≤ <
≤ <
≤
′
′
′
(4.1)
4.3 Section Proportion
4.3.1 Classification of Sections
I-sectional shapes can be classified in four categories based on different fabrication processes or their structural behavior as discussed below:
• A steel I-section may be a rolled section, also known as an I-beam (Figure 4.5a) with or without cover plates, or a built-up section, also known as a plate girder (Figure 4.5b) with or without haunches consisting of top and bottom flange plates welded to a web plate. It should be noted that the web of a rolled section always meets compactness requirements while the flanges may not. To increase the flexural strength of a rolled section, it is common to add cover plates to the flanges.
Rolled steel I-beams are applicable to shorter spans less than 100 ft. (30 m) and plate girders to lon-ger spans of about 100 to 300 ft. (30 to 90 m). Plate girder sections provide engineers freedom and flexibility to proportion the flanges and web plates efficiently. A plate girder can be considered as a deep beam. The most distinguishing feature of a plate girder is the use of the transverse stiffeners
Fish belly haunch Parabolic haunch Cross section
(a)
(b)
FIGURE 4.5 Typical steel girder sections: (a) Rolled beam with cover plate; (b) Built-up plate girder with haunches.
that provides tension-field action increasing the postbuckling shear strength. The plate girder may also require longitudinal stiffeners to develop inelastic flexural buckling strength.
• I-sections can be classified as composite or noncomposite. A steel section that acts together with the concrete deck to resist flexure is called a composite section (Figure 4.6a). A steel section disconnected from the concrete deck is noncomposite (Figure 4.6b). Since composite sections most effectively use the properties of steel and concrete, they are often the best choice. Steel–concrete composite sec-tions are used in positive moment regions and girder bridges are recommended by AASHTO-LRFD (2012), whereas noncomposite sections are used in negative moment regions (AASHTO 2012).
• I-sections can also be classified as compact, noncompact, and slender sections (AASHTO 2012, AISC 2010a). A qualified compact section can develop a full plastic stress distribution and expected to be able to achieve a level of rotational deformation ductility of at least 4. Noncompact sections only develop the yield stress in extreme fiber of compression elements before buckling locally, but will not resist inelastic local buckling at the strain level required for a fully plastic stress distribution. Slender element sections buckle elastically before the yield stress is achieved.
The slender steel sections are not permitted in bridge girders (AASHTO 2012).
• I-sections can also be classified as hybrid or nonhybrid sections. A hybrid section consisting of flanges with a higher yield strength than that of the web may be used to save materials and is being promoted more due to the new high strength steels.
The first step in the structural design of an I-girder bridge is to select an I-rolled shape or to initially size the web and flanges of a plate girder. The following sections present the basic principles of selecting I-rolled shapes and sizing the dimensions of a plate girder.
4.3.2 Depth-to-Span Ratio
For straight girders in highway bridges, AASHTO LRFD (AASHTO 2012) Table 2.5.2.6.3-1 specifies that the minimum ratio of the depth of steel girder portion to the span length is 0.033 for simply sup-ported spans and 0.027 for continuous spans; the minimum ratio of the overall depth (concrete slab plus steel girder)-to-span length is 0.04 for simply supported spans and 0.032 for continuous spans. For horizontally curved girders, the minimum depth will more than likely need to be increased by 10 to
(a)
(b)
FIGURE 4.6 Composite and noncomposite section: (a) Composite girder; (b) Nonncompposite girder.
20%. I-rolled shapes are standardized and can be selected from the AISC Manual (2010b). In the straight girders in railway bridges, the depth-to-span ratio is usually 0.05 to 0.055.
The variable cross sections may be used to save material where the bending moment is smaller and/
or larger near the end of a span (Figure 4.5b). A haunched section may be used for continuous spans.
Figure 4.7 shows a haunched steel continuous girder bridge. For haunched I-girders, the depth-to-span ratios are typically taken as 0.05–0.06 at the piers and 0.025–0.033 at the midspans. However, the manpower required for welding and fabrication may be increased. The cost of manpower and material must be bal-anced to achieve the design objectives. The designer should consult local fabricators to determine common practices in the construction of a plate girder. Figure 4.8 shows typical depth-to-span ratios.
Plate girders must have sufficient flexural and shear strength and stiffness. A practical choice of flange and web plates should not result in any unusual fabrication difficulties. An efficient girder is one that meets these requirements with the minimum weight. An economical one minimizes construction costs and may or may not correspond to the lowest weight alternative (Blodgett 1996).
4.3.3 Flanges
The flanges provide bending strength. The width and thickness are usually determined by choosing the area of the flanges within the limits of the width-to-thickness ratio, b/t, and requirement as specified in the design specifications to prevent local buckling. Lateral bracing of the compression flanges is usually needed to prevent lateral torsional buckling during various load stages. The practical guidelines are as follows:
• Flanges should be at least 12 in. wide. A constant flange width for the entire length of the girder is preferred. If the flange area needs to be increased, it is preferable to change the flange thick-ness. If flange widths need to be changed, it is best to change the width at field splices only. Width increments should be in multiples of 2 or 3 in. For horizontally curved girders, the flange width should be about one-fourth of the web depth. For straight girders, a flange width of approximately one-fifth to one-sixth of the web depth should be sufficient.
• For straight girders, the minimum flange thickness should be 3/4 in. For curved girders, 1 in.
thickness is a practical minimum. The desirable maximum flange thickness is 3 in. Grade 50 and HPS 70W steels are not available in thicknesses greater than 4 in. Flange thickness should have an increment of 1/8 in. for thicknesses up to 1 in., 1/4 in. from 1 to 3 in., and 1/2 in. from 3 to 4 in. At the locations where the flange thickness is changed, the thicker flange should provide about 25%
FIGURE 4.7 A haunched steel continuous girder bridge (U.S. 50 Bridge over Sacramento River).
more area than the thinner flange. In addition, the thicker flange should not be greater than twice the thickness of the thinner flange.
• Both the compression and tension flanges should meet the following proportion requirements (AASHTO-LRFD Article 6.10.2.2) as follows:
2b 12 t
f f
≤ (4.2)
b D6
f ≥ (4.3)
tf ≥1.1tw (4.4)
0.1 I 10 I
yc yt
≤ ≤ (4.5)
• where bf and tf are full width and thickness of the flange (in.); tw is web thickness (in.); Iyc and Iyt
are the moment of inertia of the compression flange and the tension flange about the vertical axis in the plane of web respectively (in.4); D is the web depth (in.). Equation 4.2 is to ensure that the flange will not distort excessively when welded to the web. Equation 4.3 ensures that stiffened interior web panels can develop postelastic buckling shear resistance by the tension field action.
Equation 4.4 ensures that flanges can provide some restraint and proper boundary conditions to resist web shear buckling. Equation 4.5 ensures more efficient flange proportions and prevents the use of sections that may be difficult to handle during construction. It also ensures that the lateral torsional buckling formulas used in AASHTO (AASHTO 2012) are valid.
(a)
L L
0.04 L
L L
(c)
0.025 ~ 0.033 L 0.05 ~ 0.06 L
L (b)
0.7 ~ 0.8 L L
0.032 L
L 0.7 ~ 0.8 L
FIGURE 4.8 Depth-to-span ratios and span arrangements: (a) simply supported spans; (b) continous spans with constant depth; (c) continous spans with variable depth.
4.3.4 Webs
The web mainly provides shear strength for the girder. Since the web contributes little to the bending resistance, its thickness should be as small as practical to meet the web-depth-to-thickness ratio limits D/tw ≤ 150 for webs without longitudinal stiffeners, and D/tw ≤ 300 for webs with longitudinal stiff-eners, respectively (AASHTO Article 6.10.2.1). It is preferable to have web depths in increments of 2 or 3 in. for convenience. Web depths greater than 120 in. will require both longitudinal and vertical splices.
In order to avoid an excessive distortion from welding, the web thickness is preferred not to be less than 1/2 in. A thinner plate is subjective. The thickness should be sufficient to preclude the need for lon-gitudinal stiffeners. Web thickness should be constant or with a limited number of changes. It is more desirable to have one or two web sizes for a continuous girder and one web size for a simple span. Web thickness increments should be 1/16 or 1/8 in. for plate thicknesses up to 1 in., and ¼ in. increments for plates greater than 1 in.
4.3.5 Stiffeners
For built-up I-sections, the longitudinal stiffeners may be provided to increase bending resistance by preventing local buckling, while transverse stiffeners are usually provided to increase shear resistance by the tension field action (Basler 1961a and 1961b). The following three types of stiffeners are usually used for I-Sections:
• Transverse intermediate stiffeners: They are typically welded to the web and work as anchors for the tension field force so that postbuckling shear resistance can be developed. It should be noted that elastic web shear buckling cannot be prevented by transverse stiffeners. Transverse stiffeners are designed to (1) meet the slenderness requirement of projecting elements to present local buckling, (2) provide stiffness to allow the web developing its postbuckling capacity, and (3) have strength to resist the vertical components of the diagonal stresses in the web. Stiffeners connecting cross frames/diaphragms should be welded or bolted to the both flanges. Stiffeners without connecting cross frames/diaphragms are welded to the compression flange and fitted tightly to the tension flange. Stiffener plates are preferred to have even inch widths from the flat bar stock sizes.
• Bearing stiffeners: They are required at all bearing locations and at all locations supporting concentrated loads. For rolled beams, bearing stiffeners may not be needed when factored shear is less than 75% of factored shear resistance. They work as compression members to support vertical concentrated loads by bearing on the ends of stiffeners (Figure 4.5). They are transverse stiffeners and connect to the web to provide a vertical boundary for anchoring shear force from tension field action. They are designed to satisfy the slenderness, bearing, and axial compres-sion requirements. Bearing stiffeners are welded or bolted to both sides of the web. Bearing stiffeners should be thick enough to preclude the need for multiple pairs of bearing stiffeners to avoid multiple-stiffener fabrication difficulties. AASHTO-LRFD Article 6.10.11.2 requires that the stiffeners extend to the full depth of the web and as close as practical to the edge of the flanges.
• Longitudinal stiffeners: They work as restraining boundaries for compression elements so that inelastic flexural buckling stress can be developed in a web. It consists of either a plate welded lon-gitudinally to one side of the web, or a bolted angle. It should be located at a distance of 2Dc/5 from the inner surface of the compression flange, where Dc is the depth of web in compression at the maximum moment section to provide optimum design. The slenderness and stiffness need to be considered for sizing the longitudinal stiffeners. It is recommended that sufficient web thickness be used to eliminate the need for longitudinal stiffeners as it can cause difficulty in fabrication and create fatigue-prone details.