The Design of Modern Steel Bridges - TEDI

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84 The Design of Modern Steel Bridges Table 4.1 Reliability index for various failure probabilities b 2.32 3.09 3.72 4.27 4.75 5.20 5.61 Pf ¼ f( b) 10 2 10 3 The values of the reliability index b corresponding to various failure probabilities P f can be obtained from the standardised normal distribution function of cumulative densities, and are given in Table 4.1. The failure probability P f required in codes for particular classes of structures should generally be derived from experiences with past practice, consequences of failure and cost considerations. Having chosen P f, and hence b, the determination of the appropriate partial safety factors g i is an iterative process because, in addition to the coefficients of variation vxi of the variables, initial assumptions will have to be made regarding their mean values mxi, which will have to be subsequently checked against the calculated coordinates of the design point. Non-normal distribution of variables may be converted into the equivalent normal distribution by equalising the probability densities and the cumulative densities at the design point. When the action-effect variable S is the sum of several uncorrelated basic variables S1, S2, etc., e.g. dead load, live load, wind load, etc., then the statistical parameters of the combined action-effect S will be given by mS ¼ mS1 10 4 þ mS2 þ mS3 þ ... sS ¼½s 2 S1 þ s2 S2 þ s2 S3 þ ...Š1=2 If the different basic variables S 1, S 2, etc. are normally distributed, then the combined action-effect variable S will also be normally distributed. The resistance variable R is usually a product of two un-correlated basic variables of material strength (i.e. yield stress of steel or cube strength of concrete) and component strength (i.e. buckling strength of struts or plates or beams) which depends on the dimensions of the structure. If the variable R of the total resistance is a product of two variables R 1 and R 2 representing material strength and component strength, then the statistical parameters of the combined variable R are given by mR ¼ mR1 mR2 VR ¼½V 2 R1 þ V2 R2 þ V2 R1 10 5 V 2 R2 Š1=2 10 6 10 7 10 8 when m denotes the mean of the variables and V denotes the coefficient of variation of the variables given by s/m. A structure subjected to dead and live loads and designed to satisfy the relationship NR=½gf3 gmŠ5½ND gfLD þ NL gfLLŠ
will have a probability of failure P f given by Pf ¼ ½ ðmR mD mLÞ=ðs 2 R þ s2 D þ s2 L Þ1=2 Š¼ ½ bŠ, when (1) N, m, s and V represent the nominal, mean, standard deviation and coefficient of variation of the variables (2) Suffixes D, L, S, C and R represent the variables dead load, live load, material strength, component strength and total resistance (3) s R is the standard deviation of the total resistance variable, given by VR mR ¼½V 2 S þ V2 C þ V2 S V2 C Š1=2 aS aC NR (4) a is the ratio of the mean to the nominal values of the variables (5) g f3 is a partial safety factor to cover inaccurate assessment of action effects and stresses (6) g m is a partial safety factor for resistance (7) g fL is a partial safety factor for loading (8) b is a reliability index (9) is the cumulative density function; for a normal distribution of a variable, this is given by rffiffiffiffiffið1 1 2p b e x2 =2 The following example illustrates how the failure probability can be calculated for a structure designed with specified partial safety factors on loads and resistance. Consider the structure to be designed for a nominal dead load of 10 units, a nominal live load of 15 units, gfLD ¼ 1.3, gfLL ¼ 1.5, g m ¼ 1.2 and g f3 ¼ 1.1. The nominal resistance NR must not be less than 1:1 1:2 ½10 1:3 þ 15 1:5Š ¼46:86 units Let us assume that the nominal resistance NR of the designed structure is 47 units. Let us also assume the following statistical parameters of the variables: Variables a ¼ m/N V¼ s/m Dead load 1.05 0.05 Live load 1.04 0.09 Material strength 1.10 0.064 Component strength 1.15 0.091 dx Aims of Design 85
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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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Page 46 and 47: 34 The Design of Modern Steel Bridg
Page 58 and 59: Table 2.1 Properties of some commer
Page 63 and 64: Chapter 3 Loads on Bridges 3.1 Dead
Page 65 and 66: 3.3 Design live loads in different
Page 67 and 68: interstate highways are designed fo
Page 69 and 70: a coefficient that depends on the n
Page 71 and 72: Loads on Bridges 59 In Germany, a n
Page 73 and 74: Table 3.6 Typical values of U and P
Page 75 and 76: 3.5 Longitudinal forces on bridges
Page 77 and 78: wind loading on the traffic. An upl
Page 79 and 80: where r is the air density ¼ 1.226
Page 81 and 82: Loads on Bridges 69 minimum and max
Page 83 and 84: 3.8 Other loads on bridges There ar
Page 85 and 86: References Loads on Bridges 73 wher
Page 87 and 88: Chapter 4 Aims of Design 4.1 Limit
Page 89 and 90: critical components of the structur
Page 91 and 92: But this is an attempt to achieve u
Page 93 and 94: If the basic variables R and S are
Page 95: educed variables defined by oi ¼ x
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 and 146: where m ¼ Mp b 2 twsyw ty ¼ syw p
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the case of equal flanges, the tota
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emains flat until an elastic critic
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Rolled Beam and Plate Girder Design
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longitudinal compression is given b
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where Peq ¼ s1tB PE Ps Pcro Peq1
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Figure 5.27 Tension-field forces in
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5.5.6 Design of longitudinal web st
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when Le is the effective length for
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pffiffiffiffiffiffiffiffiffiffiffif
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een derived as a condition for trea
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For interaction of various stress c
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must be at least i.e. 1 IsxbB 4 2 a
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Chapter 6 Stiffened Compression Fla
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Stiffened Compression Flanges of Bo
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cross-section in which the secant s
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longitudinal compression of the lon
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webs of main girders is denoted by
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wherefrom Ms ¼ 2 3 PE P The net be
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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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206 The Design of Modern Steel Brid
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The Design of Modern Steel Bridges - TEDI

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