BEeM 101.01
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marco rosignoli...
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Prestressed Composite Bridges with Steel Corrugated-Plate Webs Marco Rosignoli, DrIng, PE
Bridge Engineering eManuals Manual 101.01 November 2015 ©
Published by Marco Rosignoli, Dr.Ing., PE – www.marcorosignoli.com Other titles by the author: Launched Bridges (ASCE Press, 1998) – p.363, ISBN: 0784403147 Bridge Launching (Thomas Telford, 2002) – p.352, ISBN: 9780727731463 Bridge Erection Machines (UNESCO Encyclopedia of Life Support Systems, 2011) – p.72, Chapter 6.37.40 Bridge Construction Equipment (ICE Publishing, best-selling US title, 2013) – p.496, ISBN: 9780727758088 Bridge Launching, 2nd Edition (ICE Publishing, 2014) – p.376, ISBN: 9780727759979 © Marco Rosignoli 2015 All rights, included translation, reserved. Except as permitted by the Copyright, Designs and Patent Act 1988, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, printing or otherwise, without the prior written permission of the author. While every effort has been made to ensure that the statements made and the opinions expressed in this publication provide a safe and accurate guide, no liability or responsibility can be accepted in this respect by the author. While any reasonable effort has been undertaken by the author to acknowledge copyright on material published, if there has been an oversight please contact the author who will endeavor to correct this in a reprint. Cover photograph reproduced with permission from ASCE This manual has been designed for two-side printing in A4 or 8.5”x11” format.
About the author: Marco Rosignoli, Dr.Ing., PE has 34 years of experience in the design, construction engineering and on-site supervision of complex bridges, the industrialization of large-scale bridge projects, and the design review and forensic engineering of major bridges and bridge construction machines. Bridge technical director for prime constructors (Impregilo, Bonatti, Dragados USA) and design firms (HNTB, HDR, Parsons Brinckerhoff), and free-lance bridge consultant for a decade, Marco Rosignoli has been working in 22 countries on four continents. He has served as designer, reviewer, technical leader or technology consultant on six cable-stayed bridges, 34 incrementally launched bridges, multiple balancedcantilever bridges, and well over 50 kilometers of light-rail and high-speed railway bridges. International expert of mechanized bridge construction and the incremental launching of bridges, Marco Rosignoli is the author of five books published worldwide, three book chapters and over 90 scientific publications and presentations. He holds 32 patents on bridge construction methods; serves as a peer reviewer for the bridge engineering journals of ASCE, IABSE, and ICE; and teaches advanced courses for bridge engineers for the ASCE Continuing Education Program and other organizations. Founder and chair of IABSE working group WG-6 Bridge Construction Equipment for the 2009-2013 term, chair of IABSE 2010 Singapore seminar State-of-the-art Bridge Deck Erection: Safe and Efficient Use of Special Equipment, founder and manager of the Linked-In group Bridge Construction Technology, and fellow IABSE, he is a registered PE in Italy and multiple states in the US.
Prestressed Composite Bridges with Steel Corrugated-Plate Webs
Introduction Limiting self-weight is a primary requirement in the design of most types of bridges. Self-weight is among the most significant loads on the bridge and its reduction creates a reserve available for live loads. Selfweight also governs the cost of construction equipment, which is a prime component of the construction cost of a bridge (1). The density of normal-weight concrete does not vary much with the concrete strength, the weight of a prestressed-concrete (PC) bridge depends linearly on the cross-sectional area, and the most influencing parameters are therefore related to the cross-sectional geometry. The influence of the different parameters has been analyzed with a database of 165 constant-depth PC box-girder highway bridges with continuous spans ranging from 19m to 96m. The depth of the crosssection governs the shear and flexural capacity and therefore increases regularly with the span, Figure 1. The cost of materials and specialized construction equipment (1) depends on the weight of the cross-section, i.e. on its area A. The moment of inertia I is the main indicator of the flexural capacity of the cross-section (2,3,4) and is inversely proportional to the cost of post-tensioning (4). The design efficiency of a PC bridge can therefore be evaluated in terms of radius of gyration of the cross-section
Figure 1: Deck depth vs. span length 6.0
depth (m)
4.0
2.0
0.0 10
40
70
span (m)
I A
(1)
Since moment of inertia and area depend on the deck width, their increase with the span is scattered; their ratio does not depend much on the deck width, and the increase of the radius of gyration with the span is therefore more regular, Figure 2.
Figure 2: Radius of gyration vs. span length 2.0
radius of gyration (m)
r
100
1.5
Set zu and z l the distance of the cross-
1.0
sectional center-of-gravity from the upper and lower edge of the deck, respectively, the maximum radius of gyration is achieved when the area is positioned at the edges
0.5 0.0 10
40
70
span (m)
100
2 rmax zu zl
(2)
and the efficiency of the cross-section can
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be expressed as flexural efficiency f
f
r2 I 1 2 rmax zu zl A
(3)
The closer f is to the ideal value of 1, the better the flexural efficiency, and the lower the quantity of post-tensioning. The flexural efficiency of the PC bridges in the database is shown in Figure 3. The cross-section of a PC box girder includes top slab, bottom slab and webs, and these three components can be treated separately to increase the flexural efficiency f . The thickness of the top slab depends on localized wheel-load bending and on the need for punching strength and adequate concrete cover, and cannot be reduced excessively. The thickness of the bottom slab is often governed by technological requirements of internal post-tensioning, but its design is generally less restrained. Using internal post-tensioning, the web thickness is often governed by the need to contain and deviate the tendons, and in a narrow box girder the Figure 3: Flexural efficiency vs. span length webs can reach 30% of the cross-sectional area (5).
flexural efficiency
1.0 0.8 0.6 0.4 0.2 10
40
70
100
span (m)
The webs reduce the flexural efficiency of the cross-section with a small contribution to the moment of inertia because of their position close to the neutral axis. The webs also increase the cost of materials and labor, as they are the most difficult element to form and to cast of the crosssection. The webs are necessary for shear transfer, and their shear efficiency depends on the mechanical properties of the material used for the webs.
The efficiency of a material can be evaluated with its strength-to-density ratio, em f . A concrete with compressive strength fc ' 45N mm2 and density c 25kN m3 has compressive efficiency 2 em 1.8 103m and much smaller tensile efficiency. A steel plate with yielding strength fy 355N mm
and density s 77kN m3 has tensile efficiency em 4.6 103 m at first yielding and is 2.5 times more efficient than concrete in optimum work conditions. The composite bridges with concrete slab and steel girders are therefore more efficient than the PC bridges, as their masses are more distant from the gravity axis, and the materials perform better. Prestressing strand, however, is definitely more efficient. Commercial 7-wire strand reaches em 21.5103 m at the 0.1%-load, and a cable is generally the most effective way to use steel in tension.
By relating the efficiency of materials in their optimal work conditions to the efficiency of prestressing steel, 45MPa concrete reaches 8.4%, and medium-grade steel plates reach 21.4%. A PC bridge is mainly designed for bending (4). Prestressing is the most efficient and cost-effective solution to control the edge tensile stresses generated by the bending moment, and in a continuous beam this requires the presence of two flanges to compress without instability. The use of reinforced concrete for the two flanges offers compressive strength at low cost, and the use of external post-tensioning allows web design for principal stresses without geometry restraints. Once the flexural demand has been met, tendon deviation reduces the tangential stresses in the webs, with beneficial effects on web thickness and the flexural efficiency of the cross-section.
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The efficiency of concrete can be improved by increasing the strength with the same density (i.e., by using high-performance concrete) or by reducing the density with the same strength (i.e., by using lightweight concrete). The next step in the research of maximum flexural efficiency consists in abandoning the concrete webs and using two or more steel I-girders to connect the concrete slabs. Steel webs offer the same shear capacity as concrete webs with only 5-8% of their weight, tendon deviation diminishes the shear demand on the webs, the flexural efficiency of the cross-section increases because of a better mass distribution, and these sections are also fast and easy to build.
Prestressed composite box girders with corrugated-plate webs The prestressed composite bridges may be grouped into two categories, the main difference between them consisting in the transmission of shear. The space-frame decks eliminate material not working in the Mörsch shear lattice, Figure 4, and the diagonals between the concrete slabs may be made with steel shapes, steel pipes with or without concrete infill acting compositely, or precast PC members (6,7,8). The box girders with steel-plate webs benefit from the higher shear efficiency of the steel plates compared with concrete webs. The combined use of external prestressing, concrete slabs and steel-plate or trussed webs results in efficient cross-sections that make the most out of prestressing and are light and easy to build. Compared with a conventional PC box girder, the cross-sectional area diminishes without affecting the moment of inertia much, and the flexural efficiency increases. On a 40m span, a PC box girder with internal tendons requires about 0.55m3 of concrete per square meter of deck surface, which decreases to 0.45m3/m2 with the use of external tendons. A prestressed composite box girder with steel-plate webs requires only 0.35m3/m2 and is 25-35% lighter. Concrete is located at the edges of the cross-section, the radius of gyration increases, and the flexural efficiency increases with quadratic ratio. The cross-sectional area of concrete to be compressed decreases, and prestressing diminishes because of the combined effect of smaller area and higher crosssectional efficiency. The contribution of materials specializes. The concrete slabs resist bending thanks to prestressing, whose deviation reduces shear to values that can be resisted with light steel-plate or trussed webs. Each material works in uniform rather than triangular stress pattern (the concrete slabs are uniformly compressed, the web plates resist uniform shear stress, the truss diagonals resist axial compression and tension, and the prestressing tendons are subject to axial tension), which further enhances the efficiency of design. Figure 4: Space-frame deck (photo: VSL)
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A prestressed composite box girder with steel-plate webs is a hybrid between a PC box girder and a nonprestressed steel-composite deck that takes the best from the two technologies. Compared with a nonprestressed composite deck with two or multiple I-girders, the webs are thinner, the flanges are smaller, less cross frames and no lateral braces are necessary, the weight of steelwork is only 15-20%, the unit cost of steelwork is similar, field assembly is simpler, the geometry tolerances are less stringent, and the maintenance costs are lower. Compared with a PC box girder, the absence of concrete webs saves labor and post-tensioning and accelerates construction, reinforcement is simpler and easier to fabricate, and casting cell and erection equipment (1) are less expensive.
flexural efficiency
The flexural efficiency of prestressed composite box girders with steel-plate webs has been evaluated by subtracting the concrete webs from the cross-sectional area and moment of inertia of the bridge database, and by increasing the net area thus determined by 5% to account for the weight of the steel webs. The results are represented with Figure 5: Flexural efficiency vs. span length black dots, Figure 5, while the grey dots represent the data in Figure 3. The increase 1.0 in flexural efficiency is substantial, and the result is particularly interesting on longer 0.8 spans because of the progressive increase in the web depth with the span. Better 0.6 structural performance and savings in labor, concrete, reinforcement, 0.4 prestressing and construction equipment balance the cost of steel webs and the risks 0.2 10 40 70 100 of innovation, and open new perspectives in the design and construction of mediumspan (m) span bridges. The most intuitive way to replace the concrete webs of a PC box girder with steel webs is the use of stiffened-plate webs. The deck has a conventional aspect, as the prestressing tendons are hidden within the box cell. The behavior of the individual materials is well known, and the combined use of reinforced concrete slabs and steel I-girders has been amply tested in hundreds of composite bridges. However, this simplicity is only apparent, and several reasons discourage the use of this structural solution. The steel webs resist a significant portion of the prestressing force applied to the composite section. Under an axial force F, strain compatibility at the web-slab nodes governs the distribution of the axial force between the concrete slabs, of total area Ac , and the steel girders, of total area As . The initial portion of post-tensioning resisted by the steel girders is
Fs
As F E As c Ac Es
(4)
where Ec is the elastic modulus of concrete at stressing. The effects of concrete creep can be evaluated with an age-adjusted modulus that decreases with time, and therefore Fs increases with time and soon reaches 20-25% of the total prestressing force. This force is wasted and requires additional stiffeners to prevent web buckling, which increases the fabrication cost of the webs and may reduce their fatigue life due to cracks initiating at the stiffener welds (9). If the webs were devoid of axial stiffness, the webs would resist only shear force, and the concrete slabs would resist the entire bending. The axial stiffness of the webs must therefore be diminished without affecting their shear capacity, and this requires anisotropic behavior. Orthotropic plates with different
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stiffness in the orthogonal directions are commonly used in steel bridges. Webs and bottom flange of steel box girders are heavily stiffened in the direction of the principal compressive stresses, and less stiffened in the orthogonal direction. This concept may be extended to the axial stiffness of a plate by folding the plate to reduce its capability of resisting axial stress without affecting the shear capacity. Trapezoidally- or sinusoidally-corrugated steel or aluminum plates are commonly accepted in naval and aeronautical applications (10). Several fuselages and wing structures have been built with undulated or corrugated plates. Skins of F-15 and wings of AV-8B and F-22 adopt corrugated panels (11). Aircraft designers have realized long ago that the corrugated panels have larger buckling strength in the direction perpendicular to the corrugation. Corrugated metal panels have long been recognized as excellent shear carrying members. This is attributed to two characteristics of the panels: the transverse stiffness provided by the corrugation depth, and the in-plane stiffness due to narrow spaced folds that act as vertical stiffeners (12,13). In civil structures, the advantages of structural anisotropy may be exploited by replacing the stiffenedplate webs of steel or composite girders with corrugated-plate webs (14,15,16,17). Vertical corrugation by cold folding creates the in-plane flexibility necessary to minimize the longitudinal axial capacity of the web panels, and amplifies the transverse flexural capacity needed to resist transverse bending and crosssectional distortion and to prevent buckling of web panels devoid of welded stiffeners, Figure 6. Figure 6: Corrugated-plate webs (photo: Doka)
The idea of using corrugated-plate webs in civil structures was first introduced for the steel beams of buildings, where the web thickness varies from 2mm to 5mm, and the height-to-thickness ratio of the web panels varies between 150 and 260 (18). In the past three decades there has been increasing interest in prestressed composite box-girder bridges with steel corrugated-plate webs, where the web thickness varies between 8mm and 12mm, and the height-to-thickness ratio of the web panels typically ranges between 220 and 375, and has reached 445. The corrugated-plate webs have symmetrical regular shape with constant wavelength in the longitudinal direction. Load tests on specimens of corrugated plates, finite-element numerical analyses and studies on the behavior of corrugated-plate webs in prestressed composite bridges have been conducted in Britain, Canada, China, Egypt, France, Germany, Hungary, Italy, Japan, Sweden and the USA, and the first actual
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bridges confirmed many advantages of the prestressed composite box girders with steel corrugated-plate webs over the stiffened-plate ones (19,9,2).
The longitudinal flexibility of the webs minimizes the initial and time-dependent loss of prestress into the webs. Fewer tendons are needed to prestress the slabs, and fewer shear connectors are needed at the web flanges to transfer longitudinal interface shear. Internal tendons in a slab compress only that slab and do not affect the other slab.
The state of stress in the central region of the web panels is an almost pure shear one. The thickness of the web plates is chosen for shear strength, and the corrugation is designed to control buckling. The webs may often be made with 8mm plates, while stiffened-plate webs are rarely thinner than 12mm. In addition to savings in weight, this opens new perspectives for the efficient use of high-grade steels.
Shear is mostly carried by tendon deviation, the residual shear is carried by the webs, bending is carried by the concrete slabs, torsion is carried with hollow-section behavior, resistance to crosssectional distortion (20) is higher and more uniform throughout the length of the bridge, and the interaction between bending and shear is minimal. Every material works with optimal stress distribution.
The transverse stiffness of the corrugated plates avoids the need for welded stiffeners, reduces the number of cross frames and diaphragms within the box cell, and saves material and labor. These savings, along with the reduced plate thickness, are often enough to cover the cost of plate corrugation.
Higher transverse stiffness, closely spaced folds, and the boundary restraint provided by the concrete slabs increase web stability from buckling. Buckling depends on shear only, while in a stiffened-plate web it depends on combined shear and axial stresses. Longitudinal axial stresses do arise in the corrugated-plate webs in proximity of the concrete slabs, and vertical axial stresses arise in the support regions of the bridge; these local stresses, however, do not affect web stability.
The sensitivity to premature buckling due to geometry defects is definitely lower. The defects of planarity are compared with the amplitude of the folds instead of the plate thickness, and their effects are 10-20 times lower. Local distortion due to welding is limited to the edges of the panels, which are restrained by the concrete slabs. The plastic strains due to cold folding are uniform over the web depth and do not affect the elastic equilibrium of the panel.
The flexibility of the web panels facilitates fabrication and field assembly and avoids the need for trial assembly and tight geometry tolerances. The web panels are overlapped and fillet-welded on either side; in-place casting of the concrete slabs compensates for the residual geometry irregularities.
Prestressed composite box-girder bridges with corrugated-plate webs have been built by incremental launching on 40-55m spans and as balanced cantilevers on longer spans, Figure 6. Corrugated-plate webs have been used in extradosed and cable-stayed bridges, and research is in progress to extend their use to long-span arches. Combined with the use of high-performance concrete for the rib slabs, corrugated-plate webs could lighten the arch ribs, diminish the cost of foundations and temporary support systems prior to crown closure, avoid labor-intensive staged casting of concrete webs, and accelerate construction.
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State of stress in the corrugated-plate webs In a prestressed composite box girder, vertical loads generate bending, shear, torsion and distortion. Bending and shear are calculated with conventional beam analysis, but because of the orthotropic behavior of the webs, the state of stress within the cross-section is analyzed with specific criteria. The application of axial force and bending activates the specialized response of the different elements of the cross-section. Because of the axial flexibility of the webs, the concrete slabs resist most of the axial force and bending (21). The contribution of the webs to the flexural capacity of the composite section was found to be small in finite-element analysis and negligible in real applications. High flexural efficiency of the cross-section, uniform stress patterns in every material, no migration of prestress between the concrete slabs, and minimal prestress shedding to the webs are the immediate consequences. The axial flexibility of a corrugated plate results from horizontal flexural deformations in the plate and therefore depends on the thickness of the plate, the depth of the folds, and the shape of the corrugation. Sinusoidal, zigzag and trapezoidal corrugations are technically possible. The cost of cold corrugation depends on the fold angle and density, and trapezoidal fold profiles offer a clear economic advantage. Web plates with trapezoidal corrugation are also stiffer in the transverse direction, and easier to splice by overlapping at the longitudinal folds; they have therefore become standard practice in bridge applications. Figure 7: Corrugation geometry
The restraint that the concrete slabs exert on the horizontal flexural deformations of the web plates depends on the width and thickness of the steel flanges, on the effective width and thickness of the concrete slabs, and on the reinforcement ratio of the latter. All these parameters influence the stress distribution within the cross-section.
Longitudinal axial stress The response of a composite box girder with corrugated-plate webs to a longitudinal post-tensioning force depends on web anisotropy and the longitudinal restraint exerted by the concrete slabs. Horizontal bending occurs in the central region of the web panels, and with the symbols in Figure 7, the effective longitudinal modulus of elasticity of the corrugated plate is (22) E x ,eff
a f b f tw c f 3b f h f
2
Es
(5)
The benefit of formulating an effective longitudinal modulus is that analysis methods for flat plates can be applied to the solution of problems of corrugated panels. The corrugation angle is typically between 30° and 32°, although actual bridge applications range from 25° to 38°. For a typical web panel with tw 8mm , bf c f 300mm and corrugation angle 30 , the effective longitudinal modulus is
E x ,eff 0.0013 E s , i.e. three orders of magnitude smaller than E s . This reduction in the axial stiffness governs the axial stress distribution within the composite cross-section: the webs resist minimal longitudinal bending, and in the absence of prestressing, two equal and opposite axial forces in the concrete slabs balance the external moment (23). Marco Rosignoli, Dr.Ing., PE | Bridge Engineering eManuals © All Rights Reserved
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For the same reason, a prestressing force applied to a slab remains within the slab and does not affect the webs and the other slab. Finite-element analysis and load tests showed that corrugated-plate webs absorb a minimal fraction of the prestressing force applied to the cross-section, and the axial forces in the slabs can be calculated based on the geometry of a cross-section composed of concrete slabs only. Because of plate corrugation, the axial deformations of the slabs generate horizontal bending in the folds. The peak horizontal bending generated at a vertical fold line by a longitudinal axial stress c in the concrete slabs is (24)
mz
bf a f
Es tw3 c 3b f c f 72h f Ec 1 2
(6)
After corrugation, the web plates are welded to flange plates, which are ultimately connected to the concrete slabs. Strain compatibility generates longitudinal axial stress in the web regions close to the flanges; the axial stress is higher in the longitudinal folds than in the inclined folds. With the typical dimensions of the web plates for box-girder bridges, the longitudinal axial stress fades rapidly in 10-15% of the panel depth and is negligible in the central 70-80% region of the panel (24). The local axial stress due to shrinkage of concrete, slab shortening at the application of prestress and over time, live loads and thermal gradients is the smaller the more constant the web corrugation is. The longitudinal axial stress at the edges of the web panels does not seem to cause fatigue, as indicated by the rarity of problems at the web-slab nodes of non-prestressed composite bridges (25,26). Tests performed on corrugated plates welded to the flanges by intermittent welds showed that the strength of the connection is critical for stable shear transfer. Intermittent welding may overload the welds and cause premature failure, and is not recommended. Tests performed on corrugated plates continuously welded to the flanges on one side only did not show premature weld failure. The transverse restraint exerted by the flanges generates transverse bending in the vertical direction, which fades within 10-15% of the panel depth because of the progressive onset of unrestrained horizontal bending related to the axial flexibility of the plate. The peak vertical bending at the web-flange weld is higher than the peak horizontal bending at the folds.
Tangential stress As in a conventional PC box girder, the webs resist most of the shear. The shear force V applied to each web can be assumed as uniformly distributed over the web depth, so that the uniform tangential stress is V tw hw , where tw is the plate thickness and hw is the web depth measured clear between flanges. Since also the vertical axial stress z is negligible far from the support regions of the bridge, the state of stress in the web panels is one of pure shear. The two principal stresses 1 and 2 are inclined at 45°, two inclined forces F1 1tw and F2 2tw balance one another longitudinally along the vertical fold, and the vertical resultant Fz 1tw 2 resists the external shear force (27). In the transverse plane, the outward push of the compression fiber is balanced by the inward pull of the tension fiber. The flange plates do not contribute much to the flexural capacity of the composite section and are designed to be as small as possible. Web corrugation widens the flanges, their thickness is reduced to a minimum, and the shear connectors are closely spaced to restrain the compression flange and avoid local buckling. With the symbols in Figure 7, the ratio w of the initial plate length to its final length after corrugation is
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w
bf c f bf a f
(7)
The average shear angle in a several-wavelength web panel under pure shear is Geff , where the effective shear modulus (22) is Geff Gs w . For a typical web panel with tw 8mm , bf c f 300mm and corrugation angle 30 , the effective shear modulus is Geff 0.933Gs , the effective longitudinal modulus is E x ,eff 0.0013 E s , and the reduction in the longitudinal axial stiffness is two orders of magnitude larger than the reduction in the shear stiffness. The corrugated-plate webs are slightly more flexible in shear than the stiffened-plate webs because of w 1 and the use of thinner plates, but the total shear deflection is still small when compared to the flexural deflection of the composite section, as deviation forces of polygonal tendons balance most of the vertical shear. Load tests and finite-element analysis (19) showed that the reduction in Gs obtained by replacing a stiffened-plate web with a corrugated-plate web corresponds to the value of w . Finite-element analysis also showed that the shear force resisted by the concrete slabs is negligible in most of the deck but increases near discontinuities of the shear force (2,4). In correspondence with a discontinuity V in the shear force, the tangential stress discontinuity tends to deform the web as a lozenge, i.e. to produce a sudden angular discontinuity Geff in the web. The flexural and shear stiffness of the concrete slabs opposes sudden angular discontinuities in the webs, and local systems of vertical Figure 8: Web-slab interaction at shear discontinuities forces arise at the web-slab nodes to transfer the shear forces necessary to bend the slabs until a common curvature is reached, Figure 8. These local forces may require additional connectors and slab reinforcement; finite-element analysis helps in investigating the local stresses in web panels, web-slab nodes and the slabs as well as in validating approximated methods for their calculation. Different shear deformations of slabs and webs occur at the application points of localized vertical loads such as support reactions and tendon deviation forces. Different shear deformations occur in the support regions of nonprestressed composite bridges as well, but the web plates are thicker, and V is due to superimposed dead and live loads only (self-weight is not resisted with composite action). The vertical axial stresses at the web-slab interface depend inversely on the square root of the web area, and are therefore higher in thinner webs. The assumption that plane sections remain plane is not realistic in the prestressed composite box girders with steel corrugated-plate webs. It may be adopted for preliminary design and to analyze longitudinal bending and shear in the continuous beam; stress analysis in the shear discontinuity regions, however, requires more refined approaches.
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Torsion-distortion interaction In a PC box girder, the effects of eccentric loads deviate from Saint-Venant’s theory of torsion, and the cross-sectional deformation is a combination of rigid torsional rotation and distortion. These effects depend respectively on the torsional stiffness of the cross-section, which limits its capacity to rotate about its axis, and on the interaction between the frame stiffness of the cross-section and the in-plane flexural stiffness of webs and slabs, which together oppose distortion (20). In most practical cases, pier diaphragms control distortion at the support sections, tendon deviators stiffen the cross-section along the span, and warping is disregarded under the assumption that plane sections remain plane. The distortion of a prestressed composite box girder with steel corrugated-plate webs under eccentric loads is conceptually similar, but the effects are pronounced by the smaller in- and out-of-plane flexural stiffness of the webs. The transverse flexural stiffness of corrugated-plate webs is much smaller than the stiffness of the concrete slabs, the cross-section distorts as a parallelogram hinged at the web-slab nodes, and control of distortion by in-plane frame action is minimal. The webs are also designed to not resist inplane bending, and distortion is primarily resisted by lateral deflections of the concrete slabs. Pier diaphragms, tendon deviators and auxiliary cross frames within the box cell are therefore designed to stabilize the cross-sectional geometry. To study distortion, a load F applied to the top slab eccentrically from bridge centerline can be decomposed into two symmetrical loads F 2 and two anti-symmetrical loads Fh , Figure 9: Cross-section symbols equal in value and opposite in sign, applied to the top web-slab nodes of the crosssection. Symmetrical loads cause longitudinal bending and shear. Because of the minimal frame stiffness of the crosssection, external forces are only counteracted by shear forces in the webs. The shear force produced by Fh in each web, assumed as hinged to the concrete slabs and with the symbols of Figure 9, is Vw Fhc h . This force is higher than the shear force produced by a torsional moment T Fha , and corresponds to the shear force due to a moment T Fh a b .
In a box girder with vertical corrugated-plate webs, the torsional shear in the webs is twice the torsional shear determined with the Saint-Venant’s theory of torsion for an undeformable cross-section. In a trapezoidal box girder, the lateral force in the top slab can be obtained (28) by adding a force Vt Fha h to the shear force resulting from the torsional moment, and the lateral force in the bottom slab can be obtained by adding a force equal to Vb Fhb h . As a result of torsion-distortion interaction, the cross-section is subject to a rigid in-plane rotation due to the torsional moment, and to lateral deflections of the concrete slabs in the opposite directions due to distortion. Diaphragms and cross frames are therefore necessary to control lateral displacements and rotations in the concrete slabs, slab edge decompression, and transverse bending in the corrugated-plate webs (3,4). Although the transverse flexural stiffness generated by plate corrugation is relatively low, the folds are shallow and the transverse axial stress in the web regions close to the slabs may reach the yield point of steel at the plate surface. Transverse bending propagates to the web-slab nodes and may require additional flange connectors and slab reinforcement, especially in the bottom slab.
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Because of the fundamental role of external post-tensioning in reducing the shear force applied to the webs, the tendon deviators are distributed for optimal control of shear, and distortion is controlled with additional diaphragms or cross frames. An analysis method proposed for the effects of launch bearing misalignment in incrementally launched bridges (2,3,4,20) can be easily adapted to the study of the warping stresses in the cross-section based on the effective longitudinal modulus, equation (5), and the restraint provided by pier diaphragms, tendon deviators and cross frames to the lateral deflections of the concrete slabs.
Web-slab nodes The bottom web-slab node involves a few design challenges to assure durability. Longitudinal corrosion lines have been frequently observed at the deck paving level in U-spans with edge girders connected by a concrete slab at the bottom flanges. In a prestressed composite box girder the situation is similar, although only condensation water can gather within the box cell. Corrosion prevention discourages from placing triple contact points (air, concrete, and steel) near main structural members. Therefore, while the top web-slab nodes may be detailed with conventional criteria, the bottom nodes present specific design challenges, Figure 10. The simplest solution (Scheme A) consists of casting the bottom slab over the bottom flange and is adversely affected by a triple contact point on a main structural member. A longitudinal stiffener may be welded to the web at the top of the slab Figure 10: Bottom web-slab node (Scheme B) to move the triple contact away from the web. In both cases, the field splices in the bottom flange interfere with construction by incremental launching (2,3,4). The slab may also be cast under the bottom flange (Scheme C). This moves the triple contact point away from the web, reduces the weight of steelwork, and offers a better aesthetic result; however, embedded props are necessary to support the web panels on the soffit form prior to casting of the bottom slab. Concrete pouring and vibration may be simplified by inclining the webs (Scheme D), which also assures better concrete adhesion to the bottom flange and better performance of shear connectors. Additional cost savings derive from a narrower bottom slab. The web-slab nodes are designed for transfer of longitudinal interface shear, transverse bending and vertical axial loads in the deck regions subject to shear discontinuities. Headed stud connectors may be used when the concrete slab is cast over the bottom flange; the connectors for suspended slabs must also resist the weight of the slab and are therefore embedded more deeply into concrete (29,30,31). The web-slab connections of prestressed composite box girders are subject to specific design requirements. The corrugated-plate webs are workshop welded to the flange plates and painted prior to delivery. The concrete slabs generate the flexural capacity of the composite section, and the steel flanges have therefore minimal area, for economic reasons and to minimize the restraint exerted on the axial deformations of the concrete slabs. Wide flange plates are necessary to accommodate the web corrugation and to contain concrete during slab casting. The plate thickness is reduced to a minimum, and vertical buckling of the compression flange
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into the web (9) caused premature failure of tested specimens. The shear connectors are therefore designed for diffused load transfer and to control buckling of the compression flange. The connectors transfer longitudinal interface shear, vertical axial force, and biaxial bending. Weight of suspended bottom slab and vertical forces at the longitudinal shear discontinuities in the box girder tend to detach the slabs from the webs. Cross-sectional frame action and the restraint exerted by the webs on lateral slab deflections when the cross-section distorts cause transverse bending at the interface. In-plane lateral deflections of the slabs cause bending about the vertical axis. These effects are more pronounced at the bottom web-slab node because of the higher lateral flexibility of the bottom slab (4). Connectors are designed with conventional criteria. Interface shear transfer depends on the residual shear force, which depends on prestress deviation. Prestressing may be internal in the slabs, external within the box cell, or a combination of the two. The most effective design approach is balancing the shear force due to permanent loads and one-half of live loads with the deviation of polygonal tendons, so that shear forces in the webs and interface shear transfer at the web-slab nodes are only related to the presence or absence of live loads (27). In incrementally launched bridges, the integrative post-tensioning applied at the end of launch (2,3,4) is designed to force the state of stress in the box girder under permanent loads toward an axial compression one. A prestressed composite Figure 11: Inner view of box cell box girder is more flexible than a PC deck, incremental launching construction discourages from the application of cambers, and high degrees of compensation for the final prestressing (32) are also necessary for control of midspan deflections. External polygonal tendons can be used as the only form of post-tensioning or can be combined with internal tendons in the slabs. Additional prestress may be needed to control the edge stresses and to provide ultimate flexural capacity. Internal midspan tendons in the bottom slab and cap tendons over the piers are more efficient than short external tendons because of the larger eccentricity, smaller anchor blisters, and less tendon congestion within the box cell. Varying prestressing in the slabs does not cause hyperstatic effects, as the prestressing forces applied to a slab do not migrate toward the webs and the other slab. Even when only a few polygonal tendons are used, the shear reduction is significant and longitudinal interface shear transfer at the web-slab nodes is rarely a major design issue. Most design standards for composite bridges specify the maximum spacing of flange connectors in relation to slab and flange thickness and height of connectors. Tight spacing is typically specified for headed stud connectors on 1012 mm flange plates for better control of buckling in the flange regions far from connectors. Headed stud connectors, therefore, are not much fit for use on thin flange plates, as tight spacing conflicts with slab reinforcement and prevents the concrete from flowing during slab casting. Shape connectors (L-connectors with through bars that transfer vertical tensile force and transverse bending, or channel connectors) are more practical, stabilize the flange plate more effectively, and facilitate supporting the web panels on the soffit form before casting of the bottom slab. Shape connectors are workshop welded to the flange in correspondence with the web folds to stiffen the flange and to prevent buckling at the application of post-tensioning. The longitudinal spacing of shape
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connectors ( a f and b f alternate, as in Figure 7) varies between 250mm and 400mm, with smaller values near the support sections of the deck. Some design standards limit the dimensions of the welds of shape connectors to flange in relation to the plate thickness. The minimum distance between the ends of shape connectors and the edges of the flange plate is sometimes limited to 25mm. When the top flange is wide to facilitate the operations of a rolling form table for the concrete slab, Figure 11, the width of shape connectors should not exceed the depth of the web folds much. Because of combined geometry requirements, thin flange plates suggest combining regularly-spaced shape connectors with integrative headed stud connectors. Some design standards specify a reduced shear capacity for headed stud connectors subject to axial tension, and most design standards specify the nominal shear capacity of L-shapes with hoops and channels.
Stability of corrugated-plate webs From the 1960s onward, nearly all of the research work on corrugated-plate webs was addressed to study the shear capacity. In the absence of instability, a steel plate subject to pure shear can be designed for strength. The nominal tangential stress at yielding is determined with the Von Mises yield criterion from the characteristic yield strength of steel, y fy
3 , and the resistance factor s specified by the
applicable design standards for shear failure is used to determine the design tangential stress at yielding,
y ,d s y . The stiffened-plate webs of non-prestressed composite bridges are often designed to meet stability rather than strength criteria (33,34). When different buckling modes are possible, each mode is identified based on its critical tangential stress cr ,i . The design value for the critical tangential stress is then determined based on the resistance factor cr ,i 1 specified by the applicable design standards for that mode,
cr ,i ,d cr ,i cr ,i , and the smallest of the factored design values identifies the design critical tangential
stress of the web panel, cr ,d min cr ,i cr ,i . The typical design approach for a corrugated-plate web of a prestressed composite box girder is to define the plate thickness based on the ultimate factored shear demand, the factored tangential stress at yielding y ,d and the plate dimensions commercially available, and then to determine the shape and amplitude of the corrugation so that the design critical tangential stress for buckling cr ,d exceeds y,d . This approach allows reaching the structural safety represented by the reliability index of the design standards with marginal differences in the corrugation costs. When achieving this goal is unpractical or uneconomic, the range 0.8 cr ,d y 1 often results in efficient use of steel. The buckling behavior of a corrugated plate is more complex than the buckling behavior of a stiffened plate. Built-up girders with corrugated-plate webs subject to pure shear show three modes of web instability in relation to the aspect ratio of the web sub-panels and the depth of the folds: local buckling, global buckling and interactive buckling.
Local buckling Local buckling corresponds to the shear instability of a flat web sub-panel supported at two vertical folds. Local buckling affects trapezoidal and zigzag corrugations, while sinusoidal corrugations are unaffected. The corrugated plate acts as a series of flat unstiffened sub-panels with large aspect ratio ( 6 hw bf 16 in practical cases) that support one another at the folds and are restrained at the flanges. In a prestressed
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composite box girder, the restraint exerted by the flanges corresponds to fixity due to the relative stiffness of the concrete slabs. Buckling in a single sub-panel is considered as the failure mode for local buckling. Local buckling is reversible and preponderant in webs with coarse corrugation, does not cause significant risks of general instability, and has a post-critical domain. Local buckling in a sub-panel is similar to buckling in a stiffened plate and can be studied with the classical equations for isotropic plates. Because of the dense corrugation profile, the corrugated-plate webs are typically stiffer than the traditional stiffened-plate webs, and their resistance to local buckling is therefore higher. The elastic critical tangential stress for local buckling of a sub-panel fixed at the edges and with infinite length ( hw bf 6 ) may be defined with the Timoshenko’s theory of stationary potential energy (35) t T 4.826E s w bf
2
(8)
In equation (8), the longitudinal width b f of the sub-panel is replaced with the inclined width c f if bigger. The vertical folds provide a flexible restraint to the sub-panels, and the elastic critical tangential stress for local buckling is therefore lower than the Timoshenko’s formulation. The following equation has been proposed to account for the lower level of edge restraint (36) tw bf
2
cr ,l 0.88 T 4.247E s
(9)
Nonlinear finite-element analysis and tests on specimens of corrugated plates led to a second equation for the elastic critical tangential stress for local buckling, which includes the effects of the aspect ratio of the corrugation (9,37)
cr ,l
t 0.904kcr ,l E s w bf
2
(10)
The buckling coefficient kcr ,l for local buckling is a function of the aspect ratio hw bf of the sub-panel and of the support conditions at the boundaries. For simple support at the vertical folds and fixity at the flanges it is
kcr ,l 5.34 2.31
2
b b 3.44 f 8.39 f hw hw hw bf
3
(11)
and for fixity at the four edges it is
kcr ,l
b 8.98 5.60 f hw
2
(12)
The average of the critical tangential stresses calculated with simple support and fixity at the vertical folds has been found to closely match the results of finite-element analysis (9). The local buckling strength of the corrugated plate depends on the aspect ratio of the sub-panels and on the plate thickness, and is independent of the corrugation profile. Finite-element analysis tends to overestimate the results of load tests because of geometry imperfections in the corrugated plates. Inelastic local buckling may occur when cr ,l 0.8 y , and the inelastic critical tangential stress can be calculated as
cr ,l ,in 0.9 cr ,l y
(13)
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This semi-empirical equation (38) includes the effects of geometry imperfections and residual stresses related to plate corrugation and welding to flanges (39,9).
Global buckling In the global buckling mode, shear instability is characterized by diagonal buckling over several corrugation sub-panels (18). This mode of instability is critical in webs with dense corrugation, or when a small corrugation angle results in shallow folds. The regular, wide wavelength deformation is progressive in the onset and development if in absence of local buckling, and is similar to that of an orthotropic plate. If local buckling occurs in the post-critical domain, the corrugated plate may lose the stability necessary to reach shear yielding. Easley (40) investigated global buckling of corrugated-plate panels with orthotropic-plate buckling theory and tests on 0.4mm aluminum plates. Tests on steel plates carried out in France (23), the UK (19) and the USA (9) confirmed Easley’s results. The elastic critical tangential stress for global buckling is determined with the Ritz method (40,41,42) by considering the corrugated plate as an orthotropic plate with a thickness equal to the corrugation depth hf
cr ,g kcr ,g
Dz0.25Dx0.75 tw hw2
(14)
The web corrugation profile acts as uniform stiffening in the transverse direction of the panel. The transverse flexural stiffness of a unit length of corrugated plate with trapezoidal profile is
Dx E stw h2f
3b f c f
12 a f b f
(15)
while for zigzag corrugation it is
Dx E stw h2f
cf 12 a f
(16)
For the same sub-panel width and corrugation angle, the trapezoidal corrugation has higher resistance to global buckling than the zigzag corrugation because of the increased moment of inertia contributed by the sub-panels parallel to the girder length. The longitudinal flexural stiffness is reduced by the corrugation ratio w , equation (7), for both types of corrugation Dz 0.092 E s
tw3
w
(17)
The buckling coefficient kcr ,g for global buckling solely depends on the restraint exerted by the flanges and varies from 36.0 for simple support to 68.4 for fixity according to Easley (43), from 31.6 for simple support to 59.2 for fixity according to Elgaaly (9), and from 32.4 for flexible steel flanges to 60.4 for stiff concrete slabs according to Bergfeld (44). Simple support at the flanges neglects the restraint to warping exerted by the concrete slabs (45). Finite-element analysis showed that the concrete slabs exert a substantial restraint action and greatly increase the elastic critical tangential stress for global buckling. The no-warping model ( kcr ,g 60.4 ) has been recommended for the analysis of the corrugated-plate webs of prestressed composite box girders (46).
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Inelastic global buckling may occur when cr ,g 0.8 y , and the inelastic critical tangential stress (44,9) can be calculated as
cr ,g ,in 0.9 cr ,g y
(18)
Interactive buckling Interactive buckling was identified in tests on corrugated plates (44) carried out in Sweden. It results from the interaction between local and global buckling modes, is a sudden phenomenon characterized by a sharp snap and steel plasticization along the folds, and is critical in webs with folds of intermediate depth. The following equation has been proposed for the critical tangential stress cr ,i for interactive buckling 1
cr ,i
1
1
(19)
cr ,l cr ,g
Inelastic local and global critical stresses, equations (13) and (18), are used in equation (19) when the corresponding elastic critical stress exceeds 80% of the yield point of steel. The interaction among the three buckling modes and the tangential stress at yielding is shown in Figure 12 for an 8mm plate with trapezoidal corrugation, fy 350N mm2 , hw tw 250 , kcr ,g 60.4 and 30 . El Metwally (47) has proposed an interaction equation inclusive of shear yielding to predict failure of corrugated-plate webs within the entire practical range of corrugation parameters 1 cr ,i
k
1 cr ,l
k
1 cr ,g
k
1 y
k
(20)
unfactored tangential stress (MPa)
The recommended values for the exponent are k 2 for trapezoidal corrugation and k 3 for zigzag corrugation. Prior to reaching the shear yielding point of the panel, the post-critical strength of corrugated-plate webs depends on the width of the sub-panels. The ultimate shear capacity of wide subpanels may be twice their local buckling capacity, while narrow or shallow Figure 12: Non-factored critical tangential stress corrugations offer minimal post-critical 600 domain due to global buckling. local (10) 500
local (13) global (14) global (18) interactive (19) yielding
Numerous researches showed that nonlinear finite-element analysis well predicts the failure loads and the failure 300 modes of corrugated plates. Numerical 200 analysis of interactive buckling and the post100 critical domain requires consideration of geometric and material nonlinearities 0 0 200 400 600 800 because of the significant influence of the bw (mm) plastic strains. The Ramberg-Osgood strainhardening model provides more accurate results than the elastic-perfectly plastic model. Arc-length iterative algorithms may be used for the incremental loading of nonlinear models to overcome the snapthrough and snap-back numerical convergence problems often associated with nonlinear buckling analysis. 400
Block distributions of yield strains have been used to study the effects of mechanical folding of the plates. Because of the cold-forming process used for corrugation, yield strains and the degree of strain-hardening
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at the folds of the corrugation profile are higher than in the flat regions of the panels (48). Folding does not seem to have significant effects on the ultimate strength of the panels.
Web crippling and yielding due to patch loading Patch loading of corrugated-plate webs throughout a flange is rarely a problem in bridges built in their final position, as the support regions of the webs are detailed for load transfer from the bearings. In incrementally launched bridges, (2,3,4) dispersal of the launch support reactions within the webs generates local vertical axial stress, and the lower region of the web panels is subject to a triaxial state of stress. Lateral launch guides and the flexural stiffness of the bottom slab facilitate control of transverse bending, the state of stress in the webs may be assumed as a plane one, and the Huber-Mises stress may be directly compared with the yield point of steel. In the absence of bearing stiffeners, the vertical patch-load capacity of the webs must be assessed in terms of stability as well. Length of launch bearings and bottom slab haunches lengthen the patch load, which typically includes several folds of the corrugation. The support reaction is therefore applied to longitudinal sub-panels, inclined sub-panels, and several fold lines. Tests and extensive finite-element analysis identified two failure modes of corrugated-plate webs under vertical patch loads (9,49). The first mode (web crippling) includes a collapse mechanism in the loaded flange and local bending or crippling of the web. The second mode (web yielding) does not include a flange collapse mechanism, and failure is due to web yielding followed by crippling (50). When a short patch load is applied directly to the bottom flange, web crippling occurs when the load is under the longitudinal sub-panels of the corrugation, while web yielding mostly occurs when the load is under the inclined sub-panels or the folds. Additional researches (49,51) showed that nonlinear finite-element analysis well predicts the failure loads and the failure modes of corrugated plates under patch loading. Crippling does not cause instantaneous failure of corrugated-plate webs (49,52). Patch load capacity increases nonlinearly with the flange deflection, and the ultimate capacity is 10-80% higher than the first crippling load. Local detailing of the bottom slab, web and flange thickness, load distribution and position, and yield strength of the webs are the most effective factors for the control of web crippling (48,53). The following equation has been proposed for the ultimate patch load capacity of a corrugated-plate web (53) L Pr 15.6wtbtw fy 1 w 240
(21)
where t b is the thickness of the bottom flange and Lw is the length of the patch load along the web edge, both expressed in millimeters. In a prestressed composite box girder, the ultimate capacity for web yielding is calculated based on axial yielding of an effective web length resulting from 45° dispersal of the support reaction throughout the bottom web-slab node. In the most complex cases, more accurate distribution of the vertical axial stress may be attained with finite-element analysis of models including bearings and bottom web-slab node (4). In-plane bending and shear diminish the vertical patch-load capacity of a corrugated-plate web. The following interaction has been proposed (51) for combined patch loading and in-plane bending, where P is the patch-load capacity in the presence of the bending, M is the bending capacity in the presence of patch load, Pr is the ultimate patch-load capacity in the absence of bending, and Mr is the ultimate flexural capacity in the absence of patch loading
P Pr
k
M Mr
k
1
(22)
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I-girders with flat stiffened-plate webs may be checked with k 2 (54). A more conservative value k 1.25 has been proposed for corrugated-plate webs devoid of bottom slab (9). The flexural and shear stiffness of the bottom web-slab node justifies the use of k 2 for a prestressed composite box girder. The following interaction has been proposed for combined patch loading and in-plane shear (9), where Vr is the ultimate shear capacity of the cross-section
P Pr
k
k
V 1 Vr
(23)
The stiffness of the bottom web-slab node justifies the use of k 1.8 in a prestressed composite box girder.
Resistance factors Different resistance factors are used for the different buckling modes of a corrugated-plate web to reflect the different post-critical domains; allowance is also made for geometry imperfections in the box girder and the corrugation folds, and for residual stresses in the folds in relation to the corrugation process. The resistance factors for the fillet welds of web to flanges, the flanges, and the shear connectors are specified by the applicable design standards for non-prestressed composite bridges. French researchers (39,55) proposed the use of cr ,g 0.50 for global buckling and of cr ,i 0.67 for interactive buckling. These resistance factors were used for the design of the Pont de la Corniche (39). A lower resistance factor for global buckling, cr ,g 0.33, was used for the design of the Charolles Bridge. Further research could lead to relaxation or refinement of the resistance factors, especially in relation to global buckling. When global buckling occurs, the shear strength decreases and the corrugated plate must develop large diagonal deformations required for stretching the folds prior to being able to transfer diagonal tension through membrane action. Tests confirmed the existence of a post-critical domain for global buckling, but the deformations necessary to achieve post-critical stability may be excessive for bridge structures. The resistance factors for global and interactive buckling are therefore lower than the resistance factor for local buckling, whose post-critical domain is stable and well known. The resistance factor for local buckling (38) may be taken as Figure 13: Factored critical tangential stress
cr ,l s 0.91 The use of lower resistance
factored tangential stress (MPa)
250
200 150 local (10) local (13) interactive (19) yielding global (14) global (18)
100 50 0 0
200
400
bw (mm)
600
800
factors rarely has economic implications because the critical tangential stresses for global and interactive buckling are higher than the critical tangential stress of stiffened-plate webs. If necessary, deeper folds may increase the critical tangential stresses for global and interactive buckling with minimal additional costs, related only to a shorter final plate length. The design tangential strength d at the
ultimate limit state is taken as the smallest of the factored critical tangential stresses calculated for yielding and local, global and interactive buckling for the given corrugation parameters. Starting from Figure 12, the interaction among the three buckling modes and shear yielding is shown in Figure 13 for
cr ,g 0.50 , cr ,i 0.67 and cr ,l 0.91 . The design of a corrugated-plate web typically starts for
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d y 0.8 and the practical consequence of the application of resistance factors is narrowing the choice of b f for a given set of corrugation parameters.
Design of corrugated-plate webs A corrugated-plate web is designed for a given depth hw (from the general geometry of the composite section) and for a known factored shear force (from torsion-distortion interaction and the residual shear after tendon deviation). The corrugation angle is typically between 30° and 32°. In an open section the a f b f ratio may be influenced by the need to limit the width of bottom flange overhang (56), but this is rarely an issue in a prestressed composite box girder as the concrete slabs stiffen both flanges. The sub-panels may have the same width in the longitudinal and inclined directions ( bf c f ) to achieve the same aspect ratio and the same critical tangential stress for local buckling. This, however, is rarely necessary as local buckling governs only for very wide sub-panels. The depth-to-thickness ratio of typical corrugated-plate webs for box girder bridges is in the 200 hw tw 450 range, and the sub-panel aspect ratio is 6 hw bf 16 . The following statistical regression has been proposed (24) tw h 0.00828 0.00044 w hw bf
(24)
For efficient use of steel it should be d y 0.8 , and therefore it remains to find t w and b f . The following procedure has been proposed (19). 1. Determine t w from the factored shear force, the factored tangential stress at yielding, and the plate thickness commercially available. 2. Assume an initial value of bf tw . 3. Determine the critical tangential stress for local buckling with equation (10). The factored critical tangential stress typically exceeds the tangential stress at yielding and rarely governs design. 4. Input a start value for the critical tangential stress for interactive buckling into equation (19), for example cr ,i ,d 0.9 y 5. Based on
cr ,i
and
cr ,l
, determine
cr ,g
from equations (19) and (18).
Working on equations (10) and (11), d can be expressed as a function of bf tw and hw tw . Design charts with bf tw in abscissa and hw tw in ordinate can be drawn for the desired ratio d y and the chosen corrugation parameters. Design charts with hw tw in abscissa and d y in ordinate can be drawn for different values of bf tw to determine the most efficient corrugation parameters in the
0.8 d y 1 range based on a chosen fold angle. Figure 14 has been obtained with the corrugation parameters and the resistance factors of Figures 12 and 13. Design charts with b f in abscissa and d y in ordinate can also be drawn for different values of hw tw , Figure 15. The design charts may be based on non-factored tangential stresses for analysis of the physics of instability, or on factored stresses for design purposes. The critical tangential stresses for the different buckling modes can be enveloped with the following equation
d cr ,l cr ,l k cr ,g cr ,g k cr ,i cr ,i k s y k
1 k
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(25)
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The tangential stress thus determined is always smaller than the individual stresses on the right-hand side of the equation, and the exponent governs the shape of the transition. A large exponent makes the curve to tightly follow the six bounding equations (local buckling below and above 0.8 y , global buckling below and above 0.8 y , interactive buckling and factored tangential stress at yielding). The dotted line in Figure 15 has been calculated with k 10 . Figure 14: Design chart 1.2 bf/tw=80 bf/tw=40 bf/tw=60 bf/tw=20
1.0
td/ty
0.8 0.6
0.4 0.2 0.0 0
100
200
300
400
500
600
700
800
hw/tw
Figure 15: Design chart 1.2
1.0
td/ty)
0.8 0.6 hw/tw=400 hw=tw=300 hw/tw=200 hw/tw=200; equation (25) k=10
0.4
0.2 0.0 0
200
400
600
800
An optimum profile (57) is the one that gives the desired shear strength with the least cost. Optimization may be based on the least volume of steel, on the least number of folds, or on a combination of the two criteria. The width of the sub-panels affects the buckling modes significantly. For narrow sub-panels, the web buckles in the global mode and the buckling wave involves multiple sub-panels. For wide subpanels, the web buckles in the local mode and instability affects only one sub-panel. For sub-panels of intermediate width, yielding or interactive buckling controls web failure. The post-buckling behavior depends on how close the web profile is to the region governed by shear yielding as opposed to the regions governed by global and local buckling (18). Since local buckling offers stable post-critical behavior, the post-buckling strength of the web increases as b f gets larger. Figure 13 shows that
most of the sub-panel width range is governed by either shear yielding or local buckling. This suggests the use of equal longitudinal and inclined width for the sub-panels. The corrugated-plate webs of prestressed composite box girders are unlikely to fall in the region governed by global buckling, and can therefore be designed with the higher resistance factors of shear yielding and local buckling. bw (mm)
Fabrication of corrugated-plate webs Problems of large-scale fabrication of 8-10mm corrugated-plate webs should be addressed to provide sufficient robustness and durability. Higher cost of fabrication is generally perceived to be the main disadvantage of built-up corrugated-plate webs (36), although for a fabricator with joint-tracking welding equipment (58) the cost difference should be small, as the vertical stiffeners are eliminated and the field splices are much simpler. The webs may be fillet welded to the flanges on one side only (9), and I-girders with corrugated-plate webs can therefore be economical when compared with stiffened-plate girders. The flange plates are welded to the edges of the corrugated web panels after their pressing to shape. Shear connectors may be welded to the flanges prior to welding the latter to the web panels. Web
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Prestressed Composite Bridges with Steel Corrugated-Plate Webs | BEeM 101.01 – November 2015
dimensions depend on commercial standards, and it is reasonable to assume that steel plates of width hw up to about 3m, thickness t w up to 12mm, and length up to 15m, are available for corrugation. Continuous, highly-automated manufacturing processes have been developed for fabrication of built-up I-girders with 2-3mm corrugated webs for building and crane applications (59). At the beginning of the assembly line, a de-coiler feeds the web plate into a stretch leveler followed by the corrugation unit. The depth of the sinusoidal corrugation ranges between 30mm and 45mm, and the depth of the web may reach 1500mm. While the finished web is transported through the assembly line continuously, both flanges are placed into position and fixed Figure 16: Pressing equipment to the web by means of hydraulic clamps. Together they pass through the welding unit for continuous one-sided fillet welding under submerged arc. The entire manufacturing process takes place at a speed of 2m per minute. Cold pressing is likely to be the least expensive method for forming the corrugation of 8-12mm web plates for bridge applications. The webs for the Cognac Bridge in France were corrugated by flipping the plate over after each pressing. Eventually, one complete 4-fold corrugation was cold-pressed at a time by placing the plate on rollers to avoid stretching, Figure 16. Presses for steel plates up to 4m-wide are available in Europe. The pressing force Fp for the generation of four cylindrical plastic hinges in the plate can be estimated by replacing the yield stress of steel with the tensile strength fu to allow for strain hardening (36)
Fp fu
tw2 hw af
(26)
The plastic work for a complete corrugation is ( in radians)
Wp futw2 hw
(27)
For fu 510N mm2 , tw 8mm , hw 1000mm , 30 and bf 300mm , the pressing force and energy are Fp 126kN and Wp 17.1 kJ . The cycle time is mainly governed by the time taken to move the plate one wavelength between each stroke. Even if the cycle time is as short as 10 sec, the mean power required for pressing is only 17.1 10 1.7 kW . At this rate, corrugation of a 14.5m-long plate would take about 2 min and provide a 13.5m length of web (12 complete corrugations with 1120mm wavelength, w 1.072). Heavier presses would be needed for a 12mm web plate, 2.0m wide. As b f is likely to be proportional to t w , the pressing force is proportional to tw hw and so would be Fp 378kN in this case. The angle is generally between 25° and 32°. This enables a generous bend radius, such as 10 tw , to be used for the folds, which also reduces the stress concentration in the flange-to-web welds at the folds. The flange plates are welded orthogonal to the corrugated webs, and the box girders with inclined webs are obtained with haunches in the concrete slabs at the web-slab nodes. Submerged-arc joint tracking equipment (58) may be used for welding; profiled gas-shielded fillet welding is often used for the 3mm corrugated webs used in buildings. Conventional weld control technology is applicable to these welds. Handling and transport requirements and the wavelength of the folds govern the optimum length of the web segments, which often governs the modularity of the slab segments of the prestressed composite Marco Rosignoli, Dr.Ing., PE | Bridge Engineering eManuals © All Rights Reserved
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box girder. The web segments used for the Charente Bridge in France were 12m long. Web segments ranging from 10m to 14m were used for the Charolles Bridge in France in combination with 12m slab segments. The web segments used for the Ginzan-Miyuki Bridge in Japan were 5.5m long in combination with 11m slab segments. The web segments for balanced cantilever bridges are typically shorter to match the length of the casting cell of the form travelers (1). Web segments can be spliced by overlapping and double fillet welding or by slip-critical bolted connections, with or without cover plates. Double fillet welds are designed for shear and the horizontal bending (3) due to axial shortening of the box girder at the application of prestress. The flange plates are rarely spliced. Field splicing is simpler and less expensive than for the I-girders of non-prestressed composite bridges. Procedures for cambering corrugated webs (36) have been proposed; cambers are not used in prestressed composite box girders built by incremental launching (2,3,4).
Construction of prestressed composite box girders Prestressed composite box-girder bridges are typically constructed by incremental launching or balanced cantilever casting (1). The casting yard organization for incremental launching construction (2,3,4) of a prestressed composite box girder is similar to that for a conventional PC box girder, as casting and prestressing requirements govern in both cases. Field splicing of corrugated-plate webs is fast and inexpensive, and short web segments may be used to simplify handling. Whatever the optimum length of the web segments may be, the length of the slab segments is kept as constant as practical to avoid frequent repositioning of the casting cell bulkhead. Field splices in the webs and construction joints in the slabs may be staggered longitudinally, although web segments cantilevering out from the rear deck end complicate the use of rear thrust systems for launching. Staggered casting of the slab segments is an effective yard organization for the incremental launching construction of prestressed composite box girders (4). In the rear casting cell, the web segments are positioned on adjustable supports. Web flexibility simplifies plan and vertical alignment, and in-place casting of the bottom slab relaxes the geometry tolerances in the steelwork and simplifies field splicing. After alignment, the webs are fillet welded or bolted to the rear end of the completed deck. The top slab may be cast in the same cell or in a front casting cell after segment extraction to stagger the two operations for better labor rotation. The steel flanges may be welded or left discontinuous. Steel pier diaphragms and tendon deviators are assembled in the rear casting cell to simplify handling and connection. The use of L-shape connectors with through bars complicates cage prefabrication for the bottom slab. Channel connectors are preferable for this purpose. Cage fabrication is complicated by the restrained work environment and the need to insert the bars under the bottom flanges of the webs. Concrete vibration under the bottom flanges requires particular care to assure concrete penetration between the connectors and absence of defects and honeycombs. The soffit form includes two full-length extraction rails under the webs (4) and a central lowering form table. Cambering is not used and the extraction rails are set along the local projection of the launch trajectory. Steel-PTFE plates are aligned over the extraction rails to diminish the frictional resistance of the casting cell during segment extraction. In the front casting cell, the top slab is cast over the top flanges of the steel webs. The casting cell includes two outer forms for the side wings and a central form table supported on the bottom slab. The
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form table is extracted backward after launch and repositioned for the new slab segment. Concrete pouring in the two casting cells is staggered over time to optimize labor rotation. Internal launch tendons (4) are anchored at the rear construction joint to simplify strand insertion and tendon stressing from the rear anchorages. Internal tendons are spliced by coupling or overlapping and may be tensioned after a short curing time. If the front casting cell is sufficiently far from the launch abutment, internal launch tendons may cross 2 or 3 slab segments. External polygonal launch tendons are anchored by splicing in double anchor beams, and temporary deviators made of steel or concrete are necessary at the top slab level for deviation of antagonist tendons (4). Incremental launching is likely to be the most competitive construction method for prestressed composite box girders with simple geometry, 40-50m spans and a few hundred meters of length. Launching the completed deck is simpler and more rational than launching the steel girder and completing the cross-section by casting the concrete slabs in-place, although the deck is much heavier during launch. Compared with the launch of the steel girder of a non-prestressed composite bridge, the deck is several times heavier, the webs are thinner, and the launch stresses may therefore be critical. The launch nose is typically quite long (2,3,4) to reduce the peaks of negative bending and shear in the front deck region. A long launch nose is not particularly expensive with a lightweight deck cross-section, and the steel corrugated-plate webs themselves can be used as a launch nose. The bottom slab is cast full-length in the casting yard to achieve a regular launch surface for the deck, and the top slab of the front span is cast in-place on launch completion. The front U-girder thus obtained is lighter than the completed deck, and structurally more efficient under negative bending. Heavier top flanges are used for the webs in the launch nose region, and temporary lateral braces and cross frames are applied to enhance stability. The resistance to lateral torsion-flexure buckling of I-girders with corrugated-plate webs is different from that of I-girders with stiffened-plate webs (60,61), and the critical moment is slightly higher. The equations specified by the design standards for the equivalent moment factors of plane webs also seem to apply for the analysis of the effects of moment gradients. However, the research available on lateral torsion-flexure buckling of I-girders with corrugated-plate webs is minimal, and finite-element analysis is recommended for analysis of launch nose stability. When the bottom slab is wide and the front U-girder is therefore heavy, the use of a front cable-stayed system (4) may be less expensive than a custom launch nose. This solution requires a load distribution diaphragm under the mast, and the localized load applied to the deck disturbs the envelopes of launch bending and shear. The deflection of the front cantilever may be recovered with a hydraulic pantograph applied to the front deck end.All the thrust Figure 17: Local eccentricity of support reactions systems (3,4) for PC bridges may be used for launching. Rear thrust beams applied to the bottom slab are particularly effective, as the launch force and the frictional resistance of the launch bearings remain within the bottom slab and do not generate interface shear transfer at the bottom web-slab nodes. When Eberspächer launchers (4) are used at the abutment, PLC control of the low-speed final approach of the support saddles avoids sudden application of vertical loads to the deck. The bottom flanges are
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located over the bottom slab to take advantage of slab thickness for longitudinal dispersal of the support reaction within non-stiffened webs. Haunches may be used to further spread the load and to improve web stability with a smaller panel depth. Web corrugation makes the support reactions eccentric in the transverse plane, and the bottom slab must therefore resist local bending, Figure 17. Lateral launch guides are used at every pier to avoid additional load eccentricity. Haunches in the bottom slab increase the transverse flexural stiffness and improve bottom flange stability far from the connectors. This scheme of web-slab node also improves corrosion protection by moving the triple contact point far from the web-flange weld. The absence of cambers avoids angle breaks in the launch surface, and fewer modes of instability can arise in the web panels. Elastic stability of a corrugated plate is better than the one of a stiffened plate because the distance between the folds is much shorter than the typical spacing of vertical stiffeners. However, the corrugated-plate webs are thinner, and longitudinal dispersal of the support reactions often requires the use of long launch bearings (4). The high flexural stiffness of the box girder reduces the rotations of the support sections; in spite of this, rocking launch bearings are often necessary to spread the support reactions uniformly. Articulated bearings (1) soon become too flexible and expensive, and steel skids topped with polished stainless-steel sheets for low-friction sliding of Neo-flon plates are placed onto longitudinal batteries of interconnected jacks to create hydraulic hinges (1,4). Control valves between the two hydraulic launch bearings at each pier can be used to generate torsional hinges. The longitudinal axial stress in the bottom slab is significant in the negative bending regions. Although the central region of the web panels is subject to negligible axial stress, the flanges and the adjacent web regions undergo the same longitudinal strains as the concrete slabs. The local stresses are easy to calculate under the assumption of strain compatibility; since they do not contribute to the elastic equilibrium, they may be reduced by cutting the flanges at a regular distance. The flanges may also be left discontinuous at the field splices, with additional labor savings. This solution was adopted in the 80m balanced-cantilever spans of the Pont de la Corniche in France (39,62,63). The prestressed composite box girders built by incremental launching resist full self-weight shear prior to the application of polygonal tendons at the end of launch. The tangential stresses in the webs are high during launch, and additional tangential stresses due to torsion and distortion should be avoided as much as possible with accurate alignment of the launch bearings. Launch bearings on hydraulic jacks offer the additional advantage of real-time monitoring and equalization of the support reactions.
Case studies Prestressed composite box girders are also fit to balanced cantilever construction on medium spans. In varying-depth bridges, steel corrugated-plate webs increase the cross-sectional flexural efficiency significantly. The 496m Pont de la Corniche in France (39,62,63) includes with five 80m spans and two 48m end spans. The top slab is 14.5m wide and the width of the bottom slab increases from 5.15m at the piers to 7.2m at midspan, for a constant web inclination of 17°. The overall depth of the cross-section is 2.5m at midspan and 5.5m at the piers. The clear depth of the corrugated-plate webs between flanges ranges from 1081mm at midspan to 4011mm at the piers. Plate thickness is 8mm for typical web segments, 10mm near the piers, and 12mm near the tendon deviation diaphragms. The webs are welded to 350x14mm flanges equipped with 150x15 L-shape connectors spaced 430mm and 370mm ( bf and af , respectively) and 200mm wide, and the depth of the transverse folds is 220mm. Fold wavelength is 1600mm, and the web segments are 3280mm long for 80mm overlap at the field
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splices. The long-term flexural efficiency of the composite cross-section, equation (4), is f 0.625 at midspan and f 0.788 at the piers. Longitudinal prestressing consists of three families of tendons: rectilinear 12T15 cantilever tendons anchored at top slab joints during balanced cantilever construction, rectilinear 12T15 continuity tendons in the bottom slab at midspan, and polygonal 19T15 tendons for service loads, tensioned on deck completion. The following quantities of materials were reported per square meter of deck surface: 0.57m of 35MPa concrete, 26kg of T15 strand, 124kg of reinforcing steel, and 39kg of steelwork. Construction on falsework is a viable option for short bridges that do not allow the investment required by incremental launching technology. A three-span prestressed composite box girder was built on falsework at Cognac, France on a 42.9m main span and two 32.5m end spans (64,65). The top slab is 11.7m wide, the bottom slab is 4.17m wide, and the total depth of the cross-section is 2.285m. The thickness of the top slab varies from 0.23m to 0.33m. The folds in the 8mm web plates have longitudinal and inclined width of 353mm and are 150mm deep. The net depth of the web panels is 1770mm, and the transverse inclination of the steel webs is 40°. The flange plates are 250x10mm at the top and 330x10mm at the bottom, and are equipped with 250mmwide, 100x9 L-shape connectors spaced 353mm at midspan and 159mm in the support regions. The short-term flexural efficiency of the cross-section is f 0.659 . Eight 19T15 external tendons anchored at the end diaphragms required the use of concrete deviation diaphragms. The following quantities of materials were reported per square meter of deck surface: 0.41m of 40MPa concrete, 15.3kg of T15 strand, 67.3kg of reinforcing steel, and 31.5kg of steelwork. A 30m, twin box-girder span was built in Japan (66,67,68) with 70° skew angle. The deck is 14.8m wide and the bottom slab of each box girder is 2.1m wide. The box girders are 1.9m deep, and the height of the corrugated-plate webs is 1183mm. Internal and external tendons were used for longitudinal prestressing. So far, few prestressed composite box girders with steel corrugated-plate webs have been built by incremental launching. Construction of the Maupre’ Bridge in France gave researchers (69,70,71,72,73) the opportunity of comparing different types of prestressed composite box girders. The bid design proposed two solutions: a stiffened-plate prestressed composite box girder built by incremental launching, and a PC space frame built as balanced cantilevers. Among numerous alternatives, the solution of Figure 17 was chosen for incremental launching construction of a seven-span continuous deck with spans ranging from 40.9 to 53.6m and a total length of 327m. With a top slab width of 10.75m, the cross-sectional depth is 3.05m for a 17.6 span-to-depth ratio. The Maupre’ Bridge was competed in 1987. A 610mm diameter, 20.6mm-thick steel pipe replaced the bottom slab. Concrete filling stiffened the pipe walls against instability and improved dispersal of the launch support reactions. The steel girder includes two 45°-inclined corrugated-plate webs shop-welded to the bottom pipe, so that complete V-units were shipped to the site for assembly. The webs are 8mm thick, the longitudinal and inclined panel width is 284mm, the fold depth is 150mm, and the fold wavelength is 1050mm. The 300x10mm top flanges are equipped with 100x10 L-shape connectors with constant 284mm spacing. The top slab is 10.75m wide with 2.80m side wings, and its thickness ranges from 0.20m to 0.30m at the flanges. Because of the triangular shape of the cross-section, the migration of launch support reactions generates transverse tensile stresses in the slab strip between the flanges. Rectilinear HDPE-coated T15 mono-strands were used for transverse top slab post-tensioning. The deck was built by welding the field splices of 10-13m girder segments and by casting the corresponding segments of top slab and pipe infill. Pipe filling required the use of a vibrating feeding
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pipe. On filling completion, the hole left by the vibrating pipe was filled with high-pressure grout to create a residual radial compression in the concrete infill. The triangular cross-section is not subject to distortion during launch. In the absence of a bottom slab, the flexural stiffness is relatively small, similar to the one of a non-prestressed composite bridge. Large flexural rotations at the supports required the use of hydraulic launch bearings (4), and the concrete-filled bottom pipe provided a major contribution to the dispersal of support reactions. Launch prestressing was applied with 12 or 14 permanent horizontal external 6T13 tendons located near the top slab and anchored from pier to pier to anchor beams protruding from the top slab. Transverse post-tensioning in the concrete slab was increased at the anchor beams for enhanced dispersal of the anchor loads. Prior to tensioning the launch tendons of the new segment, steel props supported on the bottom pipe were used to reduce torsion in the rear anchor beam. The concrete slab of the front span was cast on launch completion to use the lighter front deck portion as a launch nose. On launch completion, prestressing was completed with four 19T15 external tendons anchored to the end diaphragms of the bridge and deviated by steel pier diaphragms and steel saddles welded to the bottom pipe at the span thirds. Polygonal post-tensioning balanced more than 80% of the permanent loads and recovered most of self-weight deflections. The tendons were jacked vertically at the central pier diaphragm to recover part of the friction losses in the 330m tendons. The use of a concrete-filled pipe instead of a bottom slab assured dispersal of the support reactions and avoided secondary transverse stresses. It also posed new problems, related to the long launch bearings, the transverse deck stability during launch, and the need to assure a reliable connection between the steel pipe and the concrete infill for control of local buckling. The following quantities of materials were reported per square meter of deck surface: 0.25m of concrete, 1.5kg of transverse strand, 5.7kg of launch strand, 8.5kg of polygonal strand, 58.6kg of reinforcing steel, and 95.2kg of steelwork. Steelwork includes 47.8kg of corrugated-plate webs, 28.3kg of steel pipe, 2.3kg of connectors, and 16.8 kg of diaphragms. The Ginzan-Miyuki Bridge (74) was built by incremental launching in Japan. The 5-span box girder has a total length of 210m and includes four 45.5m spans and a 27.4m span. The cross-section includes a 9.3mwide, 0.30m-thick top slab and a 3.9m-wide bottom slab whose thickness varies from 0.25m at midspan to 0.50m in the support regions. The deck has a constant depth of 3.0m with a span-to-depth ratio of 15.2. Plate thickness in the 2210mm web panels varies from 9mm to 12mm, the plate folds have longitudinal and inclined width of 300mm, and the fold depth is 150mm. External prestressing tendons are as long as two spans to reduce friction losses at the deviation points. The bridge includes 19 segments with constant 11m length. The 33m-long front section of the top slab was cast on launch completion to avoid the use of a launch nose, and the weight of the bottom slab required the use of a front cable-stayed system. The steel mast was located 2.2m behind the leading pier when the front deck end was landing at the new pier. Comparisons between the dynamic deck behavior predicted by finite-element analysis and the vibration modes measured with dynamic tests on the bridge confirmed a significant influence of the shear strains on the vertical modes (75). However, the vibration frequencies of the external tendons are much higher, and this prevents resonance effects. Starting from these first applications, several prestressed composite box-girder bridges with steel corrugated-plate webs have been built in Japan in the last 15 years (24). The 198m Hondani Bridge has a 97.2m central span, 11.4m-wide top slab, 6.2m-wide bottom slab, and box girder depth varying from 6.4m at the piers to 2.5m at midspan. The bridge used concrete dowel connectors (76,77,78,79).
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Symbols a
distance between web mid-fibers measured at the top slab mid-fiber
af
longitudinal projection of the fold in a corrugated plate
A
cross-sectional area
Ac
cross-sectional area of the concrete slab
As
cross-sectional area of the steel girder
b
distance between web mid-fibres measured at the bottom slab mid-fibre
bf
width of the longitudinal folds in a corrugated plate
c
sloped depth of the web
cf
sloped width of the folds in a corrugated plate
Dx
longitudinal flexural stiffness of a cross-sectional plate
Dz
transverse flexural stiffness of a unit length of web
em
efficiency of material
Ec
elastic modulus of concrete
Es
elastic modulus of steel
E x ,eff
effective longitudinal modulus of elasticity of a steel corrugated plate
f
ultimate stress
fu
ultimate strength of steel
fy
characteristic yield strength of steel
F
force
Fh
anti-symmetrical force
Fp
pressing force for plate corrugation
Fs
longitudinal axial force in the steel girder
Geff
effective shear modulus of steel
Gs
shear modulus of steel
hf
depth of the fold in a corrugated plate
hw
net height of the web plate measured clear between flanges
I
moment of inertia about the centre of gravity of the cross-section
kcr ,g
buckling coefficient for global buckling
kcr ,l
buckling coefficient for local buckling
Lw
length of patch loading along the web edge
mz
peak horizontal bending at a vertical fold line
M
bending moment
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Prestressed Composite Bridges with Steel Corrugated-Plate Webs | BEeM 101.01 – November 2015
Mr
ultimate flexural capacity of the cross-section
P
axial force in a member
Pr
ultimate patch-load capacity
r
radius of gyration of the cross-section about the centroidal axis
tb
thickness of the bottom slab or flange
tw
web thickness
T
torsional moment
V
shear force
Vb
lateral shear force in the bottom slab
Vt
lateral shear force in the top slab
Vw
shear force in the web
Wp
plastic work for a complete corrugation
zl
distance of the centroid from the lower edge
zu
distance of the centroid from the upper edge
corrugation angle
difference between two values
cr 1 resistance factor for buckling failure
s 1 resistance factor for shear failure
specific weight, load factor, average shear angle
w
development ratio of the corrugated-plate web
Poisson’s Ratio
f
flexural efficiency of the cross-section
1,2
principal axial stresses
c
axial stress in a concrete slab
z
vertical axial stress
tangential stress
cr
critical tangential stress for buckling
cr ,d
design critical tangential stress for buckling
cr ,g
critical tangential stress for global buckling
cr ,i
critical tangential stress for interactive buckling
cr ,l
critical tangential stress for local buckling
T
Timoshenko’s critical tangential stress
y
nominal tangential stress at yielding
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y ,d
design tangential stress at yielding
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
Rosignoli, M. (2013). Bridge Construction Equipment. ICE Publishing. Rosignoli, M. (1998). Launched Bridges. ASCE Press. Rosignoli, M. (2002). Bridge Launching. Thomas Telford. Rosignoli, M. (2014). Bridge Launching, 2nd Edition. ICE Publishing. Rosignoli, M. (1997). Prestressed composite box girders for highway bridges. IABSE Structural Engineering International, (7) 4, pp. 278-283. Boudot, J., Radiguet, B. and Thao, P.X. (1987). The Sylans and Glacières Viaducts. Contribution of the French Group. AIPC–IABSE Congress, Paris-Versailles. Martin, R.A.; Raspaud, B.; Richard, P. (1982). Le pont de Bubiyan au Koweit. Travaux, November. Rosignoli, M. (2001). Trusses instead of solid webs? Concrete International, May. Elgaaly, M.; Seshadri, A. (1998). Steel Built-up Girders with Trapezoidally Corrugated Webs. Engineering Journal, AISC, 1st Qtr., pp. 1-11. Rothwell, A. (1968). The Shear Stiffness of Flat Sided Corrugated Webs. Aeronautical Quarterly, Vol. 19, Pt. 3, pp. 224-234. Wu, C.L.; Duan, S.H. (2008). Buckling Behaviour of Composite Laminated Corrugated Panel with Sinusoidal Profile. Part 1: Equivalent Stiffness Terms. Aircraft Strength Research Institute of China. El Metwally, A.; Loov, R.E. (2003). Corrugated steel webs for prestressed concrete girders. Materials and Structures/Materiaux et Constructions, Vol. 36, March, pp. 127-134. Sherman, D.; Fisher, J. (1971). Beams With Corrugated Webs. Proceedings of the 1st Specialty Conference on Cold-Formed Steel Structures. University of Missouri-Rolla, pp. 198-204. Cheyrezy, M.; Combault, J. (1990). Composite bridges with corrugated steel webs – achievement and prospects. IABSE Symposium, Mixed Structures: Including New Materials. IABSE Reports, Brussels, Belgium, pp. 479-484. Elgaaly, M.; Dagher, H. (1990). Beams and Girders with Corrugated Webs. Proceedings of the SSRC Annual Technical Session, Lehigh University, pp. 37-53. Hamada, M.; Nakayama, K.; Kakihara, M.; Saloh, M.; Ohtake, F. (1984). Development of Welded IBeam with Corrugated Web. The Sumitomo Research, No. 29, pp. 75-90. Heywood, P. (1987). Corrugated Box-Girder Webs Lower Bridge Weight and Cost. Engineering News Record, December, 32. Sayed-Ahmed E. Y. (2001). Behaviour of steel and (or) composite girders with corrugated steel webs. Canadian Journal of Civil Engineering, Vol. 28, pp. 656-672. Cafolla, J., Johnson, R.P. (1997). Corrugated webs in plate girders for bridges. Proceedings of the Institution of Civil Engineers – Structures and Buildings, May. Rosignoli, M. (1998). Misplacement of launching bearings in PC launched bridges. ASCE Journal of Bridge Engineering, November, pp. 170-176. Hussain, M.I.; Libove, C. (1977). Stiffness Tests on Trapezoidal Corrugated Shear Webs. Journal of the Structural Division, ASCE. St. 5, pp. 971-987. Yoda, T.; Ohura, T. (1993). Torsional behaviour of composite PC box girders with corrugated steel webs. Journal of Structural Engineering. 39A: pp. 1251-1258. Moreau, P.; Thivans, P. (1983). Composite structures steel–prestressed concrete. IABSE Proceedings, February. Mori, S.; Miyoshi, T.; Katoh, H.; Nishimura, N.; Nara, S. (2005). A study on local stresses of corrugated steel webs in PC bridges under prestressing. Osaka University.
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