Influence line analysis is essential for bridge design. Classical influence line theory for continuous girders goes back to Müller-Breslau in the late 18th century. Currently, iterative analysis is commonly used since this method is rather simple to implement with the displacement methods. Both methods give the same results and we can use either of these methods.

However, the author is a very big fan of old design charts. One of these design charts was developed by Dr. Watanabe^{1}. His methods require only one single input, span length ratio, and it is very easy to check the input mistake. His methods give not only for moment and shear but also for deflection and rotation. Of course, his methods have some limitations, mainly due to trying to minimize the number of inputs.

1) Both side spans are the same length and all inner spans are also the same length. The span arrangement is defined as span length ratio, (center span length)/(side span length). This is the only input needed to calculate influence lines for moments, shears, deflections, and rotations.

2) All girders are prismatic and flexural rigidity EI is constant for all sections.

3) All supports are hinge/roller supports. This means the supports are free to rotate but fixed for deflection.

4) The effect of shear deformation is not considered.

5) All values are calculated at the 1/8th locations.

Let’s start with a three-span example because “Hanc marginis exiguitas non caperet”.

Figure 1. 3-span bridge continuous girder for the example.

Calculate design moment, shear, deflection and rotations at the various locations for the three-span continuous girder as shown. The given loads are p (dead) = 2.0 t/m, P (live+impact) = 30 ton, p (live+impact) is 0.4 t/m, 0.3 t/m and 0.5 t/m for span 1, 2 and 3.

All influence lines are calculated from only one single input 67.5/45 = 1.5 and detailed calculations are as attached.

This method gives exactly the same results as any influence line analysis program because no further assumptions were introduced in the derivation. This method is very useful for the preliminary design or design checks, however, can not be applied to haunched girders or non-symmetric spans.

Influence Lines of 1 Span Beam. xlsm

Influence Lines of 2 Span Beam. xlsm

Influence Lines of 3 Span Beam. xlsm

Influence Lines of 4 Span Beam. xlsm

Influence Lines of 5 Span Beam. xlsm

Example Calculation of 3 Span Beam.pdf

[1] Dr. Watanabe, Noboru, Theory, and calculation of bridge influence lines, 1967.

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