From the previous tip geometric nonlinearity explained, Dr. Seungwoo Lee compared various approaches to approximate geometric nonlinearity by showing examples of how they varied the structural effects of the studied cantilever beam. One of the easier and more accurate approaches was through linearized finite displacement analysis in midas Civil, which was then compared with the hand calculation that considered stiffness reduction due to the compression of the cantilever beam. In this article, Dr. Lee shows the stiffness strengthening effects caused by tension in the cantilever beam, and demonstrates through example how that contributes to one of the key concepts in the suspension bridge analysis.
Figure 1 shows a loaded cantilever beam with axial tension, with P = 112.5 kips. T = 9726 kips, L = 148.708 ft. The column has diameter d = 10' 6". We want to calculate its maximum displacement and the moment at its base.
Figure 1. Cantilever beam with axial tension (from Mechanics of Materials by Goodno and Gere).
From linear analysis, the end moment is M = 16730 kip-ft. However, from geometric nonlinear analysis, midas Civil gives the following result shown in figure 2, where the end moment is somewhat reduced (13855.5 kip-ft).
Figure 2. (Top) Deflection in ft, and (bottom) base moment in kip-ft of the cantilever beam.
To check the computer output using hand calculation, first we assume that the structural system shown in figure 1 is equivalent with that in figure 3.
Figure 3. Cantilever beam with spring support (from Shegg study).
The increase in stiffness caused by axial tension can be represented by a function of T/L (page 164 of midas Civil analysis reference). Therefore the new stiffness of the system is shown below,
Both the end displacement and the base moment well match with the computer output. Compared with the ordinary linear analysis, the bending moment is reduced by 17%. In other words, the axial tension in the beam decreases its base moments and end deflections, and this effect is called tension stiffening.
We can calculate bending moments from simple linear analysis (M = 16730 kip-ft), but the output is somewhat larger than that from non-linear analysis (M = 13827 kip-ft). In this short, stiff structure, these effects may not be significant (17%), however, this effect may be huge in the long span suspension bridges.
If the stiffness of the beam is nominal like truss/cable elements, the cantilever in figure 1 is unstable in the linear analysis and the vertical deflection would be infinite. In the nonlinear analysis, the cantilever has vertical stiffness as the function of T/L can help resist load P. This concept is key for the suspension bridge analysis.
Figure 4. Suspension bridge (from Britannica)
In the suspension bridge, the girder is supported vertically by main cable tension force through hangers which can be assumed as inextensible. With linear analysis, we have to deal with (inaccurately) large moments, which result in design with large girder sections. This would cause self-weights being too heavy and finally fail the design of long span suspension bridge. Compared with the column/arch design, in which the nonlinear analysis is for safety issue, it is rather for economy or feasibility issue for the design of suspension bridges. Therefore, we do need nonlinear analysis to design long span suspension bridges.
Below is an example of doing real suspension bridge analysis using midas Civil.
Figure 5. Example bridge (from Steel bridge III by Hirai).
Figure 5 is a so called A-Bridge which was selected by Dr. Hirai as an example. This A-Bridge is medium sized and has a main span of 808 m (2650 ft).
Figure 6. Girder moment comparison, with blue line represents the bridge shape, solid red line represents live load moments considering geometric nonlinearity, dashed red line represents live load moments from ordinary linear analysis.
In figure 6, the blue line represents the bridge shape, the solid red line is the live load moments from the linearized finite displacement analysis which is the simplest method considering tension stiffening effects from the cable, and the dashed red line is the live load moments from the ordinary linear analysis. The maximum moment from linear analysis is almost 5 times than that of the nonlinear analysis, this dramatic difference shows why we should not design long span suspension bridges with only linear analysis.
Similar to the cantilever example, the above example shows we cannot consider the cable tension stiffening effects from linear analysis. The girder supports itself by its own stiffness, as well as by the vertical component of the main cable elongation, and they are both nominal. Recall that the transverse stiffness component of truss element is zero in linear analysis, therefore, if the main cable is horizontal, it cannot contribute any stiffness to the girder deflection in the linear analysis.
This concept was realized by Melan and the nonlinear equation itself was formularized in 1888, however, we had to wait sometime until brilliant engineer Moisseiff solved this equation and designed the Manhattan bridge using his own solution. The Manhattan bridge was opened to traffic is 1909 and bridge engineers call it the 1st "modern" suspension bridge.