The Design of Modern Steel Bridges - TEDI

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142 The Design of Modern Steel Bridges For webs stiffened with vertical stiffeners only Pcro ¼ 2p2 B ðDx DyÞ 1=2 , ignoring H In this case Dx ¼ Et 3 /12[1 2 ], and Dy ¼ EIsy/a, when t is the thickness of the web and is the Poisson’s ratio, equal to 0.3. Hence Pcro ¼ 6E B t 3 Isy a We may design a transverse stiffener as a strut with an effective axial load Peq such that the ratio of this effective axial load to the Euler buckling load of the strut, P E, is equal to the ratio of the longitudinal force on the web to the orthotropic elastic critical buckling load P cro derived from above, i.e. wherefrom Peq PE ¼ s1tB Pcro Peq ¼ s1tB PE We should also take account of the fact that the destabilising forces in the web do not cause any axial stress on the transverse stiffeners. If a strut with an initial imperfection i is subjected to a destabilising force Peq that magnifies the initial imperfection, but does not cause any axial stress, the limiting value Peql of this force that will initiate yielding on an extreme fibre will be given by which leads to where PE Peq1 i PE Peq1 Pcro Peq1 ¼ Py Z þ Py PE Z ¼ iy r2 r ¼ radius of gyration y ¼ extreme compressive fibre distance A ¼ cross-sectional area Py ¼ squash load ¼ syA: 1=2 y ¼ sy Ar2 Peql will obviously be greater than the strut strength Ps of a normal axially loaded strut with initial imperfections i. We can thus apply a reduction factor P s/P eql to evaluate P eq on the transverse stiffener, thus leading to
where Peq ¼ s1tB PE Ps Pcro Peq1 ¼ s1tB PE ð2:5ksÞ ð5:37Þ Pcro ks ¼ 4Ps p 2 Py Z þ Py PE Using the expressions for P cro derived earlier, we have: (1) for webs stiffened longitudinally and vertically ðaÞ if m 0:25 > n Peq ¼ s1tB n 2m1=2 ð2:5ksÞ ¼1:25s1tBksnm 1=2 ðbÞ if m 0:25 < n Peq ¼ s1tB n3 m þ n4 ð2:5ksÞ ¼2:5s1tBks m þ n4 (2) for webs with transverse stiffeners only Peq ¼ s1tB p2 EIsy B 2 ¼ 4:1s1ks Rolled Beam and Plate Girder Design 143 aIsy t B 6E 1=2 a t 3 Isy 1=2 ð2:5ksÞ n 3 ð5:38Þ ð5:39Þ ð5:40Þ In the first edition of this book, the critical buckling load of a transversely stiffened web was derived from an assumed buckling mode of saw-tooth pattern, consisting of straight longitudinal strips of web between transverse stiffeners and the latter deflecting alternately inwards and outwards. The critical buckling load P cro was derived from this buckling mode as p 4 EIsya 4B 3 This expression would predict that Pcro would increase with any increase in the spacing a of the transverse stiffeners, all other parameters remaining the same. This is obviously unrealistic. The assumed saw-tooth buckling mode is really invalid, as the straight longitudinal strips are assumed to be of negligible flexural stiffness and would thus be unable to resist any applied longitudinal compressive loading. A study of the elastic critical buckling solutions for many stiffened panel geometries in References [10] and [11] indicates that the magnitude of the critical shear stress of the panels is numerically very similar to the critical longitudinal compressive stress. Thus, sl above can be taken as the sum of the
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The Design of Modern Steel Bridges
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The Design of Modern Steel Bridges
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Contents Preface vii Acknowledgemen
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Preface Bridges are great symbols o
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Acknowledgements The figures in Cha
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2 The Design of Modern Steel Bridge
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10 The Design of Modern Steel Bridg
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Table 2.1 Properties of some commer
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Chapter 3 Loads on Bridges 3.1 Dead
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3.3 Design live loads in different
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interstate highways are designed fo
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a coefficient that depends on the n
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Loads on Bridges 59 In Germany, a n
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Table 3.6 Typical values of U and P
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3.5 Longitudinal forces on bridges
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wind loading on the traffic. An upl
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where r is the air density ¼ 1.226
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Loads on Bridges 69 minimum and max
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3.8 Other loads on bridges There ar
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References Loads on Bridges 73 wher
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Chapter 4 Aims of Design 4.1 Limit
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critical components of the structur
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But this is an attempt to achieve u
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If the basic variables R and S are
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educed variables defined by oi ¼ x
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will have a probability of failure
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Aims of Design 87 g m ¼ g m1 g m2
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Table 4.2 Failure probabilities of
Page 103 and 104: Chapter 5 Rolled Beam and Plate Gir
Page 105 and 106: as possible. But deep and thin webs
Page 107 and 108: eam, with an effective area equal t
Page 109 and 110: and stresses caused by lateral defl
Page 111 and 112: Figure 5.3 Beam cross-section. In t
Page 113 and 114: Rolled Beam and Plate Girder Design
Page 115 and 116: Table 5.2 Effective length factors
Page 117 and 118: stress is equal to p 2 E/(Le/r) 2 )
Page 119 and 120: equation (5.16) has been adopted fo
Page 121 and 122: The strain energy of the elastic re
Page 123 and 124: Figure 5.6 Buckling of plates in co
Page 125 and 126: Figure 5.8 Buckling of plate under
Page 127 and 128: techniques. It may be noted that th
Page 133 and 134: 5.4.3 Effect of residual stresses R
Page 135 and 136: Aw ¼ 40 mm 2 , allowing for a smal
Page 141 and 142: Basler and Thurlimann were the firs
Page 143 and 144: Ostapenko/Chern gave the following
Page 145 and 146: where m ¼ Mp b 2 twsyw ty ¼ syw p
Page 147 and 148: the case of equal flanges, the tota
Page 149 and 150: emains flat until an elastic critic
Page 153: longitudinal compression is given b
Page 157 and 158: Figure 5.27 Tension-field forces in
Page 159 and 160: 5.5.6 Design of longitudinal web st
Page 161 and 162: when Le is the effective length for
Page 163 and 164: pffiffiffiffiffiffiffiffiffiffiffif
Page 165 and 166: een derived as a condition for trea
Page 167 and 168: For interaction of various stress c
Page 169 and 170: must be at least i.e. 1 IsxbB 4 2 a
Page 171 and 172: Chapter 6 Stiffened Compression Fla
Page 173 and 174: Stiffened Compression Flanges of Bo
Page 175 and 176: cross-section in which the secant s
Page 177 and 178: longitudinal compression of the lon
Page 179 and 180: webs of main girders is denoted by
Page 183 and 184: wherefrom Ms ¼ 2 3 PE P The net be
Page 195 and 196: Chapter 7 Cable-stayed Bridges 7.1
Page 197 and 198: Cable-stayed Bridges 185 cables con
Page 199 and 200: The following is a list of cable-st
Page 201 and 202: (a) Fan system (b) Harp system (c)
Page 203 and 204: 7.3.2 Ropes causing lateral compres
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The spiral winding of wires around
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polyethylene or even steel tube. Or
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See Fig. 7.7. Imagine a cable-stay
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It may be noted that: Et is higher
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Reference Cable-stayed Bridges 201
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