Design And Construction Of The Luis I Bridge Engineering Essay

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Opened in October 1886, the Luis I Bridge is a wrought iron trussed arch bridge which spans the Douro River between Porto and Vila Nova de Gaia in Portugal. Designed by the Belgian engineer Th ophile Seyrig, in collaboration with L opold Valentin; it was, at the time of its construction, the longest arch span in the World, at 172m [1], and remains the longest spanning iron arch to this day. It represents the swan song of wrought iron bridges as at the start of the 20th century stronger steel of more consistent quality would almost entirely replace the use of wrought iron in bridge construction.

The new bridge was commissioned in 1881 to replace an ailing suspension bridge at the same location; Seyrig designed a double-deck arch bridge with one deck at the top of the arch resting on piers and the second deck at the level of the abutments, hanging from tendons (Fig. 1). The two decks have seen a variety of loading over their life; originally both decks were designed to carry road traffic, the lower deck briefly carried trolleybuses but is now a single carriageway road, the upper deck was converted in 1905 to carry trams and widened in 1931 to add a second track.

2 Aesthetics

The aesthetic analysis of a bridge is largely subjective, Fritz Leonhardt attempted to rationalise the aesthetic design of bridges in 1982 with the publication of his book, Br cken, which sets out ten key points that should be considered during design.

2.1 Fulfilment of Function

This relates to how well the bridge divulges the way it works; in the Luis I bridge the arch is the main structural element through which forces are carried. This is apparent from looking at the bridge as the arch is the most substantial member. Truss structures in general are particularly revelatory about the way they carry loads, and the Luis I bridge is no exception.

The structural honesty of the bridge is called into question where the arch meets the masonry abutments; it appears as though the top member of the arch passes straight into the abutments but the abutments are not able to resist the high moment this would generate, so the top member of the arch must be lightly stressed at its extreme ends. On closer inspection it can be seen that the final diagonal members on both sides of the arch are of more substantial cross-section (Fig. 2) in order that they are able to carry all of the force in the top flange of the arch down to the pinned connection at the end of the bottom flange.

Figure 2: Forces transferred to bottom flange

Sometimes, one aspect of a bridge s aesthetics must be sacrificed in favour of another; in the Luis I bridge the truss which forms the upper deck is of continuous depth along its length, but the approach spans are notably longer than the sections which make up the main span. Functionally, the engineer could have designed the approach spans to be deeper than the main span but this interruption to the horizontal line of the deck would have been to the detriment of the aesthetics of the structure as a whole.

2.2 Proportion

As discussed above, the upper deck is deeper than it needs to be; proportionally this contrasts sharply with the lower deck which is significantly more slender despite spanning an equal distance. This difference is not adequately explained by the reasons already discussed. The upper deck may well have been designed to cope with a higher loading than the lower deck; studies have shown that, prior to the conversion of the upper deck to light rail, the upper deck carried nearly double the traffic of the lower deck [2]. Furthermore, at the time that Seyrig was designing the Luis I bridge he had just finished the construction of the Maria Pia Bridge (1877), designed in conjunction with Gustave Eiffel, which was to carry a train line over the same river. It is possible that the upper deck of the Luis I bridge was designed to carry train loads should it be converted to that purpose in the future, as indeed it was. The structural systems for the upper and lower decks differ greatly which may contribute to the disparity in their depth; the lower deck is a lattice through truss with traffic running within the truss itself whilst the upper deck is a brown deck truss where the deck is placed on top of the truss girder. The state of stress within the two decks also differs as the bottom deck is used to tie the arch and therefore is subject to a high initial tensile load; the advantage of wrought iron as a construction material was its affinity for tensile loads and it may be that this too contributes to the reduced deck depth.

The rise to span ratio of the arch is 1:4; this is mainly dictated by the dimensions of the gorge in which the bridge sits, but the result is an arch of typical masonry proportions [3] which offers the impression of stability.

2.3 Order within the Structure

A sense of order is given to the bridge by the repetition of the truss elements throughout the elevation. Although the lower deck uses a slightly different type of truss, the crossed elements are still present to maintain the order within the bridge.

When viewed closer up, the members are seen to be composed themselves of multiple elements, and from oblique angles the criss-crossing of these members can appear disordered (Fig. 3).

The piers and tension roads which support the two decks line up to reduce the number of vertical lines and split the bridge into equal portions. They are sufficiently close together so as not to make the arch appear superfluous, but no so close as to crowd the bridge with vertical lines.

2.4 Refinements of Design

Refinements refer to the subtle details within the bridge which can have a momentous effect on the overall appeal of the structure. In the Luis I bridge the piers taper towards the top (Fig. 4) which adds perspective by making the towers appear less stocky and prevents the optical trick of the piers appearing to be wider at the top than they are at the bottom.

In the approach spans where the vertical space beneath the upper deck is greatest, the deck spans a greater distance in order to maintain the aspect ratio of the spaces under the deck.

The aspect ratio of the crossed bracing in the arch is also maintained; where the divergent parabolic curves, which make up the top and bottom flanges of the arch, spread apart the distance between uprights is increased to keep the crosses filling a roughly square shape. As well as being aesthetically pleasing, this serves the structural purpose of keeping the members inclined at an angle where they can perform at maximum efficiency.

As previously mentioned, the individual members which make up the trusses are themselves trussed box sections (see Fig. 2), this gives the structure lightness, both in terms of its overall weight and also aesthetically by reducing the ratio of solid to voids and making the members seem more slender. However, this lightness comes at the expense of order.

Figure 3: Disorder Figure 4: Tapering piers

2.5 Integration into the Environment

Pivotal to the aesthetic success of a bridge is how well is fits into its environment; the arch form used for the Luis I bridge is particularly well suited to use in the deep gorge, and fills the space well. Despite the size of the structure, it looks comfortable in its environment.

The girder which forms the upper deck has no obvious end but instead gives the impression of merging into the hillside; this makes the bridge seem like an integral part of the gorge.

2.6 Colour of Components

Though originally unpainted (Fig. 5) the bridge now has as grey-blue finish which allows the bridge to blend well into the sky, this has the effect of making the messiness of the truss less obvious and contributes to the members looking more slender.

The widening of the upper deck in 1931has led to the creation of a dark line of shadow which serves to draw the eye away from the deep truss underneath.

Figure 5: Original design without paint

2.7 Aesthetic Conclusions

The Luis I bridge is a structure of great beauty and much consideration has evidently been given to aesthetics in its design. Despite this, as no point has structural efficiency been forfeited for purely aesthetic reasons. The structural performance of the bridge will form the next section of this paper.

3 Structural Behaviour

In 1881 the Portuguese government invited the tender for a new bridge over the Douro River; the principal challenge of the scheme was that there could be no intermediate piers placed in the river. This was due to high water depths of more than 12m, insecure ground conditions and a high tidal range in the river [4] which would have made construction exceptionally difficult. A number of schemes were proposed and the winning scheme, designed by Th ophile Seyrig, consisted of a trussed parabolic arch of wrought iron construction, 172m in span, supporting two truss girder decks (Fig. 6). Seyrig was familiar with the use of wrought iron having worked closely with Gustave Eiffel in the design of other wrought iron bridges such as the Maria Pia bridge (1877). In this new venture he sought to produce a design which would take full advantage of the mechanical properties provided by wrought iron.

Figure 6: Elevation

The arch is connected to the upper and lower decks, by piers and tendons respectively, in only four places; as a result of this the arch is subject to bending moments even when the decks are uniformly loaded. Wrought iron is a material which performs well in tension and it is evident the designer expected the material in the lower flange of the arch to be in tension at all times.

A refinement of the Maria Pia design was the use of the lower deck to tie the arch and so reduce horizontal loading of the poor quality ground at the abutments. A further departure from precedent was the use of divergent parabolic curves to create an arch more slender at the apex, where it is 7m in depth, than at the supports (17m). The change was made because of problems encountered during the construction of the Maria Pia bridge, which has a crescent arch; whilst the first sections of the arch were being built out from the abutments it had proved troublesome to provide adequate support for them using cables and staging had had to be employed [4]. In the Luis I bridge the arch is much deeper at the supports therefore allowing the first sections to be erected more securely and at less cost, it was a technique which would be used nearly 40 years later during the construction of the Sydney Harbour Bridge (1923).

The long deep gorge through which the Douro flows is characterised by high winds; the open truss system used for the Luis I bridge reduces the loading effect of the wind by limiting the area on which the wind can act. Eiffel often used tubular sections where possible in his bridges to increase the aerodynamic performance of his designs [5], but Seyrig chose not to do so in the design of the Luis I bridge, presumably to make the connections more straightforward.

The connections are riveted together, in practice this mean that the joints have some moment capacity but as the elements will still act predominantly axially, the connections in the truss can be modelled as pins without introducing too much error into the analysis. At the time of the bridge s construction, there was much debate over the relative merits of pinned or riveted connections in bridge construction [6]; whilst the riveted truss was of superior efficiency, pinned trusses could be assembled faster and cheaper using simple tools and techniques.

The connection to the abutments is by way of a rotational joint at the extreme ends of the lower flange of the arch (Fig. 7). This means that the arch can be considered a two-pin arch and will be analysed accordingly.

Figure 7: Foundation connection

In 2004 a study was undertaken to assess the current state of the bridge [1] and some samples were removed and tested. It is usual to apply measured material properties, where available, in bridge assessment rather than conservative characteristic values; tensile tests on removed sections of wrought iron from the bridge yielded a tensile strength of 397Mpa. Testing to find compressive strength was not performed so a value of 270MPa will be assumed.

4 Construction

Seyrig was a pioneer in the erection of iron bridges, to the point that he wrote a paper on the subject which was presented at the Institution of Civil Engineers (ICE) in 1881 [6]. In it, Seyrig details his conviction that the construction methods employed in the erection of iron bridges has the largest impact on their overall economy, safety and durability.

For the Luis I bridge, as with the Maria Pia bridge, Seyrig chose to employ a method of construction which least required the use of extraneous appliances, namely erection by overhang. In this technique the permanent structure of the bridge itself is used to support the construction of more remote sections. The prototype for this method of bridge construction was the Requejo Bridge designed by Jos Ribera (Fig. 8).

Figure 8: Requejo Bridge, Spain

In the Luis I bridge the approach spans were first constructed on both sides of the river until the upper deck girder protruded about 30m beyond the main piers which mark the start of the arch. The girders were pushed out on a set of four rollers which sat on top of each pier (Fig. 9).

Figure 9: Rolling apparatus

The arch was then built out as a series of premade sections which were tied back with steel-wire ropes to a point on the upper deck girder. The whole arch was constructed using only two ropes on each side of the arch, so it was necessary to be able to quickly move a cable once it has been superseded by a cable further along the arch; for this purpose the cables were connected simply to the top flange of the arch using a rounded shoe (Fig. 10) under which the continuous rope was fed.

Whilst most of the sections were erected with all of their components in place, the last few panels were put up with the top flange and some of the diagonal bracing removed in order that they should be as light as possible. Once the two halves of the arch had met and the central connecting piece inserted, the missing components were then added to the lightened sections.

Figure 10: Cable to arch connection

The work was performed to such accuracy that in plan the two halves of the arch met exactly, but in elevation both sides were around 350mm too high. This was done deliberately as it was decided that there was potential for the two halves to be too low in which case it would have been very difficult to raise them. Provision was made for lowering the arches to their correct position by the removal of a certain number of cast iron wedges which had been placed beneath the cable connections.

Once the two halves of the arch had been connected it was important to slacken off the steel cables immediately as a drop in temperature could have caused the cables to shorten and induce stresses into the arch.

With the arch in place the dwarf piers could then be erected and the upper deck girder placed on top. Exactly the same process was used for the construction of the Maria Pia bridge and is shown schematically in Fig. 11. The lower deck would have been added last, simply by spanning between the wrought iron tendons, temporary intermediate cables may have been added to reduce the hogging moments caused by cantilevering out.

Figure 11: Erection by overhanging

5 Loading

The Luis I bridge was built before design standardisation had fully emerged; consequently it was probably designed to whatever loading the engineer deemed to be reasonable. It was also built at a time when the horse drawn carriage was the predominant means of transport; Karl Benz built the first true automobile in 1885. For the purposes of this report the bridge will be analysed under its current loading conditions in accordance with BS-5400 [7].

Partial load factors, as detailed in Table 1, will be applied to nominal loads then combined to give the worst possible loading conditions.

Table 1: Partial load factors [8]

Load Type Partial Load Factor (?fl)

Stressing Relieving

Dead 1.05 1.0

Super-imposed Dead 1.75 0

Live Traffic 1.5 0

Wind 1.1 0

5.1 Dead Loads

The structural elements of the bridge are of wrought iron construction with a density of ? = 7700kg/m2. The total weight of the bridge is equal to 29841kN [9] which is approximately distributed as shown in Table 2.

Table 2: Unfactored dead loads

Arch 76kN/m

Upper Deck 31kN/m

Lower Deck 23kN/m

5.2 Super-Imposed Dead Loads (SID)

Super-imposed dead loads are the non-structural static loads on the bridge such as road finishes, lighting and street furniture. They have a high load factor (1.75) to reflect the strong likelihood of them changing over the lifetime of the bridge; they may also be removed completely should the bridge be subject to major works, though were this the case, traffic loads would almost certainly be reduced. Suggested loads given in Table 3 correspond to a 200mm layer of asphalt road surface.

Table 3: Unfactored SID

Upper Deck 38kN/m

Lower Deck 28kN/m

The values are different because the two decks are of different width; the upper deck is 8m wide and the lower deck is 6m.

5.3 Live Traffic Loads

The lower deck carries road traffic; at 6m wide it can be considered to have two notional lanes. Eq. (1) gives the live traffic loading per metre per lane (HA):

w=151(1/L)^0.475 (1)

L is the loaded length which in this case is 172m so the resultant unfactored load over two lanes is 26.2kN/m. A knife edge load (KEL) of 120kN should also be added, placed to generate maximum additional stress.

In this instance HB loading has not been considered as the access routes to the lower deck would be impassable by very large vehicles and the newer, high-level bridge close by, which is crossed by a dual carriageway, would be the more suitable route.

The upper deck carries light rail traffic, each train has an unfactored weight of 2000kN [2] and a length of 70m. The trains move very slowly on the bridge such that dynamic effects can be discounted.

5.6 Worst Case Loads

For the arch, worst case bending moments occur when the arch is non-uniformly loaded; this corresponds to fully factored dead, SID, and live loads on one half and unfactored dead loads only on the other side (Fig. 12). For the upper deck, two trains passing at quarter span have been considered.

Worst case shear loads would be caused by fully factored dead, SID and live loads at all points on the bridge.

Figure 12: Worst case loading arrangement

6 Analysis

In this section, the worst case loadings calculated previously will be applied to the structure to ascertain whether the resultant stresses are within the tolerances of the materials.

6.1 Arch

The main structural component of the bridge is the trussed arch. For the purposes of this report it will be modelled as a two pin arch, with the loading arrangement in Fig. 12 simplified to four point loads (Fig. 13).

Figure 13: Simplified arch loads

By taking moments about the point A, the vertical reactions are found to be: VA = 21691.2kN and VB = 14644.8kN.

6.1.1 Flexibility Analysis

To find the horizontal thrust produced by the arch a flexibility analysis was performed by releasing the horizontal reaction at B and applying the unit load method to find the resultant displacement at B (?B,H) and the flexibility coefficient (a11). Eq. (2) can then be used to find the value of horizontal thrust:

?_(B,H)+a_11 H=0 (2)

?B,H and a11 are found by integrating the moment in the arch with respect to the arc length which is quite complex, but the problem can be simplified by assuming that the I value of the arch changes around its profile such that I = I0sec(?), where I0 is the second moment of area at the apex of the arch [10]. Ultimately it can be shown that the value of horizontal thrust is given by Eq. (3), where a is the horizontal distance from A to the point at which the force is acting, h is the height of the arch, L is the span and W is the magnitude of the force. Multiple forces can be superposed together to get a final value of thrust of 21946.9kN.

H_1=(5W_1 a)/(8hL^3 ) (L^3+a^3-2La^2 ) (3)

6.1.2 Line of Thrust

The calculated data for loads and reactions were used to plot a thrust line for the arch under worst case loading conditions (Fig. 14).

Figure 14: Thrust line

From this plot, the moment at any point in the arch can be calculated as the eccentricity of the thrust line multiplied by the horizontal force. The moments in the arch are shown in Fig. 16; maximum sagging moment is 148.8MNm and occurs at 36m from A, maximum hogging moment is 125.9MNm and occurs at 131m from A.

For the purposes of this report, it will be assumed that bending forces in the arch are resisted by the top and bottom flanges, whilst the diagonal bracing resists shear forces; any axial forces are shared amongst all the members. The force in the flange required to resist the maximum moment detailed in Fig. 15 is equal to the moment divided by the depth of the truss which yields a force of 14.2MN.

Figure 15: Moment in arch

This load results in stresses of 133.2Mpa in each of the four arch girders; tension in the lower girders and compression in the upper girders, which is well under the material capacity.

Axial compression due to the arch shape must also be considered; by resolution of the reactant forces in the supports, it can be shown that an axial compression of 30MN is carried in the arch. Split amongst the total area of wrought iron available in the section, this results in an additional compressive stress of 74.7Mpa.

In the tension flange this acts as a relieving stress which reduces the overall stress to 58.5Mpa (tension). In the compression flange the stresses sum up to give a total stress of 207.9Mpa, which is approaching but still below the material compressive strength of 270Mpa.

Metallic members are often susceptible to buckling under high compressive loads. Eq. (4) was used to find the load required for the arch members to buckle.

F_e=(p^2 EI)/?L_eff?^2 (4)

The effective length was taken to be the span between diagonal bracing elements as it was assumed that the cross bracing would provide sufficient confinement to prevent buckling over a longer length. The load at which buckling would occur was found to be 136MN which corresponds to a stress well above the compressive strength of the material, so failure would never occur through buckling.

?f3 values were not considered in the loading calculations for the arch as the analysis methods used will result in quite high error, the extra capacity within the material, as shown above, accounts for the lack of accuracy in the analysis techniques.

6.1.3 Shear in Arch

As well as bending moments, the loads on the arch also induce shear forces which are carried in the diagonal bracing members. Worst case shear theoretically occurs under maximum loading possible which would be 13488kN applied at the four point load locations on the arch. Moments under this loading scenario were calculated using the thrust line method and then shear forces were found by differentiation of the moments. The result, shown in Fig. 16, predicts a maximum shear force of 7242.8kN located at 35m from point A.

The shear force is resisted by the diagonal bracing elements which act together, one in tension and one in compression. The force in each bracing member must be 5121.4kN which corresponds to tensile or compressive stresses of 194.7MPa.

Figure 16: Maximum shear in arch

6.2 Temperature Effects

Particularly in redundant structures like two pin arches, small strains caused by temperature changes can induce significant stresses into the structure as the structures tend to be less flexible. As the Luis I bridge is a trussed structure there should not be a high temperature difference between its elements, but overall temperature changes should be considered.

In the arch, a rise in temperature would result in the arch trying to expand; confined by the piers, this would cause moment in the arch which would be carried as tension in the top flange and compression in the bottom flange. This would act as a relieving action from the dead and live loading so should not cause a problem. A drop in temperature, on the other hand, would result in additional compressive stresses in the top flange which is already highly compressed.

The upper deck is exposed to the most direct sunlight, and the solid road surface puts the underside into shade so there may be a high temperature gradient which would result in stresses. The variation in temperature throughout the section in the morning period is shown in Fig. 17 where 0 C corresponds to ambient temperature.

Figure 17: Temperature difference in upper deck

The thermal expansion coefficient (a) for wrought iron is 12 strain/ C, using e=a?T the strain due to the temperature gradient is shown in Fig. 18. Multiplication of these values by the Young s modulus of 185GPa gives the stresses also detailed in Fig. 19.

Figure 18: Strains (left) and stresses (right)

The rollers on top of the main piers, as discussed in section 4, now act as roller bearings which allow the deck girder to lengthen and so relive some of these stresses. The stresses reduce by the average stress value which in this case is 6.6MPa; this now produces the stress profile shown in Fig. 19.

Figure 19: Additional temperature stresses

The stresses in Fig. 19 correspond to a constant moment over the length of the upper deck. As the deck is continuous over the piers there is no requirement to consider an additional moment to ensure the moment at the supports remains equal to zero.

6.3 Wind Effects

Porto lies on the Atlantic coast of Portugal and so it can be assumed that it is subject to quite high winds, the bridge itself also sits in a gorge which will have a funnelling effect on the wind. The arch itself is trussed so as to catch little wind, but the decks, when high sided vehicles pass over them, will have a large projected area and so may be subject to high wind loading. This is particularly true of the lower deck because it is a through truss so the open structure offers no advantage. Suspended as it is by tension rods, the lower deck may be highly susceptible to wind induced effects.

Assuming a mean hourly wind speed of 34m/s, akin to the speeds found on the Atlantic coast of the UK, the maximum wind gust (vC) on the bridge can be found from Eq. (5) to be 52m/s, where K1 and S2 are factors according to BS-5400 and S1 is a funnelling factor taken to be 1.1.

v_C=vK_1 S_1 S_2 (5)

Horizontal wind load can now be found using Eq. (6), A1 is taken as the projected area assuming high-sided trucks are crossing the bridge. When the deck is fully loaded the truss is obscured so the drag coefficient can simply be calculated using the b/d ratio. The result is a lateral force of 1.6MN which must be resisted by the deck.

P_t=0.613?v_C?^2 A_1 C_D (6)

Without knowing the under-structure of the lower deck it is difficult to assess how this load is carried, but it is assumed that a cross braced truss runs underneath the deck and prevents the deck from flexing laterally.

The wind can also result in dynamic effects such as galloping and flutter; these effects tend to most affect suspension bridges because of their inherent flexibility. The lower deck of the Luis I bridge, which is suspended by tendons, would be the most likely to suffer from these effects but some aspects of its design provide stiffness against them. The tendons are able to carry compression as well as tension, and are cross braced to provide torsional stiffness; coupled with the truss acting longitudinally this gives the bridge stiffness in all of the planes in which the effects of aerodynamic instability might act. There are also vast amounts of riveted connections within the bridge to provide damping against vibrations.

7 Fatigue

The Luis I bridge is over 100 years old and has therefore been subject to a high amount of loading cycles, it seems prudent therefore to give some consideration to its fatigue performance. The bridge is located close to the sea and so is considered to be in a marine environment; wrought iron is regarded as having a lower resistance to corrosion than other common construction materials of the time like cast iron [11], corrosion is worst around potential moisture traps like connections where poor maintenance can lead to interfacial corrosion (Fig. 20). The riveted connections are also prone to fatigue failure because cracks can form during fabrication and the punching action can result in local work hardening around the rivets.

Figure 20: Interfacial corrosion

In a study performed by Fernandes et al, samples of material, including a riveted connection, were removed from the bridge and analysed to find their mechanical properties [2], also performed were crack growth studies, notch toughness testing and an analysis of metallurgical content. This data was used to find the number of loading cycles the various components of the bridge would be able to withstand.

By assuming that only trucks cause fatigue loading and that one truck represents one cycle of loading it was calculated that the bridge had exhausted just 10% of its fatigue life and that remaining fatigue life was greater than 100 years. The study also considered the use of the upper deck for light rail and concluded that one train was the equivalent of four loading cycles and that residual life was less than 10 years. Consequently the bridge was retrofitted and reinforced before the new metro line was allowed to pass over it.