Baidar Bakht, Leslie G. Bridge Analysis Simplified. McGraw-Hill Book Co. Bakht and L.
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Where they are supported at close centres, i. Concrete slabs are also used for widely spaced girders, as shown in Figure 6b. In such cases, the economies that can be achieved by reducing dead weight justify the extra cost of using variable depth slabs, as shown.
However, considerable ingenuity has been devoted to improving this form of construction. Examples include: Use of full depth pre-cast units, with pockets to accommodate the shear connections. Grouting is used to complete the shear connection and complete the concrete between neighbouring slabs, Figure 7a. Use of full depth pre-cast units, with high-strength friction-grip bolts, Figure 7b and c.
Use of glass reinforced plastic permanent formwork. Use of stiffened steel plate as external reinforcement. The plate is attached to the concrete by conventional shear connectors to form a composite slab. Use of pre-cast planks as permanent formwork. Depending on the detail of the reinforcement within these planks, they may or may not contribute to the resistance of the completed slab. Apart from certain types of slab with participating pre-cast planks, all reinforced concrete slabs are isotropic and may be analysed by simple plate methods.
Analysis needs to consider the various modes of behaviour of the slab. Contribution to the overall longitudinal bending of primary girders The effective breadth of the slab is included in the modulus of the girders.
The stress resultants, compression at midspan, tension at supports, can readily be determined from the global analysis. The midspan regions are likely to be satisfactory in compression; the support regions usually require additional reinforcement, which should be placed within the effective breadth of the slab. Contribution to the overall bending of cross-girders where they are present The treatment is similar to that for the longitudinal girders.
Contribution to the overall behaviour of structure, e. Usually the slab is replaced by equivalent beams in the grillage analysis, as shown in Figure 8. Not less than eight beam strips are recommended for each span to ensure adequate modelling of the structure.
Calculation of the bending stiffness of the beam strip is straightforward - the slab is assumed to be uncracked and fully effective. It is also necessary to model the torsional stiffness of the slab - this is best done by distributing the total torsional rigidity equally between the transverse beam strips and the longitudinal twin girders, i.
Local bending action to transfer wheel and other local loads to the main superstructure Analysis is required only of the slab local to the wheel load. Most practical situations can be reduced to standard cases and evaluated using standard influence charts .
Figure 9 shears schematically a typical chart Pucher Influence chart. The patch leads are applied to the chart in a way that maximises the volume under the influence surface. The volume is then evaluated numerically.
The simplification of support conditions to permit use of standard charts normally leads to a conservative assessment of worst moments. Once the methods of analysis have determined the overall combination of moments, axial forces and shears on the slab, its adequacy in compression and shear can be checked and the reinforcement detailed in the conventional way.
It is customary for some compression reinforcement to be required in the regions of highest moment. The principal motivation for optimisation has been to develop the cheapest deck that will achieve a satisfactory fatigue life. The former has led to the use of thicker deck plate and more fatigue resistant welds; the latter has led to a particular form of welded connection. The outcome of this practical optimisation has been the development of the standard "European" orthotropic deck that is described below.
Although the functions and the resulting stresses of the component parts of a steel-deck bridge are closely interrelated, it is necessary for design purposes to treat separately the three basic structural systems, as follows : System I. The main bridge system, with the steel deck acting as a part of the main carrying members of the bridge.
In the computation of stresses in this system in girder-type bridges the entire cross-section area of the deck, including longitudinal ribs, may be considered effective as flange. System II. The stiffened steel-plate deck, acting as the bridge floor between the main members, consists of the ribs, the floor beams, and the deck plate as the common upper flange. The major contribution of Pelikan and Esslinger  is in the prediction of the behaviour of System II, the continuous orthotropic plate on flexible supports.
System III. The deck plate, acting in local flexure between the ribs, transmitting the wheel loads to the ribs. The local stresses in the deck plate act mainly in the direction perpendicular to the supporting ribs and floor beams and do not add directly to their other stresses.
The governing stresses in the design of the deck are obtained by superposition of the effects of Systems I and II. For a reliable fatigue life a sound full penetration weld is essential, Figure 10b. This can be achieved by a square cut to the end of the stiffener, providing that qualified weld procedures are adopted and a consistently good fit is achieved between the two plates. The latter requires careful manufacture of the trough stiffener and adequate jigs and clamps on the panel welding line.
Another important detail is the stiffener to cross girder or diaphragm connection shown in Figure 10c. The stiffener is continuous through an opening in the cross girder to ensure full continuity.
Only the webs of the stiffener are welded to the cross girder; this improves the fatigue performance of the stiffener.
The most practical method of analysis is that of Pelikan and Esslinger. It assumes that the deck system is a continuous orthotropic plate, rigidly supported by its main girders and elastically supported by the floor beams. The design procedure is divided into two stages: In the first stage it is assumed that the floor beams, as well as the main girders, are infinitely rigid.
In the second stage, a correction is applied, considering the floor beams as elastically supported. The reactions of the plate on the floor beams are replaced by a load proportional at each point to the deflection of the floor beam. The total moments are found by superposition, due to the influence of dead and live loads assuming rigid supports and live loads assuming elastic floor beams.
Points of particular note are : The effective breadth of a discretely stiffened plate is less than that of a fully continuous orthotropic plate. Its determination must recognise that only the deck plate is continuous. Trough shaped stiffeners are not fully effective in torsion because of cross-section distortion. Guidance is available on suitable methods of accommodating this reduction in rigidity. In principle, the design of the deck should be verified separately for static strength and fatigue resistance.
For static strength the individual components of the deck need to be checked for the following stresses, in combination: Longitudinal stresses from participation in overall bending of the superstructure. Transverse stresses from participation in bending of the cross girder. Longitudinal stresses and shear stresses from bending of the stiffened plate between cross girders.
Transverse bending of the deck plate between trough webs. For fatigue loading the critical regions are those identified in 3.
In practice, adequacy is demonstrated by experience rather than by calculation of the very complex elastic stress fields. Usually, the flanges of bridge cross-sections are relatively wide with respect to their spans. The effects of shear lag need, therefore, to be included in the bending analysis. Shear lag effects cause the stress distribution over the cross-section to be non-linear. The maximum stress values occur at the flange-to-web junctions.
The effective width is defined by the condition that the stresses at the flange-to-web junction, according to engineering bending theory, must be identical to the maximum stresses calculated by applying the mathematical theory of elasticity.
The effective width bm is defined as the width of a rectangular area of height dx max, and having the same area as that enclosed by the distributed curve stress area. The effective width, Figure 11a is calculated by the following equation. Most practical situations can be reduced to standard cases and evaluated using standard charts and tables, Figure 11b.
Bridge analysis simplified