In the deflection theory, the girder deflection is assumed as some kinds of exponential function as below
The details of this formula will be discussed later. However, it should be mentioned that the term Hp is horizontal cable tension force due to the live load (or any additional loads other than dead loads) and shall be calculated by iteration.
Timoshenko assumed the girder deflection as a trigonometric function shown below,
This formula looks much simpler comparing those of deflection theory, but it requires some calculations to determine each ai.
Modern displacement methods assume the girder deflection as a cubical function shown below,
Timoshenko’s trigonometric methods give theoretically more exact solution than deflection theory and the convergency is faster, however, the required calculation itself is more voluminous compared to deflection theory. It was 1943 and computers might not be available to bridge engineers and this method was not easy to apply to real projects.
They employed some minor simplifications and finally, Dr. Kawada in Japan improved full trigonometric methods and applied them to the mega suspension bridge projects in Japan. Dr. Kawada is the chairman of Kawada Industries, Inc., he is a respected suspension bridge designer, fabricator, and builder. One of his books was translated into English, History of the Modern Suspension Bridge: Solving the Dilemma between Economy and Stiffness.
The attached excel calculates maximum girder moments considering live load and temperature changes simultaneously in a non-linear way. Applicable only to three-span, two-hinged suspension bridges and can consider only distributed live loads, not concentrated live loads (In the long-span suspension bridges, the effects of concentrated loads are not controlling).
The detailed numerical theory itself is also quite interesting and will be discussed later.
Figure 1. Girder moment comparison (Linearized Finite Displacement vs Trigonometric Method)
Figure 1 shows the girder moment comparison between the Linearized Finite Displacement and Trigonometric Method. The model is again the so-called A-Bridge and details are attached. The maximum difference is around 15% and the main reasons are
In the trigonometric method, the effects of temperature changes are included, while not in the linearized finite displacement method.
The assumed shape functions are different. (Trigonometric vs cubical).
Theoretically, the trigonometric method gives a more exact solution than the linearized finite displacement method. We can not consider the non-linear effects of temperature changes in the linearized finite displacement method.
From the author’s view, both methods give solutions with enough precision for practical application. The main shortage of trigonometric methods is, it is not easy to consider for wind/seismic loads. But, as can be seen from the excel worksheet, it is extremely easy to analyze for live loads with temperature.
This method is very useful at least in the preliminary design or design checks. With modern computers, the effectiveness of the trigonometric method shall be re-evaluated.