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where eik;zD is the phase shift of the incident wave as it passes through the layer of thickness D. The solution of a thick layer can be solved by cascading layers of thicknesses equal to D (Fig. 3.1.2). Suppose the total thickness of the layer is d with d D. Then we can divide d into many layers with each layer of thickness D that obeys both conditions (i) and (ii). That is, each layer is "thin" enough such that single scattering applies. Nevertheless, D still has to be thick enough to obey condition (i). Then we have the recurrence relation

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(3.1.19)

------------------------------ -b------------------------------- ---

.::=0

Figure 3.1.2 Studying large thickness problem by cascading: Total thickness is d and each layer thickness is D.

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Slippage, rotation, shrinkage, or the lack of continuity in a timber joint is dif cult to allow for in a computer analysis of a timber truss. Multiple-chord trusses will invariably appear stiffer when analyzed, even when all joints are free to rotate in the computer model. Structural engineers are aware that it is very dif cult to produce a true hinge or a true xed joint in the actual structure. Joints in timber trusses may act somewhere between the two, causing a very different distribution of forces than produced by the analysis. How does one model a half-lapped and notched joint in an indeterminate frame What about the problem of describing the intersecting member of a multiple chord truss where half of the member section in each direction passes through a joint and all pieces nailed together with wrought iron nails

where the subscript n denotes the nth layer solution. Thus letting the zeroth layer solution be ('l/J)o = ~)inc, we have, from (3.1.19),

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Using a line segment is more appropriate in the case of prior trends, since the trend has already changed and there is no reason to clutter your chart with a line that continues going past the trend s end In order to choose whether or not you want to extend the line or not, you use the Extend option This option is available for a number of drawn objects, but the principle is the same for all of them: Extend makes the drawn object continue forward in time, all the way to the rightmost edge of the chart Notice how the pointer changes into a hand when you move the mouse over the drawn object This is the signal that you can right-click to modify that particular object To access Extend, right-click on any trendline, which creates the popup menu shown in Figure 710, and choose Extend.

if27fnaD) n ik. nD k e" 'ljJinc(Z = 0) iz Using D = din and taking the limit of large n with fixed d, we have f 27fnad)n ik nD hm 'l/J n = hm 1 + k 'l/Jinc(Z = O)e "

. ()

n~CX)

n---+CX)

(3.1.20) iz The net effect of cascading for the coherent wave is that the net change occurs in the exponent. The result of (3.1.20) is an effective propagation constant K of the medium such that (3.1.21) so that lim (-~J)n

= exp (

(3.1.22)

Highway bridge inspection, rehabilitation, and replacement programs have involved many structural engineers in the evaluation of older, sometimes historic, highway bridges. Many times these bridges are found de cient because of deterioration of structural elements, the increased magnitude of service loads, inadequate lane width, or a geometry or con guration of the highway or the bridge itself that makes the passage of vehicles unsafe at normal speeds. The structural engineer plays a pivotal role in the determination of possible methods to retain a historic bridge structure. Many two-lane bridges have been converted to one-lane bridges with addition of an adjacent span to carry traf c in the opposite direction. The lacing and cover plates of built-up members and the eyebolts and pinned connections of older steel truss spans provide the structural engineer with the opportunity to study design details that are uncommon today. Repairs and reinforcement can be made to deck, abutments, and superstructure. When this cannot be justi ed, truly historic spans may be removed from service by rerouting traf c to a replacement span. Many historic bridges have been successfully adapted for reuse for pedestrians in areas such as parks and downtown redevelopment projects.

(3.1.23)

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