Performance based design of steel arch bridges using practical inelastic nonlinear analysis

Journal of Constructional Steel Research 59 (2003) 91–108 www.elsevier.com/locate/jcsr

Performance based design of steel arch bridges using practical inelastic nonlinear analysis Seung-Eock Kim ∗, Se-Hyu Choi, Sang-Soo Ma Department of Civil and Environmental Engineering, Construction Tech. Research Institute, Sejong University, 98 Koonja-dong Kwangjin-ku, Seoul, South Korea Received 20 August 2001; accepted 13 February 2002

Abstract A performance based design method of three-dimensional steel arch bridges using practical inelastic nonlinear analysis is presented. In this design method, separate member capacity checks after analysis are not required, because the stability and strength of the structural system and its component members can be rigorously treated in analysis. The geometric nonlinearity is considered by using the stability function for beam-column members and the geometric stiffness matrix for truss members. The Column Research Council (CRC) tangent modulus is used to account for gradual yielding due to residual stresses. A parabolic function is used to represent the transition from elastic to zero stiffness associated with a developing hinge of beam-column members. The load–displacements predicted by the proposed analysis compare well with those given by other approaches. A case study has been presented for the steel arch bridge with 61 m span. The analysis results show that the proposed method is suitable for adoption in practice.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Performance based design; Inelastic nonlinear analysis; Geometric nonlinearity; Material nonlinearity; Steel arch bridge

1. Introduction The steel design methods are Allowable Stress Design (ASD), Plastic Design (PD), and Load and Resistance Factor Design (LRFD). In ASD, the stress computation is ∗

Corresponding author. Tel.: +82-2-3408-3291; fax: +82-2-3408-3332. E-mail address: [email protected] (S.-E. Kim).

0143-974X/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 9 7 4 X ( 0 2 ) 0 0 0 1 9 - 6

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based on a linear elastic analysis, and the inelastic nonlinear effects are implicitly accounted for in the member design equations. In PD, a linear plastic-hinge analysis is used in the structural analysis. Inelastic nonlinearity and gradual yielding effects are approximated in member design equations. In LRFD, a linear elastic analysis with amplification factors or a direct nonlinear elastic analysis is used to account for inelastic nonlinearity, and the ultimate strength of members is implicitly reflected in the design interaction equations. However, despite popular use of conventional design methods in the past and present as a basis for design, the methods have their major limitations. The first of these is that it does not give an accurate indication of the factor against failure, because it does not consider the interaction of strength and stability between the member and structural system in a direct manner. It is well-recognized fact that the actual failure mode of the structural system often does not have any resemblance whatsoever to the elastic buckling mode of the structural system. The second and perhaps the most serious limitation is probably the rationale of the current two-stage process in design: elastic analysis is used to determine the forces acting on each member of a structure system, whereas inelastic analysis is used to determine the strength of each member treated as an isolated member. There is no verification of the compatibility between the isolated member and the member as part of a structural system. The individual member strength equations as specified in specifications are unconcerned with system compatibility. As a result, there is no explicit guarantee that all members will sustain their design loads under the geometric configuration imposed by the structural system. To solve these problems, performance based design should be carried out. Performance based design uses inelastic nonlinear analysis that can sufficiently capture the limit state strength and stability of a structural system and its individual members, so that separate member capacity checks encompassed by the specification equations are not required. It is expected that the use of performance based design method will simplify the design process considerably. The main difference between performance based design method and conventional methods is that performance based design method can predict the structural system strength, whereas others can predict only member strengths. Over the past 30 years, research efforts have been devoted to the development and validation of several inelastic nonlinear analysis methods. Inelastic nonlinear analyses may be grouped into the second-order plastic-zone and the second-order plastic-hinge analyses. The second-order plastic-zone solution is known as the ‘exact solution’, but cannot be used for practical design purposes [4,5]. This is because the method is too intensive in computation and costly due to its complexity. Secondorder plastic-hinge analyses, practical analyses, for the space frames were developed by Orbison [11], Ziemian et al. [14], Prakash and Powell [13], Liew and Tang [9], and Kim et al. [7,8]. Recently, a number of studies of steel arch bridges have been performed by Chatterjee and Datta [2], Nazmy [10], and Pi and Trahair [12]. However, these studies are based on nonlinear elastic analysis to evaluate the effects of various design parameters influencing the strength and stability of steel arch bridges. The purpose of

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this paper is to present a performance based design method of steel arch bridges using a practical inelastic nonlinear analysis.

2. Practical inelastic nonlinear analysis 2.1. Stability functions accounting for second-order effect of beam-column member To capture second-order effects, stability functions are used to minimize modeling and solution time. Generally only one or two elements are needed per a member. The simplified stability functions reported by Chen and Lui [3] are used here. Considering the prismatic beam-column element in Fig. 1, the incremental force–displacement relationship of this element may be written as

冦冧 MA MB P

冤 冥冦 冧 S1 S2 0

EI S2 S1 0 ⫽ L A 0 0 I

qA

qB

(1)

e

where S1, S2 are the stability functions, MA, MB the incremental end moments, P the incremental axial force, qA, qB the incremental joint rotations, e the incremental axial displacement, A, I, L the area, moment of inertia, and length of beam-column element and E the modulus of elasticity. The stability functions given by Eq. (1) may be written as

Fig. 1.

Beam-column subjected to double-curvature bending.

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S1 ⫽

S2 ⫽

冦 冦

π冑rsin(π冑r)⫺π2rcos(π冑r)

2⫺2cos(π冑r)⫺π冑rsin(π冑r)

if P ⬍ 0

π2rcosh(π冑r)⫺π冑rsinh(π冑r)

(2a)

2⫺2cosh(π冑r) ⫹ π冑rsinh(π冑r) π2r⫺π冑rsin(π冑r)

2⫺2cos(π冑r)⫺π冑rsin(π冑r)

if P ⬎ 0

if P ⬍ 0

π冑rsinh(π冑r)⫺π2r

2⫺2cosh(π冑r) ⫹ π冑rsin(π冑r)

(2b) if P ⬎ 0

where r ⫽ P / (π2EI / L2), P is positive in tension. The force–displacement equation may be extended for the three-dimensional beam-column element as  EA 0 L

 冦冧 P

MyA

MyB MzA T



0

0

0

EIy EIy S 0 L 2L

0

0

EIy EIy S 0 L 1L

0

0

0

S1

0

S2

⫽

MzB

0

0

0

0

EIz EIz S3 S 0 L 4L

0

0

0

EIz EIz S4 0 S L 3L

0

0

0

0

0



 冦 冧 d

qyA



qyB qzA

(3)

qzB f

GJ L 

where P, MyA, MyB, MzA, MzB, and, T are axial force, end moments with respect to y and z axes and torsion respectively. d, qyA, qyB, qzA, qzB, and, f are the axial displacement, the joint rotations, and the angle of twist. S1, S2, S3 and S4 are the stability functions with respect to y- and z-axes, respectively. 2.2. CRC tangent modulus model associated with residual stresses The CRC tangent modulus concept is used to account for gradual yielding (due to residual stresses) along the length of axially loaded members between plastic hinges. From Chen and Lui [3], the CRC Et is written as Et ⫽ 1.0E for Pⱕ0.5Py

(4a)

S.-E. Kim et al. / Journal of Constructional Steel Research 59 (2003) 91–108

冉 冊

P P Et ⫽ 4 E 1⫺ for P ⬎ 0.5Py Py Py

95

(4b)

2.3. Gradual yielding and force–displacement relationship of beam-column member The tangent modulus model is suitable for the member subjected to axial force, but not adequate for cases of both axial force and bending moment. A gradual stiffness degradation model for a plastic hinge is required to represent the partial plastification effects associated with bending. We shall introduce the parabolic function to represent the transition from elastic to zero stiffness associated with a developing hinge. The parabolic function h is expressed as: h ⫽ 1.0 for aⱕ0.5

(5a)

h ⫽ 4a(1⫺a) for a ⬎ 0.5

(5b)

where a is a force-state parameter that measures the magnitude of axial force and bending moment at the element end. The term a may be expressed by AISC-LRFD and Orbison, respectively. 2.3.1. AISC-LRFD Based on the AISC-LRFD bilinear interaction equation [6], the cross-section plastic strength of the beam-column member may be expressed as a⫽

8 My 8 Mz P 2 My 2 Mz P ⫹ ⫹ for ⱖ ⫹ Py 9 Myp 9 Mzp Py 9 Myp 9 Mzp

(6a)

a⫽

My Mz P 2 My 2 Mz P ⫹ ⫹ for ⬍ ⫹ 2Py Myp Mzp Py 9 Myp 9 Mzp

(6b)

2.3.2. Orbison Orbison’s full plastification surface [11] of cross-section is given by a ⫽ 1.15p2 ⫹ m2z ⫹ m4y ⫹ 3.67p2m2z ⫹ 3.0p6m2y ⫹ 4.65m4z m2y

(7)

where, p ⫽ P / Py, mz ⫽ Mz / Mpz (strong-axis), my ⫽ My / Mpy (weak-axis). Initial yielding is assumed to occur based on a yield surface that has the same shape as the full plastification surface and with the force-state parameter denoted as a0 ⫽ 0.5. If the forces change so the force point moves inside or along the initial yield surface, the element is assumed to remain fully elastic with no stiffness reduction. If the force point moves beyond the initial yield surface, the element stiffness is reduced to account for the effect of plastification at the element end. When softening plastic hinges are active at both ends of an element, the slopedeflection equation may be expressed as

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冦冧冤 P

EtA 0 L

MyA

MyB MzA

⫽

MzB T

where

0

0

0

0

0

kiiy kijy 0

0

0

0

kijy kjjy 0

0

0

0

0

0

kiiz kijz 0

0

0

0

kijz kjjz 0

0

0

0

0

冉

qyA qyB qzA

(8)

qzB

GJ L

0

冥冦 冧 d

f

冊

S22 EtIy kiiy ⫽ hA S1⫺ (1⫺hB) S1 L

(9a)

EtIy kijy ⫽ hAhBS2 L

(9b)

冉 冉

冊 冊

S22 EtIy kjjy ⫽ hB S1⫺ (1⫺hA) S1 L

(9c)

S24 EtIz kiiz ⫽ hA S3⫺ (1⫺hB) S3 L

(9d)

kijz ⫽ hAhBS4

冉

EtIz L

(9e)

冊

S24 EtIz . kjjz ⫽ hB S3⫺ (1⫺hA) S3 L

(9f)

The terms hA and hB is a scalar parameter that allows for gradual inelastic stiffness reduction of the element associated with plastification at end A and B. This term is equal to 1.0 when the element is elastic, and zero when a plastic hinge is formed. To account for transverse shear deformation effects in a beam-column element, the stiffness matrix may be modified as

冦冧冤 P

EtA 0 L

MyA

MyB MzA MzB T

⫽

0

0

0

0

0

Ciiy Cijy 0

0

0

0

Cijy Cjjy 0

0

0

0

0

0

Ciiz Cijz 0

0

0

0

Cijz Cjjz 0

0

0

0

0

0

冥冦 冧

GJ L

d

qyA qyB qzA qzB f

(10)

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97

where Ciiy ⫽

kiiykjjy⫺k2ijy ⫹ kiiyAszGL kiiy ⫹ kjjy ⫹ 2kijy ⫹ AszGL

(11a)

Cijy ⫽

⫺kiiykjjy ⫹ k2ijy ⫹ kijyAszGL kiiy ⫹ kjjy ⫹ 2kijy ⫹ AszGL

(11b)

Cjjy ⫽

kiiykjjy⫺k2ijy ⫹ kjjyAszGL kiiy ⫹ kjjy ⫹ 2kijy ⫹ AszGL

(11c)

Ciiz ⫽

kiizkjjz⫺k2ijz ⫹ kiizAsyGL kiiz ⫹ kjjz ⫹ 2kijz ⫹ AsyGL

(11d)

Cijz ⫽

⫺kiizkjjz ⫹ k2ijz ⫹ kijzAsyGL kiiz ⫹ kjjz ⫹ 2kijz ⫹ AsyGL

(11e)

Cjjz ⫽

kiizkjjz⫺k2ijz ⫹ kjjzAsyGL kiiz ⫹ kjjz ⫹ 2kijz ⫹ AsyGL

(11f)

The force–displacement relationship of a beam-column element from Eq. (10) may be symbolically written as {fe} ⫽ [Kef]{de}

(12)

in which {fe} and {de} are the element end force and displacement arrays, and [Kef] is the element tangent stiffness matrix. 2.4. Ultimate strength of truss member A force-state parameter of truss element b is expressed as b⫽

P Pn

(13)

where Pn is the ultimate strength of a truss member, and determined by the AASHTO-LRFD [1] equations as: For tension Pn ⫽ FyA

(14)

For compression Pn ⫽ 0.66lFyA for lⱕ2.25

(15a)

0.88FyA for l ⬎ 2.25 l

(15b)

Pn ⫽

for which l is l⫽

冉 冊

L 2Fy rs π E

(16)

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where Fy, A, and E are yield stress, gross cross-sectional area, and Young’s modulus. L is unbraced length and rs is radius of gyration about the plane of buckling. Then the force–displacement relationship of a truss member may be expressed as

冦 冧 冤 冥冦 冧 P

MyA

MyB MzA MzB T

EtA 0 0 0 0 0 m L

0

0 0 0 0 0

⫽ 0 0

0 0 0 0 0

0

0 0 0 0 0

0

0 0 0 0 0

0 0 0 0 0

d

qyA qyB qzA

(17)

qzB f

where the unit function m is determined as: m ⫽ 1.0 for bⱕ1.0

(18a)

m ⫽ 0.0 for b ⬎ 1.0

(18b)

Eq. (17) may be symbolically written as {fe} ⫽ [Ket]{de}

(19)

in which {fe} and {de} are the element end force and displacement arrays, and [Ket] is the element tangent stiffness matrix. 2.5. Element stiffness matrix of beam-column and truss The end forces and end displacements used in Eqs. (12) and (19) are shown in Fig. 2(a). The sign convention for the positive directions of element end forces and end displacements of a member is shown in Fig. 2(b). By comparing the two figures, we can express the equilibrium and kinematic relationships in symbolic form as {fn} ⫽ [T]T6×12{fe}

(20a)

{de} ⫽ [T]6×12{dL}

(20b)

{fn} and {dL} are the end force and displacement vectors of a member expressed as {fn}T ⫽ {rn1 rn2 rn3 rn4 rn5 rn6 rn7 rn8 rn9 rn10 rn11 rn12}

(21a)

{dL}T ⫽ {d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12}

(21b)

{fe} and {de} are the end force and displacement vectors in Eqs. (12) and (19). [T]6 × 12is a transformation matrix written as

S.-E. Kim et al. / Journal of Constructional Steel Research 59 (2003) 91–108

Fig. 2.

Element end forces and displacements notation.

 ⫺1 0 0

 

0 0 0 1 0

0 0

0 0 

0

0 ⫺

1 0 1 0 0 0 L

1 0 L

0 0

0

0 ⫺

1 0 0 0 0 0 L

1 0 L

1 0

0

1 0 L

0 0 1 0 ⫺

1 0 0 L

0 0

0

1 0 L

0 0 0 0 ⫺

1 0 0 L

0 1

[T]6×12 ⫽ 

99

  

(22)

 0 0 0 1 0 0 0 0 0 ⫺1 0 0  Using the transformation matrix by equilibrium and kinematic relations, the force– displacement relationship of a member may be written as {fn} ⫽ [Kn]{dL}.

(23)

[Kn] is the element stiffness matrix expressed as [Kn]12×12 ⫽ [T]T6×12[Ke]6×6[T]6×12.

(24)

where [Ke] is [Kef] for a beam-column element and [Ket] for a truss element. Eq. (24) can be subgrouped as [Kn]12×12 ⫽

冋

册

[Kn]1 [Kn]2

[Kn]T2 [Kn]3

(25)

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where

[Kn]1 ⫽

[Kn]2 ⫽

冤 冥 冤 冥 冤 冥 a 0 0

0 0

0

0 b 0

0 0

c

0 0 d

0 ⫺e 0

0 0 0

f 0

0

0 0 ⫺e 0 g

0

0 c 0

0 0

h

⫺a 0

0

0

0

0

0

⫺b 0

0

0

c

0

0

⫺d 0

0

0

0

⫺f 0

0

0

0

e

0

i

0

0

⫺c 0

0

0

j

⫺e 0

(26a)

(26b)

a 0 0 0 0 0

0 b 0 0 0 ⫺c

[Kn]3 ⫽

0 0 d 0 e 0 0 0 0 f 0 0

(26c)

0 0 e 0 m 0

0 c 0 0 0 n

For beam-column members a⫽

Ciiz ⫹ Cijz Ciiz ⫹ 2Cijz ⫹ Cjjz EtA ,c⫽ ,b⫽ L L2 L

(27a–c)

d⫽

Ciiy ⫹ Cijy Ciiy ⫹ 2Cijy ⫹ Cjjy GJ ,e⫽ ,f⫽ L2 L L

(27d–f)

g ⫽ Ciiy, h ⫽ Ciiz, i ⫽ Cijy, j ⫽ Cijz, m ⫽ Cjjy, n ⫽ Cjjz

(27g–n)

For truss members, a⫽

EtA ,b⫽c⫽d⫽e⫽f⫽g⫽h⫽i⫽j⫽l⫽m⫽n⫽0 L

(28a–n)

Eq. (25) is used to enforce no sidesway in the member. If the member is permitted to sway, an additional axial and shear forces will be induced in the member. We

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101

can relate this additional axial and shear forces due to a member sway to the member end displacements as {fs} ⫽ [Ks]{dL}.

(29)

where {fs}, {dL}, and [Ks] are end force vector, end displacement vector, and the element stiffness matrix. They may be written as {fs}T ⫽ {rs1 rs2 rs3 rs4 rs5 rs6 rs7 rs8 rs9 rs10 rs11 rs12}

(30a)

{dL}T ⫽ {d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12}

(30b)

册

(30c)

冤 冥

(31)

[Ks]12×12 ⫽

冋

⫺[Ks]

[Ks]

⫺[Ks] [Ks] T

where

[Ks] ⫽

0

a ⫺b 0 0 0

a

c 0

0 0 0

⫺b 0 c

0 0 0

0

0 0

0 0 0

0

0 0

0 0 0

0

0 0

0 0 0

For beam-column members a⫽

MyA ⫹ MyB P MzA ⫹ MzB ,b⫽ ,c⫽ L2 L2 L

(32a–c)

For truss members, a ⫽ b ⫽ 0, c ⫽

P L

(33a–c)

By combining Eqs. (23) and (29), we obtain the general beam-column element force–displacement relationship as {fL} ⫽ [K]local{dL}

(34)

where {fL} ⫽ {fn} ⫹ {fs}

(35)

[K]local ⫽ [Kn] ⫹ [Ks]

(36)

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3. Design principles 3.1. Design format Performance based design follows the format of LRFD. In AASHTO-LRFD [1], the factored load effect does not exceed the factored nominal resistance of structure. Two kinds of factors are used: one is applied to loads, the other to resistances. The LRFD has the format h

冘

giQiⱕfRn

(37)

where Rn is the nominal resistance of the structural member, Qi the force effect, f the resistance factor, gi the load factor corresponding to Qi and h a factor relating to ductility, redundancy, and operational importance. The main difference between current LRFD method and performance based design method is that the right side of Eq. (37), (fRn) in the LRFD method is the resistance or strength of the component of a structural system, but in the performance based design method, it represents the resistance or the load-carrying capacity of the whole structural system. In the performance based design method, the load-carrying capacity is obtained from carrying out inelastic nonlinear analysis until a structural system reaches its strength limit state such as yielding or buckling. The left side of

冘

Eq. (37), (h giQi) represents the member forces in the LRFD method, but the applied load on the structural system in the performance based design method. 3.2. Resistance factor AASHTO-LRFD specifies the resistance factors, f, for the strength limit state shall be taken as follows: 1.0 for flexure, 0.95 for tension yielding, and 0.9 for compression, respectively. The proposed method uses a system-level resistance which is different from the AASHTO-LRFD specification using member level resistance factors. When a structural system collapses by forming plastic mechanism, the resistance factor of 1.0 is used since the dominant behavior is flexure. When a structural system collapses by member yielding, the resistance factor of 0.95 is used since the dominant behavior is tension. When a structural system collapses by member buckling, the resistance factor of 0.9 is used since the dominant behavior is compression. 3.3. Serviceability limit The most common parameter affecting the design serviceability of steel bridge is the deflection. The performance based design follows AASHTO-LRFD specification. Service live load deflections may be limited to L / 800 where L is the span length of a steel arch bridge. At service load state, member yielding is not permitted anywhere in the structure to avoid permanent deformation under service loads.

S.-E. Kim et al. / Journal of Constructional Steel Research 59 (2003) 91–108

Fig. 3.

103

Space frame of six-story.

4. Verification 4.1. Six-story space frame Fig. 3 shows Orbison’s six-story space frame [11]. The yield strength of all members is 250 MPa (36 ksi) and Young’s modulus is 206,850 MPa (30,000 ksi). Uniform floor pressure of 4.8 kN/m2 (100 psf) is converted into equivalent concentrated loads on the top of the columns. Wind loads are simulated by point loads of 26.7 kN (6 kips) in the Y-direction at every beam-column joints. The load–displacement results calculated by the proposed analysis compare well with those of Liew and Tang’s (considering shear deformations) and Orbison’s (ignoring shear deformations) results (Tables 1, 2, and Fig. 4). The ultimate load factors calculated from the proposed analysis are 2.057 and 2.066. These values are Table 1 Analysis result considering shear deformation Method

Proposed

Plastic strength surface LRFD Ultimate load factor 1.990 Displacement at A in the 208 mm Y-direction

Proposed

Liew’s

Orbison 2.057 219 mm

Orbison 2.062 250 mm

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Table 2 Analysis result ignoring shear deformation Method

Proposed

Proposed

Orbison’s

Plastic strength surface LRFD Ultimate load factor 1.997 Displacement at A in the 199 mm Y-direction

Orbison 2.066 208 mm

Orbison 2.059 247 mm

Fig. 4.

Comparison of load–displacement of six-story space frame.

nearly equivalent to 2.062 and 2.059 calculated by Liew and Tang and Orbison, respectively. 4.2. Truss with double braced panel Fig. 5 shows a two-dimensional truss with double braced panel subjected to a concentrated load at point A. The stress–strain relationship is assumed to be elastic– perfectly plastic with a yield stress of 250 MPa (36 kips) and elastic modulus of 200,000 MPa (29,000 ksi). W14 × 82 is used for all members. The load–displacement results from the proposed and the step-by-step analysis are compared in Fig. 6. The proposed and step-by-step method calculates the ultimate loads of 6012 KN (1351 kips) and 6020 KN (1353 kips), respectively. The difference in the ultimate loads between two approaches is less than 0.13%.

S.-E. Kim et al. / Journal of Constructional Steel Research 59 (2003) 91–108

Fig. 5.

Fig. 6.

105

Truss with double braced panel.

Load–displacement of truss with double braced panel.

5. Design example 5.1. Configuration of steel arch bridge Fig. 7 shows a steel arch bridge which is 7.32 m (24 ft) wide and 61.0 m (200 ft) long. The stress–strain relationship was assumed to be elastic–perfectly plastic with elastic modulus of 200,000 MPa (29,000 ksi) and the yield stress of 248 MPa (36 ksi). The square box section of 24 × 24 × 1 / 2 was used for the arch rib. The wide flange section of W21 × 101 was used for the tie. The wide flange section of

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S.-E. Kim et al. / Journal of Constructional Steel Research 59 (2003) 91–108

Fig. 7.

Steel arch bridge.

W8 × 10 was used for the vertical truss members. The wide flange section of W10 × 22 was used for the lateral braces. 5.2. Load combination The dead load, live load, and impact load specified in AASHTO-LRFD [1] were considered as design loads. The concentrated dead loads and live loads of HS-20 were applied on each joint. The load factors of 1.25 for the dead load, 1.75 for the live load, and 0.30 for the impact load were used. Fig. 8 shows the design load considering the load factor. 5.3. Result of analysis The load–displacement curve of the proposed analysis at the mid-span of the tie is shown in Fig. 9. The steel arch bridge encountered the ultimate state when the applied load ratio reached 1.20. The system resistance factor of 0.95 was used since the frame collapsed by tension yielding at the vertical truss member. Since the ulti-

Fig. 8.

Load conditions of steel arch bridge.

S.-E. Kim et al. / Journal of Constructional Steel Research 59 (2003) 91–108

Fig. 9.

107

Load–displacement of steel arch bridge at midspan.

mate load ratio l resulted in 1.14( ⫽ 1.20 × 0.95) which was greater than 1.0, the member sizes of the system were adequate. The maximum deflection by the service load was calculated as 73 mm (2.87 in) at mid-span. The deflection ratio was L / 835 which satisfied the deflection limit of L / 800.

6. Conclusion The performance based design method using practical inelastic nonlinear analysis for three-dimensional steel arch bridges has been developed. The concluding remarks are as follows: (1) A practical inelastic nonlinear analysis method for three-dimensional steel arch bridges has been developed. (2) The proposed method can practically account for all key factors influencing behavior of frame members and truss members: gradual yielding associated with flexure, residual stresses, and geometric nonlinearity. (3) The proposed analysis is adequate in assessing the strengths when compared with the other approaches. (4) The proposed performance based design method overcomes the difficulties due to incompatibility between the elastic global analysis and the limit state member design in the conventional LRFD method. (5) The proposed method does not require tedious separate member capacity checks, including the calculations of K-factor, and thus it is time-effective.

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(6) The proposed method can account for inelastic force redistribution and thus may allow some reduction of structure weight.

Acknowledgements This work presented in this paper was supported by funds of National Research Laboratory Program (2000-N-NL-01-C-162) from Ministry of Science & Technology in Korea. Authors wish to appreciate the financial support.

References [1] AASHTO. AASHTO LRFD bridge design specification. AASHTO; 1998. [2] Chatterjee PK, Datta TK. Dynamic analysis of arch bridges under travelling loads. Eint J Solids Struct 1995;32(11):1585–94. [3] Chen WF, Lui EM. Stability design of steel frames. Boca Raton: CRC Press, 1992 [380pp.]. [4] Clarke MJ, Bridge RQ, Hancock GJ, Trahair NS. Benchmarking and verification of second-order elastic and inelastic frame analysis programs. In: White DW, Chen WF, editors. SSRC TG 29 Workshop and Nomograph on Plastic Hinge Bas Methods for Advanced Analysis and Design of Steel Frames. Bethlehem, PA: SSRC, Lehigh University; 1992. [5] El-Zanaty M, Murray D, Bjorhovde R. Inelastic behavior of multistory steel frames. Structural Engineering Report No. 83, University of Alberta, Alberta, Canada ;1980. [6] Kanchanalai T. The design and behavior of beam-columns in unbraced steel frames. AISI Project No. 189, Report No. 2, Civil Engineering/structures Research Lab., University of Texas, Austin, Texas; 1977. 300pp. [7] Kim SE, Park MH, Choi SH. Direct design of three-dimensional frames using practical advanced analysis. Eng Struct 2001;23/11:1491–502. [8] Kim SE, Park MH, Choi SH. Practical advanced analysis and design of three-dimensional truss bridges. J Construct Steel Res 2001;57/8:907–23. [9] Liew JY, Tang LK. Nonlinear refined plastic hinge analysis of space frame structures. Research Report No. CE027/98, Department of Civil Engineering, National University of Singapore, Singapore; 1998. [10] Nazmy AS. Stability and load-carrying capacity of three-dimensional long-span steel arch bridges. Comput Struct 1997;65(6):857–68. [11] Orbison JG., Nonlinear static analysis of three-dimensional steel frames. Report No. 82-6, Department of Structural Engineering, Cornell University, Ithaca, New York; 1982. [12] Pi Yong-Lin, Trahair NS. Inelastic lateral buckling strength and design of steel arches. Eng Struct 2000;22:993–1005. [13] Prakash V, Powell GH. DRAIN-3DX: base program user guide, version 1.10. A computer program distributed by NISEE/computer applications, Department of Civil Engineering, University of California, Berkeley; 1993. [14] Ziemian RD, McGuire W, Dierlein GG. Inelastic limit states design part II: three-dimensional frame study. ASCE J Struct Eng 1992;118(9):2550–68.