler case III and IV would be due to the rotational resistance of thebearing.

In order to evaluate the results obtained from the FEA, a study of the Euler casesI, III and IV was necessary. Since the system column-bearing-beam introduces abending moment and a lateral force acting at the column top, post-buckling analysisof the Euler cases was needed. The load-displacement curves for the first two Eulercases can be treated analytically, with relative ease, using energy considerations. Theform of the buckling mode for Euler case III generates very large expressions whenthe load-displacement curve is sought, so it is rational to solve it numerically usingan FE-software. Since for the Euler cases, the critical loads and their correspondingmodes are well known, it would be possible to check the load-displacement curvesgiven by the FE-solution.

The buckling modes are different in these basic cases. In the first and second casesit is the column top which is displaced most. In the third, it is at a point at about0.7L from the base and in the fourth it is at the middle of the column. To comparethe load-displacement curves for different modes another displacement parameter,

8

3.3. A CONTINUATION METHOD

the vertical displacement of the column top, should be used.

Figure 3.2: The support conditions of the basic Euler cases.

3.3 A Continuation Method

The main difficulty in the post-critical analysis of structural systems is due to thestiffness matrix of the discretized system becoming singular if a critical load1 or alimit load2 is reached.

In ABAQUS, a method for the determination of load-displacement relations in thepost-critical regime called the Modified Riks Method is implemented [2]. As thename implies, it is a modification of a method devised in order to solve for the post-critical response of conservative, structural systems, proposed by E. Riks. Thereare many possible modifications to this method [5, 12] and the following is only anoutline of its main ideas. The method is also known as the arc-length method.

Riks method is a load control method. This means that the displacements of thesystem freedoms are continuously determined for successive increments of the loadsacting on the system. In a static FEA, a set of N displacement variables are deter-mined by the simultaneous solution of a set of N linear or non-linear equilibriumequations. The main idea of Riks method is to add the load parameter, λ, to the un-known variables, t, of the system [12]. Hence, the magnitude of all the loads actingon the system, must depend only on this load parameter. Since large deformationsare likely to occur during the post-critical analysis, the equilibrium equations aregeometrically non-linear. The implementation in ABAQUS also allows for materialnon-linearities [2].

1A bifurcation point.2A limit point.

9

CHAPTER 3. THE LIMITING CASES

Now, a set of N + 1 unknown variables are to be determined but there are still onlyN equations at hand;

fi(t : λ) = 0, i = 1, 2, ..., N (3.1)

t = t1, t2, ..., tN (3.2)

These equations (3.1) describe N curves in the space RN+1 which is spanned by theN displacement parameters and the load parameter. In order to specify a particularpoint on one of these curves, an extra equation must be introduced. In generalterms, it can be expressed as:

fN+1 = g(t : λ)− η = 0, i = 1, 2, ..., N (3.3)

By varying η, the surface g(t : λ) is translated in RN+1. If the surface intersects thecurves of (3.1), a sequence of points along these curves are generated. Then η canbe regarded as a path parameter.

For each of the curves (3.1), now parameterized by η, the arc-length is defined by thelength of the curve travelled as η is varied. This is the measure used in ABAQUS tofollow the progression of the solution. All the displacements are given as functionsof the arc-length.

Equation (3.3) represents an N -dimensional surface in RN+1 and for a given set ofvalues for the path parameter, the solution can be written as:

x =

[tλ

]=

[t(η)λ(η)

], x ∈ RN+1, η0 < η < η1. (3.4)

By adding the load parameter to the set of unknowns, the banded structure of thestiffness matrix is destroyed. The system matrix is no longer a stiffness matrix sinceRiks method augments one row and one column for the load parameter. This isalso the reason why this method allows for post-critical analysis. Even though thepart of this matrix, which really is related to the stiffness of the structural system,becomes singular at a critical load or a limit load, the system matrix generally doesnot.

The set of N + 1 equations defined by equations (3.1) and (3.3) are still non-linear,at least in a geometrical sense, so the solution must be determined in an iterativemanner [12]. In ABAQUS, the iterative scheme used is Newton’s method [2].

3.4 The Limiting Cases — Result and Discussion

The load-displacement relations of the basic Euler cases were obtained by the use ofRiks method. The column was constrained to motion in the plane. A small load wasadded to initiate the critical behavior. The element type and mesh density used werethe same as in the column–beam model. Actually, the mesh density for the beamelements to use in the column-beam model was determined by a convergence teston the column model presented here. Using the B32 (Section 5.4) quadratic beam

10

3.4. THE LIMITING CASES — RESULT AND DISCUSSION

elements of ABAQUS, an element length of 1m gave limit loads very near thosecalculated by Euler’s formula. However, both in the beam–column model and inthe column model here, an element length of 0.05m was used in order to get smoothdisplacement fields. The results, or the limiting cases, are presented in Figure 3.3.

0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Equilibrium paths (vertical displacement of the column top as the displacement parameter) for boundary conditions corresponding to the Euler cases I, II, III and IV

Vertical displacement of the column top [m]

Axi

al fo

rce

[MN

]

Clamped−FreeClamped−GuidedClamped−HingedClamped−Clamped

Euler case IV

Euler case III

Euler case II

Euler case I

Euler’s column formula

Figure 3.3: The equilibrium paths, determined using Riks method, of the basic Eulercases with the vertical displacement of the column top as displacementparameter. Comparison with Euler’s column formula.

The validity of the results may be checked by comparing with the buckling loadsgiven by Euler’s column formula;

PE =π2EI

L2E

, (3.5)

where;LE = 2L, L, 0.7L and 0.5L (3.6)

for the boundary conditions: clamped–free, clamped–guided, clamped–hinged andclamped–clamped, respectively. Indeed, when introducing the values for the param-eters L, E and I, chosen for the column (Section 4.1), the results of using Riksmethod were well met by the results obtained by the use of equation (3.5).

11

Chapter 4

The System - A Parametric Study

4.1 Geometry

The geometry of the system was chosen with a unidirectional pot sliding bearing inmind. The idea of the system design was to enable the buckling of the column withinthe capacity of the bearing. This would cohere with the design principles used inpractical situations. In engineering calculations, the bearing dimensions are chosenso that the bearing has higher strength than the structural elements, connected viathe bearing.

In Sweden, the approval of bridge bearings is governed by VV. Two manufacturersof pot bearings, TOBE and MAGEBA, have the approval of VV. For this study, thegeometry of the pot bearings manufactured by the company TOBE, have been used.The dimensioning load of the bearing was chosen to 1MN. This would resemble thesmallest standard bearing manufactured by TOBE. The choice of a small bearingwas necessary to limit the size of the FE problem. The geometry of the bearingwas simplified as far as possible, without leaving out details which affect the overallbehavior of the bearing (Figure 4.2). The features of interest in this study were thealmost free rotation and the translation and sliding friction of the pot bearing.

The bending stiffness of the column was determined to give an Euler III criticalload well below the dimensioning load of the bearing. For simplicity, a square cross-section was chosen and the length of the column was set to 5m. The Euler IV criticalload was set to 0.75MN, which then gave a column side of 120mm when Young’smodulus, E = 32GPa.

For the beam, a span length of 10m and a rectangular cross-section with the width300mm were chosen. The deflection of the beam had to be bounded by the rotationalcapacity of the bearing, 2%, and a criterion on small deformation. Taking rotationof the column top into consideration, the deformation of the beam had to be kepteven lower. The rotation of the column top was not known, but it should becomesmall compared to the rotation of the beam prior to buckling. Therefore, boundingthe rotation of the beam over the bearing to 1% seemed reasonable. The smalldeformation criterion was chosen to limit the maximum deflection of the beam to

13

CHAPTER 4. THE SYSTEM - A PARAMETRIC STUDY

Figure 4.1: The system geometry.

L/500, where L is the length of one span. This criterion gave a beam height of930mm. To model the bearing with reasonable simplicity, a number of simplificationswere made. The outer sealing ring was omitted since its only purpose is to keep dirtout of the bearing. The inner sealing ring was left out since the coarse discretizationused in the model would not suffice to model the fluid-like behaviour of the rubber.Details concerning the mounting of the bearing were omitted. In the model, thebearing was mounted using constraints and the guide was introduced by constrainingthe whole model to motion in the plane of the system.

The piston was modelled as a cylinder. In a real bearing, the side of the pistonis rounded or chamfered to allow it to rotate without constraining the pot. In thestudy however, this was dealt with by making the radius of the cylinder a little bitsmaller than the largest radius of the original piston.

The PTFE sheet (Figures 2.1 and 4.2) was omitted since the deformation of this5mm thick plastic sheet is negligible with respect to the overall deformation of thesystem. Instead, the contact between the top surface of the piston and the slidingplate was given the frictional properties of steel–PTFE.

Figure 4.2: Simplifications on the bearing geometry, left figure copied from [13].

4.2 Material Properties

4.2.1 Material properties for the beam and the column

The beam and the column are modelled using pure concrete with a Young’s modulusEconcrete = 32GPa and Poisson’s ratio νconcrete = 0.25. As the study was based on

14

4.2. MATERIAL PROPERTIES

dynamic methods, a density ρconcrete = 2500kg/m3 was chosen. Reinforcement wasomitted since the purpose was purely theoretical and the non-linear effects of cracks,relaxation and creep would make the model unnecessarily complicated.

4.2.2 Material properties for the bearing

Two materials were used in the modelling of the bearing - steel and rubber. Thesteel was given a Young’s modulus of Esteel = 210GPa, Poisson’s ratio νsteel = 0.3and the density ρsteel = 7850kg/m3.

Rubber is a hyperelastic material. The elastic properties of such materials aredescribed in terms of a strain energy potential. The strain energy potential definesthe strain energy stored in the material per unit of reference volume (volume in theinitial configuration) as a function of strain at that point in the material [2]. Thestrain energy potential can be expressed in many forms, holding parameters whichhas to be determined by experiment. The form used in this study, was the Mooney–Rivlin form. The reason for this choice was that it was difficult to find values forthe elastic parameters for the rubbers used in pot bearings.

U = C10(I1 − 3) + C01(I2 − 3) +1

D1

(Jel − 1)2 (4.1)

U is the strain energy per unit of reference volume; C10, C01 and D1 are temperaturedependent material parameters; I1 and I2 are the first and second deviatoric straininvariants; Jel is the elastic volume ratio, defined as:

Jel =J

J th

where J is the total volume ratio and J th is the thermal volume ratio:

J th = (1 + εth)3

For an incompressible material, the last term of equation 4.1 is omitted. In ABAQUShowever, incompressibility is defined by setting D1 = 0 which may seem strange at

the first glance, since1

0→∞.

Material properties for natural rubber at room temperature were found in [10]. Here,the strain energy, derived for a membrane in biaxial stretch, is written on the form;

F =1

2V

[s+(Tr(B)− 3)− s−(Tr(B−1)− 3)

](4.2)

where V is the volume of the membrane, B is the stretch1 tensor and s+ and s− aretemperature dependent parameters. The expression is derived with the purpose of

1Stretch is defined as B = L0+∆LL0

and the structure of the stretch tensor is equivalent to thatof the strain tensor.

15


studying rubber elasticity under constant temperature so the third term is omitted.[10] gives approximate values for the temperature dependent parameters;

s+ = 3bar s− = −0.3bar

with the comment: ...these values may be used to obtain fairly good results at roomtemperature.

By comparing equations (4.1) and (4.2) one finds:

C10 =1

2V s+ C01 = −1

2V s−

4.3 The Coefficient of Friction

In order to create a useful model for the purpose of this study, a number of sim-plifications on the tribology of the bearing had to be done. Wear is a factor whichis very difficult to predict. The effect of wear on a tribological system has to bemeasured [7, 9]. In the tribological system at hand, the main effect of wear is anincreasing coefficient of friction (Figure 2.2). However, wear is a long-term effect, soit would not cause any difficulties in this context. The collapse load is due to trafficand will be applied over a time period of seconds.

Temperature (◦C) Maximum coefficient of frictionµs,1 µdyn,1 µdyn,1/µs,1 µs,T µdyn,T µdyn,T/µs,T

−35 0.030 0.025 0.833 0.050 0.040 0.800−20 0.025 0.020 0.800 0.040 0.030 0.750

0 0.020 0.015 0.750 0.025 0.020 0.800+21 0.015 0.010 0.667 0.020 0.015 0.750

µs,T static friction at subsequent cyclesµdyn,T dynamic friction at subsequent cycles

Table 4.1: Upper limits for the coefficient of friction in long-term tests. Reproducedfrom [7].

4.3.1 Friction within the bearing

The complexity of the mechanical system constituted by a pot bearing makes an-alytical design methods rather complicated. The design of pot bearings is basedon tests. Therefore it has been difficult to find specific values for the coefficient offriction for the tribological systems within the bearing. EN 1337-5 states that thevalue 0.2 should be used as the maximum characteristic coefficient of friction for thesteel–steel contact, when determining the rotational resistance caused by frictionbetween the piston and the pot.

The model of the pot bearing was intended to give a nearly free rotation, and themain details of interest were the sliding surfaces of the PTFE–steel contact. All the

16

4.3. THE COEFFICIENT OF FRICTION

contacts within the bearing were given a static coefficient of friction of 0.02. Theresult was a bearing which was too stiff, but still almost function like a hinge. Thetesting of the bearing model in this aspect, is accounted for in section 5.6.

4.3.2 Temperature and contact pressure dependence

When defining the tangential interaction between the piston and the sliding plate,further simplifications were needed. The buckling of a bridge column is likely tooccur as a result of large traffic loads combined with a large initial displacement ofthe column top. As mentioned earlier, the traffic load is applied over a short periodof time. Therefore, the temperature dependency must not be included explicitly inthe analysis of the limit load.

0 20 40 600

0.02

0.04

0.06

0.08

0.1

0.12

Contact pressure [MPa]

Sta

tic

coe

ffic

ien

t o

f fr

icti

on

The contact pressure dependency of the friction coefficient

0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

Contact pressure [MPa]

Fri

ctio

na

l sh

ea

r st

ress

[M

Pa

]

The contact pressure dependency of the frictional shear stress

o

T = -20 C

o

T = -35 C

o

T = -10 C

o

T = 0 C

o

T = 21 C

o

T = 21 C

o

T = 0 C

o

T = -10 C

o

T = -20 C

o

T = -35 C

Figure 4.3: The contact pressure dependency of the static coefficient of friction. Therings mark values taken from [7], through which cubic splines have beeninterpolated

The contact pressure dependency could be eliminated by the following discussion.Tests of the tribological system PTFE–steel–grease show that the static coefficient

17


of friction decreases with increasing contact pressure. The available2 friction forceor, shear stress, over the surface is proportional to the coefficient of friction. Bymultiplying the measured values of the static coefficient, with the contact pressure,a function of available friction force, or shear stress, is obtained. This function isshown in Figure 4.3, together with the measured values of the static coefficient offriction given by the MPA [7]. The available friction grows with the contact pressure,at all the temperatures accounted for.

Since at all temperatures of interest, the available friction force is increasing withthe contact pressure, the contact pressure dependency could also be eliminated fromthe contact definition of the model.

4.3.3 Static and kinematic coefficient of friction

The results of the study of the coefficient of friction done by the MPA, as reportedin [7], does not hold much information about the kinematic coefficient of friction forthe PTFE–steel contact. Some upper bounds are given (Table 4.1) and these haveserved as the upper bound also in the study at hand. The results from the testsdone by the MPA, show that the ratio of dynamic and static coefficient of frictionvaries between 0.67 and 0.83. In the bearing model, this ratio was set to 0.75. Forthe internal contact surfaces, it was set to 1.00.

4.4 Parametric Studies

The coefficient of friction for the steel–PTFE contact studied here, depends on anumber of variables with different time spans. In order to gather the effect of allthese dependencies for a short-term analysis, it is easier to perform a parametricstudy where the coefficient of friction itself is varied. The only variable which hasa short-term effect on the coefficient of friction is the contact pressure dependency.Having shown that this effect is favorable for the phenomena studied here (section4.3.2), this approach can be justified.

Critical loads for the column was to be determined by applying a line load on oneof the two spans. This choice of load would impose an imperfection, suitable forinitializing the expected buckling modes, as the load was applied. The first partof the study was to determine the critical load for the static coefficients of friction;µs = 0, 0.005, 0.02 and 0.05. The second part was a study on the influence of thebending stiffness of the beam. For this purpose, three beam cross-sections, designedto meet the criteria on deflection; L/250, L/500 and L/1000, were used with thestatic coefficient of friction; µs = 0, 0.005 and 0.02.

2The word available is used here since the friction force itself will not begin to develop unlesssome external, or internal force, is trying to initiate a relative displacement between the contactsurfaces.

18

Chapter 5

Modelling the System inABAQUS/CAE

ABAQUS/CAE is a graphical interface used to define and analyze FE-models. Thebasic procedure used to model the system was:

1. The model geometry was defined. The parts of the model: the beam, thecolumn and the parts of the bearing were defined as purely geometrical entities.

2. Material properties and the cross-sectional properties of the beam and thecolumn were defined.

3. The model was assembled and all the necessary surfaces and node sets weredefined.

4. Two analysis steps were defined. The first step was a static step, used toensure that the contact pairs were in contact before the load was applied. Thesecond was an implicit dynamic step, during which the load was applied.

5. Contact pairs and surface interactions were defined

6. Loads and boundary conditions were applied.

7. The structure was meshed.

In order to make the model work, some intermediate steps had to be taken. Themain problem was to make the contact pairs well defined. Two parts of the model,the rubber plate and the piston, were only constrained by surface interaction.

5.1 Surface Interaction

The interaction between surfaces is defined using the *CONTACT PAIR option.A contact pair consists of a master and a slave surface or node set. The *TIE-constraint is the simplest form of surface interaction where two surfaces are tied to

19

CHAPTER 5. MODELLING THE SYSTEM IN ABAQUS/CAE

each other, constraining the nodes on the slave surface to move with the nodes onthe master surface. This is very useful when meshing complicated geometries, asthe geometry can be divided into more easily meshed parts.

To have surfaces making contact during analysis is somewhat more complicated.First of all, ABAQUS has to determine whether the surfaces of a contact pair arein contact or not. Secondly, the forces transmitted via the contact pair has to becalculated. The two basic methods used for these purposes are: the penalty methodand the Lagrange multiplier method. They can be used to define both normal andtangential interactions between a contact pair.

The penalty method is based on the use of non-linear mathematical springs [14].The idea of the method is to add a penalty term to the system energy. This penaltyterm has the same principal structure as the stiffness term of the energy but thedisplacements are replaced by the relative distance between the contact surfaces andthe stiffness factors are replaced with penalty parameters. Using this method, theconstraint is never met exactly. The slave surface will penetrate the master surfaceto an extent determined by the penalty parameter. A large penalty parameterwill lead to small penetrations and vice versa, but also to more severe convergenceproblems.

As the name implies, the Lagrange multiplier method enforces the contact constraintby adding a term which holds the constraint and a Lagrange multiplier to the energy.In this way, the constraint can be enforced exactly. The Lagrange multiplier isintroduced as an extra unknown variable, and gives an extra equation when theenergy is varied. It constitutes the reaction force between the points of contact [14].

In ABAQUS, normal surface interactions are defined by the Lagrange multipliermethod in different forms. The interactions may be defined as hard or softened.The hard contact definition is a direct implementation of the Lagrange multipliermethod, whereas the softened contact definition is somewhat more complicated.The softened pressure–overclosure relation can be defined with an exponential law,in tabular form or as a linear function.

Using the Lagrange multiplier method for tangential surface interaction, more vari-ables are introduced, increasing the computational time [2]. Therefore, it shouldonly be used when the stick-slip behaviour must be modelled with great accuracy.The penalty formulation permits some relative motion of the surfaces even thoughthey should be sticking. To control the relative motion between the surfaces, the“allowable elastic slip” is defined. The setting of this parameter tells ABAQUS howto vary the penalty parameters. A large allowable elastic slip corresponds to a smallpenalty parameter. The effect is a faster solution with the drawback of greater rel-ative motion between the surfaces when they should be sticking. If the allowableelastic slip is taken very small, the penalty parameters are set very high. This maylead to convergence problems due to ill-conditioning of the system matrix.

In the beginning of a step in which contact is to be established, some tricks need tobe performed in order to constrain the parts. This is done by creating an artificialover-closure of the contact surfaces. The over-closures must be introduced with some

20

5.1. SURFACE INTERACTION

(a) (b)

Figure 5.1: (a) The dashed lines show how the initial overclosure was introduced.(b) The division of the bearing parts in order to refine the mesh at thesurfaces. The thick lines follow the outer lines of the original geometries.

care. At the start of a step, the slave nodes can be adjusted so that they lie exactlyon the master surface. To make these adjustments without corrupting the geometry,the geometry must be defined with the adjustments in mind (Figure 5.1, left). Itturned out that the best way to establish the contacts was to hold all the parts ofthe system constrained in all DOF. The node adjustments were then performed byABAQUS and the model was well defined as the loads were applied in subsequentsteps.

A well defined surface interaction is highly dependent on a rather fine mesh. In orderto mesh the parts of the bearing without using the refinement needed for the surfaceinteraction, the parts were divided into surface parts and bulk parts (Figures 5.1,right and 5.2). As these parts were tied together, conflicting constraints appeared.The contact constraints were defined between surfaces which contained nodes whichwere already used in other constraints. In the ABAQUS/Standard manual this isreferred to as over-constraints. An over-constrained model may yield inaccurateresults or convergence difficulties [2]. The only way to get around this problem is tochoose the slave and master surfaces carefully. The definition of tangential surfaceinteraction with non-zero friction coefficients introduces asymmetric terms as thesurfaces are sliding relative to each other. This can have a negative effect on theconvergence rate if the frictional stresses have a substantial influence on the globaldisplacement field and the magnitude of the frictional stresses is highly solutiondependent. ABAQUS has a special algorithm for the solution of an asymmetricsystem of equations.

21


Z

Y

X

1

2

3

Figure 5.2: The parts used to mesh the bearing model.

5.2 Hyper Elastic Material Models

Solid, rubber-like materials are referred to as elastomers and in the ABAQUS/Standardmanual, their elastic behavior is called hyperelastic. Elastomers are non-linear elas-tic materials. Their elastic properties are often highly dependent on temperature.This group of materials are almost incompressible, having values for Poisson’s rationear 0.5.

The ABAQUS/Standard 6.3 manual states that the incompressibility of an elastomershould be included if the material is highly confined. In the model used for this study,the rubber plate has been assumed to be fully incompressible. This assumption wasthe result of difficulties finding the necessary material parameters for the rubber.The assumption was justified, by reasoning that the overall behavior of the bearingcould be tested quite easily.

5.2.1 Hybrid elements

The basic structural elements which are used for small deformation analysis arederived from assumed displacement fields [4]. They are sometimes referred to asdisplacement elements. For a fully incompressible material, it is possible to add ahydrostatic pressure without altering the displacements. A nearly incompressiblematerial exhibits very large changes in pressure for small changes in displacement.This makes a displacement-based solution too sensitive numerically. This singularbehavior in the system is removed by treating the pressure stress as an independentlyinterpolated solution variable, coupled to the displacement through the constitutivetheory and the compatibility condition. The coupling is implemented by the use of aLagrange multiplier [2]. The result of this mixed formulation of the energy functionalis a hybrid-element, which should be used when modelling incompressible or nearly

22

5.3. DAMPING

incompressible materials.

5.3 Damping

Structural systems seldom produce damping ratios greater than 0.02. Therefore, itwas assumed that an undamped system may be used for this study. Since the solepurpose of this study was to study the critical behavior of the column, damping wasregarded as a superfluous parameter, making the interpretation of the results moredifficult.

5.4 Modelling the Beam and the Column

Both beam and column were modelled using quadratic beam elements (B32). Theseelements are well suited for the purpose of the study since they can model thedeformation of the system well and at a low computational cost. For these elements,all the quantities related to the Euler-Bernoulli beam theory, are available withoutany need for post-processing. Their rotational degrees of freedom makes it veryeasy to extract the rotation along a group of elements or at a node. The B32 beamelements are defined using the Timoshenko beam theory, which means that they takethe shear stiffness (assumed to be negligible in the Euler-Bernoulli beam theory) intoaccount.

5.5 Modelling the Bearing

All the bearing parts were modelled with linear solid elements. Using quadraticelements may give rise to oscillations and spikes in the contact pressure [2]. Asmentioned earlier, the bearing had to be divided into a number of smaller parts toenable a fine mesh on the contact surfaces and a coarse mesh on the bulk material.The parts which built up the bearing are shown in Figures 5.2 and 5.3.

23


Z

Y

X

1

2

3

ZY

X

12

3

(a) (b)

Figure 5.3: The meshed, geometrical parts of the bearing seen from the top (a) andthe bottom (b).

5.6 The Rotational Resistance of the Bearing Model

The rotational resistance of the bearing model was tested on a simply supportedbeam with a TOBE 1000 fixed bearing at one end. The beam was 5m long and hada square cross-section with side length 0.4m. To rotate the bearing piston, a lineload was applied to the beam.

The reactive (or restoring) moment of the TOBE pot bearings can be calculatedaccording to the following formula [13]:

Mφ = 138kφttd3 (kNm) (5.1)

which is valid if there are no horizontal loads transferred via the bearing.

Mφ: Reactive moment (kNm)

kφ: Value of angular change

tt: Temperature factor

d: Diameter of rubber plate (m)

24

5.6. THE ROTATIONAL RESISTANCE OF THE BEARING MODEL

Table for the value of angular change forVmax = 30N/mm2 pressure against the rubber padAngular change (%) 0.5 1 1.5 2

kφ 2.3 3.2 4.1 5.0

Table 5.1: Values of angular change for 30N/mm2 pressure against the rubber pad.Reproduced from [13].

Table of the temperature factorsTemperature (oC) tt

20 1.00 1.0

-20 1.3-30 1.5-40 2.1

Table 5.2: Values of the temperature factor for the rigidity of the rubber. Repro-duced from [13].

Using equation (5.1), the restoring moment should be MTOBE = 2.539kNm for arotation of 0.5%. However, when the test model was subjected to a rotation of0.5%, the restoring moment was found to be MABAQUS = 7.795kNm. At the middleof the beam, the maximum bending moment was 239kNm, so the bending momentdiagram was very similar to that of a simply supported beam, subjected to a lineload, but the bearing model was too stiff. Sources for this error may have been:

• The mesh density used on the rubber plate.

• The incorrect representation of the elasticity of the rubber, caused by assumingan incompressible material.

• A faulty definition of the friction within the bearing.

The effect of this overestimation of the rotational resistance of the bearing is thatthe spring constant at the column top will be too large, resulting in an overestimatedbuckling load.

The most likely source for the discrepancy between the restoring moments of theFE-model and the tests made by TOBE was the parameters in the hyperelasticmaterial model. Since these parameters were not available, no further efforts tomake the bearing model behave more naturally were made.

25


Z

Y

X

1

2

3

Figure 5.4: The simply supported beam used to test the restoring moment of thebearing model.

26

Chapter 6

Results

In the first section of this chapter, a comparison of the critical loads for differentvalues of the static coefficient of friction is given. The second section holds theresults from varying the bending stiffness (essentially the area moment of inertia)of the beam. Additional results, such as reaction forces and displacements of thecolumn top and the middle of the beam, are given in appendices B and C.

6.1 Limit Loads

The critical loads are presented together with the limiting cases defined in Chapter3. The results of these simulations were well met by the assumptions stated at theonset of this study. With µs = 0, the result was a limit load between the Euler casesII and III. The non-zero static coefficients of friction gave a limit load between theEuler cases III and IV. All the values of µs gave limit loads which were closer to theupper limit, Euler case III for µs = 0 and Euler case IV for non-zero µs. This was theresult of the bearing model being too stiff (section 5.6). Figures A.5, A.8 and A.11,show the horizontal reaction force at the column base together with the availablefriction force. The available friction force was calculated by multiplying the verticalreaction force with the static coefficient of friction. Since no horizontal forces wereintroduced along the column, the absolute value of the horizontal reaction forceat the column base was the same force, which the friction force had to overcomeso as to prevent the column top from sliding. In the beginning of the applicationof the collapse load, the slope of the available friction force was greater than thatof the horizontal reaction force for all non-zero µs. During the course of collapse,the available friction force had increased to a value which could not be met bythe horizontal reaction, leading to the expected result. Figures A.3, A.6, A.9 andA.12 show the displacements and rotations over the connection between beam andcolumn. For non-zero µs, there was a small discrepancy in the relative motionbetween the column top and the middle of the beam. This was due to the inevitableelastic slip introduced with the penalty formulation of the tangential interactionbetween the piston and the sliding plate.

27

CHAPTER 6. RESULTS

Many of the diagrams presented here show some periodic variation over time. Sincethe analysis were performed using a dynamic approach, some dynamic effects mustbe accounted for. If the column is assumed to be infinitely stiff, the first eigenfre-quency of the beam can be estimated using the following expression [3]:

ωn = π2

√EI

ρAL4(6.1)

where all the quantities refer to the beam. L is the length of one span. Then, theperiod corresponding to the first eigenfrequency can be determined as:

Tn =2

π

1√EI

ρAL4

(6.2)

The periods corresponding to the beam heights hb = 0.74m, 0.93m and 1.17m are;Tn = 0.084s, 0.066s and 0.053s. This is best seen in Figure B.2.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Axial force − vertical displacement of the column top

Axi

al fo

rce

[MN

]

Vertical displacement [m]

µs = 0

µs = 0.005

µs = 0.02

µs = 0.05 Euler case IV

Euler case III

Euler case II

Euler case I

Figure 6.1: Axial force against vertical deflection at the column top. The curves forµs = 0.02 and µs = 0.05 coincide.

28

6.2. VARYING THE BENDING STIFFNESS OF THE BEAM

6.2 Varying the Bending Stiffness of the Beam

Figure 6.2 shows the main result of this part of the study. As expected, a smallerbending stiffness of the beam gave a lower limit load, due to the larger imperfectionintroduced by a weaker beam. For µs = 0.005, the variation in bending stiffness(Ibeam) resulted in a rather large variation in the limit load.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Limit load − Area moment of inertia, I beam

Area moment of inertia, Ibeam

[m4

]

Lim

it lo

ad

[M

N]

µs=0

µs=0.005

µs=0.02

Euler case IV

Euler case III

Euler case II

Euler case I

Figure 6.2: Limit load as function of the area moment of inertia of the beam.

29

Chapter 7

Discussion

The primary purpose of this thesis was to determine the limit load for a column,clamped at the bottom and with a sliding pot bearing at the top. The secondarypurpose was to study the influence of the relation between column and beam stiffnesson the critical load. This study was performed in order to initiate a more thoroughinvestigation of this problem. Nevertheless, the result of this thesis indicates that thecode rule is over-simplified and that a deeper understanding of the friction developedin pot sliding bearings most likely will allow for a more cost-efficient design of bridgecolumns.

7.1 General Remarks

I am more than willing to admit that the results of my efforts are mainly due toa trial and error process. This study, is the first to address this problem so itdoes not claim to deliver a final solution to the problem, but indications on howthe reality behind this problem may look. Nevertheless, the results presented hereshould motivate a continued analysis of this problem.

The main problem for researchers in the field of civil engineering and structuralmechanics in Sweden today, is finding economical resources for real live experiments.Therefore, computer simulations may become a feasible alternative in the future.However, the Swedish master degree in civil engineering today, does not hold thetheoretical foundation needed to perform these simulations since useful and reliableresults are highly dependent on a firm understanding of the numerics implementedin the commercial FE softwares. Obviously, these simulations should be performedby engineers, specialized in computational methods. This does of course not makethe FEM less useful in the design of structures, but I wish to make the distinctionbetween simulating experiments and using the method for linear, small strain designpurposes clear.

31

CHAPTER 7. DISCUSSION

7.2 FE simulations

The main result of this study is that the limit load of a column, which is connectedto a beam via a pot sliding bearing, may be substantially greater than that proposedby VV. Simple arguments based on the small deformation (Euler-Bernoulli) beamtheory and the definition of friction, lead to the conclusion that if the availablefriction force is large enough the critical load is an intermediate case of the Eulercases III and IV. This has been confirmed by the FE analysis performed. The limitloads determined, using the model described in this thesis, are overestimated dueto the rotational resistance of the bearing model being too large. One observationwhich speaks in favor for the assumptions used here, is that the equilibrium pathsobtained for µs = 0.02 and µs = 0.05 coincide. If the static coefficient of friction islarge enough, no sliding will occur and the lateral connection between column andbeam may be regarded as rigid.

The limit load of the column is dependent on the deformation of the whole system.A stiff beam gives a higher critical load than a less stiff beam, since with a less stiffbeam the imperfection or initial displacement of the column becomes larger. Therelation between limit load and beam area moment of inertia is non-linear (Figure6.2). The effect of a less stiff beam becomes noticeable when the static coefficient offriction is close to zero. In such cases, the difference in critical load can vary greatly.Hence, for small values of the static coefficient of friction, the deformation of thewhole system becomes highly relevant.

Common for all the limit loads determined, is a decrease in the slope of their load-displacement curves near the limit load. For non-zero µs, this occurs around 0.75MN(Figure 6.1). By solving the Sturm-Liouville problem (Appendix C) obtained for ageometrically perfect column with constant bending resistance, clamped at the baseand hinged with a rotational spring at the top, one finds:

Pcr,ABAQUS = 0.767MN

for the column used in this thesis. The rotational spring coefficient was determinedusing the results of Section 5.6. Thus,

kABAQUS =MABAQUS

θ=

7.795 · 103

0.005= 1.559 · 106Nm/rad.

This suggests that the beam has a stabilizing effect on the column as the axial forceapproaches the limit load.

The effect of the bearing model being too stiff can be seen by using the restoringmoment given by TOBE (Section 5.6) in the equations of appendix C. Then:

kTOBE =MTOBE

θ=

2.539 · 103

0.005= 5.087 · 105Nm/rad.

andPcr,TOBE = 0.644MN.

32

7.3. SUGGESTED FURTHER RESEARCH

4 5 6 7 8 9 10 11 12 13 140.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9Axial force − vertical displacement of the column top

Axi

al f

orc

e [

MN

]

Vertical displacement [mm]

µs = 0.05

Euler case IV

Euler case III

ABAQUS

TOBE

Figure 7.1: Comparison between the limit load given by the FE-model with µs =0.05 and the critical load (dashed lines) for a geometrically perfect col-umn, clamped at the base and with a rotational spring at the top. Thespring stiffness is that of the bearing model used in the FE-model andthat given by TOBE.

Summarizing, the results of this thesis are:

• With a coefficient of static friction, µs > 0.5%, the limit load is an intermediatecase of the Euler cases III and IV.

• The limit load dependency on the deformation, due to bending, of the super-structure is such that a weaker superstructure yields a lower limit load.

7.3 Suggested Further Research

A normative design rule for the buckling of columns with pot sliding bearings, couldprobably be formulated using a column, hinged at both ends with rotational springsover both hinges. At the base, the spring constant is dependent on the properties ofthe substructure and/or a bearing. The spring constant for different bearing typesand sizes are already available from tests, performed by the bearing manufacturers.The main issue in formulating such a rule seems to be the available friction force.

33

CHAPTER 7. DISCUSSION

For this reason, the minimum possible value of the static coefficient of friction mustbe determined. Another important aspect to study, is the influence of deformation ofthe superstructure. This study is limited to a collapse load. A general design rule forcolumns with pot sliding bearings, must of course be based on a study, incorporatingdifferent load sets, including all the load cases which should be regarded in bridgedesign. Another possibility is a rule based on the clamped–guided column. Themaximum bending moment for these boundary conditions is much smaller than thatfor a clamped–free column (Figure 1.2) but does not differ very much from thoseof the clamped–hinged and the clamped–clamped columns. The rotational capacityof the foundation and the bearing, and the proper geometrical imperfection mustof course be included. In order to improve the bearing model, the proper materialand tribological parameters must be assessed. Yet another aspect of this problemmay need some attention, and that is the relation between the column cross-sectionand the width of the bearing. In this thesis, this relation was unnatural since thecolumn cross-section was smaller than the width of the bearing. Then the guidingof the column top is unquestionable. In a real application, however, the columncross-section may be much wider than the bearing width, so that the guidance ofthe column top may become only partial.

7.4 Modelling Issues

In order to rationalize the work for those who will dig deeper into this problem, somethings about the modelling of the bearing in ABAQUS/CAE need to be said. First ofall, contact modelling is computationally expensive. In the beginning I used a 32-bitCPU machine with 1GB ram. With this machine, solving for about 15000 unknownstook around 15–20 hours. After a while, a 64-bit CPU machine with 4GB ram waslent to me and the same jobs took around 5–7 hours to complete. In order to makeparameter studies on systems of this kind, good computer resources are necessary.Using a more powerful computer would enable a more extensive manipulation of theparameters which control the accuracy of the solutions. I have set these parametersso as to get a solution within reasonable time. The mesh densities and geometricalsimplifications of the bearing parts were also chosen with computational effort inmind. Apart from this, the main difficulties have been due to:

• The initial contact between parts which are only constrained by contact withother parts.

• Over-constraints between parts with several constraints.

The initial contact was solved by using over-closures and the adjust option in the*CONTACT PAIR option in ABAQUS as described in section 5.1, ”Surface Inter-action”. Regarding the over-constraints, one has to be very careful when definingcontact pairs. I had to use both surfaces and node regions to eliminate the over-constraints. If a group of nodes are defined using a surface, nodes on the surface andon the edge of the surface are included. If one wants only the nodes on the surface,

34

7.4. MODELLING ISSUES

one has to use a node region or a more refined way of choosing nodes than via theGUI (Graphical User Interface). The use of node regions resulted in warnings fromABAQUS/Standard regarding the accuracy in stress. Since I was only interestedin the mechanism of the bearing, I neglected those warnings but in a more detailedmodel with a finer mesh, this issue may have to be addressed.

35

Bibliography

[1] BRO 2004, VV Publ 2004:56. Vagverket, 2004.

[2] ABAQUS, Inc. ABAQUS/Standard 6.3 Manual, 2002.

[3] Clough, R. W., and Penzien, J. Dynamics of Structures, second ed. McGraw Hill, Singapore, 1993.

[4] Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. Conceptsand Applications of Finite Element Analysis, fourth ed. John Wiley & Sons,New York, 2002.

[5] Crisfield, M. A. Non-linear Finite Element Analysis of Solids and Struc-tures, Volume 1: Essentials. John Wiley & Sons, New York, 1991.

[6] Dym, C. L. Stability Theory and its Applications to Structural Mechanics.Dover Publications, New York, 2002.

[7] Eggert, H., and Kauschke, W. Structural Bearings. Ernst & Sohn, Berlin,2002.

[8] El-Naschie, M. S. Stress, Stability and Chaos in Structural Engineering: AnEnergy Approach. Mc Graw Hill, London, 1990.

[9] Jacobson, S., and Hogmark, S. Tribologi—Friktion, notning ochsmorjning. Liber Utbildning AB, 1996.

[10] Muller, I., and Strehlow, P. Rubber and Rubber Balloons, Paradigmsof Thermodynamics, Lect. Notes Phys. 637. Liber Utbildning AB, Springer-Verlag.

[11] Oden, J. T., and Martins, J. A. C. Models and computational methodsfor dynamic friction phenomena. Computer Methods in Applied Mechanics andEngineering 52, 1–3/september (1985), 527–634.

[12] Riks, E. Some computational aspects of the stability analysis of nonlinearstructures. Computer Methods in Applied Mechanics and Engineering 47, 3/De-cember (1984), 219–259.

[13] Spennteknikk AS. http://www.spennteknikk.no/HjemmesideEng/hovedeng.html,Online Internet: January 9, 2006.

37

BIBLIOGRAPHY

[14] Wriggers, P. Computational Contact Mechanics. John Wiley & Sons, NewYork, 2002.

38

Appendix A

Limit Loads

For each of the coefficients of static friction; 0, 0.005, 0.02 and 0.05, additionalresults from the FE-simulations are presented in the following order:

• The equilibrium path or, the axial force against the vertical deflection of thecolumn top.

• The reaction forces at the base of the column.

• Displacements and rotations over the bearing, that is at the middle of thebeam and at the top of the column.

39

APPENDIX A. LIMIT LOADS

A.1 µs = 0

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Axial force − vertical displacement of the column top; µs = 0

Axi

al fo

rce

[MN

]


µs = 0

Euler case III

Euler case II

Euler case I

Figure A.1: Axial force vs. vertical deflection at the column top, µs = 0.

40

A.1. µS = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

Vertical reaction force at the column base; µs = 0

Ver

tical

rea

ctio

n fo

rce

[MN

]

Time [s]

Vertical reaction forceLimit load

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−2

0

2

4

6

8

10

Horizontal reaction force at the column base; µs = 0

Hor

izon

tal r

eact

ion

forc

e [k

N]

Time [s]

Horizontal reaction forceLimit load

Figure A.2: Reaction forces at the base of the column, µs = 0.

41


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−50

0

50

100

150

Horizontal displacement; µs = 0

Hor

izon

tal d

ispl

acem

ent [

mm

]

Time [s]

Beam middleColumn topLimit load

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−6

−4

−2

0

2

4x 10

−3 Rotation; µs = 0

Rot

atio

n [r

ad]

Time [s]


Figure A.3: Displacement and rotation at the column top and the at the middle ofthe beam, µs = 0.

42

A.2. µS = 0.005

A.2 µs = 0.005

0 0.005 0.01 0.015 0.02 0.025 0.030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Axial force − vertical displacement of the column top; µs = 0.005

Axi

al fo

rce

[MN

]


µs = 0.005

Euler case I

Euler case II

Euler case III

Euler case IV

Figure A.4: Axial force vs. vertical deflection at the column top, µs = 0.005.

43


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Vertical reaction force at the column base; µs = 0.005

Ver

tical

rea

ctio

n fo

rce

[MN

]

Time [s]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−1

0

1

2

3

4

5

Horizontal reaction force at the column base; µs = 0.005

Hor

izon

tal r

eact

ion

forc

e [k

N]

Time [s]

Horizontal reaction forceAvailable friction forceLimit load

Figure A.5: Reaction forces at the base of the column, µs = 0.005.

44

A.2. µS = 0.005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.5

0

0.5

1

1.5

2

2.5

3

Horizontal displacement; µs = 0.005

Hor

izon

tal d

ispl

acem

ent [

mm

]

Time [s]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.02

0

0.02

0.04

0.06

0.08

Rotation; µs = 0.005

Rot

atio

n [r

ad]

Time [s]


Figure A.6: Displacement and rotation at the column top and the at the middle ofthe beam, µs = 0.005.

45


A.3 µs = 0.02

0 0.005 0.01 0.015 0.02 0.025 0.030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Axi

al fo

rce

[MN

]


µs = 0.02

Euler case IV

Euler case III

Euler case II

Euler case I


46

A.3. µS = 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1


Ver

tical

rea

ctio

n fo

rce

[MN

]

Time [s]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−5

0

5

10

15

20


Hor

izon

tal r

eact

ion

forc

e [k

N]

Time [s]



47


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.5

0

0.5

1

1.5

2

2.5


Hor

izon

tal d

ispl

acem

ent [

mm

]

Time [s]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.05

0.1

Rotation at the middle of the beam and at the column top; µs = 0.02

Rot

atio

n [r

ad]

Time [s]



48

A.4. µS = 0.05

A.4 µs = 0.05

0 0.005 0.01 0.015 0.02 0.025 0.030

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Axi

al fo

rce

[MN

]


µs = 0.05

Euler case IV

Euler case III

Euler case II

Euler case I


49


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1


Ver

tical

rea

ctio

n fo

rce

[MN

]

Time [s]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−10

0

10

20

30

40

50


Hor

izon

tal r

eact

ion

forc

e [k

N]

Time [s]



50

A.4. µS = 0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.5

0

0.5

1

1.5

2

2.5


Hor

izon

tal d

ispl

acem

ent [

mm

]

Time [s]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.02

0.04

0.06

0.08

0.1

0.12

Rotation; µs = 0.05

Rot

atio

n [r

ad]

Time [s]



51

Appendix B

Varying the Bending Stiffness ofthe Beam

For each of the coefficients of static friction; 0, 0.005 and 0.02, additional results forthe beam bending stiffness (Ibeam); 0.01m4, 0.02m4 and 0.04m4 are presented. Theresults are presented in the following order:

• The equilibrium paths for the three different values of Ibeam along with a limitload vs. beam area moment of inertia diagram.

• The rotation over the bearing and the rotation of the column top.

• The displacements at the middle of the beam and at the top of the column.

• The reaction forces at the base of the column.

53

APPENDIX B. VARYING THE BENDING STIFFNESS OF THE BEAM

B.1 µs = 0

0 0.002 0.004 0.006 0.008 0.010.2

0.25

0.3

0.35

0.4

0.45

Axial force − vertical displacement of the column top; µs = 0


Axi

al f

orc

e [

MN

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Euler case III

Euler case II

0 0.02 0.040.35

0.36

0.37

0.38

0.39

0.4

0.41


[m4

]

Lim

it lo

ad

[M

N]

(a) (b)

Figure B.1:(a) Axial force vs. vertical displacement at the column top for differentbeam bending stiffness, µs = 0.(b) Limit load vs. the area moment of inertia of the beam, µs = 0.

54

B.1. µS = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.002

0.004

0.006

0.008

0.01

Rotation of the bearing; µs = 0

Time [s]

Be

ari

ng

ro

tati

on

[ra

d]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−6

−4

−2

0

2

4x 10

−3 Rotation of the column top; µs = 0

Time [s]

The rotation of the column top [

rad

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(b)

Figure B.2: (a) Bearing rotation (θbeamMiddle − θcolumnTop), µs = 0.(b) Rotation of the column top (θcolumnTop), µs = 0.

55


0 0.2 0.4−20

0

20

40

60

80

100

120

140

160

180

Ibeam

= 0.01m4, µs = 0

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


0 0.2 0.4−20

0

20

40

60

80

100

120

140

Ibeam

= 0.02m4, µs = 0

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


0 0.2 0.4−20

0

20

40

60

80

100

120

Ibeam

= 0.04m4, µs = 0

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


Figure B.3: Displacement of the column top and the middle of the beam, µs = 0

56

B.1. µS = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

Vertical reaction force at column base; µs = 0

Time [s]

Ve

ritc

al r

ea

ctio

n f

orc

e [

MN

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−2

0

2

4

6

8

10

Horizontal reaction force at column base; µs = 0

Time [s]

Horizontal reaction force [

kN

] Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(b)

Figure B.4: (a) Vertical reaction forces at the base of the column, µs = 0.(b) Horizontal reaction forces at the base of the column, µs = 0.

57


B.2 µs = 0.005

0 0.005 0.01 0.015 0.020.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Axial force − vertical displacement of the column top, µs = 0.005


Axi

al f

orc

e [

MN

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Euler case IV

Euler case III

0 0.02 0.040.5

0.55

0.6

0.65

0.7

0.75

0.8


[m4

]

Lim

it lo

ad

[M

N]

(a) (b)

Figure B.5:(a) Axial force vs. vertical displacement at the column top for differentbeam bending stiffness, µs = 0.005.(b) Limit load vs. the area moment of inertia of the beam, µs = 0.005.

58

B.2. µS = 0.005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.04

−0.03

−0.02

−0.01

0

0.01

Rotation of the bearing, µs = 0.005

Time [s]

Bearing rotation [ra

d]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.04

−0.03

−0.02

−0.01

0

0.01

Rotation of the bearing, µs = 0.005

Time [s]


d]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(b)

Figure B.6: (a) Bearing rotation (θbeamMiddle − θcolumnTop),µs = 0.005.(b) Rotation of the column top (θcolumnTop), µs = 0.005.

59


0 0.2 0.4 0.6

−20

0

20

40

60

80

100

120

140

160

Ibeam

= 0.01m4, µs = 0.005

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


0 0.5 1−0.5

0

0.5

1

1.5

2

2.5

3

Ibeam

= 0.02m4, µs = 0.005

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Ibeam

= 0.04m4, µs = 0.005

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]Beam middleColumn topLimit load

Figure B.7: Displacement of the column top and the middle of the beam, µs = 0.005

60

B.2. µS = 0.005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Vertical reaction force at column base , µs = 0.005

Time [s]

Ve

ritc

al r

ea

ctio

n f

orc

e [

MN

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−5

0

5

10

15

Horizontal reaction force at column base, µs = 0.005

Time [s]


kN

] Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Available friction forceLimit load

(b)

Figure B.8: (a) Vertical reaction forces at the base of the column, µs = 0.005.(b) Horizontal reaction forces at the base of the column, µs = 0.005.

61


B.3 µs = 0.02

0 0.005 0.01 0.015 0.020.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Axial force − vertical displacement of the column; µs = 0.02


Axi

al f

orc

e [

MN

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Euler case IV

Euler case III

0 0.02 0.040.77

0.78

0.79

0.8

0.81

0.82


[m4

]

Lim

it lo

ad

[M

N]

(a) (b)

Figure B.9:(a) Axial force vs. vertical displacement at the column top for differentbeam bending stiffness, µs = 0.02.(b) Limit load vs. the area moment of inertia of the beam, µs = 0.02.

62

B.3. µS = 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

Rotation of the bearing; µs = 0.02

Time [s]


d]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.02

0.04

Rotation of the column top; µs = 0.02

Time [s]

Ro

tati

on

of

the

co

lum

n t

op

[ra

d]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(b)

Figure B.10: (a) Bearing rotation (θbeamMiddle − θcolumnTop), µs = 0.02.(b) Rotation of the column top (θcolumnTop), µs = 0.02.

63


0 0.5 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Ibeam

= 0.01m4, µs = 0.02

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


0 0.5 1−0.5

0

0.5

1

1.5

2

2.5

Ibeam

= 0.02m4, µs = 0.02

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]


0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Ibeam

= 0.04m4, µs = 0.02

Time [s]

Hor

izon

tal d

ispl

acem

ent [

mm

]Beam middleColumn topLimit load

Figure B.11: Displacement of the column top and the middle of the beam, µs = 0.02

64

B.3. µS = 0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Vertical reaction force at column base; µs = 0.02

Time [s]

Ve

ritc

al r

ea

ctio

n f

orc

e [

MN

]

Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Limit load

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−1

0

1

2

3

4

5

Horizontal reaction force at column base; µs = 0.02

Time [s]


kN

] Ibeam

= 0.01m4

Ibeam

= 0.02m4

Ibeam

= 0.04m4

Available friction forceLimit load

(b)

Figure B.12: (a) Vertical reaction forces at the base of the column, µs = 0.02.(b) Horizontal reaction forces at the base of the column, µs = 0.02.

65

Appendix C

The intermediate Euler cases

Solution for the critical load of the intermediate case of the Euler cases III and IV.ML = kv′L is the moment produced by a rotational spring of stiffness k at the topof the column. v′L = v′(L) is the rotation of the column top.

Figure C.1: The structural system of the intermediate case of the Euler cases IIIand IV.

The cross-sectional bending moment is given by;

M(x) = −M0 + Rx + Pv(x), (C.1)

where M0 is given by the moment equilibrium equation:

M0 −ML −RL = 0, M0 = ML + RL = kv′L + RL. (C.2)

67

APPENDIX C. THE INTERMEDIATE EULER CASES

According to the Euler-Bernoulli beam theory;

M(x) = −EIv′′(x) (C.3)

which yields the following Sturm-Liouville problem;

v′′(x) + κ2v(x) =1

EI

(kv′L + RL−Rx

), (C.4)

where;

κ =

√P

EI(C.5)

with the boundary conditions:

v(0) = v′(0) = v(L) = 0, v′(L) = v′L. (C.6)

The solution to the homogeneous part of this differential equation is well known;

vh(x) = A sin(κx) + B cos(κx) (C.7)

and the particular part is easily found by substituting;

vp(x) = a + bx (C.8)

in equation (C.4) and identifying the coefficients a and b. Then, the particular partof the solution is given by:

vp(x) =1

κ2

(kv′L + RL

EI

)− 1

κ2

(R

EI

)x =

kv′L + RL

P− Rx

P(C.9)

and the complete solution can be written:

v(x) = vh(x) + vp(x) = A sin(κx) + B cos(κx) +kv′L + RL

P− Rx

P(C.10)

In order to solve for the four unknowns A,B,R and v′L, the boundary conditions(C.6) are introduced in equation (C.10), yielding the following system of equations:

Ax = b (C.11)

where:

A =

0 1 −LP

kP

κ 0 1P

0sin(κL) cos(κL) 0 − k

P

κ cos(κL) −κ sin(κL) 1P

−1

, (C.12)

x =

ABRv′L

(C.13)

andb = 0. (C.14)

68

Non-trivial solutions are obtained only if det(A) = 0, which yields the followingequation for κ:

(κ2LkL − κ2EI

)sin(κL) +

(2κkL + κ3EIL

)cos(κL)− 2κkL = 0 (C.15)

Finally, the critical load is determined by substituting the κ given by solving equa-tion (C.15) into equation (C.5) and solve for P .

Adding a rotational spring at the base of the column changes the boundary condi-tions. The bending moment distribution is the same and thereby, the non-homogeneouspart of the differential equation remains a second order polynomial. By solving forR in equation (C.2), one finds:

M0 −ML −RL = 0, R =M0 −ML

L(C.16)

and the differential equation can be written:

v′′(x) + κ2v(x) =M0

EI− M0 −ML

EI

x

L, (C.17)

v(0) = v(L) = 0, v′(0) =M0

k0

, v′(L) = −ML

kL

. (C.18)

The solution to this equation is found in the same manner as in the previous case:

v(x) = A sin(κx) + B cos(κx) +M0

P− M0 −ML

P

x

L. (C.19)

By substituting the boundary conditions (C.18) one finds the system matrix:

A =

0 1 1P

0κ 0 −(

1PL

+ 1k0

)1

PL

sin(κL) cos(κL) 0 1P

κ cos(κL) −κ sin(κL) − 1PL

(1

PL+ 1

kL

)

, (C.20)

for the unknowns:

x =

ABM0

ML

. (C.21)

The equation for κ is now:

(κ2LkL − κ2EI − 1

k0

[κ4E2I2L− κ2EIkL

])sin(κL) +

+(2κkL + κ3EIL +

kL

k0

[κ3EIL

])cos(κL)− 2κkL = 0, (C.22)

where the bracketed terms are the contributions due to the second rotational spring.Equation (C.22) can be used to determine the critical loads for Euler cases II(hinged–hinged), III (clamped–hinged), IV (clamped–clamped) and all the inter-mediate cases.

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