Stability and load-carrying capacity of three-dimensional long-span steel arch bridges

Compuars 0

PII: soo45-7949(97)oooo1-1

STABILITY AND LOAD-CARRYING THREE-DIMENSIONAL LONG-SPAN BRIDGES

dl Srruerures Vol. 65, No. 6, pp. 857-868, 1997 1997 Elasvier Science Ltd. All rights reserved Printed in Great Britain cws-7949197 $17 w +mo

CAPACITY OF STEEL ARCH

A. S. Nazmy Department

of Civil and Environmental

Engineering, U.S.A.

University

of Maine,

Orono,

ME 04469-5711,

Abstract-The behavior of several models of three-dimensional long-span steel arch bridges is investigated for evaluating the effects of various design parameters on both the strength and stability of these special structures. The major concerns in the design of a long-span steel arch bridge, from the structural safety point of view, are the yield and buckling failures. Different design parameters may affect the failure load for either type of failure in various ways. This study investigates how changes in certain design parameters would affect the behavior of steel arch bridges, which could lead to an optimum design of this type of bridge structures. The effects of the plate girder stiffness and arch bracing stiffness as well as the rise-to-span ratio and inclination of the arches towards each other are examined in this study. Both critical buckling load and the load-carrying capacity of each design alternative are investigated using the finite element method. All design alternatives are based on the latest AASHTO code for highway bridge design. It is concluded from this study that the inclined arch bridge using the maximum practical rise-to-span ratio (which is about 0.25) is the most favorable design. In addition, the increase in the stiffness of the plate girder does not reduce the bending moments in the arch ribs. However, providing a lateral bracing system with sufficient stiffness greatly reduces the out-of-plane bending moments and increases the load-carrying capacity and the critical buckling load of a long-span arch bridge. 0 1997 Elsevier Science Ltd

1. INTRODUC3ION

is connected to them at the intersection, the bridge is called a half-tied through-arch bridge. Steel arches can span a distance of 60-600 m and the arch rise can be as high as 90 m. A number of studies have been performed in the past to investigate the elastic stability of steel arch bridges. Harrison [2] studied the in-plane stability of a parabolic arch under the effect of partial loading and concluded that partial loading can significantly reduce the buckling load of an arch bridge. However, the reduction in the critical load is less for a pinned supported arch. Austin and Ross [3] investigated the effect of rise-to-span ratio of two-dimensional symmetrically-loaded arches on their in-plane buckling load and found that in the case of a parabolic arch carrying uniformly distributed vertical load, the maximum buckling load was reached at rise-to-span ratio of 0.25 for two-hinged arches and 0.3 for two-fixed arches. Shukla and Ojalvo [4] compared the stability of a deck-type arch bridge with that of a through-arch bridge and their results showed that the out-of-plane buckling load for the through-arch bridge was three to four times as large as the buckling load of a deck-type arch bridge. Wen and Medallah [S] showed that the buckling loads of deck-type arch bridges can be influenced by the presence of the deck, the bracing system between the ribs and the deck, the type of column connections and the stiffness of the deck. They suggested that in computing the buckling loads of arch bridges, the

bridges constitute a major component in the long-span bridge category, which includes cablestayed and suspension bridges. However, arches transmit the applied loads to the supports primarily through axial compression in the arch ribs while bending moments are relatively small. This leads to savings in construction material when compared with other types of bridge superstructures. However, it makes arch bridges uniquely susceptible to buckling, especially with the growing use of high strength steel which produces slender arch ribs. The current AASHTO standard specifications for highway bridges [l] recognizes this problem and requires that the live load and impact load be magnified by a certain factor to account for the in-plane buckling of the arch ribs. However, the specifications fail to recognize the role of the bridge deck in determining the buckling load. Therefore, it is crucial to understand the behavior of this bridge type under service loads considering both strength and stability criteria and to identify the design parameters that are critical to both yielding and buckling failure modes, The deck of an arch bridge is generally made of a concrete slab supported by steel stringers. The deck can be built above the arch, as in the so-called deck-type arch bridges, or below the arch and supported by cables, as in the tied through-arch bridges. If the deck level intersects the arch ribs and Arch

857

858

A. S. Nazmy

entire bridge system must be considered and not just the arch ribs. Several studies have also been conducted to investigate the effect of lateral bracing systems on the ultimate strength of steel arch bridges [6-81. These studies concluded that the ultimate lateral strength of an arch with diagonal bracing alone may be governed by the buckling load of the diagonals at the edge panels of the bracing system and that adequate cross beams are needed at the edges of the bracing system to prevent this premature buckling from taking place. In almost all the investigations described above, the bridge models were either two-dimensional or very simplified and idealized three-dimensional models and no attempt has been made to evaluate both the stability and load-carrying capacity of real three-dimensional arch bridge models designed according to the latest AASHTO specifications

PI. In the present study, several three-dimensional models of steel rib long-span arch bridges with concrete decks, designed according to the latest AASHTO specifications, are investigated to ascertain how changes in the design parameters would affect the load-carrying capacity as well as the buckling load of these arch bridge models. The models include deck-type arch bridges and half-tied through-arch bridges. The main objective of the study was to identify the critical design parameters that greatly influence the strength and stability of long-span arch bridges using realistic and comprehensive three-dimensional models.

.;14,12,

2.

DESIGN OF THE ANALYTICAL

24012 m=288 m

111 Lr

1

MODEL

Two analytical models of parabolic steel arch bridges were designed according to AASHTO design specifications [ 1] to serve as prototypes for the present study. The first prototype is a half-tied through-arch bridge with an arch span of 340 m and a rise-to-span ratio of 0.16 (see Fig. 1). The arch ribs are inclined inward with a spacing between the ribs of 6 m at the crown and 10.6 m at the base. At the deck level, the spacing between the ribs is 9.3 m to accommodate two lanes of traffic on the roadway that goes between the arch ribs. The ribs were designed to support the gravitational loads (dead and live) and were made of a built-up thin-walled tubular rectangular section 76 cm wide, 275 cm deep and 51 mm thick, with a constant size throughout the arch length to simplify modeling. The current AASHTO code [I] was used along with A36 steel (Fyleld= 36 ksi, or 250 MPa) to size the members and the combined stress equation was used in the design of the arches. The value of the equation was kept to almost 0.8 to provide additional strength for supporting wind and seismic loads. The plate girders carrying the deck were two continuous 932 mm deep wide-flange beams (W36 x 210) supported by 75 mm diameter high strength cables placed every 12 m. The arches and the deck were both braced laterally by K-bracing systems as shown in Fig. 1. At the nodes where cables are attached to the arch ribs, rigid beam elements representing cross-diaphragms capable of resisting out-of-plane bending moments connected the two arches together. At the deck level, 855 mm deep wide-flange beams (W33 x 201) were placed every 6 m as floor beams to

340m

ELEVATION Fig. 1. The half-tied through-arch bridge model.

J2,,14,,

41

q

L ‘I

859

Three-dimensional long-span steel arch bridges

L

1

340m

* 1

ELEVATION Fig. 2. The deck-type arch bridge model.

transfer loads from the stringers to the plate girders. The K-brace elements of the deck had very large cross sectional areas to provide the full diaphragm action provided by the concrete slab. The second prototype used in this study is a deck-type arch bridge with an arch span and rise similar to those in the first prototype, but with the deck level being 7 m above the arch crown (see Fig. 2). This model, which has similar bracing systems and almost similar cross sectional properties to the half-tied through-arch prototype, has been developed to examine the difference in structural behavior between deck-type arch and half-tied through-arch bridges. One difference between the two prototypes is that the elements connecting the deck system to the arch ribs are high-strength tension cables in the half-tied through-arch prototype and beam-column elements rigidly connected to both the deck and the arch ribs in the deck-type prototype. The two prototypes are used to investigate how changes in some design parameters would affect the performance of the bridge when subjected to gravitational and wind loads. These loads are computed based on the latest AASHTO specifications [l]. Only symmetrical loads were considered in this parametric study for simplicity. The dead weight of the arch rib was computed to be 28 kN m- ’ uniformly distributed. The total dead load applied on each plate girder is 36.5 kN m - ’ which includes the weight of a 180 mm concrete slab, a 25 mm wearing surface, future pavement, a 45 cm parapet and railing

that spans along the edges of the bridge. Due to the bridge span, the equivalent lane loading was used in designing the bridge girders and arch ribs since it was more critical than using the AASHTO HS20 truck loading. Preliminary computations indicated that wind loads would affect the design more than the live loads. Therefore, load groups two and five in AASHTO were investigated. With design wind speed of 145 km hr- ‘, AASHTO requires that a 2.92 kPa base wind load intensity be applied on the arches and a 1.96 kPa base wind load intensity be applied on the girders. The depth of the arch ribs was 275 cm making the wind load on the arch equal to 8.03 kN m-‘. The exposed area of the plate girder, floor system, parapet and railing summed up to 155 cm, resulting in a wind load of 3.04 kN m- ’ on the deck. 3. EFFECT

OF VARIOUS PARAMETERS LOAD-CARRYING CAPACITY

ON THE

This study examined the effect of six design parameters on the load-carrying capacity of longspan arch bridges. These parameters include: (1) stiffness of the bridge deck main girders; (2) inclination of the arch ribs towards each other; (3) arch rise-to-span ratio; (4) stiffness of the arch bracing system; (5) support conditions of the arch ribs; and (6) location of bridge-deck with respect to the arch, i.e. comparing a deck-type arch bridge to a through-arch bridge.

860

A. S. Nazmy

Increasing the stiffness of the deck main girders may affect the bending moment in the arch ribs, while inclining the arch ribs may provide a better resistance to lateral loads but may also change the in-plane and out-of-plane bending moments in the ribs under gravitational loads. The cases of inclined as well as perfectly vertical ribs will be studied. Varying the rise-to-span ratio will affect the internal forces in the arch ribs. There is an optimum rise-to-span ratio at which the thrust line is close to the neutral axis of the arch, resulting in low bending moments in the arch ribs. This investigation will evaluate the performance of several models with different rise-to-span ratios under gravitational loads in order to evaluate the currently used rise-to-span ratio. The bracing system plays a vital role in the overall strength of an arch bridge. This is especially true when the arch ribs are inclined. This study investigates how the bracing system affects the load-carrying capacity of an arch bridge under lateral wind loads. The type of arch supports will affect the performance of the arch ribs. A pin-supported arch will need smaller foundations than a fixed arch since it will have no moments at the base; however, it might develop slightly higher axial thrust and is more flexible than a fixed arch bridge. This portion of the study will investigate the effect of changing the arch support conditions on the bridge performance under both gravitational and lateral loads. The last design parameter investigated in this study is the location of the bridge deck. The two types of bridge designs examined were a deck-type arch bridge and a half-tied through-arch bridge. The two types of arch bridges behave differently when lateral and gravitational loads are applied. This study will determine what these behavioral differences are and how they affect the load-carrying capacity of each bridge type. 3.1. Method of analysis The computer program IMAGES-3D has been utilized for computing the load-carrying capacity of the three-dimensional long-span arch bridge models investigated in this study. The program, which runs efficiently on a personal computer, is based on the finite element method. It performs a linear static analysis by solving the stiffness equation: IF) = Kl{~~

(1)

for the nodal displacement vector {u}, where {F) is the nodal load vector and [K’j is the stiffness matrix of the entire structure, in global coordinates, constructed from the stiffness matrices of its individual members by the standard assembly procedure [9, lo]. However, before assembling the global stiffness matrix, a transformation of each element’s stiffness matrix from local to global coordinates must be performed by using the standard

transformation

formula:

(2) in which [kc]is the element stiffness matrix in its local coordinates, [k] is its stiffness matrix in global coordinates and [r] is its rotation matrix. Two types of finite elements were used is this study to represent the members of the bridge models. Space truss (or axial-force) elements were used for the lateral bracing members and for the cables that connect the deck system to the arch ribs in the half-tied through-arch-model, while space frame (or beam-column) elements were used for all the other members of the bridge models. The element stiffness matrix in the member local coordinates, as well as the corresponding rotation matrix, for both space truss and space frame elements, is readily available in several structural analysis references [9, IO] and is presented here only for the sake of completeness. 3.1.1. For space truss elements. The element stiffness matrix in local coordinates, considering only one translational displacement in the direction of the member at each end, is given by:

[k&r=

y[

_‘]

(3)

; l]*Xz

where A is the cross sectional area of the member, L is its length and E is its material modulus of elasticity. The corresponding rotation matrix for this element is given by [IO]:

Irk7= [ cos0 ex cos0 ey cos0 ez cos0 e*

cos0

er cos0 8,

1

2x6 (4)

where ox, 0, and f& are the angles between the direction of the member and the global X-, Y- and Z-axes, respectively. It should be noted here that the stiffness matrix of this space truss element in global coordinates, obtained from eqn (2), will be of order 6 x 6, representing only the three global translations at each end of the element. 3.1.2. For space frame elements. The element stiffness matrix in local coordinates, considering three translations and three rotations at each element end, is in the form [9, IO]: k(l,l) k(2,l)

k(l,2)

...

k(2,2)

...

k(l,l2) k(2,12) : 1

k(li,l)

k(li,2)

1..

k(li,l2)]

(5) 12x12

in which k(l,l)

= k(7,7) = -k(l,7)

= -k(7,1)

= EA/L (6a)

861

Three-dimensional long-span steel arch bridges k(2,2) = k(8,8) = -k(2,8)

= -k(8,2)

= 12EZJL’ (6b)

k(3,3) = k(9,9) = -k(3,9) = -k(9,3)

= 12EZJL’

(6~)

k(2,6) = k(6,2) = k(2,12) = k(12,2) = -k(6,8) = -k&6)

= -k(8,12)

= -k(9,5)

= 6EZ;/L* (6d)

= -k(12,8)

k(3,5) = k(5,3) = k(3,ll)

where ox, 8, and 0z are the angles between the direction of the member and the global X-, Y- and Z-axes, respectively. The above definitions for rl through r9 are valid for any member orientation, except for the specific cases when Or is either zero or 180”. For either of these two cases, the expressions for r 1 through r9 are: rl = r3 = r5 = r6 = r7 = r8 = 0

= k(11,3) = -k(5,9)

r2 =

= -k(9,11)

r4 = -cos

= -k(11,9)

= -6EZ,./L*

(6e)

k(4,4) = k(10,lO) = -k(4,10) = GZJL

= -k(10,4) k(5,5) = k(ll,ll)

(6f)

= 4EZJL

(6g)

k(6,6) = k(12,12) = 4EZJL

(6h)

= k(l1,5) = 2EZJL

(6i)

k(5,ll)

k(6,12) = k(12,6) = 2EZJL

(6j)

where E is the member material modulus of elasticity, A is the cross sectional area, L is the member length, Z, and Z2 are the moments of inertia of the cross section about the local principal y- and z-axes, respectively, Z, is the torsional moment of inertia of the cross section and G is the member material shear modulus. The corresponding 12 x 12 rotation matrix for this element is gjven by[lO]:

b-lsF=

[r*] 0 0 0

0 [r*] 0 0

0 0 0

0 0 [r*] 0

1

(7)

k*l

where [r*] is a submatrix of order three, in the form

L 1 r2 r5 r8

rl

[r*] =

r4 r7

r3 r6 r9

(8)

in which,

r4=

cos er

rl = cos 0,

(9a)

r2 = cos 0,

(9b)

r3 = cos 0,

(9c)

-

(lOa) (lob)

e,

UOc)

r9 = 1

(lOd)

After computing the nodal displacement vector {u}, the program also computes the member end-forces for selected elements in their local coordinates {fe} using the corresponding rotation matrix of each element as follows: {fe) =

(11)

Wel[rl~~*J

in which {u*} is the vector of nodal displacements of the element ends in global coordinates. The prototypes used in this study contained 344 nodes and 708 elements. The internal forces were computed at the arch rib base and crown for all cases studied in order to demonstrate the effect of changing the previously mentioned design parameters on the load-carrying capacity of the arch. 3.2. Effect of bridge girder stzfness on the arch rib moments This effect was studied for the half-tied througharch bridge prototype by doubling and tripling the moment of inertia of the bridge girders and computing the in-plane bending moments in the arch ribs due to gravity loads. Figure 3 shows the relationship between the girder stiffness ratio (or the ratio of the girder moment of inertia to that of the prototype) and the in-plane bending moment ratio (relative to that of the prototype) for both the fixed and pinned arch bridges. As can be seen in this figure,

cos ex cos 8,

(CO? e, + COS* e,)

r5 = J(cos*

r6= r7 =

r9 =

-

Bx + co? 0,)

cos

eu

cos

ez

(~0~2 e, + ~0~2 e, - cos ez

1

(cos2 ex + co? ez ) r8 = 0 cos

(9c)

e,

(~0~2 ex + ~0~2 ez )

(9h)

0.9

1

1.o

2.0 GIRDER

(9i)

STIFMESS

3.0

I

RATIO

Fig. 3. Effect of increasing the girder stiffness on the arch bending moments for the half-tied through-arch bridge.

862

A. S. Nazmy

lttheao*nrdplnmdrrch

_t_ 099

I

! 6 m(prototype)

7.6 m

1

I

9.3 m (veMcalribs)

6 m (prototype) 7.6 m

ARCH SPACINQ AT THE CROWN

9.3 m (verQcalribs)

ARCH SPACING AT THE CROWN

Fig. 4. Effect of arch inclination on the arch moments under gravity loads for the half-tied through-arch bridge.

increasing the girder stiffness has very little effect on the bending moments in the arch ribs. This is due to the fact that the arch rib stiffness is much larger than the girder stiffness (almost 50 times larger), which implies that even tripling the girder stiffness does not draw moments away from the arch. Increasing the girder stiffness to 10 times the prototype value or beyond resulted a decrease in the arch moments (this result is not shown in Fig. 3). However, this increase seemed to be impractical and the extra cost of steel in the girders would overshadow any savings in the arch rib steel. 3.3. Efect of arch inclination on the arch rib response The inclination of the arch ribs towards each other in the half-tied through-arch bridge prototype was changed by keeping constant the spacing between the arches at the deck level at 9.3 m and changing the spacing at the crown from 6 to 7.6 m and then to 9.3 m (perfectly vertical arches). Figure 4 shows the relationship between the bending moment ratio and the spacing between the arch ribs at the crown under the effect of gravity loads. All moment ratios are computed with respect to the corresponding values in the half-tied through-arch bridge prototype, which are listed in Table 1. The figure shows that decreasing the arch inclination has very little effect on the in-plane bending moments in the arches under the

effect of gravity loads and greatly reduces the out-of-plane bending moments as the arches approach the perfectly vertical position. In fact, the out-of-plane moments have dropped to almost zero in the vertical arches from 11.3 kN m at the base and 237 kN m at the crown in the prototype (see Table 1 for the prototype response values). Under lateral loads, the results are quite different. In this study, lateral loads on the bridge are due to wind effect. The loads, calculated according to AASHTO specifications [1], turned out to be 8.03 kN m-’ on the arch and 3.04 kN m-’ on the deck, as described earlier. Under this load, the in-plane bending moments were generally greater than the out-of-plane moments as could be seen in Table 1. Figure 5 shows the relationship between the response ratio (whether it is bending moment, axial force or displacement computed relative to the corresponding values in the half-tied through-arch bridge prototype) and the arch spacing at the crown. The figure illustrates that decreasing the arch inclination increases the in-plane moments in the arch as well as the lateral displacement at the crown. It also caused the axial force and the out-of-plane moments in the arch to increase at the base and decrease at the crown. However, the out-of-plane moment at the crown is considerably small compared with the in-plane moment (under wind load, the

Table 1. Internal forces in the half-tied through-arch bridge prototype Response quantity In-plane moment at base (kN m) In-plane moment at crown (kN m) Out-of-plane moment at base (kN m) Out-of-plane moment at crown (kN m) Axial force at base (kN) Axial force at crown (kN)

Due to gravity loads Pinned-base arch Fixed-base arch 2706 760 11.3 237 24 455 20 781

0 672 0 237 24 597 20 786

Due to lateral wind load Fixed-base arch Pinned-base arch 13 736 1287 6196 194 12 183 3715

0 1576 0 229 12 232 4537

Three-dimensional

863

long-span steel arch bridges

”

:

l-

~

#

o-

ln.pmle-nl~

_b_

outdphm

__t_

mlhlloramb

_c-

h*mlMepkementmUo

moment mtb

I

s6 m (pmtolype)

7.0 m

9.3 m (vailcal rb3)

6 m (prototype)

(II) Reqou9e

7.6 m

9.3

m (ventcalribs)

ARCH SPACING ATTHE CROWN

ARCH SPACINQ AT THE CROWN

at thebase.

(b) Response at the crown.

Fig. 5. Effect of arch inclination on the arch response under lateral wind loads for the fixed-base half-tied through-arch bridge.

in-plane moment at the crown is 1287 kN m while the out-of-plane moment is 194 kN m, as seen in Table 1). Therefore, it appears that the reduction in the axial force and the out-of-plane moment at the arch crown, achieved by using perfectly vertical arches, is over-shadowed by the increase in the in-plane moments and the axial force and out-ofplane moments at the base of the arch. In addition, the values of the out-of-plane moments at the crown are too small to control the design of the arch ribs in the lateral direction. Thus, the lateral stiffness of the arch ribs at the crown will be controlled by lateral stability demands and not by bending strength demands. This implies that the prototype, which has the largest practical inclination of the arches towards each other is the most cost-effective design. 3.4. Effect of rise-to-span ratio on the arch response The response displacements and member forces in the arch ribs of the half-tied through-arch prototype,

which has a rise-to-span (RTS) ratio of 0.16, were compared to their corresponding quantities in similar models that have RTS ratios of 0.14 and 0.20. A lower RTS ratio means a shallower arch. In each model, the ratio of the deck height above the support level to the total rise of the arch was kept constant at 0.28. Figure 6 illustrates this comparison under the effect of gravity loads. It can be seen from this figure that a bridge that has a higher RTS ratio will react more favorably than a mode1 with a lower RTS ratio. The high parabolic shape of the 0.20 RTS ratio is closer to the pressure line pattern formed by the external loads, thus lowering the bending moments in the arch rib. The results also show that the axial compression in the arch and the vertical displacement at the crown get smaller as the RTS ratio increases. However, there are also disadvantages in continually increasing the RTS ratio due to construction considerations and due to the effect of other types of loads. Previous studies on simplified arch models showed that fixed arches with very high RTS ratios subjected to lateral wind and seismic loads will have higher internal forces at the bases and that the maximum practical RTS ratio for an arch bridge is around 0.25.

3.5. Eflect response

0.14

0.16 (pmw&pe)

0.20

ARCH RISE-tO%PAN RATIO

Fig. 6. Effect of rise-to-span ratio on the arch response under gravity loads for the fixed-base half-tied through-arch bridge.

of arch bracing stzjiiess on the arch

In the earlier discussion on the effect of arch inclination on the arch rib response, it was noticed that the in-plane bending moments were always larger than the out-of-plane moments at the same arch section. This behavior was also noticed by Yabuki et al. [1 11, who concluded in their study that the out-of-plane deformations and internal forces of a steel arch bridge would remain practically unchanged as the load increases from the design values to the ultimate state values provided that

864

A. S. Nazmy

0.01 ’

0.00

0.25

0.50

LATERAL SRACINQ

I 0.75

1.00

STIFFNESSRATIO

Fig. 7. Effect of lateral bracing stiffness on the bending moments in the fixed-base half-tied through-arch bridge under wind loads. sufficient out-of-plane stiffness existed. The purpose of the study conducted herein is to examine the effect of the stiffness of the arch rib lateral bracing system on the arch response. Figure 7 illustrates the relationship between the arch bending moments under wind load effect vs the arch lateral bracing stiffness. All the values are normalized with respect to the in-plane bending moment at the base of the half-tied through-arch prototype (which is 13736 kN m, as shown in Table 1). In the case of no lateral bracing (zero stiffness ratio), the out-of-plane bending moment was approximately 4.8 times the amount of in-plane bending moment at the base. The arch bracing stiffness was then increased to about one quarter of the stiffness used in the prototype. As can be seen in Fig. 7, the out-of-plane bending moment dramatically dropped while the in-plane bending moment increased about 33% Increasing the stiffness of the arch bracing system thereafter had a small effect on further reducing the out-of-plane moment in the arches. The results of this study show that providing a sufficient stiffness for the lateral bracing system will greatly improve the behavior of long-span arch bridges under lateral loads. 3.6. Effect of support conditions of the arch ribs on the response Table 1 shows the response of the half-tied through-arch prototype to gravity loads as well as to wind loads for both cases of fixed-base arch ribs and pinned-base arch ribs. It can be seen in this table that

under gravity loads, the bending moments at the crown of the pinned arch are either equal to or less than the corresponding values in the fixed arch, while axial forces are very slightly higher in the pinned arch. Under lateral wind loads, the pinned arch develops larger moments and thrust at the crown than the fixed arch. Similar observations can be made for the deck-type arch bridge by examining the response values in Table 2. Knowing that the design at the crown will most likely be governed by stability rather than by strength, in addition to the fact that the base bending moments are zero for the pinned arch, one may conclude that a pinned arch will be more economical than a fixed arch. However, if the stability of the arch ribs controlled their design and if the buckling load for the pinned arch was less than that for the fixed arch, which could be the case for deck-type arch bridges, then a fixed arch would be preferable. 3.7. Effect of bridge deck location with respect to the arch on the response Under the effect of gravity loads, the internal forces at the base of the deck-type arch bridge, where the deck level is above the arch crown, were slightly higher than the corresponding values in the half-tied through-arch bridge, while the internal forces at the crown of the first type were smaller than the corresponding values in the second type, as could be noticed by comparing the results in Tables 1,2. When the fixed-supports at the arch bases were replaced by hinged supports, the results showed the same trend. In the deck-type arch bridge, the deck stiffened the bridge at the crown. In the half-tied through-arch bridge, where the deck was located at the lower portion of the bridge, the deck stiffened the bridge near the base supports. Since the internal forces at the base are generally larger than the forces at the crown, a half-tied through-arch bridge would be preferable to a deck-type arch bridge if a constant cross section for the ribs is used. However, many bridge designs use nonprismatic members to save on steel material. A deck-type arch bridge would allow the designer to reduce the arch rib size at the crown more than the through-arch bridge if the bridge is designed to resist gravity loads only. Under the effect of lateral wind loads, the axial force at the base of the deck-type arch bridge was almost 36% larger than the corresponding value in

Table 2. Internal forces in the deck-type arch bridge prototype Response quantity In-plane moment at base (kN m) In-plane moment at crown (kN m) Out-of-plane moment at base (kN m) Out-of-plane moment at crown (kN m) Axial force at base (kN) Axial force at crown (kN)

Due to gravity loads Fixed-base arch

Pinned-base arch

3033 250 15.6 192 24 508 20 287

0 320 0 190 24 566 20 416

Due to lateral wind load Fixed-base arch Pinned-base arch 12 675 396 6071 95 16 627 1661

0 503 0 118 17 245 2202

Three-dimensional

the half-tied through-arch bridge. However, the latter bridge had a higher axial compression at the crown as well as higher in-plane and out-of-plane moments at both the base and crown, as can be seen from the results in Tables 1,2. The axial thrust at the crown of the through-arch bridge was more than twice the corresponding value in the deck-type arch model. Therefore, under the combined effect of both vertical and lateral loads, we cannot favor one bridge type over the other based on strength only. Other factors such as topography of the bridge site and its soil conditions might control the choice of the arch bridge type.

4. EFFECT OF VARIOUS PARAMETERSON THE ELASTIC STABILITY

This investigation examined the effect of five design parameters on the buckling load of long-span half-tied through-arch bridges. These parameters include: (1) support conditions of the arch ribs; (2) rigidity of the arch rib-to-deck connection; (3) stiffness of the arch bracing system; (4) inclination of the arch ribs towards each other; and (5) arch rise-to-span ratio. Previous research has shown that the buckling strength of a pinned arch can be as low as one half of the buckling strength of a fixed arch when subjected to vertical loads. These results need to be re-evaluated in light of the fact that for half-tied through-arch bridges, the deck can laterally brace the arch ribs. In addition, the degree of rigidity of the connection between the arch ribs and the bridge deck may greatly influence the buckling load. The lateral bracing system could affect the stability of an arch bridge. There are several research papers that have examined this effect for simplified arch bridge models. This study will investigate how the bracing system affects the critical buckling load of a comprehensive three-dimensional model of an arch

k& =

t”

0

0

0

10

0

0

0

0

o-1 0

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long-span steel arch bridges

0

4.1. Method of analysis The computer program ABAQUS, which is a general purpose finite element program, has been used in this investigation to determine the elastic buckling load of the half-tied through-arch bridge model. The program determines the buckling load by the well-known eigenvalue method [12, 131, which will be briefly described here for the sake of completeness. In buckling analysis, the elastic stiffness matrix of the structure must be modified to include the effect of the interaction between the axial forces in the members and their out-of-plane deformations. This is done by adding the geometrical stiffness matrix of the structure, which is function of the axial forces in its members, to its elastic stiffness. Therefore, ABAQUS sets up an elastic stiffness matrix of the structure, KE, based on the dead load conditions and, as the live load is applied with a uniform intensity of P, per unit length, the stiffness matrix of the structure will change to the total stiffness, Kr, as follows: K, = K, + KG

where KG is the geometrical stiffness matrix (also called the first-order incremental stiffness matrix or initial stress stiffness matrix) evaluated at deformations resulting from the change of loading from the dead load to the total applied load and is proportional to the added live load of intensity P,. The elastic stiffness matrix of the structure, KE, is constructed from the elastic stiffness matrices of its individual members, which were given earlier in this paper for space truss and space frame elements, by the standard assembly procedure. Similarly, the geometrical stiffness matrix of the structure, KG, is also assembled from the geometrical stiffness matrices of its individual members, which are given as follows: 4.1.1. For space truss elements. The element geometrical stiffness matrix in local coordinates, considering three orthogonal translations at each end, is given by [14]:

0

0

o-1 10

o-1

0

0

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bridge that has inclined as well as perfectly vertical ribs. Finally, the effect of the rise-to-span ratio on the critical buckling load of half-tied through-arch bridges will be examined.

0

0

0 o-1

(12)

0 1

(13)

where P is the axial force in the member and L is its length. 4.1.2. For space frame elements. The element geometrical stiffness matrix in local coordinates, considering three translations and three rotations at

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A. S. Nazmy

each element end, is given by [14]: 0

0

;

0

0

0 0

0 0

6 J 0

0

L

0

EL2 2

0

0

0

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-lo ok

klw = ;

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(14) in which P is the axial force in the member and L is its length. It should be noted from the above two equations (eqns (13) and (14)) that if the structural element carries an axial compressive force, its geometrical stiffness will have a reducing effect on its total stiffness. If the live load P, is gradually increased, then at a certain value of the load buckling will occur. Since the geometric stiffness, KG, is proportional to the live load, then the total stiffness matrix at the load which represents the dead load plus 1 multiples of the live load can be obtained from eqn (12) as: K, = K, + AK,

(15)

and the equations governing the linear incremental behavior of the structure (or rate of load-displacement relationship) at that state is given by (KE + 1K,)du = dF

(16)

A basis for obtaining a critical load of a structural system is the vanishing of dF in the above equation, i.e. buckling would take place when the structural system experiences an increase in displacement without any increase in the load. Thus, eqn (16) becomes: (KE + LK,)du = 0.

(17)

Equation (17) represents an eigenvalue problem which, when solved, will provide the eigenvalues lli that represent the multipliers to the live load P,. The lowest eigenvalue 1, will correspond to the critical buckling load. By adding A,P,, to the dead

intensity, one gets the value of the critical buckling load per unit length. 4.2. Efect of arch support type on the critical buckling load

The critical buckling load of the fixed-base half-tied through-arch prototype calculated by ABAQUS was found to be qC= 148 kN m - ‘. If we put this quantity in the nondimensional form (q,L’/EI), where L is the arch span = 340 m; E is Young’s modulus of elasticity = 200 GPa and Zis the moment of inertia of arch cross section = 0.292 m4, we get a value of 99.6. This value is almost 15% higher than the one given by Austin and Ross [3] for a parabolic arch that has a rise-to-span ratio of 0.16 and whose buckling mode is in-plane and antisymmetrical. Wen and Medallah [5] found that when transverse bracing is provided at the crown of the arches, which is the case for the models in the present study, the dominant buckling mode was the in-plane antisymmetrical mode. This justifies the use of Austin and Ross’s results for evaluating the results of the present study. The 15% increase in the critical buckling load obtained in this study is attributed to the presence of the bridge girders which are rigidly connected to the arch ribs. The buckling load of the pinned-base prototype as computed by ABAQUS was approximately 97% of the critical load of the fixed-base prototype. This was an unexpectedly high result. Austin’s data showed that the critical load of a pinned parabolic arch can be as low as 50% of the critical load of a fixed parabolic arch. However, further tests in the present study have shown that the

Three-dimensional

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long-span steel arch bridges

Table 3. Critical buckling loads for different rib-to-deck connection types Connection type

Critical buckling load (kN m-‘f Pinned-base arch Fixed-base arch 116 137 148

Loose connection Pinned connection Rigid connection

high buckling load of the pinned prototype can be attributed to the rigid moment connection between the arches and the girders. 4.3. Effect of rib-to-deck connection rigidity on the critical buckling load Table 3 shows how the degree of rigidity of the connection between the deck and the arches would affect the buckling load of the fixed-base prototype and its pinned-base counterpart. As the connection changed from loose to pinned and then to rigid, the critical load kept increasing. Due to the bracing elements at the arch level and the lateral stiffness of

the deck, the critical buckling load was insensitive to the support conditions of the arches when the arch-to-deck connection was rigid (only 3% difference between the fixed-base and pinned-base buckling loads). Therefore, when considering stability, there is virtually no advantage to choosing a fixed arch over a pinned arch for the half-tied through bridges if the arch-to-deck connection is totally rigid. 4.4. Eflect of arch bracing st@ness on the critical buckling load

Varying the stiffness of the arch bracing system elements in the prototype showed how the buckling load of the brace elements may control the buckling load of the entire bridge. Table 4 shows this effect for both the prototype and the bridge with perfectly vertical arches. Clearly, a sufficient arch bracing system is vital to the buckling resistance capabilities of the arch bridge. These results agree with the findings of Kuranishi and Yabuki [8] concerning the threshold of bracing stiffness. In the case of the prototype, the threshold value was half of that used in the prototype. When the stiffness was below the threshold value, the critical buckling load substantially decreased and was governed by the stiffness of the brace elements. The stiffness of the bracing elements influenced the critical buckling load of the perfectly vertical arch model more than the prototype.

81 98 144

4.5 EfSect of arch inclination on the critical buckling load

Almost no change in the critical buckling load was observed when the arch inclination changed. The difference in the value was less than 0.2% when the crown separation changed from 6 m (the prototype) to 7.6 m and to 9.3 m (vertical arches), indicating that the critical buckling load is insensitive to the inclination of the arch. Therefore, the model that has the optimum design cannot be determined by the buckling failure criterion alone. As discussed earlier, the prototype acts more favorably under wind load. However, the perfectly vertical arch model takes advantage of the high strong axis stiffness and low weak axis stiffness of the rectangular tube section more effectively when the bridge is subjected to gravitational loads. 4.6. Effect of rise-to-span ratio on the critical buckling load

The analysis showed that as the arch rise-to-span (RTS) ratio increased from 0.14 to 0.16 (the prototype) to 0.2, the critical buckling load increased from 130 to 148 to 185 kN m- ‘. However, we should not keep increasing the ratio since there are construction considerations that impose a limit on the maximum practical RTS ratio, which is almost 0.25. In addition, if the RTS ratio exceeds some optimum value, the buckling load of the bridge decreases since the arch stability begins to be governed by lateral buckling and the torsional rigidity becomes an important factor in the stability of the bridge as demonstrated by Shukla and Ojalvo [4] for simple arch bridges.

5. CONCL~J~IONS Based on a comprehensive analysis of several three-dimensional steel arch bridge models that resemble real long-span arch bridges designed according to the latest AASHTO Specifications and

Table 4. Critical buckling loads for different bracing stiffness values Arch bracing stiffness ratiot

Critical buckling load (kN m) Prototype model Vertical arch model

0.33 0.50 0.66 1.oo 1.33 twith respect to the stiffness in the prototype. CA.7 65/&D

33 148 148 148 148

19 127 148 148 148

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A. S. Nazmy

in which the bridge deck and arch ribs are acting integrally to resist gravitational and lateral wind loads, the following conclusions and recommendations concerning the strength and stability of long-span arch bridges can be made: (1) Bending moments in the arch ribs are not sensitive to increasing the bending stiffness of the bridge girders. Therefore, there is no reason to increase the bridge girder stiffness beyond what is actually needed to support the girder moments. (2) Inclination of arches towards each other is an important design parameter that influences the arch internal forces under vertical and lateral loads, although it does not affect the critical buckling load of the bridge. Perfectly vertical arches perform better than inclined arches under gravitational loads. However, in the presence of lateral loads, using the largest practical inclination of arches towards each other becomes a more favorable design. (3) The rise-to-span ratio of an arch bridge is another important parameter that affects both the strength and stability of the bridge. As the RTS ratio increases, the bridge buckling load and its resistance to vertical loads increase. However, due to construction considerations, the effect of lateral loads on the bending moments at the base of fixed arches and the lateral instability of arches with very high RTS ratios, a practical value of 0.25 should not be exceeded. (4) There is a threshold for the stiffness of arch bridge lateral bracing system at which the critical buckling load is governed by the stiffness of the arch rib itself and not by the stiffness of the bracing elements. Below that threshold stiffness, the bridge buckling load drops substantially. The provision of sufficient stiffness for the lateral bracing system can also reduce the out-of-plane bending moments of the arch ribs dramatically when subjected to lateral loads. (5) A pinned-base arch bridge generally develops less bending moments in the ribs than a fixed-base arch bridge for both deck-type and half-tied through-arch bridges. In the latter type, the buckling load is almost the same for pinned or fixed arches and, therefore, the pinned arch becomes more economical. However, for a deck-type arch bridge the buckling load for the pinned arch is less than that for fixed arch and, therefore, fixed arches become more economical if stability of arch ribs controlled their design. (6) By comparing the performance of deck-type and half-tied through-arch bridges under the

combined effect of vertical and lateral loads, one cannot favor one bridge type over the other based on strength only. Other factors such as topography of the bridge site and its soil conditions might control the choice of the arch bridge type. (7) The critical buckling load of half-tied through-arch bridges can be increased by as much as 28% (for fixed arches) and 78% (for pinned arches) when the connections between the arch rib and the deck are changed from pinned to fully rigid connections. Acknowledgements-The

author would like to acknowledge the contribution of Mr Carey Ngai, a former student at Polytechnic University of New York, for preparing the computer models used in this investigation. This contribution is greatly appreciated REFERENCES

1. American Association of State Highway and Transportation Officials, Standard Specifications for Highway Bridges, 15th edn. AASHTO, Washington, DC, 1992. 2. Harrison, H., In-plane stability of parabolic arches. Journal of the Structural Division, ASCE,

1982, 108,

195-205. 3. Austin, W. J. and Ross, T. J., Elastic buckling of arches under symmetrical loading. Journal of rhe -Structural Division. ASCE. 1976.102. 1085-1095. 4. Shukla,’ S. anh Oj&ot ‘M., Lateral buckling of parabolic arches with tdting loads. Journal of the Structural Division, ASCE, 1971, 97, 1763-1773. 5. Wen, R. K. and Medallah, K., Elastic stability of deck-type arch bridges. Journal of Structural Engineering, ASCE, 1987, 113, 757-768. 6. Komatsu, S. and Sakimoto, T., Ultimate load-carrying capacity of steel arches. Journal of the Structural Division, ASCE, 1977, 103, 2323-2336. 7. Sakimoto, T. and Komatsu, S., Ultimate strength of arches with bracing systems. Journal of the Structural Diuision, ASCE, 1982, 108, 1064-1076. 8. Kuranishi, S. and Yabuki, T., Lateral load effect on steel arch bridge design. Journal of Structural Engineering, ASCE, 1984, 110, 2263-2274. 9. Weaver, W. Jr. and Gere, J., Matrix Analysis of Framed Structures, 2nd edn. Van Nostrand, New York, 1980. 10. Fleming, J. F., Computer Analysis of Structural

Systems. McGraw-Hill,.New York, 1989. 11. Yabuki. T.. Vinnakota. S. and Kuranishi, S., Lateral load e&t dn load-carrying capacity of steel arch bridge structures. Journal of Structural Engineering, ASCE, 1983, 109, 24342449. 12. Timoshenko, S. and Gere, J., Theory of Elastic Stability, 2nd edn. McGraw-Hill, New York, 1961. 13. Ziegler, H., Principles of Structural Stability. Blaisdell, Waltham, MA, 1968. 14. Nazmy. A. S. and Abdel-Ghaffar, A. M., Three-dimensional nonlinear static analysis of cable-stayed bridges. Computers and Structures,

1990, 34, 257-271.