bE ý là e + TY 4 _ - döng SCHLAICF " do Hartmut SCHEEF ¬ mm CONCRETE - BOX-GIRDER BRIDGES
International Association for Bridge and Structural Engineering IABSE
Trang 3FOREWORD
The box girder is today the most widely used superstructure in concrete bridge construction That fact
justifies the suggestion made by the Commission ill of the [ABSE that a comprehensive survey be
written concerning this particular bridge type The authors proceed from the assumption, however, that its contents will first be drawn upon when all possible design alternatives for the particular bridge pro- ject have been thoroughly examined, and the box girder has been proven appropriate Their aim is less that of encouraging the one-sided propagation of box-girder bridges but rather much more that of con- tributing to the improvement of the quality of such bridges They hope to contribute to this by exten- sively relieving the engineer of the study of today's hardly surveyable mass of literature on the subject so that he can better devote that time to the actual design of the bridge That explains why this paper is kept short, why in particular cases the reader is referred to the literature, and why subjects not per- taining to the central theme are only touched upon and not handled exhaustively
For greater clearness, the survey follows the sequence of a practical bridge design process by dividing itself into three main parts, namely, “Design”, “Structural Analysis”, and “Dimensioning and Structural Detailing”; each section with its individual numbering and literature list
This survey directs itself especially to the design engineer, which manifests itself, for example, in the
fact that the construction methods are handled only briefly and in the section “Design”, because they decisively influence the design at the very beginning
Major contributions to Section il, “Structural Analysis”, were made by Prof Dr.-Ing Kurt Schafer, a col-
league of the authors in the Institut fiir Massivbau at the University of Stuttgart In this section the
attempt is made to portray the cafculation of the box-girder sectional! forces resulting from eccentric vehicle ioads with consideration of the folded plate action or profile deformation so comprehensively that it is not only easily understood but also rapidly applicable in the design office This thereby elimi- hates the often-discussed, controversial question as to whether the effort involved in the “exact” calcu- Jation of this loading case is actually worthwhile or whether an estimation of the transverse load distrib-
ution would not suffice
The authors would like to take this opportunity to thank Professors R Favre, Lausanne, and C Menn, Zirich, for their critical examination of the paper They are indebted to Mrs 1 Paechter and Mrs E Schnee for their conscientious preparation of the manuscript; and Mr E Kluttz for his empathetical translation of their German original into English
Trang 4
Copyright © 1982 by
International Association for Bridge and Structural Engineering
Trang 5TABLE OF CONTENTS Part| DESIGN 4 2 3 4 Part lì 1 2 3 Terms, Symbols Introduction Historical Development Over AliDesign .- - 41 Design Principles
4.1.1 The Role and Sequence of the Design Process ae
41.2 RemarksastoForm . - +22 eee eee 4143 Costs 42 Construction Methods 43 Superstructure 434 General 43.2 Longitudinal Direction 443.3 Transverse Direction - 4.4 Complete System and Supports 45 Substructure - - 45.1 Abutments 45.2 Plers 45,3 Foundations Lieratue - Introducton Loads and Extemal Influences Structural System 31 FinalStaite 3.22 During Construction Critical Loadings and Sectional Forces 41 Longitudinat Direction 4.2 Transverse Direction Simplified Structural Analysis of the Superstructure 51 Genesral 5.2 Analysis Procedure 5.3 Analysis in the Longitudinal Direction 5.3.1 Sectional Forces du STAUCTURAL ANALYSIS
8 to Loads plus Restraints
5.3.2 Time-Dependent Alteration of the Sectional Forces by Creep 5.4 Analysis in the Transverse
5.4.1 Sectional Forces Acting on the Flanges 64.2 Analysis as a Frame 5.4.3 Transverse Bending Direction .- Moments in the Haunch of Variable-Depth Girders 5.5 Folded PlateAction .- 55.1 Fundamental Concept
5.5.2 Solution by Means of the Analogy of a Seam on an Elastic Foundation
5.6 Multipfe-Cell Box Girders
Trang 67 Abưtments ee wae 72
8 Piers 20.0 ee ee eee tae vơ tae tee ca 72
81 Loadings .- wae te sa " tae 72
82 Effective Length nee ca tee wae 73
83 Moments According to 2 Order Theory © 2-0 ee ee ng 74 9 Foundation 75 10 Literature 78 Part Il] DIMENSIONING AND DETAILING 1 Introduoetion TỪ ỒỮ ee eee 2 General Detailing Principles 3 Prestressing -
3.1 The Level of Prestress cư 3.2 Tendon Profile in the Transverse Directi 3.2.1 Top Flange 3.2.2 Prestressing of the Webs 323 Bottom Flange .- 3.3 Tendon Profile in the Longitudinal Direction 331 IntheWebs
33.2 In the Top and Bottom Flanges
3⁄4 Transfer of Concentrated Prestress Forces
3.5 Construction Joints and Coupling Joints 4 Dimensioning of the Top and Bottom Flanges 5 Dimensioning and Reinforcement of the Webs
5.1 Dimensioning for Shear, Torsion, and Transverse Bending " tee see
6.2 Web Reinforcement ee
6 Transverse Diaphragms - SH HH he kh nơ
Abutments, Piers and Foundations "—= 8 Bridge Bearngs ee ee B81 BearingTypes 2 eee ee 82 Installation and Maintenance - 83 Design of Bearings 9, Bridge Finishes .-.- 91 Expansion Joints wae 92 Bridge Railing .-.- 9.3 Roadway Surface 94 Drainsge 10 Literature Note:
Parts t, It, and Ill form a whole and are only divided for organisation reasons Should the reader be referred to a figure, a section, or a reference in one of the other parts, he Will find that the Roman numeral of the other part of the text is placed before the Arabic number; for example, Figure 1,7 or
Trang 7Part | DESIGN 1, TERMS, SYMBOLS Terms, Symbols 1 + re) rhe ? / Ty 7 iv? | L SECTION a-a
Fig 1 Sections through a typical simple box-girder bridge The Elements of a simple Box-Girder Bridge Foundation 1 plate 2 pile plate 3 bored piles 4 driven piles Substructure 5 6 7 9 9 10 1 12 13 14 18 16 17 18 box abutment spill-through abutment columns, piers {with 2 or more hearings) breast wall wing walt back wall edge beam end diaphragm bridge seat support walls bridge seat beam access chamber bearing (can be fixed or allow movement) expansion joint Superstructure 19 20 21 22 23 24 25 26 2a 28 29 30 31 transverse diaphragm {at abutments, within the span
and over the piers) with opening box-girder web
top slab (area between the webs)
Trang 8
2 Design
2, INTRODUCTION
Though box-girder bridges are indeed often not the only solution to a bridge project they are, however, seldom the absolutely false one and really only excluded in the case of very small spans or sharply skewed bridges This universal applicability they owe, from the point of view of load-carrying, to their indifference as to whether the bending moments are positive or ne- gative and to their torsional stiffness; from the point of view of economy, to their suitability for a factory-like construction sequence; and finally, from the point of view of form, to their sleek lines with which they fit into every landscape and surroundings
He, though, who expects a structure to reflect the flow of forces within it through ite outer shape will regret the above-mentioned neutral load-carrying behaviour of the box-girder bridges, especially those of today As a result of the need to construct the bridges economically, the.development has tended towards bridges with constant depths, even for varying spans Box- girder bridges therefore deserve special care and attention with respect to pleasing proportions and conscientious shaping of their details
The above mentioned characteristics have made the concrete box-girder bridges the most widespread bridge type today, The fact that this develop- ment will continue for some time justifies this survey concerning a topic that is certainly much too specialised for he who does not like to see his design possibilities restricted Therefore let one be reminded that the box girder is only a part of the entire bridge structure and that the directly supported box girder continuous over the supports is only the standard case The box girder can also be found in portal frame bridges, arch and bow bridges, and cable-stayed and suspension bridges of all kinds {Fig 2)
TTT 7N EELV, Atmadh, mT Wy „im gg»
beam portdl” frame arch / bow suspension / cable- stayed
Trang 9Historical Development 3
3, HISTORICAL DEVELOPMENT
The first bridges of reinforced concrete were built as were their predecessors of stone, They were arch bridges with a gravel fill for the road surface between the two bridge parapets over the arch, Later the gravel fill was replaced by a transverse roadway slab, and the transition to a box girder was achieved
The world’s first reinforced concrete bridge, ari arch bridge, was built in 1875 t1 Probably the first box-girder bridge was Hennebique’s
Rigor gimento Bridge in 1911, a 3-hinged arch (see Fig 3) pe la SECTI0N a-q |SECTION b-b 1 4 50 E†-s.o—+-laso =TTT)
Due to improvements in the quality of concréte and steel as well as a better
understanding of the material behaviour of reinforced concrete, the arch lost
more and more on importance as a load-carrying system to that of the beam loaded in bending The longest span simply reinforced bridge to-date was
puilt in 1939 at Ville-Neuve-St George, a three -celled, thin-walled, vary-
ing-depth box girder of three spans with a 78 m middle span Fig 3 Risorgimento Bridge
It was only with the development of high-strength prestressing steel that it became possible to span longer distances, The first prestressed concrete
bridges, most of 7 or j-cross-section, were built towards the end of the
1920’s, The great breakthrough was achieved only after 1945, The Sclayn bridge over the river Maas, which was built by Magnel in 1948, was the first continuous prestressed box-girder bridge with 2 spans of 62, 70 m(see Fig 4) In the following years the ratio of wages to material costs climbed sharply This thereby shifted the emphasis of development to the construction method, section a-a Ị = 82.70 m | TT \prestressing cable —!— (without bond} 6,75
Trang 104 Dasign
Important development stages were the following:
- cantilever construction in situ: the bridge over the river Rhine at Worms in 1953 with a main span of 114 m, the Hamana bridge in Japan in 1978 with a main span of 230 m, the cable-stayed Brotonne bridge in 1978 with a main span of 320 m
- cantilever construction with precast elements: the Chillon Viaduct in 1969 with a main span of 104 m, the cable-stayed bridge over the Columbia River in the USA in 1980 with a’main span of 300 m
- construction with a travelling scaffolding: in situ (bridge on the Kettiger
Hang in 1959, with equal spans of 39 m each) or precast (Rio Niteroi Bridge in 1974 with equal spans of 80 m)
- incremental launching method: Rio Caroni Bridge in 1962 with equal spans of $6 m and the Taubertal Bridge in 1965 with spans of 54 and 60 m
The box-girder cross-section evolved structurally from the hollow cel] deck bridge or'the T-beam bridge The widening of the compression zone that be- gan as a structural requirement at the central piers was in the end extended throughout the entire length of the bridge because of the advantageous trans- verse load-carrying characteristics
The first box-girder cross-sections possessed deck slabs that cantilevered
out only slightly from the box portion (see Fig 5, a - e) With prestressed
concrete, the length of the cantilever could be increased, The high form- work costs caused a reduction in the number of cells (see Fig 5, f and g) In order to reduce the construction loads to the minimum possible or to require only one longitudinal girder in the working state even with multiple
traffic-lanes, the one-celled built-up cross-section constructed in modular
fashion emerged as the last development (see Fig, 5h ) This allowed the Semorile Viaduct to be built by first incrementally launching the U-shaped
portion of the box girder and adding the deck slab afterwards by means of
precast elements and insitu concrete infilling (see Fig 6) Some precast elements were built in before launching to provide stiffness In the case of the Eschachtal Bridge (Fig 7), the closed box girder was first constructed with only short cantilever deck slabs using the balanced cantilever method with erection girder In the follow-up stage of constructions, the main can- tilever deck portion was added in insitu concrete and supported by precast concrete struts, The Kochertal Bridge and the Liesertal Bridge were simi-
larly constructed, In the case of the West Gate Bridge in Melbourne however,
Trang 11Historical Development 5 Fig 5 Development of the box-girder cross-section Fig 6 Semorile Viaduct [3] Fig 7 Eschachtal Bridge [4 longitudinal prestressing strand girder component insitu concrete joint
Trang 12
6 Design
The concrete box girder with streamlined cross-section has also been success- ful recently in the case of cable-stayed bridges, Its high dead load favorably influences the dynamic stress amplitudes of the cables and the necessary long- itudinal prestressing steel A torsionally stiff box girder is required to handle the torsional moments incurred by attaching the cables to the bridge’ s centerline (see Fig 9) Attaching the cables to both sides allows a much- reduced section depth A case.in point was the Columbia River Bridge where the side cells of the approach span cross-section (Fig 10 ) were only requi- red in the area near the cable-stayed portion of the bridge, and transverse diaphragms were added only where the cables were attached to the cross- section
A further area of application for box girders arose from noise and automobile emissions control for inner city elevated highways (Fig 11) Fig 10 Columbia River Bridge [6]: a) cable-stayed bridge, b) approach span glazed openings for natural lighting
Fig 11 Closed box girder for inner city elevated highways (7
Even though the structural development of the box girder cross-section is prob-
Trang 13Design Principles T7
4 OVER ALL DESIGN 4,1 Design Principles
4,1.1 The Role and Sequence of the Design Process
The over-all structural and architectural design is the most important and, for its designer, the most stimulating and beautiful stage in the creation ofa structure Problems during detailed analysis and-design and during construc- tion as well as defects arising during its use canbe traced in most cases to
a faulty overall design Design entails finding an optimum’ compromise among
the particular objective and subjective boundary conditions of an individual
structure That is why no bridge, even a box-girder one, can be like another
Much latitude remains for creative fantasy and considerations of quality and responsibility, The design engineer must be conscious at all times of the fact that he is irrevocably changing an area’s environment with his bridge Therefore, not only considerations of stability, serviceability, and economy count, but of equal importance are the bridge shape, the bridge’ s harmoni- zation with the surrounding landscape, and its impact on the preservation of
the quality of life of man and nature, `
The optimum solution, always and exclusively a subjective evaluation, can only be found through the comparison of many alternative solutions with a different assessment of the individual boundary conditions in each case, The entire planning process from the sketched design to the planning of the con- struction is a cyclical process of increasing refinement (Fig 12) It is to be emphasized that the development of the structural details should be inclu- ded in the earliest phase of the design process, whereas the computer calcu- lations range at its end and should confirm only whether the roughly deter- mined dimensions suffice, Construction is carried out according to drawings,
not calculations Above all, it would be a great mistake to believe that the
computer could relieve the engineer of the design of a bridge
4.1.2 Remarks as to form
In contrast somewhat to arch, suspension, or cable-stayed bridges, the box-
girder bridge fits into almost every surrounding, be it varied or monotonous {8] Just because of its simplicity, the observer finds every imbalance in its proportions and uncleanness in its lines to be disturbing
Realizing well enough that, fortunately, it is not possible and perhaps even detrimental to put forth generally valid rules for aesthetics, a few remarks on the subject will nevertheless be hazarded here, It has served its purpose well if the fact that a separate section in this paper is devoted to architectural design and form promotes the sharpening of the consciousness among engineers with respect to this subject That will even be achieved in the event that the reader finds the following guidelines or orientation aids to be false and in this manner critically grapples with the questions of form (Compare especially
Trang 148 Design ECONOMY |- of the piers of the foundation SKETCHED DETAILS} MODEL ANALYSIS DIMENSIONING =exhgust COMPUTATION emissions OF QUANTITIES |-waste water climate Skercuey pes
Fig 12 Planning: a continuous refining process
- Order in the structural system: Retain the selected structural system (beam, arch, frame, suspension or cable~stayed) and only combine it with another system if the topografical boundary conditions change signi- ficantly along the length of the bridge (e.g widened river bed or approach) - Harmony: Strive for balanced proportions among the length of the spans,
the construction depth, and the depth of the valley; and between the
supporting and supported structural elements, The span/depth ratio , 1/d,
alone is no guaranty for a light and elegant appearance (Fig 13)
Trang 15Design Principles 9
Fig 13 Although 1/4, is greater than L/4, , (b) appears lighter than (a)
- Simplicity and clearness: Allow the function of the structural element and the necessity of the material to be perceived, and avoid unnecessary frills - Integration into the environment: Either subordinate the bridge to its
surroundings; or, if the landscape is monotonous or the surroundings are
disorderly built up, make a feature of the bridge
- Pay attention to the order of scale between the bridge and its surroundings and the bridge and the individual person A long bridge, especially iftit passes through hilly terrain, should never be set out in a straight alignment but harmonically pick up the movement of the terrain and the lay-out of the
curves of the highway
An odd number of spans which decrease in length in the direction of the abut- ments are found to‘be pleasin (Figs 14a, “e and i) Veryirreguier-spen-—-
lengtli® produce a feeling of uneasiness (Pig 14 b), Many spans of equal length produce a boring effect, and the valley appears to be walled in (Figs
14¢ and d), although a design in the form of Figure 14d with very slender in- dividual piers can be a viable solution The harmonic division of a line accord-
ing to the "golden section" commonly used in antiquity is limited by various
Trang 16
10 Design-
Should the bridge gradient follow a trough or a crest, i.e the upper charac- teristic line is curved, then a satisfying appearance can be best achieved by means of a constant depth, d, of the box-girder for the entire bridge Especially for inner city elevated highways, the depth should be as small as
possible and the corresponding reduction in the lengths of the spans be
accepted, a) dy +
7 STs
GE u
Fig, 15 Shadow effect caused by Fig 16 Fascia Beam: the form (b) the cantilevered deck slab causes the bridge to appear
sleeker than (a)
The length,a,of the cantilevered deck slab and the form given to the fascia beam and railing determine the side view of the bridge as follows:
- For a/d< 1 no shadow effect is created, but for a/d> 3 the shadow is
very noticeable, and for a/d >2 it can be usefully employed as an element of form (Fig 15), For varying bridge depths, however, the shadow should not be allowed to extend beneath the soffit of the box girder
- A deep, if necessary white-painted, fascia beam reduces the perceived depth of the box girder (Fig 16), Suitable values are the following [ 8, 10}:
d/d =1/5 to 1/4 but always < 1/3 d/l “ 1/20 to 1/80 for long bridges
4d, > 200 mm
An inclined fascia beam appears even lighter because of the angle at which the light falls on it (Fig 17) In addition, the wind is also deflected to ad-
vantage, In the case of bridges near residential areas, the fascia beams
can be placed higher to act as noise barriers (Fig 18), Varying inclina- tion cause light-dark effects that give the bridge a sleeker appearance With an appropriate design such deep fascia beams can be used as load- bearing members together with the box-girder
pa m—
Fig 17 An inclined fascia beam Pig 18 Fascia beam as noise barrier Great care should be employed with the shape and structural details of the guard rail Sufficient protection for the pedestrian plus an unhindered
view for the driver of the motor vehicle are achieved with a hefty cross-
beam mounted onto slender vertical standards placed at intervals of
approximately 150 mm
Trang 17Design Principles 11
Especial care should also be taken with the proportions of the piers with respect to the bridge superstructure, If the piers are too slender, the super- structure will appear too heavy, especially if the bridge cross-section is increased in depth from a minimum at mid-span to a maximum over the piers There the piers should never be any narrower than the box- girder bottom flange which they support Round piers appear boring compared with prismatic ones with at least six sides, The abutments should be placed high in the embankment and be only marginally exposed,
In order to assess the appearance of the bridge, it is important to compose a
“picture' of it This means making sketches at first and later drawings to
scale, Along with a plan view, cross-section, and elevation, at least one per- spective drawing is necessary in order to obtain an impression of the bridge in its surroundings With an inclined view one can best determine if the super-
structure and substructure are in harmony with each other, Of great help in
this respect is the photomontage, which is really not so costly, or a model The model must include a sufficient portion of the surroundings Not only should one not attempt to evoke a favourable impression of the structure by choosing an unrealistic standpoint from which to observe the bridge,but one should design a bridge that is found to be harmonically proportioned even when viewed from
the worst possible perspective
4.1,3 Costs
In letting the contract for a bridge one should not overlook the fact that the total costs include not only the costs of construction but also the follow-up costs of maintenance and the ensuring of proper functioning of the bridge for at least 50 years Unfortunately, even in the rich industrialized nations of the world, bridges are predominantly let on the basis of the lowest bid One is seldom prepared to pay a bit more for a good architectural design and form and tong-lasting quality
The growth of costs shown in Fig, 19 are valid for Central Europe, Depending upon the regional construction materials at hand, the qualifications of the wor-
kers, the ratio of wages to material costs, the extent of mechanization of the
construction industry, and the climate; the particular cost analysis yields dif- fering results, The course of costs over time portrayed in Fig 20 shows that
an economical solution is only achieved today if the labour-intensive work is li-
mited through repetition of the same operations or construction under factory conditions Cost optimizations with specification of the most favourable span length or the like, as is so often found in papers are questionable for the most part, because they can never take every possible parameter into consideration They often only consider material expenditure.Only if the ratio of labour costs
to material costs is less than 1 do savings of material gain significance
One can observe from Fig, 19 that inflated calculations of savings of mild steel
reinforcement, approximately half of which is required as minimum reinforce- ment anyway, do not change the total cost of the bridge to any great extent Lar-
Trang 1812 Design 8% 50% 86% 100% seven areata sec — TÑ— tín sơi 30% 00% 500 costs % sof superstructure | ¿00 labour, ge Hie _TỦQ% 639`¬-.400%, 300 [dai ell foundations | concrete + steel amt 200 its 23:0 100 (18% 68% 1 0% essing 1960 970 — T980 concrete steel time normal teinforcement
Fig 19 Cost breakdown:average values of Fig 20 Growth of costs in the
17 typical examples (from [14] ) : industrialized ‘nations For a bridge with average span lengths and a span/depth ratio, 1/d © 18, one
can proceed from the following values for the superstructure (from [14]): 3 2, ; - m concrete per m_ bridge surface area, expressed as an average thickness 0,45-1[m} 470,38 + ng {m - quantity of reinforcement (mass of steel/m®? concrete) normal reinforcement = 110 ‘kg/m?
prestressing steel for the case of predominantly
continuous prestressing tendons © ( 4,5 + 0, 5-] [m] hee /m 3} } - average cost of falsework and formwork = 60 % of the expenditure for
concrete and steel, 4.2 Construction Methods
The method of construction influences the design and its details in both the longitudinal and transverse direction, In the laSt 20 years construction me- thods have experienced a stage of rapid development They appear today, however, to have reached the end of this stage Box-girder bridges are con-
structed today on stationary falsework (Fig 21), if at all, only in the extreme
ease of a small number of spans or when the superstructure is not at a great height above the ground For some of the large bridges highly mechanized constructions methods are used, By means of construction rhythms and many
repetitive construction operations, these methods reduce construction time
and formwork (Figs, 22 and 23),
For the foreseeable future one can expect that long bridges will be construc- ted using either launching girders or segmental cantilever construction, For
medium -length bridges between 200 m and 500 m the incremental launching
Trang 19Construction Methods 13 3,00 concreting po formwork release position | “fier | | 5,15 1 2,00 T 9, | 2,00 † 5,15 fig 31 Basic formwork for a box-girder cross-section “tk At=35-som |
ce) cantilever construction with launching gantry ( Siegtal Bridge )
Trang 2014 Design CONSTRUCTION SPAN LENGTH BRIDGE LENGTH METHOD 20 40 60 80 140 400 1 CANTILEVER qux PRECAST LAUNCHING GIRDER INCREMENTAL LAUNCHING
Fig 23 Classification according to construction method, span length, total
bridge length, and construction progress (from { 151) 4,3 Superstructure
4.3.1 General
The box girder often is more advantageous than say aTT -beam due to - its high bending stiffness combined with a low dead load, yielding a
favorable ratio of dead load to live load;
- its high torsional stiffness which allows freedom in the selection of both the supports and bridge alignment; and
- the possibility of utilizing the space inside the box girder
Several of the following aspects apply to both the box girder and 1 -beam,
however `
The superstructure should always pe designed as a complete entity, However, in order to provide a better overview, the longitudinal and transverse direc- tion are handled separately here
4.3.2 Longitudinal Direction
Because of excessive bending deformations even under constant loads and in order to avoid cracking under repeated Joading, most box- girder pridges are prestressed, As short a span a8 20 m is more economical and possesses more favorable load-carrying characteristics when partially prestressed than if it were simply reinforced, Under 20 m is a box-girder cross-section no longer sensible anyhow, The limit for mild steel reinforced bridges of a single span lies at approximately 35 m; for more than one span at a maximum of 60 m Today practically all box-girder bridges are prestressed
Trang 21Sup ture
Especially important for the form and the dimensioning of the bridge is the selection of the bridge depth Up to a span length of about 90 m, a constant bridge depth is sensible [16, 17}, whereby beginning with a span length of approximately 50 m it is expedient to increase the thickness of the bottom flange over the piers on the inside of the box girder where this can not be seen from the outside For span lengths, 1, in the middle range and constant bridge depth, d, the following ratios are normally used:
- mild steel reinforced: single span 1/d“~ 17
multiple spans 1/d“ 18
- prestressed: single spans 1/4 ~ 31 multiple spans 1/d = 25
If not done earlier for aesthetic reasons, it is structurally and economically
advantageous to vary the bridge depth in the longitudinal direction beginning with span lengths of about 60 m onwards For span lengths over 150-m this cannot be avoided, According to [18], the depth, dg, over the piers should vary so that it is about 3 times as jarge as at mid-span, dyn The depth of the box girder should vary in the longitudinal direction between the piers and mid-span in such a way that the forces in the tensile and compressive chords increase linearly and therefore the shear forces in the webs remain roughly constant throughout the span Suitable ratios are:
1/dy = 33 to 50 dg ~ 12 to 20
Should the bridge be placed high above the valley, aesthetics dictate that, de- pending upon the width of the piers, the ratio dg /dg is better chosen some- what smaller than the structural analysis indicates as the optimum
Much larger slenderness ratios can be achieved in the case of suspension and cable-stayed bridges, as the depth of the main girder is not determined by the total span length but by the spacing of the hangers, which act as spring supports For small hanger or cable intervals of from 6 m to 15 m, which also prove suitable for the free cantilever construction of cable-stayed bridges, the depth of the main girder when suspended from both sides is in principle determined only by its strength in the transverse direction, When the girder is suspen- ded from the center,the necessary torsional strength or torsional stiffness plays a decisive role as well
Trang 22
16 Design
For curved bridges, where in addition to longitudinal bending moments My, torsional moments ,Mp,are also necessary to satisfy equilibrium, the box- girder cross-section is especially advantageous The angle of curvature,œ, (Fig, 25) is the governing criterium for the ratio of My /Mg For a< 30° it is sufficient to calculate M, as for a straight bridge and My for the curved bridge, i.e to neglect the coupling effect of the two upon each other (e g
(21, 22})
4.3.3 Transverse Direction
Figure 26 portrays the factors influencing the cross-sectional form are all approximately of the same importance, They USE -pedestrian -automobile strain PROPORTIONS -length of cantilever -web inclination -dimensions -longitudinal / transverse aifinese -nUHUes -later widening -widened section BRIDGE FINISHES + FORM fascia beam ~guard rail in, - ie -web inclination -view from below DESIGN OFA BOX -GIRDER CROSS-SECTION POSSIBLE CROSS-SECTIONS -single-cetl ~mulfiple - cel! sconstant or varying -wlth/without diaphragms CONSTRUCTION METHOD ~astationary falaework -incremental launching -farrawork girder SUPPORTS =pler wall with multiple bridge bearings -several individual piers -eingte middle pier ~euspended from bridge centerline -suepended from both sides of crose- section -free cantilever ~launching girder ~precast elements
Fig 26 Influences on the design in the transverse direction
The use to which the cross-section is put determines its form, especially the width of the top slab No standards exist for the necessary width of foot- bridges That must be determined according to the expected number of pedestrians for the particular location in question {23} In order to provide ample space for at least 2 baby carriages plus one pedestrian and to impart a feeling of safety and well-being to pedestrians, a minimum width of 3.5 m between the hand rails should be provided Should the bridge possess light, open balustrades or carry pedestrians over major traffic arteries, the width should be chosen expecially liberally
Trang 23Superstructure 17 [te T 6,00 m 6,00m ahaa al | k¬ si
Fig 27 Railway clearance Fig 28 Double-track railway: Place the webs as nearly as possible underneath the paths of the loads
and height of the box girder as well (Fig 29) The details of the slab carry- ing the tracks depend in addition upon whether the ballast bed is continued over
the bridge or whether the sleepers are placed on vibration-absorbing pada
sitting directly on the structure
For highway bridges the bridge should maintain the normal highway cross-section (e.g Fig 30) Should later widening of the highway be planned, the additional lanes should be constructed for the initial bridge stage and the areas not planned for initial commissioning blocked off, This avoids expen- sive reworking of the bridge in the future
=< ar
+ omlem—mmberok
Trang 24
18 = Design
The box-girder cross-section shown in Fig 31 has proven itself with regard
to its form and structural characteristics Many variations of this “standard cross-section " are possible E 13,00m al
Fig 31 Typical single-cell box-girder cross-section (1/d © 20;see Fig 43)
If the available depth of the girder,d;is greater than from 1/6-to 1/5 of the
bridge width ,boay (© roadway slab width), a single-cell box girder is in order, If a/b lab < 1/6, a 2-cell or multiple-cell box girder is more sen-
sible (Fig sah 25 ¥or wider bridges the vehicle loads acting on the canti- jever slab can be distributed longitudinally by means of a pronounced edge
beam, enabling the cantilever length to be increased (Fig 33)
Fig 32 Double-cell box girder with Fig 33 Load-distributing edge support in the middle beam
The number of cells should be kept as small as possible even for wider bridges with a small depth in order to minimize problems in construction As can be seen in Fig 34, no substantial improvement in the transverse load distribu-
tion is achieved with 3 cells and beyond, For economic reasons, today more
than 2 cells are rare
If 2 or more box girders are placed next to each other it is advantageous to connect their top and bottom flanges in order to achieve a better transverse load distribution (Fig 35 b) If only the top flanges are connected, they will be hightly stressed due to bending moments in the transverse direction with-
out being able to effectively distribute the stresses in this direction (Fig 35 a)
Trang 25Superstructure = 19 ———- 100 KN single load at A =-~— 100 kN uniformly distributed over ail webs ye kN ¡03 Al y 7 T 7 "E † ‘| 06 7 ; | Ai 20 To Hệ hy _} 0 tt — 63 & no.of celS 8 ‡ 02 D -sL —-# -~===r-sa-g~ [u2b———m 304
Fig 34 Longitudinal bending stresses o at mid-span in relation to the number
of cells; (example for 1/d 18 from [26]) Tả “a
Fig 35 The coupling of box girders
The combination of extremely wide roadway decks with slender piers can be accomplished with a box girder by supporting the cantilevers with precast
struts (Fig 36) This is also advantageous for the construction,
At the inner supports of continuous long span bridges it is usually necessary to increase the thickness of the bottom flange to take the compression stresses (Fig 37) The center of gravity is thereby lowered towards the bottom flange,
80m ————+|
Fig 36 Kochertal Bridge at Geislingen Fig, 37 Thicker bottom flange
for a 4-lane Autobahn (from (27) Valley Creek Bridge, over the support (Pine
Trang 26
40 Design
affecting favorably the moment arm of the prestressing force However, the increased stiffness also causes an increase in moments over the supports
The bottom flange should be gradually thickened over a distance of 1/10 times
the span on both sides of the inner support,
With varying bridge depth and inclined webs, the box-girder soffit is wider in the span than over the supports (Fig 38) In order to simplify the formwork
in this case, vertical webs are therefore often selected (Fig 39) cross-section cross-section Em ——— over the supports - in the span ~ a 10,92m ry een m2 TT over the supports bbs
Fig 38 Felsenau Bridge, single- Fig 39 Oléron Bridge with ver- celled box girder with tical webs and two sepa- ‘ inclined webs rate box girders
Especially in the case of inner city elevated highways, exit lanes often begin on the bridge, requiring a widening of the bridge top slab If the widening is not too great, it can be handled by extending the length of the cantilever por-
tion (Fig 40a), Its soffit must however be carefully designed (Fig 41) If the
widening of the roadway is not symmetric, an equal cantilever length for both sides of the box girder can be achieved by offsetting the center line of the box girder (Fig 40b), The torsional moments are thereby reduced as well, The formwork costs are substantially increased however
In extreme cases (Fig, 42) the distance between the webs or the number of celis can be increased, The possible position of supports thereby influences the number of webs more in most cases than the maximum cantilever length of the top slab or the maximum span of the top slab between the webs,
a) symmetrical widening b) unsymmetrical widening; equal cantilevers achieved through a curved alignment of the box section
Trang 27Superstructure 21
a) simple design but too b) extensive formwork but in
stubby in the normal region return a well-proportioned
cantilever slab in the normal region
Fig 41 Extension of the cantilever slab ~ >> " é section q-d x mm , -~ oo Fig 42 Widening for exit lanes The following: - advantageous proportions - minimum dimensions
- aesthetic and structural suggestions
are intended to facilitate the preliminary design but should in no way restrict
the scope of the design possibilities (Fig 43) [* slab tae TT
8pecia] form for the coup- ling of prestressing tendons Fig 43 Design aids for the cross-sectional dimensions
< tr ngg
Trang 28
22 Design
- slab thickness tạ > 1/30, otherwise the slab must be stiffened in the compression zone by means of transverse ribs at intervals of a = 1g
(This also applies for tạ},
- minimum dimensions: deck slab ty = 200 mm; ti 2 200 mm webs ty > 300 ram or (200 + 2-Banet? bottom slab ts 2 150 mm
(with the utmost of care in construction, e.g pre~
cast elements, these values can be decreased by
as much as 50 mm)
The ratio 1,:1, shown in Fig 43 is very much dependent upon the transverse bending stiffness of the slabs and the webs as well as the maximum possible eccentricity of the vehicle loading as “defined by the sidewalk For normal cross-sectional dimensions, 80 % to 90 % of the fixed-end moment of-the can-
tilever slab is transmitted into the web and 20 % to 10 % into the deck slab
between the webs The optimum cantilever length 1, lies between 2,0m and 3.5m 1, should be so selected that no negative moments in the span occur in the deck slab between the webs
In order to minimize the width of the pier and the span length of the bottom
slab between webs, the webs should be given an inclination of from 4: 1 te
3:1 for a constant bridge depth The transverse tensile forces in the deck slab must be accepted thereby
In order to facilitate rainwater run-off, the roadway should be transversely inclined as follows:
for a straight alignment: 2 2.5%
for curves, depending upon the
radius of curvature 2.5% to 6 %,
For narrow bridges the transverse gradient can be achieved by varying the road surfacing thickness In general though, the deck slab should be given a gradient itself The possibilities to achieve this are shown in Figure 44,
a} rotation of the b) a rhombic cross- ce) differing web cross-section section with vertical heights and a about the axis webs of equal height horizontal bottom
through point A slab
Fig 44 Transverse gradient of the roadway (from [14])
Trang 29Comptete System and Supports 23
Diaphragms hinder the construction of box- girder bridges in most cases They are sensible in the span only for the case of very long spans, if then Over the piers they can take the form shown in Figs 45 and 46, depending upon how the cross-section is supported They can be omitted if the bridge bearings are placed directly underneath the w his makes it more difficult to replace
“a bearing however ~
2
section 11 a) most common form; b) for a central support e) for very stiff
with access opening box girders lotally thickened webs suffice Fig 45 Diaphragms over the piers | _ _t = i 1 1 = — 1 4 1
a) small longitudinal displacement b) very large longitudinal displacement Fig 46 Design of the diaphragm dependent upon the movement allowed by
the bridge bearing 4.4 Complete System and Supports
The superstructure is acted upon by both horizontal and vertical loads which
must be carried into the foundation by way of the bearings, piers, and abut- ments (Fig 47)
‘Transverse horizontal forces acting on the superstructure and piers include
mainly wind but also earthquake forces, In exceptional cases, impact forces
from vehicles or ships and ice pressure should be considered,
Trang 3024 Design ——————— | fa stucturai system for verticot loads a pier abutment % |, abutment structural! system for tronsverse horizontal toads what ra - ° + -Re i { narrow box ! on piers ° + + ~Ð- ¿~t- ¬ 7 + + FF Ap : : wide box on piers z ~ : ' ' ° ~~ ha hp THỌ j ‘ TS pers of stitt | texible individual fincese of columns cotumns ° -= 3 narrow box on + -t- : central cotumns ° ae fF =a
twin columns necessary every = 100m of the bridge
{because of torsional moments)
Fig 47 Typical longitudinal and transverse support schematic for a medium- "length (L* 300 m) box-girder bridge with short or medium high piers Up to a bridge of medium length, the fixed point should’be at one of the abut- ments, as the abutment is heavy anyway and can take the horizontal forces more economically than the piers In this case the bridge also needs only one expansion joint (Figs 47 and 48a) For long bridges it can be more advantage- ous to place the point of fixity somewhere near the middle of the bridge in order to halve the horizontal movements to be accomodated by the bearings on the piers and by the two expansion joints at the bridge ends (Figs 48b and c), This point of fixity should lie at a pier with a large vertical load in the case of short _piers For.a bridge with tall piers, several piers near the middle ofthe bridge
may be monolithically connected with the superstructure, yielding a floating -support,The stability of the entire system should, however, be examined,
Piers constructed integrally with the superstructure are not only advantageous
because the vulnerable bearing can be done away with,but also because they
then possess the shortest effective Euler buckling length (Fig 49) One should therefore take advantage of the deformability of tall:piers — ————————_—_—_.SSsSBS T
If the final fixed points are not located at the abutments or the bridge construc-~ tion is advanced from both abutments, the fixed points must be moved during the construction (Fig 50) This requires a careful determination of the initial
Trang 31fixed paint Complete System and Supports 25 point of fixity ' 4 i #È | fo pro “TT1 b) ma cì Fig 48 Support possibilities in the longitudinal direction tị — tr l Lối -3E1 ' MrMỤ GÐ Mr LỆ MoeM,*0 MacMur0 (without friction} l/=0,5L e07L el geal
Fig 49 Influence of end restraint on the effective Euler buckling length đà fixed I fixed free ITT Fig 50 Variation of support conditions during construction: For extremely long bridges it can be more economical to provide intermediate joints in the superstructure and therefore several fixed points, as the bearings become more complex and costly the heavier the loads they are required to handle and the larger the movements they are required to accomodate sirnul-
taneously, i.e the higher and wider the "trees" of Fig, 51 for the particular
Trang 3226 Design tt 1 Q 100 200 300 400 500m fixed point fp a wt jint a k ° oA without joint k + + oe - +18 2 joints SCT TT Lt ee rrr rrr rrr rrr b pp ps TTT TTT Ty TT TT Tre c load carried by: g movable support ` 5 TA fixed support R ' 3 monolithic connectionimaintenance ‘free) \ Sor ® ty : 5 | | ! ẹ i ˆ | e 1 & i Ị 9 la b ;© L 1 I
temperature- induced movement
Fig 51 Load vs movement depending upon the support condition in the lon-
gitudinal direction (from [29])
>> TTR ST To increase the number of joints
because of an unsubstantiated fear of differential settlements to the point that the effect of continuity is lost (Fig 52) should, however, be rejected The best detailed expan- sion joint will always be much worse than the continuous superstructure In addition, the structural reserves of the continuous system are lost by the multiple-span statically deter- minate system
Trang 33Complete System and Supports 27
The torsionally stiff box girder enables the_ skewed abutment and pier walls to
be avoided for the case that the bridge must cross over an obstruction at a skew (Fig 53), / section a-a section b-b J Tig, 53 Skewed crossing with a torsionally stiff bridge constructed at a right angle
Torsionally stiff box-girder bridges built on sharp horizontal curves and d_sup-
Ported by at least 3 individual columns located under the middle of the cross-
section are stable even without the bridge ends being fixed (Fig 54), This permits very transparent support systems, which are especially welcomed in
the inner city, Nevertheless, in order to be able to use the normal expansion
jeints, the bridge ends are fixed torsionally stiff
alternative
alternative or |
ixed
Fig 54 Support of sharply curved box-girder bridges
In the case of long lightly curved bridges, the columns can be alternatingly placed off-center to the bridge axis (Fig 55a) if the bending stiffness of one or all of the columns is not to be utilized (Fig 55b), Of course a fixed end as shown in Fig 53 suffices for short bridges
Trang 34
28 Design
As curved bridges experience horizontal deformations due to prestressing and creep parallel to the bridge axis and deformations due to shrinkage and tempe- rature radially from the fixed point, bearings capable of accomodating move- ment in any direction are to be preferred over roller bearings,
This requirement of freedom of movement in any direction produces difficul- ties for the free bridge ends Therefore in most cases the movements there are restricted to one direction, and the piers or abutments are designed for the unavoidable restraint stresses (Fig 56)
_
w ~- — 0 N , ahs
Vy
Fig 56 Longitudinal movement in the direction of the bridge axis can be
brought about by horizontal restraint forces (from [9))
For bridges with transverse or longitudinal gradients the bearings can be placed as shown in Figs 57 and 58 The transverse diaphragms should be placed vertically Beginning with radients g greater than 3°, the gradient
must be considered Ốc ¬
Fig 57 Placement of Fig 58 Placement of bearings for a longitu- bearings for a dinal gradient avoiding horizontal transverse gra- forces on the piers
Trang 35Substructure 28 4.5 Substructure The loads of the superstructure are carried into the soil by the following: - abutments - piers - foundations 4.5.1 Abutments
The abutment provides the transition between the earth embankment and the bridge superstructure Its wing walls secure the embankment and its back wall holds a space free for displacements of the superstructure, The super- structure is supported here by bearings mounted on the bridge seat into which the loads are carried into the support walls and they in turn into the founda- tion and the soil
In Fig 1, two abutment types are sketched For small bridges the spill- through abutment is sufficient A box abutment large enough for access ingide is normally provided for large bridges The earth fill behind the abutment should be well compacted As the abutments even for small bridges possess
large dimensions, they should be covered as much as possible by the slope or
embankment, The sloped area underneath the superstructure ends should be covered with dark paving stones, as no plants will grow there, The view into the access chamber, where the bearings and any drainage lines are located, should be concealed by the end diaphragm The end diaphragm should, how- ever, leave a gap of approximately 100 mm between it and the soffit of the
box girder,
4,5,2 Piers
For low bridges, especially those for elevated highways, two individual columns are more transparent and therefore more advantageous than pier walls, It can be more aesthetically pleasing to set the columns back somewhat from the outside edge of the bridge cross-section (Fig 59b) and to forego” the more efficient carrying of the loads directly from the webs into the columns (Fig 59a).In any case, good uniform soil conditions are necessary
in order to prevent differential settlement between the columns standing so
close to each other Otherwise a common foundation should be provided, on
piles if necessary Should it not be necessary to handle any torsion moments,
a single column underneath the middle of the cross-section appears more ele-
gant than two (Figs 59c to e), However, this is only sensible for one or two-
celled cross-sections (Fig 59d), By-inclining the webs more a smaller
bottom slab is thereby possible (Fig 59e) The single columns should be de-
signed sufficiently robust, as columns that are too slender do not convey a feeling of safety underneath a large box-girder (Fig 59f) Round, elliptic, or octagonal cross-sections should be preferred above square or rectangular ones because of their better appearance (Fig 60a) Even for extremely wide, multiple-celled cross-sections, not more than 3 individual columns together
should be used, as the skewed view through them produce an unsettling effect
Trang 3630 Design
b} đ) f)
Fig 59 Pier arrangement
Fig 60 Cross-sectional shapes for piers (from [9] }
Fig 61 Comparison between pier and twin columns for multiple-celled box girders
Trang 37
Substructure 31
For high bridges over valleys the piers should also he designed with one or two-celled box cross-sections (Fig 63) The walls should be at least 200 mm to 300 mm thick in order to permit the use of slipforms or climbing formwork a}
Fig 63 Box piers Fig 64 Pier design
The piers should be lightly tapered from the bottom upwards, at least for one dimension (Fig 64a) Stiffening diaphragms are not normally required or desired because of thermal stresses, The top of the pier (Fig 64b).should be designed compatible with the construction method of the superstructure, In the longitudinal direction of the bridge the width of the top of the pier depends upon the type of bearings selected, the span between the bearings, and the dimensions of the jacks necessary to replace the bearings It varies between
1,20 m and 2.0 m
4,5,3 Foundations
The simplest type is the shallow spread foundation which is used if good soil conditions are found at a shallow depth The foundation slab is constructed on top of an 0,1 m thick sub-base, The slab thickness should be so chosen that shear and punching reinforcement can be avoided, The bottom of the founda- tion must lie below the level of frost penetration
Should adequate soil conditions be found only at lower levels, an excavated pile foundation or one on drilled or driven piles should be selected In gene- ral this type of foundation is provided with a pile cap with which the pier is connected Depending upon the loads to be carried, the piles are either dri-
ven (for loads up to * 1 000 KN) or drilled (up to * 9 000 KN) In the case of
pile foundations for single columns, it is better to carry the forces directly into the foundation by means of large drilled piles or wells rather than for-
cing them to travel a roundabout way through a pile cap on top of several small individual piles and its complicated reinforcing arrangement
(Fig 65)
If unfavourable soil conditions are encountered
or the foundations are into a hillside, it can be necessary to carry out open excavations down to a suitable soil layer or to sink caissons,
The pier can then either be placed directly on Fig 65 (a) is more the bottom of the excavation or on top of the
Trang 3834 Structural Analysis
understand The detailed explanation included herein should alleviate this difficulty Formulae are derived which are not only suitable for hand cal- culation but also for small computer programs, With enough experience so gathered in this area, one can later on appropriately estimate the effect of
this loading and forego the exact calculation,
2, LOADS and EXTERNAL INFLUENCES
Those loads to which a bridge is subjected in the structural analysis are generally specified in the applicable code of practice for a particular country, As the design loads should not be given consideration independent of their respective safety concept and as the live loads vary greatly indeed from country to country (Fig 1), only the general basis for the consideration of loadings is given here as follows: Hoof Mpeg MN $2258 8 33 8 &
Fig 1 Comparison of various loading assumptions (from [ 1)
- Dead loads of the structure, gy: normal reinforced eqnerete Yo = 25 KN/m?3,
lightweight reinforced concrete Yrg™ 16 to 2l kN/m
~ Dead loads of the inishes, go: roadway surfacing (asphalt) y * 22 kN/mŠ,
concrete) y* 25 kN/m3, bridge railing * 0.22 KN/m,
guard railing ~ 0,25 - 0.35 kN/m
~ Live loads: In most codes a single-vehicle loading with a series of wheel
foads,Q,,and a uniformly distributed load , q; (main lane, secondary lanes),
is given Railway bridges are loaded with the appropriate loading configura- tion for each track (e.g UIC 71 [ 2]) Impact is considered by increasing portions of the live loads by means of system-dependent impact factors The machinery employed in constructing the bridge should also be included as live loads In contrast to the assumed vehicle loadings however, these loads act with 100 % of their actual magnitude, a fact to be taken into account in the safety concept for the design
Trang 39
Loads and External Influences TS TE h +t - Lam Jot afb | 35
Fig 2 Notation used for the determination of wind load from Figure 3 W_ in kN/m M_in kNm/m factors "from without live load 2 ty q-h 2O,°q-b - cụ = Fig 3b
Bo = ‘wa cwab oh 3c
= 2 [with live load 3d
5 # =%4-cwq [h+h HE =-icug:b ch cự =Fig 3b
* ® - cy = Fig 3e
ø ithout live load -
: 2 without live loa = cựq ch = cụ-q :b th cự = ig af
a S -
8 £ [with live load Cy =Fig 3f
z BA i = Rey 2'cewgtn+ lh+h) = 08-q-b -h q Xz = Fig 3f = Fi a) 1,6 - PIN 1.4 SAN, b) ` 1,2 1,0 3 6 9 ——b.ib;Ÿ0,5 e) ¬ fl 10 === 4 1 os ow _———+—-——> 3 6 12 b dh 30° 40° 50° 60° 70°
Trang 40
36 ‘Structural Analysis
- Wind loads: With the cross-sectional dimensions (Fig 2) and the dynamic pressuré,
q, obtained from the wind velocity, v {m/sec], according to
the formula q [kN/m?] ws v2/1600, the wind loads, W and M, on the entire cross-section can be determined from Figures 3a to f and the localized
wind loads, w, from Figures 3 g toi Information concerning the maximum
wind velocity can be taken from the applicable Code or obtained from a local weather station If constant contact is maintained with the weather station during construction the design wind loads for the erection phase can be re- duced, depending upon the expected duration of the particular erection,any-
where from 0.2 max W(< 1 day) to0.7 max W(>3 days),
The critical wind velocity causing such vortex-shedding phenomena as flutter’ or galopping vibrations can be estimated with Figure 4 and the formula (from [4]) w.*2mpt-a|1+('‡ˆ - 95) 228 | where Í_ = fundamental natural B frequency of flexural An = SECTION & vibration for the ": criti hape TV/ To “LT To3» H03b HH 0,15-0,5b fas fp = fundamental natural nitical shap frequency of torsional ơ"T- Saeeeđ 1 - 015b 0,5 vibration for the critical shape “=——(013b ~~ U7 101-0,2b P t¬ 015 - 0,3b 0,7 m = mass per unit of length — 0,11b ——- 0.12b r= radius of gyration Fig 4 Factor a derived from wind b= bridge width
tunnel tests a* factor from Fig 4 derived from wind tunnel tests
A more exact analysis requires wind tunnel tests in most cases, Bridges of normal dimensions continuous over fixed supports are generally not critical Suspension or cable-stayed box girders could be, however,
‘yi These forces act horizontally on the ints of the bridge and are degressively
dependent upon the entire vehicle loading; i.e the longer and wider the bridge the smaller is the chance that all of the vehicles simultaneously accelerate or brake
(According to the German Code, Fy * 3/20-L+ b
but Fy is at least = 0.3 - Q ) roadway [EN] (1 £200 m),
heavy vehicle’