The optimal tendon profile and its associated duct geometry for posttensioned box girders are investigated. A computational algorithm has been developed to determine an ideal shape for the tendon and duct. The algorithm is based on a diagonal cross point in the Magnel quadrilateral and uses computational geometry instead of graphical drawing. Thereafter, an ideal parabolic and linear tendon profiles can be calculated from which the covering duct could be shaped. To check the optimality of the results, an automatic examination of Magnel diagrams in various cross sections is incorporated in the algorithm. This enables a unique prestress level to be selected that suits all sections. Then, the mideccentricity of the two crossing points with a common prestress line is chosen as a design eccentricity in each cross section. The optimal duct shape is determined based on such automatic inspections. In a numerical example, the linear, parabolic, and optimal duct geometries are compared and drawn. It is concluded that both linear and parabolic duct shapes can be very close to an optimal shape. In a numerical example, an optimal, box girder with linear open access-type tendons is designed that can withstand extra blast load when explosion occurs.

Steel is used in concrete bridges as a reinforcement material. Therefore, the protection against corrosion for increasing lifetime of the bridge is an important issue and remains an active field of research [

(a) Externally posttensioned: inside of box girder cross section. (b) Externally posttensioned beam: girder outside web.

It is obvious that the accessibility to the steel tendon as shown in Figure

Recently, protection of the bridges against explosions due to, for example, car bombs is studied seriously, and one of the damages is shown in Figure

Permanent deformation of box girder after explosion.

Regarding the extra blast load, the bomb power or explosive amount (equivalent TNT mass) and also the distance between the explosion centre and the bridge are required. Then, we need to find the scaled distance from the following equation:

Then, the resulting overpressure (in kPa) due to the explosion [

Then, the extra blast load is

This paper provides a new method that is suitable for the optimal design of an accessible linear-type, posttensioned tendon and its duct. The method is based on the traditional Magnel diagram [

There are two types of prestressed concrete members. In pretensioned type, the strands are pretensioned before casting, and then it is essential that the prestress level in various cross sections should be different. In order to achieve this objective, debonding technology is adopted by which some strands are partially debonded from concrete [

However, in posttensioned beams and members, the prestress level in all cross sections is similar. The tensioning is planned after casting; therefore, the tendons are not straight and can be covered in a curved duct [

Since an ideal tendon profile determination is a design problem, it should be noted that the designers have been using a graphical design method which was developed by Magnel [

In this article, it is shown that the ideal tendon profile in the posttensioned beam could be determined by automated computer inspection of the Magnel diagrams, in the various cross sections without drawing any quadrilateral on paper. By using computational geometry, first the coordinates of the edges of the Magnel quadrilateral is calculated. Then, the coordinates of the crossing points of the two diagonals of the quadrilateral in the cross section in which maximum service bending moment occurs is determined. Similar procedure is repeated for two more cross sections, and thereby a parabolic or linear curve could be computed, knowing three eccentricities across the beam span. Further investigation is carried out to show the accuracy of this type of design.

In a numerical example for a long span box girder bridge, by examining the Magnel quadrilateral in many more cross sections and considering a prestress level that suits all the cross sections, the corresponding eccentricities are computed and a smooth curve is fitted to all the points. When comparing this optimal duct shape with the parabolic and linear ones, it does not show any significant difference.

The whole drawing process and inspection and selection are algorithmic and can be done via one computer program. It is shown that we can determine the ideal parabolic and linear duct shapes and also check their closeness to optimal duct shape, all in one algorithm. It is concluded that both parabolic and linear duct geometries introduced in this paper are simple and optimal. Therefore, we can choose an accessible and easily protectable linear tendons and duct shape which is also an optimal design. The method is based on automatic inspection of the Magnel diagram and particularly can be useful for checking the Magnel polygon in the cross section where the deviator is located and sharp changes (discontinuities) in prestress occur.

The top fibres of a beam in transfer (before service) are in tension since the beam planned to be in hogging, and if the allowable tensile transfer stress is

In (

The bottom fibres of a beam in transfer, when it is in hogging, are in compression, and if the allowable compressive transfer stress is

In (

The top fibres of a beam in service obviously are in compression because the beam is in sagging, and if the allowable compressive service stress is

In (

The bottom fibres of a beam in service (sagging) are in tension, and if the allowable tensile service stress is

In (

The four stress constraints (

In Section

The borderlines of the constraints for

The borderlines of the constraints for

The borderlines of the constraints for

The borderlines of the constraints for

The edge with the coordinates

The edge with the coordinates

Intersection of the lines (

Similarly, we can calculate

Since at

Another ideal tendon profile can also be achieved, if the Magnel diagram would be drawn in many cross sections

By simple visual inspection of all diagrams in one figure, it is very easy to check which prestress level suits all the quadrilaterals in the figure. This inspection is also possible without any drawing and can be done by intelligent computer searching. When a design prestress level was selected and named

To be on the conservative side, the average of the above could be considered as the design eccentricity for each cross section, that is,

The numerical example is taken from [

The scaled distance of this explosion will be considered as a part of service load:

The resulting overpressure (in kPa) due to the explosion is

This produces overpressure load as a result of explosion:

The maximum transfer bending moment occurring at a midspan is

The maximum transfer bending moment occurring at a midspan is

The Magnel diagram in the maximum bending moment cross section is drawn in Figure

Magnel design diagram in the midspan cross section.

In Figure

Magnel design diagram in the supporting ends.

Based on (

Position of the tendon, neutral axis, and top and bottom fibres across the span.

In order to see if these types of tendon profiles are accurate enough, we will draw the Magnel diagram in nine cross sections each with

Magnel diagrams for the cross sections: (a) (

This enabled (

Ideal tendon profile based on the Magnel diagram of 9 cross sections each

In this article, the ideal tendon profile in the posttensioned beams is investigated by multiple automatic examinations of the Magnel design diagrams. An algorithm was introduced by which one can calculate ideal parabolic and linear tendon profiles. It is shown that ideal parabolic and linear shape tendons and ducts, calculated by the formulas in this paper, are very close to the ideal and optimal tendon profile. It is recommended that both linear and parabolic ducts can be used by designers because they do not have any disadvantage. Moreover, the design is based on simple formulas without engaging with graphical work. It can also be implemented in design of continuous girders, assuming that the moments in intermediate pier supports are found by static calculations. Moreover, the blast load as a result of car bomb and so on can be considered and added to the service load. In this case, the relevant eccentricity and prestress level can be found as shown in the numerical example. This avoids the permanent deformation (Figure

As far as that author is aware, there is not any design method which is particularly suitable for design of accessible linear-type tendons and ducts in posttensioned beams. Instead, designers use traditional methods and increase the factor of safety to compensate lack of accuracy in modelling. Therefore, the method in this paper can be used for future design applications in which bridge design should be optimised to take an extra blast load as a result of explosion due to car bomb and so on. Moreover, in future by installing sensors in the open access box girder, compression conservation can be monitored and the validity of formulas in this paper can also be checked.

Area of beam cross section

Prestress conservation in service

Eccentricity

Design eccentricity in midspan

Design eccentricity in supporting ends

Allowable transfer tensile stress

Allowable transfer compressive stress

Allowable service tensile stress

Allowable service compressive stress

Beam length

Transfer bending moment

Service bending moment

Slope of the quadrilateral diagonals

Total number of beam cross sections

Prestress compressive force

Load distribution in transfer

Load distribution in service

Location of the cross section

Section modulus, top fibre

Section modulus, bottom fibre.

The author declares that there are no conflicts of interest.