The effect of residual stresses and imperfections on the shear buckling stress of plate girder webs is treated in a different manner. Instead of considering an effective section for the buckled plate, the critical buckling stress in shear as calculated from Eqn. 5.21 is divided by a suitable factor of safety to give the allowable buckling shear stress. This stress is empirically modified to allow for residual stresses and imperfections. For plate girders with practical
proportions, an economic solution can be obtained in most cases by using a thin web stiffened transversally by stiffeners as shown in Fig. 5.21.
d1
d
Fig. 5.21 Web Plate with Transverse Stiffeners
Post Buckling Stress in Shear: For transversely stiffened girders where the transverse stiffener spacing lies within the range 1 < a/d < 3, full account may be taken of the considerable reserve of post-buckling resistance. This reserve arises from the development of "tension field action" within the girder.
Figure 5.22 shows the development of tension field action in the individual web panels of a typical girder. Once a web panel has buckled in shear, it loses its resistance to carry additional compressive stresses. In this post-buckling range, a new load-carrying mechanism is developed, whereby any additional shear load is carried by an inclined tensile membrane stress field. This tension field anchors against the top and bottom flanges and against the transverse stiffeners on either side of the web panel, as shown. The load-carrying action of the plate girder than becomes similar to that of the N-truss in Figure 5.22 b.
In the post-buckling range, the resistance offered by the web plates is analogous to that of the diagonal tie bars in the truss. The total shear buckling resistance for design is calculated by adding the post-buckling resistance to the initial elastic buckling resistance.
In this case, the shear buckling factor kq, is computed from Eqn. 5.21 according to the value of α = d1/d and the slenderness parameter in shear λq as determined from Eqn. 5.28.
Steel Bridges
Fig. 5.22 Tension Field Action in Plate Girders
The calculation of the allowable shear buckling stress then depends, as illustrated in Figure 5.23, upon whether the web is:
1. thick (λq < 0.8 , region AB in Fig. 5.23) in which case the web will not buckle and the shear stress at failure will reach the shear yield stress of the web material:
qb = 0.35 * Fy ... (5.37)
2. intermediate (0.8 < λ q < 1.2, region BC in Fig. 5.23) which represents a transition stage from yielding to buckling action with the shear strength being evaluated empirically from the following:
qb = (1.5 - 0,625 λq ) (0.35*Fy) ... (5.38)
3. slender or thin (λq > 1.2, region CD in Figure 5.23) in which case the web will buckle before it yields and a certain amount of post-buckling action is taken into account empirically:
qb = (0.9 / λq ) (0.35*Fy) ... (5.39)
In all cases the calculated shear stress qact should not exceed the allowable buckling shear stress qb .
Fig. 5.23 Buckling Shear Stress
Web plate without transversal stiffeners: The web plate of a typically unstiffened plate girder has a large aspect ratio α. For such a case, the allowable buckling shear stress qb is obtained from the Eqn. 5.21 using a value of kq = 5.34 as:
For (d/t) <159/ Fy : qb = [1.5 – (d/t) Fy / 212] [0.35 Fy] < 0.35Fy…..(5.40) For (d/t) > 159/ Fy : qb = {119 / [ (d/t) Fy ] } {0.35 Fy}………..(5.41)
The forgoing equations may require relatively thick webs making the resulting design uneconomic.
Effect of Longitudinal Stiffeners on Shear Buckling
Both shear and bending strengths of a plate girder are increased by the presence of a longitudinal stiffener. Its location is, therefore, a key factor that
2.0 q
Steel Bridges
affects both. Theoretical and experimental studies have shown that the optimum location of one longitudinal stiffener is at 0.2d from the compression flange for bending and 0.5d for shear. It is important to note that these criteria for location of the stiffeners are based on elastic buckling considerations. The longitudinal stiffener may be more effective in contributing to the ultimate strength of the plate girder under combined bending and shear if placed somewhere between 0.2d and 0.5d from the compression edge of the web. In bridge design practices, 0.2d has been adopted as the standard location for a longitudinal stiffener. Theoretical and experimental studies have shown that the contribution of the longitudinal stiffener placed at 0.2d to the shear buckling stress is relatively small and is usually neglected, see Fig. 5.24.
Fig. 5.24 Effect of Longitudinal Stiffeners on Shear Buckling 5.4.4 INTERACTION BETWEEN SHEAR AND BENDING
In general, any cross-section of a plate girder will be subjected to bending moment in addition to shear. This combination makes the stress conditions in the girder web considerably more complex. The stresses from the bending moment will combine with the shear stresses to give a lower buckling load.
The interaction between shear and bending can be conveniently represented by the diagram shown in Fig. 5.25, where the allowable bending stress is plotted on the vertical axis and the allowable buckling shear stress of the girder is plotted horizontally. The interaction represents a failure envelope, with any point lying on the curve defining the co-existent values of shear and bending that the girder can just sustain. The equation representing this interaction diagram is:
Fb = [ 0.8 - 0.36 (qact / qb)] Fy ... (5.42)
The interaction diagram can be considered in 3 regions. In region AB, the applied shear stress qact is low (< 0.6 qb) and the girder can sustain the full bending stress Fb based on the effective width beff for the compression flange At the other extreme of the interaction diagram in region CD, the applied shear stress is high (= qb) then the allowable bending stress is reduced to 0.44 Fy to allow for the high shear. In the intermediate region BC the allowable bending stress is reduced linearly from 0.58 Fy to 0.44 Fy.
Shear Stress
Bending Stress
0.44 F y 0.58 F
y
0.6 q b q b
A B
C
D
Fig. 5.25 Interaction between Shear and Bending 5.5 Flange Plate Curtailment:
Welded girders offer more flexibility than design with rolled sections. Since the total design moment varies along the girder span, flange plates of varying thicknesses, and sometimes of varying widths, may be butt welded to provide a section strength that closely approximates the variation in bending moment.
Theoretical locations at which flange-plate thickness or width may be changed along the girder length can be determined as follows; Fig. 5.26(a):
1. The resisting moments of the girder with several selected flange plate areas are calculated.
2. The above values of the resisting moments are super-imposed on the graph of the total design moment. This plot is then used to determine the required length of each size flange plate.
Steel Bridges
(b) Transition in Thickness
(c) Transition in Width Welded
Fig. 5.26 Curtailment of Flange Plates
The actual changes in flange plate thickness or width are made near theoretical locations. Although a minimum steel weight results from such changes, an excessive number of changes should be avoided since the cost of making and testing the necessary butt welds increases the over-all cost of the fabricated girder. For a simple span, the flange is usually made from three plates of two sizes; a center plate covering 40 - 60 % of the span, and two plates butt-welded to the center plate.
When flange plates of different thicknesses are butt-welded, design codes require a uniform transition slope between the offset surfaces not exceeding 1 in 4, Fig. 5.26(b). If plates of different widths are joined, the wider plate must taper into the narrower plate with the same slope or with a radius of 60 cm, Fig. 5.26(c).