THE STABILITY OF ARTICULATED TIPPING TRAILER UNITS by Simon G Pickering, BEng Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy January 1994 _.... _ 11' ! '~"" ; CONTENTS Page No TABLE OF CONTENTS (i) ABSTRACT (iii) ACKNOWLEDGEMENTS (v) NOTATION (vi) CHAPTER 1 INTRODUCTION 1.1 Size of the problem 1 1.2 Trailer construction 2 1.3 Theoretical model 4 CHAPTER 2 GENERAL BACKGROUND 2.1 General description of tipping trailer units and operation 7 2.2 Loading modes 10 2.3 Design features 13 2.4 Roll over models 17 2.5 General discussion and conclusions 18 CHAPTER 3 BASIS OF THE THEORETICAL MODEL 3.1 Introduction 30 3.2 Co-ordinate system 32 3.3 Assumptions 33 3.4 Theoretical basis 37 3.5 Outline of the overall solution procedure 55 CHAPTER 4 FINITE ELEMENT ANALYSES 4.1 Introduction 64 4.2 Finite element meshes 66 4.3 Design investigations 72 4.4 Loading of finite element meshes 73 4.5 Flexibility influence and stiffness matrices 75 4.6 Discussion 78 (i) CHAPTER 5 DESIGN INVESTIGATIONS 5.1 Introduction 97 5.2 Reference chassis 98 5.3 The effect of using changes on stability 102 5.4 The effect of increasing the torsional stiffness of the cross members 103 5.5 The effect of reducing the trailing arm stiffness 104 5.6 Discussion 105 CHAPTER 6 GENERAL DISCUSSION AND CONCLUSION 6.1 Technique 152 6.2 Theoretical model 152 6.3 Implications 156 6.4 Requirements of further investigations 157 REFERENCES 159 APPENDIX A AIRBAG STIFFNESS Al Introduction 162 A2 Theoretical prediction of airbag force/weight relationship 162 A3 Experimental procedure 163 A4 Results 164 APPENDIX B TYRE STIFFNESS B1 Theory 172 B2 Experimental equipment and procedure 174 B3 Results 176 APPENDIX C COMPUTER PROGRAM AND FLOW CHART Subroutine descriptions 188 APPENDIX D PROGRAM USER GUIDE D1 Program execution 271 D2 Changing program variables 173 D3 Changing flexibility, stiffness and influence matrices 274 APPENDIX E FINITE ELEMENT ANALYSES AND FLEXIBILITY, INFLUENCE AND STIFFNESS MATRICES El Introduction 278 E2 Flexibility matrices 278 E3 Influence matrices 279 E4 Stiffness matrix 280 (ii) ABSTRACT When an articulated tipper unit is being loaded or is tipping, it is unlikely to be standing on perfectly level ground. Also, the centre of gravity of the load is unlikely to be in the centre of the body. Hence the loadscarried by the suspension and tyres on one side of thetipper will be greater than those on the other side. This uneven loading will cause the tyres and suspension on one side of the tipper unit to deform more than those on the other side. It will also cause the chassis to deform; the twisting about its longitudinal axis being the most significant mode of deformation. As a result of tl.ese deformations caused by the uneven loading, the position of the centre of gravity will be shifted even further towards the more heavily loaded side. This will cause even more uneven loading and further deformations. Under stable conditions a situation will exist at which the position of the centre of gravity, the deformations and the forces transmitted through the system are compatible. Instability, resulting inroll-over would occur ifthe overall centre ofgravity of the load, body, chassisetc. were to fall outside the area bounded by the contact of the wheel with the ground, before a stable condition was reached. Many factors influence the roll stability. To increase stability, an understanding of the influence of components of the lorry on the stability is required. In order to achieve this, a theoretical model of an articulated tipper wasdeveloped which will allow roll-over predictions to be made for a given lorry in likely attitudes. In this model dimensions and stiffness of the lorry components can be altered to assess their influence on roll stability. (iii) The previous theoretical roll-over models were based on lumped mass systems, representing various parts of the lorry inter-connected by compliant elements. Certain tlexibilities such as the tyres, suspension units, etc. could be obtained from the respective components manufacturers but the tractor and trailer chassis tlexibilities are unknown. To overcome this problem the tlexibilities were obtained from full scale static tilttests. This isa very expensive undertaking, providing a limited means in which to assess those elements of trailer design which are important in improving stability, without further recourse to more tilt tests. It was decided that the finiteelement method should be used to model the tractor and trailer, in order to determine the important deformations. Once the finite .element model is created it is relatively straight forward to make changes to the structure. Hence an assessment of component contribution to roll stability can be undertaken relatively inexpensively. Whilst a vehicle operator should always endeavour to discharge the payload with the vehicle standing on levelground, practical situationsarise where this is not possible. This may be due to the absence of level ground or poor judgement by the operator, which may result in the vehicle being tipped on a lateral ground slope. Asa result of this, the maximum ground slope angle considered for the theoretical model is limitedto eight degrees, as this position is at least twice the severity of ground slope on which a vehicle should normally be tipped. For each trailer design, the magnitude of the load, position of the load, ram lengthand ground slope can be varied in any combination. Four payloads and up to nine payload positions are considered, varying the ground slope from 0 to 8 degrees and varying the ram length from 2 to 8 meters. Also, three further chassis configurations, based on the reference chassis were modelled to investigate the contribution of important component tlexibilities on roll stability. (iv) ACKNOWLEDGEMENTS The author wishes to thank all those colleagues who have assisted him with this work. Particular thanks are due to Professor Henry Fessler and Professor Tom Hyde who provided tremendous enthusiasm, encouragement and guidance. Thanks also to York Trailers Ltd and the SERe who provided financial support for this work. The author also wishes to thank Mr Bob Davies (York Trailers Ltd) for guidance on technical issues, and toJudith Bray for her care and patience in typing this script; and Helen Geraghty and myparents for providing muchneeded support during the writing of this thesis. (v) NOTATION Forces left and right airbag forces, see Fig. 3.5 forces at the top and bottom of the right hand airbag of the 1st suspension, see Fig. 3.5 forces at the top and bottom of the right hand airbag of the 2nd suspension, see Fig. 3.5 forces at the top and bottom of the right hand airbag of the 3rd suspension, see Fig. 3.5 A R(ARt' A A Rl, Rl, AR4, ARS, ARJ columr. 0;' forces applied to the right hand airbags B- (By, BJ body force vector column matrices of all important chassis forces G (GLh Gu, GU, G Rt, GRl, GRl) column matrix containing the left and right vertical tyre forces left hinge force vector right hinge force vector L force applied to cantilever payload force vector ram force vector ! (TL, TJ column matrix containing the left and right tie bar forces wind force vector tractor self-weight vector chassis self-weight vector (vi) Position Vectors (relative to origin of X, Y, Z coordinate system) b (bn by,bJ position of the centre of gravity of body .e (Px, Py, pJ position of the centre of gravity of payload position of the .eft hinge position of the right hinge position of the bottom of ram position of the top of ram position of the centroid of wind pressure position of the top of the ram relative to the bottom of the ram Dis.placements column matrix of all important chassis displacements left hinge displacement vector right hinge displacement vector bottom of ram displacement vector y components of displacement of the upper and lower, Valand Val right hand airbag ends of 1st suspension, see Fig. 3.5 Vuand Va4 y components of displacement of the upper and lower, right hand ends of the 2nd suspension, see Fig. 3.5 VaSand Va6 y components of displacement of the upper and lower, right hand ends of the 3rd suspension, see Fig. 3.5 v, (Va"Val,vu, va4, VaS,va6)column matrix of the y components of displacements of - the right hand airbags vertical displacement of cantilever beam tip tj>(i) rotation of cantilever beam tip 8 hinge bar rotation about X axis, see Fig. 3.1 slope of the ground, see Fig. 3.1 hinge bar rotation about Y axis, see Fig. 3.1 (vii) Others a, b dimensions of the rigid frame attached to the tip of the cantilever, see Fig. 3.2 E Young's modulus I second moment of area influence matrices relating the important chassis displacements to the important chassis forces for different airbag conditions influence matrices relating the tie bar forces to the important chassis forces for different airbag conditions [1(·)0], [I(b)0], [I(C)01 [1(cI)0] influences matrices relating the ground/tyre reaction forces to the important chassis forces (or displacement) for different airbag conditions influence matrix relating the right hand airbag displacements to the important chassis forces [K] stiffness matrix relating the important chassis forces to the important chassis displacements I length of cantilever, see Fig. 3.2 length of gas contained in the airbag cylinders at the limiting pressure n exponent in the equation for polytropic compression of a gas Q constant in Equation 3.17 x, Y,Z co-ordinate system with its origin at the centre of the hinge bar inclination of the body relative to the x axis angle between the vectors! land.! b angle between the vectors! land £ vector product scalar product + vector addition vector subtraction (viii)