The Design of Modern Steel Bridges - TEDI

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166 The Design of Modern Steel Bridges This buckling mode of the whole stiffened flange will involve interactive buckling of both the longitudinal and the transverse stiffeners; this buckling mode will not only be sensitive to the initial imperfections of the stiffeners, but will be of a very ‘brittle’ nature, i.e. there will be a sudden and substantial shedding of the load at the onset of buckling. To avoid this catastrophic phenomenon, the transverse stiffeners are normally designed to be sufficiently stiff not to buckle when the longitudinal stiffeners do. Further reasons for making the transverse stiffeners sufficiently stiff are that on the top flange they have to support without large deflection any locally applied axle loadings of vehicles, and in box girders they form components of the internal crossframes or diaphragms which are provided to prevent distortion of the box cross-section. With such rigid transverse stiffeners the buckling of the stiffened flange will have one half-wave in the longitudinal direction between adjacent transverse stiffeners. The rigidities Dx; Dy and H will thus not involve the geometric properties of the transverse stiffener and will be given by where Dx ¼ EIx b 0 , Dy Et ¼ 3 12ð1 v2Þ H ¼ Gt3 6 1 þ 2 vyDx þ 1 2 vxDy þ GJx 2b 0 Ix ¼ second moment of area of a longitudinal stiffener b 0 ¼ spacing of longitudinal stiffeners t ¼ flange plate thickness v ¼ Poisson’s ratio of the flange plate Jx ¼ the torsional constant of the longitudinal stiffener vx ¼ v½b 0 t=ðb 0 t þ AsxÞŠ vy ¼ v Asx ¼ cross-sectional area of one longitudinal stiffener. 9 >= >; ð6:6Þ For an ideal orthotropic plate, vyDx ¼ vxDy. In the case of a compression flange discretely stiffened by longitudinal stiffeners between rigid transverse stiffeners, this equality is not satisfied and the contribution of Dx towards the torsional rigidity H is doubtful. Hence it is safe to take H ¼ Gt3 6 þ vxDy þ GJx 2b 0 ð6:7Þ From equation (6.4), the total critical longitudinal compressive force in the whole orthotropic panel will be scrobt ¼ p2 b Dx f 2 þ Dyf 2 þ 2H The real orthotropic flange has discrete flange stiffeners, each of crosssectional area Asx. If the width of the orthotropic flange panel between adjacent
webs of main girders is denoted by B, instead of b, the average critical stress on the flange will be given by the above expression divided by Bt þ Asx; i.e. scro ¼ p 2 B 2 t þ Asx B Dx f 2 þ Dyf 2 þ 2H The natural half-wavelength for the minimum value of s cro should be BðDx=DyÞ 1=4 ; but for the orthotropic panel buckling between adjacent transverse stiffeners, Dx and Dy are given in equations (6.6) and ðDx=DyÞ will be very large. Hence the actual half-wavelengths will be limited to the spacing L between transverse stiffeners and f should be taken as L=B. This leads to scro ¼ p 2 t þ Asx B Dx L2 þ 2 DyL 2H þ B4 B2 ð6:8Þ The minimum required stiffness Iy of transverse stiffeners, necessary to ensure that overall buckling involving the transverse stiffeners is not more critical than buckling between adjacent transverse stiffeners, may be obtained from equations (6.5) and (6.8). For this purpose we may note that in applying (6.5) we should take Dy ¼ EIy Gt3 , H ¼ L 6 GJx GJy þ þ 2b0 2L and in applying (6.8) we should take Et Dy ¼ 3 12ð1 v2 Gt3 , H ¼ Þ 6 þ vxDy þ GJx 2b0 For torsionally weak longitudinal and transverse stiffeners, i.e. open-type stiffeners, H is negligible in applying (6.5), and both Dy and H are negligible in applying (6.8). The minimum value of Iy of transverse stiffeners for such stiffening, to ensure that overall buckling is less critical than buckling between transverse stiffeners, is given by leading to Stiffened Compression Flanges of Box and Plate Girders 167 2 B 2 pffiffiffiffiffiffiffiffiffiffi DxDy Iy > B4 Ix 4b 0 L 3 > Dx L 2 ð6:9Þ This is only true for elastic critical buckling of an ideally flat stiffened panel. However, to prevent interactive buckling between local and overall modes, sudden drastic unloading and acute sensitivity to initial imperfections, Iy should be several times the above value. For an isolated strut with a maximum initial out-of-straightness of e 1 in a sinusoidal mode, the magnification m of the out-of-straightness under an axial
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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
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Chapter 5 Rolled Beam and Plate Gir
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as possible. But deep and thin webs
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eam, with an effective area equal t
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and stresses caused by lateral defl
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Figure 5.3 Beam cross-section. In t
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Rolled Beam and Plate Girder Design
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Table 5.2 Effective length factors
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stress is equal to p 2 E/(Le/r) 2 )
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equation (5.16) has been adopted fo
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The strain energy of the elastic re
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Figure 5.6 Buckling of plates in co
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Figure 5.8 Buckling of plate under
Page 127 and 128: techniques. It may be noted that th
Page 129 and 130: Rolled Beam and Plate Girder Design
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 and 154: longitudinal compression is given b
Page 155 and 156: where Peq ¼ s1tB PE Ps Pcro Peq1
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: longitudinal compression of the lon
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
Page 205 and 206: The spiral winding of wires around
Page 207 and 208: polyethylene or even steel tube. Or
Page 209 and 210: See Fig. 7.7. Imagine a cable-stay
Page 211 and 212: It may be noted that: Et is higher
Page 213: Reference Cable-stayed Bridges 201
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The Design of Modern Steel Bridges - TEDI

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