The Design of Modern Steel Bridges - TEDI

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134 The Design of Modern Steel Bridges Figure 5.23 Buckling of compressive part of web. than its plastic moment of resistance. At this stage, the compressive part of the web undergoes substantial buckling and consequently the compressive flange buckles into the web as shown in Fig. 5.23. As a result of the buckling of the compressive part of the web, the distribution of bending stress changes from the ideal linear pattern, as shown in Fig. 5.23, and the web becomes less efficient. To quantify the reduction in the bending strength of the web, the following reduction factor was suggested pffiffiffiffiffiffiffiffiffiffiffiffiffi by Cooper[9] for an I-beam of equal flanges and a web deeper than 5.7tw E=syw 1 0:0005 Aw sffiffiffiffiffiffiffiffi! d E 5:7 ð5:34Þ Af where Aw and Af are the area of the web and each flange, respectively, and d is the web depth. According to this approach there is no reduction in bending strength if d=tw is less than 137 and 165 for syw ¼ 355 N/mm 2 and 245 N/mm 2 , respectively. Cooper’s expression for reduction in bending strength can also be expressed as a reduced effective web thickness twe as follows: sffiffiffiffiffiffiffiffi! twe d E ¼ 1 5:7 0:003 þ 0:0005 Aw ð5:35Þ tw tw tw syw From the results of large-deflection elasto-plastic computer studies on the strength of plate panels subjected to in-plane bending and with different edge conditions, welding residual stresses and out-of-plane imperfections (see Section 5.4.4), the following expression for the effective width is specified in BS 5400: Part 3[2] twe ¼ 1:425 0:00625 tw dc rffiffiffiffiffiffiffiffi syw ð5:36Þ tw 355 where dc is the depth of the compressive part of the web. This expression: (1) ignores the effect of the different ratios of web to flange areas, as this effect, as predicted by equation (5.35), was in fact found to be quite small (2) is valid for girders with equal or unequal flanges (3) stipulates no reduction in the effectiveness of the web if the ratio of the depth of the compressive zone to thickness is less than 68 and 82 (or, in syw Af
the case of equal flanges, the total web depth to its thickness ratio is less than 136 and 164) for steel yield stresses of 355 and 245 N/mm 2 , respectively. These factors are marginally above the buckling coefficients discussed in Section 5.4.4 for plate panels without any in-plane restraints at the edges and subjected to applied in-plane bending stresses. 5.4.7 Combined bending moment and shear force on girders Various researchers have extended their tension field models to apply to coexistent shear force and bending moment on the girder sections. For example, in the tension field model due to Porter, Rockey and Evans[8], the elastic critical stress can be reduced to take into account interaction with bending moment by means of a formula like equation (5.25), and the plastic moment of resistance M p of the flanges in equation (5.33) can be reduced by deducting the flange bending stress from its yield stress. In Basler’s tension-field model, the shear is carried by the web only, without any contribution from the flange; but this shear capacity is only valid as long as the moment is less than the amount that can be resisted by the flanges only, i.e. less than s yfA fd. Any larger moment must be resisted by bending stresses in the web which reduces its shear capacity, until the latter becomes very small, possibly zero, when the bending moment reaches the full plastic moment of the girder, or in the case of thin web girders its elastic moment of resistance. This interaction between moment and shear, in the absence of any lateraltorsional or web buckling, can be represented by Fig. 5.24(a), where Vu represents the full ultimate shear capacity of the girder and V w represents the value ignoring any contribution from the flange. In Reference [2] it has been assumed that, if the bending and shear capacities are calculated without any contribution from the web and the flange, respectively, then a beam can resist these values of bending moment M and shear force V even when coexistent; this has been verified from many test results. M f is taken as s bA fd when s b is the limiting stress in the flange derived from equation (5.13), A f is the flange area and d is the effective girder depth between its flanges; Vw is obtained by putting Mp ¼ 0 in equation (5.32) or (5.33). It has been further assumed, supported by test results, that the bending capacity M u of the girder as calculated for lateral-torsional buckling, elastic or plastic distribution of stresses or web buckling as appropriate, can be attained even when there is a coincident shear on the cross-section provided the latter is less than 1 2 V w; similarly the full shear capacity V u given by equation (5.33) is attained even in the presence of a coexistent bending moment provided the latter is less than 1 2 M f. This interaction is shown in Fig. 5.24(b). 5.5 Design of stiffeners in plate girders Rolled Beam and Plate Girder Design 135 Stiffeners are often used to improve the buckling strength of plated structures. Deflection of the plate normal to its plane along the line of stiffening is resisted
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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
Page 95 and 96: educed variables defined by oi ¼ x
Page 97 and 98: will have a probability of failure
Page 99 and 100: Aims of Design 87 g m ¼ g m1 g m2
Page 101 and 102: 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: where m ¼ Mp b 2 twsyw ty ¼ syw p
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 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
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Cable-stayed Bridges 185 cables con
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The following is a list of cable-st
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(a) Fan system (b) Harp system (c)
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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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204 The Design of Modern Steel Brid
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