## Base Plate Moment Diagram

PLATE ANGLE BULB HALF BEAM (a) OPEN RIBS FIGURE 4.30 Types of ribs for orthotropic plates.

(b) CLOSED RIBS (d) MEMBER EZ - GIRDER FIGURE 4.31 Four members treated in analysis of orthotopic plates.

Member IV comprises girders and plate (Fig. 4.31d). Stresses are computed by conventional methods. Hence determination of girder and plate stresses for this member will not be discussed in this article.

The plate theoretically should be designed for the maximum principal stresses that result from superposition of all bending stresses. In practice, however, this is not done because of the low probability of the maximum stress and the great reserve strength of the deck as a membrane (second-order stresses) and in the inelastic range.

Special attention, however, should be given to stability against buckling. Also, loading should take into account conditions that may exist at intermediate erection stages.

Despite many simplifying assumptions, orthotropic-plate theories that are available and reasonably in accord with experiments and observations of existing structures require long, tedious computations. (Some or all of the work, however, may be done with computers to speed up the analysis.) The following method, known as the Pelikan-Esslinger method, has been used in design of several orthotropic plate bridges. Though complicated, it still is only an approximate method. Consequently, several variations of it also have been used.

In one variation, members II and III are analyzed in two stages. For the first stage, the floorbeams are assumed as rigid supports for the continuous ribs. Dead- and live-load shears, reactions, and bending moments in ribs and floorbeams then are computed for this condition. For the second stage, the changes in live-load shears, reactions, and bending moments are determined with the assumption that the floorbeams provide elastic support.

Analysis of Member I. Plate thickness generally is determined by a thickness criterion. If the allowable live-load deflection for the span between ribs is limited to i/3ooth of the rib spacing, and if the maximum deflection is assumed as one-sixth of the calculated deflection of a simply supported, uniformly loaded plate, the thickness (in) should be at least t = 0.065aVp (4.162)

The calculated thickness may be increased, perhaps Vi6 in, to allow for possible metal loss due to corrosion.

The ultimate bearing capacity of plates used in bridge decks may be checked with a formula proposed by K. Kloeppel:

where pu = loading, ksi, at ultimate strength eu = elongation of the steel, in per in, under stress fu fu = ultimate tensile strength, ksi, of the steel t = plate thickness, in

Open-Rib Deck-Member II, First Stage. Resistance of the orthotropic plate between the girders to bending in the transverse, or x, direction and torsion is relatively small when open ribs are used compared with flexural resistance in the y direction (Fig. 4.32a). A good approximation of the deflection w (in) at any point (x, y) may therefore be obtained by assuming the flexural rigidity in the x direction and torsional rigidity to be zero. In this case, w may be determined from d4w

where Dy = flexural rigidity of orthotropic plate in longitudinal, or y, direction, in-kips p(x, y) = load expressed as function of coordinates x and y, ksi

For determination of flexural rigidity of the deck, the rigidity of ribs is assumed to be continuously distributed throughout the deck. Hence the flexural rigidity in the y direction is

where F = modulus of elasticity of steel, ksi

!r = moment of inertia of one rib and effective portion of plate, in4 a = rib spacing, in

Equation (4.164) is analogous to the deflection equation for a beam. Thus strips of the plate extending in the y direction may be analyzed with beam formulas with acceptable accuracy. FIGURE 4.32 (a) For orthotropic-plate analysis, the x axis lies along a floorbeam, the y axis along a girder. (b) A rib deflects like a continuous beam. (c) Length of positive region of rib bending-moment diagram determines effective rib span s..

In the first stage of the analysis, bending moments are determined for one rib and the effective portion of the plate as a continuous beam on rigid supports. (In this and other phases of the analysis, influence lines or coefficients are useful. See, for example, Table 4.5 and Fig. 4.33.) Distribution of live load to the rib is based on the assumption that the ribs act as rigid supports for the plate. For a distributed load with width B in, centered over the rib, the load carried by the rib is given in Table 4.6 for B/a ranging from 0 to 3. For B/a from 3 to 4, the table gives the load taken by one rib when the load is centered between two ribs. The value tabulated in this range is slightly larger than that for the load centered over a rib. Uniform dead load may be distributed equally to all the ribs.

The effective width of plate as the top flange of the rib is a function of the rib span and end restraints. In a loaded rib, the end moments cause two inflection points to form. In computation of the effective width, therefore, the effective span se (in) of the rib should be taken as the distance between those points, or length of positive-moment region of the bending-moment diagram (Fig. 4.32c). A good approximation is se = 0.7s (4.166)

where s is the floorbeam spacing (in).

The ratio of effective plate width a0 (in) to rib spacing a (in) is given in Table 4.6 for a range of values of B/a and a/se. Multiplication of a0/a by a gives the width of the top flange of the T-shaped rib (Fig. 4.34).

TABLE 4.5 Influence Coefficients for Continuous Beam on Rigid Supports Constant moment of inertia and equal spans

Midspan End moments moments Reactions at C at 0 at 0 ## Renewable Energy 101

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