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|STRUCTURAL DESIGN - PART 01||
1. Structural analysis and calculations
The structural calculations of Stari Most were performed to evaluate the maximum stresses in the structure produced by the design loads and the maximum deflections produced by the loads during the working phases.
The following loads were considered: permanent loads, live loads, thermal loads, accidental loads due to the flood and to the earthquake.
The check of the structure against the earthquake actions was made determining the peak acceleration which produces the collapse of the structure.
In the fisrt part of this report some notes have been taken on the basis of a preliminary and simplified geometry of the bridge, then, in the second part, more accurate investigations have been performed on the "most likely bridge shape".
2. The thrust line applied to the arch of the Old Bridge of Mostar
The thrust line of an arch is equivalent to the pressure of the infinite number of funicular polygons joining the stresses applied, whose first and last sides are represented by the springer reactions Ra, and Rb.
This polygon is also referred to as polygon of subsequent resultants because one of its generic sides represents the straight line on which the resultant R of all the forces that precede it, including also reaction Ra, acts. The value of R comes instead from the projecting line corresponding to the polygon of forces shown in the figure.
fig. 1 - the thrust line
Thus, the ordinary stress N and the shearing stress T over a section S can be calculated by decomposing the resultant R, referred to S, that follows the ordinary line and the line that runs in parallel with the track of section S. The bending moment originates instead from any of the following expressions:
M= R d= V do = H dv,
where d, do and dv, respectively indicate the distances of the section of resultant R, of its vertical component V and of its horizontal component H from barycenter G.
The thrust line offers an immediate view of the static conditions of the arch that tends to improve with smaller deviations from the geometrical axis and with the limited bending moments M. If it coincided with the axis, each section would be subjected only to an ordinary stress, because the stress would be uniformly divided along the section and could therefore be equivalent to the safety load.
For this reason, you generally try and reach this result by changing the form of the arch axis or, when it is possible, by changing the thrust line through the distribution of the loads.
The sections appear completely compressed if the thrust line is within the strip defined by the core points of all the sections (third medium points of the rectangular section).
When all the loads are vertical, which occurs frequently, the resultant R of any section S has an horizontal component H that is constant and equivalent to the horizontal component (thrust) of reactions Ra and Rb. In this case, it is easier to calculate value M of a section S using the third expression of those given above, thus taking as reference the horizontal component. Thus, the thrust line of the arch axis, with vertical ordinates, is represented by diagram M less factor H.
On three-hinge arches, the thrust line is statically determined by the fact that the end and intermediate sides pass through the three hinges. On two hinge arches, the ends must pass through the hinges. In the previous examples and on restrained arches with rigid constraints, the thrust line cannot coincide with the arch axis. As stresses M and T would have no effect on all sections, the arch would be deformed due to the ordinary stress that causes a reduction of length of all the infinitesimal trunks ds, theoretically forming the frame. In other words, this would produce a reduction in length of the chord that is incompatible with the rigidity of the restraints.
By managing to apply appropriate stresses independent from loads (for example by appropriately moving the restraints) it is possible to confer an axial characteristic to thrust line of a specific system of loads.
It is evident that the bending thrust line is more significant in presence of higher loads, because the projecting lines of the polygon of forces change direction more rapidly in these points. Thus, with heavy loads on the reins, the thrust line appears as shown in Figure 2-a, while higher loads on the key stone produce a curve equivalent to the one shown in Figure 2-b. If a significant load is concentrated on the key stone, the thrust line is characterised by a cusp in the same point, as shown in Figure 2-c.
fig. 2 - thrust line shapes due to loads
The main consideration suggested by Heyman's theory, underlines that the loss of stability of an arch or of a vault is never a consequence of the breakage of material due to exceeded resistance limits, but a result of the geometrical configuration of the structure in relation to the funicular polygon connected to the external loads. This means that from a safety point of view it is not important to know the "true" thrust line. It is sufficient to make sure that the arch thickness contains at least one line. The above consideration can also be supported by Heyman's safe theorem, that states that "the arch is safe, if it possible to find a thrust line for the whole arch, both in balance conditions and with loads applied, provided that is within the thickness of the arch".
Given these premises, it is evident that the problem implies verifying the compliance with this requirement for one of the three infinite funicular polygons connected to the system of loads and acting on an arch restrained by the springing line.
This approach to the problem is the most plausible one, also in consideration of the fact that the following issues were reported for the examined arch:
In these conditions it is advisable to view the problem in generic terms as suggested by Heyman's interpretation of the thrust line.
For this purpose we have created several models with finite elements.
Intellectual property of this report and of the design drawings is owned by the University of Florence - Department of Civil Engineering
author of the text: Prof.Eng. Andrea Vignoli – other contributes have been mentioned in related paragraphs
© - General Engineering Workgroup -
Final Design Report
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