3p

The needed equation for determining X2 is obtained by equating the right-hand side of Eqs. (4.142b) and (4.144) and substituting S4 and 8S given by Eqs. (4.143b) and (4.145). The result, after division of both sides of the equation by n3h3/48EI. is n/32 P + X2 + 4516 P + 5/16X2 = 27/16 P - X2 + 12%4 P + 3/16X2

Solution of the equation yields 35P

With these forces known, the bending moments, shears, and deflections of the girders can be computed by conventional methods.

To examine a more general case of symmetrical framing, consider the orthogonal grid with rectangular boundaries in Fig. 4.25a. In the x direction, there are n spaces of width h. In the y direction, there are m spaces of width k. Only members symmetrically placed in the grid are the same size. Interior nodes carry a concentrated load P. Exterior nodes, except corners, carry P/2. Columns are located at the corners. For identification, nodes are numbered in one quadrant. Since the loading, as well as the framing, is symmetrical, corresponding nodes in the other quadrants may be given corresponding numbers.

At any interior node r, let Xr be the load carried by the girder spanning in the x direction. Then P - Xr is the load at that node applied to the girder spanning in the y direction. For this example, therefore, there are six unknowns Xr, because r ranges from 1 to 6. Six equa-

FIGURE 4.25 Rectangular bay with orthogonal girder grid. (a) Loads distributed to joints. (b) Loads on longer midspan girder. (c) Loads on shorter boundary girder AD. (d ) Loads on shorter midspan girder. (e) Loads on longer boundary girder AB.

tions are needed for determination of Xr. They may be obtained by the method of consistent deflections. At each interior node, the vertical displacement of the x-direction girder is equated to the vertical displacement of the y-direction girder, as in the case of the square grid.

To indicate the procedure for obtaining these equations, the equation for node 1 in Fig. 4.25a will be developed. When use is made of Eqs. (4.136) and (4.138), the deflection of girder 7-7 at node 1 (Fig. 4.25b) equals

n3 h3

Renewable Energy 101

Renewable Energy 101

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