Suspension Bridge Engineering and Design

Finca de las Ninfas, Hatillo Costa Rica

For:  Dwight Wainman, Maureen Naughton, and Alex Hruby

By:  Paul Collar
Osa Power and Water

Date:  March 22, 2010

INTRODUCTION AND OVERVIEW

Access to the residential complex under construction at Finca de las Ninfas is presently made possible through an access road (Figure 1) with a final approach too steep for routine use and intended to support construction with materials access.   The road that provides farm access outside of the property boundaries can be reached from Hatillo and Baru.  Both of these roads are four wheel drive roads owing to steepness in a number of places.  Similarly portions of the access road immediately outside the Ninfas property boundary as well as within are four-wheel drive accessible only, in all cases due to steepness.  Due to this steepness any driveway to replace the final rise to the home site must rely upon switchbacks winding around the terrain to achieve a reasonable approach grade or a bridge to span a key gorge to enable reasonable access to the home without untenably steep grades.  Figure 1 shows the existing driveway from the approximate location of the entrance gate to the building site overlain on a color-keyed topographic model of the main part of the finca.

Figure 1.  Approximate location of existing access road.  The final portion of the road, labeled ‘VERY STEEP’ is the portion of the access road that is to be decommissioned following construction because it is too steep and architectonically incongruous with the residential complex under construction.

Figure 2 illustrates two hypothetical alternate routes for a driveway without a bridge.  While these alternatives have not been rigorously examined in terms of engineering and costs, the steepness of the terrain makes it likely that both of these routes, even if possible within existing property boundaries, will require extensive use of retaining walls and anchoring structures to ensure the permanence and integrity of a driveway.  While these and other possible routes have not been rigorously analyzed as part of this engineering report and are presented here as hypothetical alternatives, it is expected that the costs of the retaining walls and the extra length of road bed required to follow contours and wrap their way up to the home site are likely to equal or exceed the cost of construction of a suspension bridge with its ancillary access and exit grades and entrance retaining wall.   But perhaps more significantly than costs, the location of these roads so near the building sites presents the geotechnical challenge of ensuring that slope stability is not compromised during their construction and through their existence and use, which may prove to be non-trivial challenges given the relatively limited area available for the final approach to the house and the steep slopes near the top.  Lastly, it is considered that the aesthetics of a suspension bridge on the mountain side of the home will be superior to that of winding driveways on either the mountain or ocean side of the home, both from the perspective of the home site looking out as well as from the perspective of the driver’s seat upon approach.

Figure 2.  Hypothetical alternative access routes that do not rely on a bridge, one of them on the western face of the mountain and the other on the eastern face of the mountain.

This report summarizes the engineering and design of a suspension bridge and the ancillary retaining wall that will be needed for the entrance ramp.  The rationale for the proposed location and analytical methods and preliminary design were previously reported in a pre-proposal dated February 19, 2010, and this final engineering report depends on findings and background information in that pre-proposal, which is attached as Appendix 1 of this report for reference as needed.

FINAL LOCATION

Figure 3 reveals the proposed location of the bridge and entrance and exit ramps on the same three-dimensional topography models shown above.   The bridge itself is proposed to span a 48-m wide gorge between two parallel ridge lines that descend from the home site.  On the home side of the bridge (exit ramp) the final access road is already defined and present cut to within 2.75 meters of its final required elevation to service the bridge at its proposed location.  The exit ramp and final home approach is a road segment that does not present a geotechnical threat to the home site and is not too steep to negotiate safely in reasonable vehicles.  The entrance ramp, on the other hand, is separated from the existing access road by a small gully 32 meters across that must itself be bridged or filled to enable an approach at a reasonable grade.   I propose a semi-circular retaining wall and compacted fill to form the entrance road bed.  Above the road, I suggest a single cut bank step and an expansion of the existing parking terrace around the shoulder before the final rise to the building site. 

Figure 3-a.  Proposed bridge route in relationship to existing access road(s) and the building site.  The proposed bridge is shown in red.  The blue segment is intended to represent the required retaining wall, but the depiction is not to scale and relative to the bridge is somewhat longer (34 m) than shown.

FINAL DESIGN AND ENGINEERING

Statics and Above Ground Design Features

The method of engineering analysis used here was previously reported and described in adequate detail (Appendix I) and is only partially repeated in the body of this document.  The basic bridge design that was proposed in the report dated February 19 is very similar to the final recommendation presented in this document.  There are a few gross differences, however, that are listed below as a prelude to the engineering discussion to follow.

1)      Rather than a 2.44 meter width originally proposed, the final design is one that has a 2.5 meter usable width and a total width from edge to edge of 2.75 meters.

2)      The final bridge length is actually only 48 meters rather than the 55 meters originally envisaged.

3)      The deck undercarriage has been beefed to increase support for final wood decking in a longitudinal rather than a latitudinal orientation.

4)      The retaining wall necessary to provide a usable access ramp to the bridge has an estimated total length of 34 meters rather than the original estimate of 30 meters.  The height of this retaining wall, originally projected to extend above the lowest point of the ground surface by as much as 5 meters is now proposed to not exceed 3.5 meters at its highest point.  A dip in the road of as much as 2 meters must be achieved across a horizontal distance of about 20 meters under the proposed wall height.

The new bridge geometry results in a increase in the normalized weight of the bridge deck but the shortening of the overall length is a self-cancelling design correction.  All changes have effects on the final statics calculations that are not easily appreciated because they are mostly self-cancelling configuration changes.  The model used for bridge analysis is given by formula (1) below, with the variable definitions illustrated in Figure 4.

                      T = w  (x² + (a² / (2 * S)) ²) ^1/2                                                                    (1)

Where T = Tension force expressed in pounds at any point along the span of the bridge, x = distance from bridge midpoint to any length (in feet) along the span where the Tensile force is to be calculated,  w = distributed weight load expressed in lb/foot, a = ½ of the total span to be bridged (feet) and S is the maximum sag, also expressed in feet.

Figure 4.  Geometry of a section of an idealized suspension bridge showing the variables that are required for the engineering analytical model presented as Equation (1).

In this case, all the design variables, w, x, and a, have changed from the pre-proposal bridge geometry.  Since the overall bridge length is now 48 meters (or 157.44 feet), the new value of a is equal to one half this span, or 78.72 feet.  Since the maximum tensile stress is developed at the very edges of the bridge deck, the value of x for this analysis has been equated to the value of a.  To calculate the weight distribution, Figure 5 shows a plan and cross section of the deck design, an undercarriage of 6-meter long panels of structural 3” I-beams (5 kg/lineal meter) welded together, overlain by galvanized steel 2” x 4” beams (1.7 kg/lineal meter) lain crossways to enable bolting of the final decking, 2” thick hard wood planks (density 1200 kg/cubic meter) lain lengthwise.  Normalized per lineal foot and expressed in pounds, the static bridge weight is therefore 146.5 pounds per foot.  Application of a 2000 pound load across a 48-meter length adds an applied load of 12.7 pounds per foot, making the total maximum dynamic load one of 159.2 lbs/ft.  Since the model is developed for a single supporting cable, and there are two such main cables, the final value of w is one half of this, or 79.6 lbs/ft.  I have rounded this up for structural modeling purposes to 85 lbs/ft to include suspender cables and miscellaneous fittings. 

Figure 5.  Plan and Cross section of the undercarriage of the deck.  Eight six-meter panels are required in total. 

The mathematical model described by Equation (1) is solved in Table 1 as a function of variable values of sag (s), itself a surrogate variable for tower height.  For the Finca de las Ninfas application, I am assuming that at the bridge midpoint the lowest point of the parabola described by the main cables (the point of maximum sag) corresponds to a height 1.5 meters above the bridge deck.  Hence, tower heights predicated on this assumption are 1.5 meters greater than the sag (s) for the point at which tensile strengths are optimally satisfied by the strength of the cable used.  By convention, a maximum of one fifth of the design breaking strength of the cable to be used is allowed for determination of optimal sag distance (and by correlation, tower height).  In the previous report we bracketed the strengths of cable diameters of 1”, 1 1/8” 1 ¼”, and 1 ½” and concluded that 1 ¼” is the size optimal for the proposed duty as suspension bridge main cables.  As previously reported this stainless steel wire rope has a rupture limit of 63 tons.  Hence its safe operating capacity (1/5 this amount) is equal to 25,200 pounds.

Table 1.  Bridge analysis modeling of equation (1) using values of w, x, and a defined in the preceding section and variable amounts of sag distance (s).  Tower height is by definition 1.5 meters greater than s.  Options shaded in yellow are acceptable heights in conjunction with 1 ¼” diameter steel main cable.  The option shaded in orange is the final recommendation for the proposed bridge.

Table 1 reveals that for sag heights of less than 11.5 feet, which corresponds to a tower height of 5 meters, the maximum tensile stress exceeds the safe operating limits of the cable selected.  In order to provide an additional measure of safety, I have settled on a recommended tower height of 6.5 meters.  If we preserve the assumption of 1.5 meters from bridge deck to the lowest part of the main cables, this is actually 13% of the total breaking strength and an 8 times overall safety margin (as opposed to standard engineering practices criterion of 20% of breaking strength, or a 5-times safety factor.  If for aesthetic reasons the owners prefer a distance of only 1 meter between the lowest point of the cable and the bridge deck, the same deck height of 6.5 meters would imply a safety margin of 8.5 times.

The final recommended bridge design presumes towers that rise 6.5 meters above the earth’s surface and main cables 1 ¼” in diameter.  Rational sizing of the suspender cables is predicated on their distribution and the weight of the deck.  If we take the variable w, 80 lbs per foot, over the 48 meter length of the bridge, this means that one half the deck weighs 5,725 kilograms.  With suspender cables deployed every one meter along this distance, each cable on each side of the bridge must support 119.3 kilograms.  If we continue to apply a criterion of 20% of breaking strength, then a 1/8” steel cable (breaking strength 1,700 pounds) barely satisfies this criterion.  However, for aesthetics and ease of working and an additional measure of strength, I recommend suspender cables of 3/8” diameter.

The final static consideration in the engineering analysis is the anchoring of the bridge.  We just established that the deck weight of the bridge is 11,450 kilograms.  Since tensile forces in the cables are transferred vertically to the ground through the steel beams used as primary support towers, the weight of these does not require anchoring.  However, unlike in the analysis of tower height versus cable width, the weight of the main cables must now be factored into the anchoring weight design.  Since this cable weighs 5 kilograms (4.3 kg, rounded up) per lineal meter, I have allocated an additional 1000 kilograms for 200 meters of cable, even though final cable length is not expected to exceed 180 meters.  Lastly I have budgeted 100 kilograms for the suspender cables.  So, the total bridge weight is estimated to be 12,550 kilograms.  Anchoring is to be achieved through buried concrete anchors located at either end of the bridge.  Hence the total bridge weight shall be bound by two anchors, and each of these must capably secure a dynamic bridge weight of 6,275 kilograms.  I have applied a mass safety factor of 2 times to result in a design anchor weight of 12,550 kilograms.  In practice, since the anchors are buried in native soil on the exit side and compacted fill on the entrance side, the actual anchoring capacity is considerably greater than the mass of the anchors themselves and may be considered to comprise a dramatic supplemental safety factor that has not been calculated formally.  Discounting the density of the reinforcement steel to be used and depending on the density of concrete alone (2400 kg / cubic meter), a total volume for each anchor of 5 cubic meters is required.  I propose concrete prisms 3 meters in length.  Width and height of 1.3 meters is adequate to provide the volume needed, but I propose final anchor geometry of 1.5 meters wide, 1.5 meters high, and 3 meters in length, which provides by mass alone a 2.6 times safety factor.

An overall graphic representation of the bridge is shown in Figure 6.

Subsurface considerations

Tower Footings

The towers must be set in a foundation that will adequately displace tensile forces vertically into the ground in a stratum sufficiently dense that no subsidence is possible.  In the absence of soil testing with borings, it is not possible to definitively state how deep the footings must be buried to achieve these

Figure 6.  Finca de las Ninfas proposed suspension bridge:  section and plan views with principal dimensions shown.

construction criteria.  For this reason soil testing where the towers are to be located is needed in advance of final footings engineering.  I estimate that a footings depth of 2.5 meters will be sufficient to ground these footings on solid subsurface soil strata, but this must be confirmed either through advance soil testing or by digging footings until revealed strata is sufficiently competent for footings.  The tentative tower footings design presented in advance of additional soils information presumes a prism of reinforced concrete 1 meter wide, 3.75 meters long, and 2.5 meters deep.  The tower footings design drawing, based on these geometric assumptions, is shown in Figure 7.

Anchors

The previously described bridge anchors shall be built with internal ½” steel plate as shown in Figure 8 together with reinforcing steel rebar, which is not shown.  For the eyes to which the main cables shall be affixed with the use of thimbles and turnbuckles, three thicknesses of plate shall be welded together to produce a triangular diagonal anchor assembly of 1.5” thick steel.  The details are shown in Figure 8.

Retention wall

While the retention wall is technically a separate structure following its own separate engineering requirements, it is integral in function to the deployment of the bridge and is required to ensure that the approach grade is reasonably gentle across the small gully that presently separates the access road from the proposed bridge entrance tower.  Deployment of a retention wall and backfill has the further benefit of making it possible to expand an area at the basis of the final slope that is likely to be a useful infrastructural addition as a parking area or to provide physical plant staging area or to provide some similar useful function.   Figure 9 presents three idealized views of the proposed retention wall.

Figure 7.  Tower footings design

Figure 8.  Anchor design.  The side plates are 1.5” in thickness, the bottom plate ½” in thickness.

Figure 9.  Retaining wall, global, section, and plan views.

Like the footings, it is not possible to undertake a rational retention wall design without knowing soil characteristics, and I intend for final design details to be informed by soil borehole tests or soils tests taken from footings excavations a decision to proceed.  Absent that information, I propose a rigorous retention design that includes two structural features to overcome the two retention wall failure modes of slippage and overturning.  I propose an anchored cantilever retaining wall for the 24-meter stretch where the wall is relatively high, with the anchor (a concrete deadman) and cantilever (poured base) providing redundant structural support to resist slippage and overturning.  The end of the cantilever undergirding of this portion of the retaining wall is proposed to coincide with the location of the bridge anchor 13 meters from the entrance towers.  The bridge anchor shall be tied into the steel armature of the retaining wall deadman and cantilever floor that is then extended (at lesser widths of cantilever poured base) for a distance of 12 meters in the very central part of the swale to ensure that at its maximum retained soil height of 3.5 meters that overturning forces are overcome by the cantilevered geometry and slippage forces overcome by the inside anchoring footer to which the wall will be supported by poured concrete tensed-steel reinforced columns.

Final wall dimensions will be driven by the swale geometry and the soil and subsurface characteristics, and it is not possible to precisely define the wall geometry at this point without geotechnical testing or excavation of soils in preparing for the work.  This is significant for purposes of cost estimation because final geometry affects both the materials and the time required to complete the installation.  The drawings in Figure 9 present my design expectations.   Installed dimensions and specifications are likely to vary somewhat.  Drainage shall be provided along the entire length of the wall to ensure that ground water does not pool behind the retaining wall and add unwanted hydrostatic pressures to the force exerted by the weight of the compacted fill alone.

Entrance and Exit Access ramps

The proposed bridge has an exit elevation 2.75 meters beneath the existing grade of the road bed on the exit ramp of the bridge.  The exit ramp must be cut down, the gradient distributed along the presently gently rising slope on the exit end.  On the entrance end, an artificial level of as much as 3.25 meters of compacted fill is needed to raise the grade so that it links well with the existing road and provides an upward slope to the final bridge approach and expected elevation dip of 2 meters, which could be eliminated altogether through a dramatic increase in retaining wall size. 

Final Miscellaneous Details

There are several remaining mechanical engineering details to complete the design and enable the completion of a work plan and materials list.  These include the gliders over which the main cables are in contact at the towers, the tee-connectors that secure suspender cables to the main cables, tower cross braces, the means of attachment of the main cables to the anchor eyeholes, and finish details.  These final engineering elements are described below.

Gliders

The tops of the 10” tower H-beams are to be machined to form gliders from the steel.  As shown in Figure 10, the beam web is to be impressed into service as the glide axis.  A saddle groove will be cut so that the profile of the beam tops will resemble arching valleys into which the main cables may be positioned.  The web will be cut in an arc with a radius of curvature of about 4”.  Along each side of the web, steel will be welded along this arc to provide for a steel channel 1 ½” in diameter into which the cable will be seated.  This saddle will be sanded and polished to a smooth rounded finish to avoid abrasion of the cable during tensing of the main cables.

Figure 10.  Glider design.  Not accurately represented is the 5” arc of curvature of the web and machined channel side additions.  This hump is to provide the arced fulcrum for the cable to lay saddled within the machined 1.5” D saddle groove.  The saddle itself will be polished steel or may possibly have a durable thick polymer to separate the cable from the saddle, contingent upon materials suitable for this stressful duty.

Tee connectors

I am in consultation with a colleague who built a similar bridge using my engineering for the lodge Bosque del Cabo.  He used a special tool and materials to crimp a connector to secure to the main cable to provide a tee-connector drop for suspender cables.  I don’t have the final details on these parts and the tool for it and am awaiting alternate recommendations from the supplier of steel cable to determine which method will best suit the project needs.  This engineering element remains undefined at this stage of the design process but is expected to be within reasonable expectations in terms of both materials and labor to install.  A preliminary drawing of the tee-connectors is presented in Figure 11 but is subject to changes depending on the full range of materials and connector styles available to adequately satisfy this function.

Figure 11.  Tee connector design 

Main cable anchors

Primary support for the main cables has not been defined.  While muscular turnbuckles remain the least costly alternative, a formal bridge socket at several times the cost is the signature part to use for this application, though it is costly. 

Tower braces

Tower braces shall be made in the form of an “X” from either carved wood or in solid structural steel braces.  In either case, the braces shall be affixed to the web of towers and will be situated in the center to ensure vertical symmetry.    The braces shall be of equal lengths and shall form perfect quadrants of 90 degrees at their intersection in the middle and shall be bolted together.  Final sizing of these bracing elements will depend on whether they are made of wood or steel.  One factor that must be weighed in the decision of whether to use carved wood or steel for these cross braces is that they will extend from 4.5 meters above the ground surface to the total height of 6.5 meters.  At that distance it is unlikely that carvings will be able to be adequately appreciated visually.  It may be that an aesthetic adornment—whether in wood or steel—would be most profitably deployed geometrically in consonance with aesthetic and adornment motifs, rather than in bass-relief, though this is a final finish detail that is adjustable according to the owner’s preferences.

Figure 12.  Main cable anchor.  The poorly drawn turnbuckle is a candidate for this duty, though a separate specially fabricated bridge socket is recommended by the wire rope supplier and will be considered as a possible more costly alternative for securing the main cables to the anchors.

Finish details

A number of finish details apply to this structure, and these are detailed below:

a.       Tower adornment.  For aesthetic integration with home design, the support towers may be boxed in with finished hard wood pieces or laminates.  Alternately, the towers may be painted in a color of the owner’s choosing.

b.      Handrail and protection.  Some form of protection is needed to ensure that pedestrians do not slip off the side of the bridge in a moment of carelessness to fall to the ground below.  This can be achieved with either a wooden handrail on either side of the bridge with vertical wood slats or alternately by a hand rail and the deployment of thin webbing of nylon or an appropriately durable polymer.  The former may integrate aesthetically if wood tower accents are selected, but the latter is considered to be less costly and to require less maintenance and to leave a more open feeling while transiting the bridge, rather than the closeted sensation that I expect will result from vertical slats

SUMMARY IMAGING

Figure 13 shows the proposed access route superimposed on the modeled three-dimensional topography of the part of the Finca de las Ninfas relevant to this study.  In Figure 14, the final design render is presented to approximate an expected final look of the proposed project. 

Several aspects of the final render are not perfect.   For instance, the image does not accurately reflect glider assembly and details.  Also, the location should be shown closer into the mountain, a perspective that is awkward to visualize because on the entrance tower, this will require excavation of the ridgeline.  Lastly, the perspective of this render, intended to be from the curve of the road, makes the bridge look larger than it will actually appear as the final render zoomed into the photograph so that the bridge would fill the frame.  As a result this perspective is one that would be seen from a distance of 20 meters or so from the entrance bridge tower, rather than the approximately 65 meters distance from the actual bend in the road.  Correspondingly, the render of Figure 14 presents a representation that is exaggerated visually.

Figure 13.  Approximate superimposition upon modeled three-dimensional topography of the existing access road (in orange) existing final steep driveway slope (dotted black), the building site, yellow box, and the approximate location of the proposed suspension bridge.

FIGURE 14.  Rendering of the final bridge design.  View is from the final bend in the road on the access road approach.  However, the view is zoomed from the point where the photo was taken, so that the perspective is in reality one from an estimated distance of about 20 meters from the bridge and is somewhat unrealistic since the Cartesian position of the vantage point is actually in the air.

CONCLUSIONS

This report summarizes final engineering and design recommendations for a suspension bridge / retaining wall structural conjuncture to enable vehicular access to the Finca de las Ninfas home under construction.  Forthcoming within coming days hours in a second part of this report I will present a materials list, work plan, time line, and itemized budget and proposal for construction of the bridge and retaining wall by Osa Water Works S.A., an engineering and construction firm, in partnership with Osa Builders, a construction firm, to build the bridge.

APPENDIX I.

 Suspension Bridge Engineering, Analysis, and Installation Pre-proposal

Finca de las Ninfas
Hatillo, Costa Rica

For:  Dwight Wainman, Maureen Naughton, and Alex Hruby

By:  Paul Collar

February 19, 2010

Overview

Driveway access to the home under construction at Finca de las Ninfas is expected to be complicated by the steep slopes of the hill upon which the home is being built.  Vehicle traffic on the road shall be restricted to electric golf carts, so overall loading and road footprint is not sufficiently large so as to support the weight and size of large motorized vehicles, at least not as routine traffic.  Nevertheless the steep gradients make it challenging to deploy an access road that is safe and does not present excessive grades without the deployment of a bridge to span one or more of the gulches along the edges of the hill slope.  This pre-proposal summarizes preliminary considerations, the optimal location for such a bridge, preliminary engineering and design, as well a formal proposal for final engineering and design and for the preparation of a work plan and budget for the construction of such a bridge.

Background and Introduction

The existing Finca de las Ninfas access road is being used to transport building materials during construction.  However the final stretch of the existing access road is extremely steep and will be neither safe nor viable for long term property access and will be decommissioned once the home is finished and an alternate access road is completed.  Preliminary road analysis by project manager Alexander Hruby has largely discounted the western side of the mountain due to excessively steep slopes and the challenges of integrating a driveway approach with the aesthetics of the facility layout.  Preliminary road planning has successfully defined an optimal final approach to the home on the eastern side of the mountain.  Likewise the major part of the eventually permanent driveway has already been developed and completed and the useable portion of this extends to within 200 meters or so to the future home site.

Based on analysis of the topography and terrain, gradients of the resultant driveway, safety, and aesthetics, the most reasonable alternative to providing a safe, easily navigable access road is through the use of a bridge to span a key gorge separating the approach of the access road and the final arrival to the home site.  The span required for the proposed bridge is one of at least 50 meters in length.  For this span the only bridge design style that would not be exorbitantly costly is a suspension bridge.

This document summarizes the preliminary engineering analysis, design, and charts a proposed path to continue through evaluation, final design, and construction of the proposed bridge.

Suspension bridge overview and analysis assumptions

The figure below shows a cross section and plan view of an idealized suspension bridge. 

Figure 1.  Section and plan views of an idealized suspension bridge.

Abstracting footings for the moment, the visible portions of the bridge are comprised by three main parts.  These include the four towers (two on either side of the span) that transfer all forces acting on the bridge vertically downward, the bridge deck upon which vehicles transit, and the steel cables that carry the weight of the bridge and its load.  Ideally the approach and exit of the bridge are such that the anchors of the cables shall be located on either side of the entrance and exit ends of the bridge.  This presumes that the road on either side of the bridge has straight stretches in line with the bridge orientation for at least the distance necessary to anchor the supporting cables.

If we examine a large-scale topographic map of the eastern side of the mountain and the most obvious location for an access road that would follow the existing land contours, it is apparent (Figure 2) that this configuration would not allow for cables to be anchored on either side of the entrance and access roads.  Instead, in both cases, the entrance and exit ramps would necessarily pass beneath at least one of the diagonal anchor cables on both sides, as shown in Figure 2.  While this is not untenable and has no adverse impact upon bridge structure nor stability, it presents an aesthetic distraction that would best be eliminated through optimized bridge orientation.

In order to avoid the aesthetic disadvantage of having the access ramps pass beneath the anchoring cables, it is possible to change the orientation of the bridge so that the exit from the bridge (on the home side of the span) corresponds to an existing straight stretch of roadway recently rough cut with a backhoe.  On the approach stretch of the access road, it will require fill and a retaining wall to fill a small gulch that presently separates the road from the proposed bridge location.  The proposed configuration is shown in Figure 3.  Note that the recommended road configuration and bridge orientation requires construction of a retention wall (or deployment of retention tiles or gabions) of around 20 meters in

Figure 2.  Large scale topographic view of the building area and the eastern side of the mountain slope showing the most obvious location for a bridge if the access road is to follow existing topographic contours.

Figure 3.  Optimal bridge location and orientation to enable anchoring cables to be deployed on either side of the road on access and exit stretches.  The yellow curve denotes the proposed location of a retaining wall to enable compacted fill to provide for the approach ramp.

length, shown in Figure 3 in yellow.  The approach road grade would be at an elevation between 0 and 4 meters above the existing land surface and would be achieved through fill and compaction with earth originating from other portions of the property, including material excavated for the towers on the approach plus the volume that would need to be removed for adequate footings for these towers.

Suspension Bridge Engineering Analysis

Equation (1) presents a model for the analysis of the stresses developed on a simple suspension bridge such as the one shown in Figure 1.  The variables listed in Equation (1) are shown in the drawing shown in Figure 4.

                T = w  (x² + (a² / (2 * S)) ²) ^1/2                                                     (1)

Where T = Tension force expressed in pounds at any point along the span of the bridge, x = distance from bridge midpoint to any length along the span where the Tensile force is to be calculate, measured in feet, w = distributed weight load expressed in lb/foot, a = ½ of the total span to be bridged (feet) and S is the sag, also expressed in feet.

Figure 4.  Geometry of a section of an idealized suspension bridge showing the variables that are required for the engineering analytical model presented as Equation (1).

This mathematical model permits the analysis of tensile stresses along the length of the bridge as a function of overall bridge length and a surrogate value of tower height, the vertical sag of the cable.  If we solve along varying lengths from one end to the midpoint of the bridge, it reveals that the highest tensile stress is developed at the edges of the bridge where a single tower couplet is supporting essentially all of the weight for that half of the structure.  The lowest tensile stress occurs at the precise midpoint of the span, where all forces are divided evenly between the towers on opposite sides of the span.   The variable corresponding to the bridge loading in the model (w) is normalized with respect to bridge length and comprises exclusively the deck undercarriage, decking, and the anticipated applied load.  The weight of the suspension cables are not included in the loading as the engineering analysis is predicated on the breaking strength of the cable under evaluation, which is a constant independent of its weight.

Ninfas Bridge Assumptions

The span length of the bridge drawn in figure 3 is one of 55 meters.  Per discussions with the project manager, preliminary calculations have been made under the assumption that the decking is to consist of tropical hard wood with a thickness of two inches and a density of 1200 grams per cubic centimeter.  I have assumed an undercarriage support consisting of two steel I-beams 3” by 3” suspended from the suspension cables by hangers fashioned from either steel rods or steel cable, with the wooden planks bolted to the steel undercarriage.  A bridge width of 2.5 meters is assumed.  Even though the anticipated loading consists of a single golf cart with an anticipated maximum weight of 500 kilograms, the loading factored into the design is one of 2000 kilograms, corresponding to the estimated weight of a fully loaded pickup truck.  The final two assumptions used for the statics calculations are that a maximum tolerable stress of 20% of the absolute breaking strength of the cable is to be tolerated and that the lowest portion of the cable in the midpoint of the bridge span reaches 1.5 meters above the deck.

Analysis

The weight that the bridge must support includes the weight of the deck planking, the steel undercarriage, and the load.  The steel is 5 kilograms per meter length, so the two lengths equal a total of 7 lb/foot.  A one foot span of wood planking weighs 85 lbs.  Lastly a 2000 kg weight, normalized to the bridge length is 24 lb/foot.  The net loading is therefore equal to 116 lbs/ft.  Since the analytical model is predicated on the strength of a single supporting cable and in fact there are two identical cables, the

Table 1.  Engineering model showing the calculated variable of maximum tensile stress (Tmax) calculated as a function of sag distance calculated at either ends of the bridge, where the maximum stresses occur.  The lowest point of the sag is assumed to be 1.5 meters above the bridge deck.  Therefore the Tower Height is in this case by definition equal to Sag + 1.5 meters.

value required for modeling the load (w) is one half this value, or 58 lb/ft.  For our 55 meter bridge, the maximum tensional stress as a function of sag (and correspondingly tower height, which is equal to sag plus 1.5 meters) is given in Table 1 for a variety of tower heights.

Having established the anticipated maximum tensile stresses for the loading distribution, we simply compare the weights with the tolerance of the cable being used to settle upon a cable diameter and tower height.  For example, one-inch steel cable has a breaking strength of 32 tons.  Therefore if we use 20% of this as our peak loading allowance, a one-inch cable will support a tensile stress of 12,800 pounds.  Hence, abstracting additional safety factors, Table 1 reveals that a tower height of 9.5 meters is required with the use of one-inch diameter cable.  Table 2 summarizes the breaking strength of cables of varying diameters and the minimum tower height required for the use of each of these cables and their varying strengths.

Table 2.  The suspension bridge analytical model summary of tower heights required for different sizes of steel cable.

Having established reasonable alternatives between tower height and cable diameter, the next rational design challenge is in the supporting towers themselves.  Since the entirety of the forces developed by the bridge is transferred vertically downward by its geometry, this means that the forces acting upon the tower(s) are under normal circumstances entirely in compression.  Since the forces are nominal with respect to the compressive strength of typical materials of construction, it means that the towers can theoretically be very thin so long as they are perfectly vertical.  In practice the width and depth of the beams used may be decided in terms of construction practicality since even a very small beam will easily support the structure statically.  The wild card is the variable effects of seismicity, so the rule of thumb ‘when in doubt, build it stout,’ is perhaps the most appropriate design axiom beyond practicality of installation.  I propose two steel 10”x10” steel I-beams tied together by a linked reinforced steel footing below ground and by one-half inch steel plate cut and fastened at the tops of the tower couplets in a design compatible with overall project aesthetics, with additional details to be defined in the final engineering stage.  A final preliminary consideration is the angle of the anchoring cable.  The steeper the angle, the greater the stresses acting on the deadman anchoring the cables.  For steep angles, a deadman is required with greater inertia, which shall vary as a function of mass, depth of burial, and soil characteristics.  For this application, topography and aesthetics are considered to carry greater design weight than the limitations (cost/geometry) of the anchoring deadman to be used.  Preliminary to final engineering calculations it is considered that the angle between the ground and the anchoring cable need not be less than thirty degrees nor greater than 45 degrees and that a final decision be based on final engineering findings and the optimal angle that can be duplicated on both sides of the bridge that ideally coincides with approach and exit stretches of the roads.

Subsurface design and engineering

There are two subsurface structures required.  The footings for the towers must be adequate to support the weight of the beams used and the maximum compressive stress transferred from an added load onto the free standing load of the finished bridge.  While final design of such footings will vary as a function of soil qualities determined by geotechnical testing, I assume that a footing depth of 2 meters is likely to be adequate.  A width of the footer of the width of the bridge plus an additional 40 centimeters from each beam is assumed will be adequate, subject to final engineering determinations, with a total footing thickness of 2 meters at the location of the beams.  Since the forces are entirely vertical, there is no need for a mass of 2 meters thickness in the mid section of the footer.  However, it is necessary for the footings to be tied together so that no differential settling is possible under any circumstances.  A tentative design proposal subject to additional engineering, is shown below.

Figure 5.  Plan view of the footings for each of the support towers to be located on opposite sides of the bridge span.

The deadman to which the anchor cable is affixed must be adequate to resist the pull from the free standing structure as well as the passage of a vehicle and its corresponding load.  Again, final configuration depends upon factors that must be considered in a final engineering stage; however a deadman design that appears preliminarily reasonable is one comprised of a prism of reinforced concrete 1 m wide, 1 m thick, and 3.5 meters long, with anchoring eye bolts welded to the reinforcing steel inside the poured concrete.

Retaining Wall

The proposed bridge orientation presumes an approach ramp that passes over a small gully that separates the present access road from the approach set of support towers.  This stretch must be backfilled to raise the grade in order to have a horizontal approach to the bridge deck.  Based on existing topography, this requires an amount of compacted fill up to a maximum depth of 3.5 meters.  A retaining wall of at least this height is required to impede downward migration by gravity and mass-wasting of the compacted fill required to comprise the road bed.

While a retaining walls structure dependent upon either gabions or retaining tiles would appear at first glance the most appropriate and perhaps elegant retaining structure in conjunction with the suspension bridge, a poured retaining wall presents certain design advantages that in the final engineering stage may prove to tip the scale in its favor, discounting for the moment aesthetic considerations.  Structural integration between the bridge footings, the retaining wall, and even the bridge deadman and a potential perpendicular deadman for the retaining wall, while more costly than a simple gabion and fill design, would provide greater structural integrity to the entire approach end and will necessarily be considered in a final engineering evaluation.

Work plan

The following series of steps and job elements are considered to be the logical progression of tasks required to complete the evaluation, secure the permit, procure materials, fabricate parts, and complete the installation.

1)      Undertake a rigorous final engineering design taking into account all possible materials of construction, orientation, and supplemental loading considerations as deemed necessary by the ownership.

2)      Submit signed off drawings and additional details for application for building permits.

3)      Purchase steel and transport to far side using backhoe while access is still possible prior to advancing stages of home construction (unless it is too late to do this).

4)      Prepare and pour the exit end of the bridge while the backhoe can access this area:  tower base footings and deadman installation and final road grade preparation (this would have to be done prior to permit issuance and carries the risk of abandonment in the unlikely event that a permit is not granted by the Municipality).

5)      Build retaining wall.

6)      Fabricate deck undercarriage fasteners, cable fasteners, steel plate tower ties, hangers, gliders, and eyehooks.  All this work is to be done in my metal-working shop in Puerto Jimenez.

7)      Excavate approach footings and fill and compact approach.

8)      Erect exit end tower beams, fasten together, pour base.

9)      Erect approach end tower beams, fasten together, pour base.

10)   Excavate, form, and pour deadman anchors on both sides of span.

11)   Extend cables and tighten

12)   Hang deck undercarriage

13)   Rig trolley for deck deployment.

14)   Bolt decking

15)   Finish details:  paint or wood veneer, webbing or paneling and hand rails, etc.

Budget 

Based on some preliminary calculations it is estimated that the bridge and retention wall may together carry construction costs in the range $75-100,000.  Pending a final engineering review and supplemental decisions, this estimate should be considered at this stage only a preliminary ballpark estimate.

Proposal

The additional work necessary to complete design and engineering for the suspension bridge and retaining wall structure includes compiling mechanics of materials information and supplemental statics analysis coupled with geotechnical information gathered from borings already undertaken plus or minus new borings at the site of the towers themselves.  Also multiple use criteria, safety margins, and supplemental footings criteria will be important considerations.  A final work plan will also be contingent upon access to the far side of the bridge.  The inability to use a backhoe on the bridge towers on the exit side of the span would complicate the effort as the steel beams would have to be hand carried and erected manually and all footings excavation undertaken by hand. 

A $4000 fee will be assessed for completed engineering and the elaboration of a detailed work plan and construction budget for both the bridge itself and both entrance and exit ramps and retaining wall.  This amount covers all fees associated with this pre-proposal as well as the engineering fees and expenses associated with completion of final engineering and design, to include the following deliverables:

1)      Completed design and engineering of both the bridge and the retaining wall, ready for a registered engineer’s stamp to proceed toward permitting.

2)      Full economic and mechanics of materials analysis of using wood decking versus sheets of expanded steel metal for decking.

3)      Work plan and time line for the installation of all features during an anticipated 12-week project duration, one half of which is expected to be shop work undertaken in my metal shop on the Osa Peninsula, the other half to be the onsite construction and installation.

4)      A full and final firm budget for the project, with all materials, labor, and supplemental costs itemized and detailed.

Terms include a $2000 deposit payable immediately and the balance of payment due upon delivery of the final engineering/design report and formal itemized bid for the bridge installation.  Of this engineering fee, $1000 will be applied toward construction should my firm be contracted to do the installation.  My target for completion of the engineering design stage and for the delivery of a final report shall be three weeks from the date of approval to proceed.