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This means that on iteration of an operator equation or an integral equation where the operator is a strongly connected diagram, the weakly connected diagrams will be reproduced from those iterations. Thus when one sets up an operator equation or an integral equation, only strongly connected diagrams need to be included. The mass operator is defined as the sum of all strongly connected diagrams, minus end connectors:

+ ...

(4.1.11)

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Structural upgrading of a bridge, if needed may involve the substitution of larger pieces in certain areas of a truss or oor structure. The lateral bracing of the bridge truss or a radial stiffening arch may have to be increased or modi ed. Wherever possible, the upgrading of an historic timber bridge should

-- -= t=\

(4.1.12) The weakly connected diagrams in (4.1.10) which contain two strongly connected elements may be summed as (4.1.13) Weakly connected diagrams are reproduced by cascading of mass operators. Continuing with this process, the Neumann series for the mean dyadic Green's function may be written in the form:

C"'\

C2S::::-.-

+ '"

FIGURE 8.5 This is a more extreme example of showing the future, since this Fibonacci time series has its next data point decades away.

C"'\

+ ...

=-+--0

[-+-- -+~+...]

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utilize members and connections similar to the original bridge fabric. Steel cover plates, stiffeners, ties and modern connectors should be avoided wherever possible. The timber bridge trusses of Reuban L. Partridge (c.1823 1900) in Ohio appear to be designed to allow for the easy replacement of web compression members and the outboard chord material. Only the tension diagonals that are notched through the multiple chord members are impossible to replace without extensive shoring or the bridge. Mr. Partridge s bridges, less than 100 feet in span, can be structurally upgraded by substituting full-length glued laminated timber members for existing bottom-chord material. This substitution will increase the tensile capacity of the bottom chord in three ways. The use of full-length glued laminated timber removes the reduction to the multichord net section due to shear block notches and butt splices. Also, higher allowable design values in tension are available with glued laminated timber materials. These kinds of solutions would bring to these projects timber design expertise capable of evaluating and analyzing historic bridges as timber structures. Historic timber structures provide an excellent laboratory for the timber industry. Information regarding the long-term performance of wood would be of great assistance to structural engineers faced with the problem of evaluating historic structures. Research opportunities such as these would add to our knowledge of modern timber engineering, as we utilize this renewable construction material, so important to our forest products and construction industries. Structural engineers with timber design experience must become actively involved in historic preservation in order to save our remaining historic covered bridges. Highway department bridge engineers must seek out consultants with timber design expertise. By working together, we can save an important part of our civil engineering heritage.

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(4.1.14) (4.1.15)

==-+--@=

Equation (4.1.15) is the diagrammatic representation of the Dyson's equation. One can verify that all the diagrams, both weak and strong, are included in the Dyson's equation. The Dyson's equation is an exact equation for the mean field. In analytical form, Dyson's equation becomes

= =(0)

In 7, you learned how to use and draw trendlines. The tool next to the trendline tool, which is the channel tool (Figure 8.6), lets you draw two parallel trendlines easily in order to bound price data in a channel pattern. Channel patterns are rarer than trendline patterns, simply because it is much easier to nd prices bouncing off a consistent support or resistance level than it is to nd prices bouncing within both support and

=(0)

(r, rI) Q(rl, 1'2)' (G(r2' 1'0 (4.1.16)

where Q(rl' r2) is the dyadic mass operator. For a statistical homogeneous medium, the mass operator is statistically translational invariant so that the mean field is also translational invariant.

(Q(rl, r2)) = Q(7'l - 7'2) (G(r, r o )) = Gc(r -r o )

National Forest Products Association, National Design Speci cations for Wood Construction, Washington, D.C.: National Forest Products Association, 1977.

(4.1.17a) (4.1.17b)

Glk)

Q(k) =

(4.1.18a) (4.1.18b)

Gc (7')

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