Prevention of buckling is important in bridge design, because of the potential for collapse. Three forms of buckling must be considered in bridge design.
Types of Buckling
The first, and most serious, is primary buckling of an axially loaded compression member.
Such column buckling may include Euler-type elastic buckling and inelastic buckling. This is a rare occurrence with highway bridges, attesting to the adequacy of the current design provisions.
A second form of buckling is local plate buckling. This form of buckling usually manifests itself in the form of excessive distortion of plate elements. This may not be acceptable from a visual perspective, even though the member capacity may be sufficient. When very thin plates are specified, in the desire to achieve minimum weight and supposedly minimum cost, distortions due to welding may induce initial out-of-plane deformations that then develop into local buckling when the member is loaded. Proper welding techniques and use of transverse or longitudinal stiffeners, while maintaining recommended width-thickness limitations on plates and stiffeners, minimize the probability of local buckling.
The third, and perhaps the most likely form of buckling to occur in steel bridges, is lateral buckling. It develops when compression causes a flexural member to become unstable. Such buckling can be prevented by use of lateral bracing, members capable of preventing deformation normal to the direction of the compressive stress at the point of attachment.
Usually, lateral buckling is construction-related. For example, it can occur when a member is fabricated with very narrow compression flanges without adequate provision for transportation and erection stresses. It also can occur when adequate bracing is not provided during deck-placing sequences. Consequently, designers should ensure that compression flanges are proportioned to provide stability during all phases of the service life of bridges, including construction stages, when temporary lateral bracing may be required.
Maximum Slenderness Ratios of Bridge Members
Ratios of effective length to least radius of gyration of columns should not exceed the values listed in Table 11.24.
The length of top chords of half-through trusses should be taken as the distance between laterally supported panel points. The length of other truss members should be taken as the distance between panel-point intersections, or centers of braced points, or centers of end connections.
Plate-Buckling Criteria for Compression Elements
The ‘‘Standard Specifications for Highway Bridges’’ of the American Association of State Highway and Transportation Officials set a maximum width-thickness ratio b/ t or D/ t for compression members as given in Table 11.25.
Stiffening of Girder Webs (ASD)
Bending of girders tends to buckle thin webs. This buckling may be prevented by making the web sufficiently thick (Table 11.25) or by stiffening the web with plates attached normal to the web. The stiffeners may be set longitudinally or transversely (vertically), or both ways.
(See Art 11.17.)
Bearing stiffeners are required for plate girders at concentrated loads, including all points of support. Rolled beams should have web stiffeners at bearings when the unit shear stress in the web exceeds 75% of the allowable shear. Bearing stiffeners should be placed in pairs, one stiffener on each side of the web. Plate stiffeners or the outstanding legs of angle stiffeners should extend as close as practicable to the outer edges of the flanges. The stiffeners should be ground to fit against the flange through which the concentrated load, or reaction, is transmitted, or they should be attached to that flange with full-penetration groove welds.
They should be fillet welded to both flanges if they also serve as diaphragms connections.
They should be designed for bearing over the area actually in contact with the flange. No allowance should be made for the portions of the stiffeners fitted to fillets of flange angles or flange-web welds. A typical practice is to clip plate stiffeners at 45 at upper and lower ends to clear such fillets or welds. Connections of bearing stiffeners to the web should be designed to transmit the concentrated load, or reaction, to the web.
Bearing stiffeners should be designed as columns. For ordinary welded girders, the column section consists of the plate stiffeners and a strip of web. (At interior supports of continuous hybrid girders, however, when the ratio of web yield strength to tension-flange yield strength is less than 0.7, no part of the web should be considered effective.) For stiffeners consisting of two plates, the effective portion of the web is a centrally located strip 18t wide, where t is the web thickness, in (Fig. 11.7a). For stiffeners consisting of four or more plates, the effective portion of the web is a centrally located strip included between the stiffeners and extending beyond them a total distance of 18t (Fig. 11.7b). The radius of gyration should be computed about the axis through the center of the web. The widththickness ratio of a stiffener plate or the outstanding leg of a stiffener angle should not exceed
where Fy yield strength, ksi, for stiffener steel.
For highway bridges, no stiffeners, other than bearing stiffeners, are required, in general, if the depth-thickness ratio of the web does not exceed the value for girder webs without stiffeners in Table 11.25. But stiffeners may be required for attachment of cross frames.
Transverse stiffeners should be used for highway girders where D/ t exceeds the aforementioned values, where D is the depth of the web, the clear unsupported distance between flanges. When transverse stiffeners are used, the web depth-thickness ratio should not exceed the values given in Table 11.25 for webs without longitudinal stiffeners and with one longitudinal stiffener. Intermediate stiffeners may be A36 steel, whereas web and flanges may be a higher grade.
Where required, transverse stiffeners may be attached to the highway-girder web singly or in pairs. Where stiffeners are placed on opposite sides of the web, they should be fitted tightly against the compression flange. Where a stiffener is placed on only one side of the web, it must be in bearing against, but need not be attached to the compression flange.
Intermediate stiffeners need not bear against the tension flange. However, the distance between the end of the stiffener weld and the near edge of the web-to-flange fillet welds must not be less than 4t or more than 6t.
Transverse stiffeners may be used, where not otherwise required, to serve as connection plates for diaphragms or cross frames. In such cases, the stiffeners must be rigidly connected to both the tension and compression flanges to prevent web fatigue cracks due to out-ofplane movements. The stiffener may be welded to both flanges, or a special bolted detail may be used to connect to the tension flange. The appropriate fatigue category must be used for the tension flange to reflect the detail used (see Art. 11.10).
Transverse stiffeners should be proportioned so that
For stiffener pairs, I should be taken about the center of the web. For single stiffeners, I should be taken about the web face in contact with the stiffeners. In either case, transverse stiffeners should project a distance, in, from the web of at least bƒ /4, where bƒ is the flange width, in, and at least D/30 + 2, where D’ is the girder depth, in. Thickness should be at least 1⁄16 of this width.
Intermediate transverse stiffeners should have a gross cross-sectional area A, in^2, of at least
but not farther than 1.5D.
If the shear stress is larger than 0.6Fv in a girder panel subjected to combined shear and bending moment, the bending stress Fs with live loads positioned for maximum moment at the section should not exceed
Fabricators should be given leeway to vary stiffener spacing and web thickness to optimize costs. Girder webs often compose 40 to 50% of the girder weight but only about 10% of girder bending strength. Hence, least girder weight may be achieved with minimum web thickness and many stiffeners but not necessarily at the lowest cost. Thus, the contract drawings should allow fabricators the option of choosing stiffener spacing. The contract drawings should also note the thickness requirements for a web with a minimum number of stiffeners. (A stiffener is required at every cross frame.) This allows fabricators to choose the most economical fabrication process. If desired, flange thicknesses can be reduced slightly if the thicker-web option is selected. In some cases, the most economical results may be obtained with a stiffened web having a thickness 1⁄16 in less than that of an unstiffened web (Art. 11.17).
Preferably, the drawings should show the details for a range from unstiffened to fully stiffened webs. During the design stage, this is a relatively simple task. In contrast, after a construction contract has been awarded, the contractor cannot be expected to submit alternative girder designs, with or without value engineering, because it is often more trouble than the effort is worth. Contractors generally bid on what is shown on the plans, risking the possibility of losing the contract to a concrete alternative or to another contractor. On the other hand, by providing contract documents with sufficient flexibility, owners can profit from the fact that different fabricators have different methods of cost-effective fabrication that can be utilized on behalf of owners.
Longitudinal stiffeners should be used where D/ t exceeds the values given in Table 11.25. They are required, even if the girder has transverse stiffeners, if the values of D/t for a web with transverse stiffeners is exceeded.
The optimum distance, ds, of a plate longitudinal stiffener from the inner surface of the compression flange is D/5 for a symmetrical girder. The optimum distance, ds, for an unsymmetrical composite girder in positive-moment regions may be determined from
where Dcs is the depth of the web in compression of the non-composite steel beam or girder, ƒDL is the non-composite dead-load stress in the compression flange, and ƒDL is the total LL non-composite and composite dead-load plus the composite live-load stress in the compression flange at the most highly stressed section of the web. The optimum distance, ds, of the stiffener in negative-moment regions of composite sections is 2Dc /5, where Dc is the depth of the web in compression of the composite section at the most highly stressed section of the web. The stiffener should be proportioned so that
Bending stress in the stiffener should not exceed the allowable for the stiffener steel. The stiffener may be placed on only one side of the web. Not required to be continuous, it may be interrupted at transverse stiffeners.
Spacing of transverse stiffeners used with longitudinal stiffeners should satisfy Eq. (11.25a) but should not exceed 1.5 times the subpanel depth in the panel adjacent to a simple support as well as in interior panels. The limit on stiffener spacing given previously to ensure efficient handling of girders does not apply when longitudinal stiffeners are used. Also, in computation of required moment of inertia and area of transverse stiffeners from Eqs. (11.21) to (11.23), the maximum sub-panel depth should be substituted for D.
Longitudinal stiffeners become economical for girder spans over 300 ft. Often, however, they are placed on fascia girders for esthetic reasons and may be used on portions of girders subject to tensile stresses or stress reversals. If this happens, designers should ensure that butt splices used by the fabricators for the longitudinal stiffeners are made with completepenetration groove welds of top quality. (Plates of the sizes used for stiffeners are called bar stock and are available in limited lengths, which almost always make groove-welded splices necessary.) Many adverse in-service conditions have resulted from use of partial-penetration groove welds instead of complete-penetration.
In highway girder bridges, AASHTO requires that the need for lateral bracing be investigated.
The stresses induced in the flanges by the specified wind pressure must be within specified limits. In many cases lateral bracing will not be required, and a better structure can be achieved by eliminating fatigue prone details. Flanges attached to concrete decks or other decks of comparable rigidity will not require lateral bracing. When lateral bracing is required, it should be placed in the exterior bays between diaphragms or cross-frames, in or near the plane of the flange being braced.
Bracing consists of members capable of preventing rotation or lateral deformation of other members. This function may be served in some cases by main members, such as floorbeams where they frame into girders; in other cases by secondary members especially incorporated in the steel framing for the purpose; and in still other cases by other construction, such as a concrete deck. Preferably, bracing should transmit forces received to foundations or bearings, or to other members that will do so.
AASHTO specifications state that the smallest angle used in bracing should be 3 21⁄2 in. Size of bracing often is governed by the maximum permissible slenderness ratio (Table 11.24) or width-thickness ratio of components (Table 11.25). Some designers prefer to design bracing for a percentage, often 2%, of the axial force in the member.
Through-truss, deck-truss, and spandrel-braced-arch highway bridges should have top and bottom lateral bracing (Fig. 11.8). For compression chords, lateral bracing preferably should be as deep as the chords and connected to top and bottom flanges.
If a double system of bracing is used (top and bottom laterals), both systems may be considered effective simultaneously if the members meet the requirements as both tension and compression members. The members should be connected at their intersections.
AASHTO ASD and LFD specifications require that a horizontal wind force of 50 lb / ft2 on the area of the superstructure exposed in elevation be included in determining the need for, or in designing, bracing. Half of the force should be applied in the plane of each flange.
The maximum induced stresses F, ksi, in the bottom flange from the lateral forces can be computed from
Cross Frames and Diaphragms for Deck Spans
In highway bridges, rolled beams and plate girders should be braced with cross frames or diaphragms at each end. Also, AASHTO specifications for ASD and LFD require that intermediate cross frames or diaphragms be spaced at intervals of 25 ft or less. They should be placed in all bays. Cross frames should be as deep as practicable. Diaphragms should be at least one-third and preferably one-half the girder depth. Cross frames and diaphragms should be designed for wind forces as described above for lateral bracing. The maximum horizontal force in the cross frames or diaphragms may be computed from
End cross frames or diaphragms should be designed to transmit all lateral forces to the bearings. Cross frames between horizontally curved girders should be designed as main members capable of transferring lateral forces from the girder flanges.
Although AASHTO specifications for ASD and LFD require cross frames or diaphragms at intervals of 25 ft or less, it is questionable whether spacing that close is necessary for bridges in service. Often, a three-dimensional finite-element analysis will show that few, if any, cross frames or diaphragms are necessary. Inasmuch as most fatigue-related damage to steel bridge is a direct result of out-of-plane forces induced through cross frames, the possibility of eliminating them should be investigated for all new bridges. However, although cross frames may not be needed for service loads, they may be necessary to ensure stability during girder erection and deck placement.
The AASHTO LRFD specifications do not require cross frames or diaphragms but specify that the need for diaphragms or cross frames should be investigated for all stages of assumed construction procedures and the final condition. Diaphragms or cross frames required for conditions other than the final condition may be specified to be temporary bracing. If permanent cross frames or diaphragms are included in the structural model used to determine force effects, they should be designed for all applicable limit states for the calculated member loads.
For plate girders, stiffeners used as cross-frame connection stiffeners should be connected to both flanges to prevent distortion-induced fatigue cracking. Although many designers believe welding stiffeners to the tension flange is worse than leaving the connection stiffener unattached, experience has proven otherwise. Virtually no cracks result from the attachment weld, but a proliferation of cracks develop when connection stiffeners are not connected to the tension flange. The LRFD specifications also recommend that, where cross frames are used, the attachment be designed for a transverse force of 20 kips (Fig. 11.9). This applies to straight, nonskewed bridges when better information is not available.
Portal and Sway Bracing
End panels of simply supported, through-truss bridges have compression chords that slope to meet the bottom chords just above the bearings. Bracing between corresponding sloping chords of a pair of main trusses is called portal bracing (Fig. 12.8). Bracing between corresponding vertical posts of a pair of main trusses is called sway bracing (Fig. 11.8).
All through-truss bridges should have portal bracing, made as deep as clearance permits.
Portal bracing preferably should be of the two-plane or box type, rigidly connected to the flanges of the end posts (sloping chords). If single-plane portal bracing is used, it should be set in the central transverse plane of the end posts. Diaphragms then should be placed between the webs of the end posts, to distribute the portal stresses.
Portal bracing should be designed to carry the end reaction of the top lateral system. End posts should be designed to transfer this reaction to the truss bearings.
Through trusses should have sway bracing at least 5 ft deep in highway bridges at each intermediate panel point. Top lateral struts should be at least as deep as the top chord.
Deck trusses should have sway bracing between all corresponding panel points. This bracing should extend the full depth of the trusses below the floor system. End sway bracing should be designed to carry the top lateral forces to the supports through the truss end posts.
Bracing of Towers
Towers should be braced with double systems of diagonals and with horizontal struts at caps, bases, and intermediate panel points. Sections of members of longitudinal bracing in each panel should not be less than those of members in corresponding panels of the transverse bracing.
Column splices should be at or just above panel points. Bracing of a long column should fix the column about both axes at or near the same point.
Horizontal diagonal bracing should be placed, at alternate intermediate panel points, in all towers with more than two vertical panels. In double-track towers, horizontal bracing should be installed at the top to transmit horizontal forces.
Bottom struts of towers should be strong enough to slide the movable shoes with the structure unloaded, when the coefficient of friction is 0.25. Column bearings should be designed for expansion and contraction of the tower bracing.