Expert Tips: Nonlinearities in Cable-Stayed Bridge Analysis

Why is cable-stayed bridge difficult to analyze? 

Cable-stayed bridges, including so-called Extradosed bridges, do not have any classical solutions. Computer based displacement method is the only way to analyze cable-stayed bridges and it is hard to check the output. Also, in the cable-stayed bridge analysis, the following three nonlinearities should be considered. 

The first nonlinearity is from cable sag effects. 

Figure 1. Cable sag (Cable Supported Bridge, 3rd ed., Gimsing and Georgakis , 2011)

Cables may look straight but cannot be, except vertical cables, due to cable self-weights. In figure 2, the cables have the same chord length L, the same elastic modulus E, and the same area A. If we ignore the sag effect, the cable axial stiffness is identical for each case as K=EA/L. However, we can feel that their axial stiffness shall not be identical, and the second case shall be more flexible. 

Figure 2. Cables with different sag. 

Therefore, how can we catch the stiffness differences in this case? We have some options.

• Catenary elements:

Catenary elements give theoretically exact solutions and intermediate nodes are not needed except loading points. For detailed element formulation, refer to midas manual. 

• Non-linear (or linearized) truss elements:

A single cable can be divided in to multiple nonlinear or linearized truss elements, as shown in figure 3. A cable shape can be considered this way, as well as its internal forces. 

Figure 3. Single cable modeling with multiple nonlinear/linearized truss elements. 

Actually this is the way we model the cables in a suspension bridge. We have to divide the main cables into many numbers of straight truss elements due to the hanger connection. We can model each cable segment using catenary element, but little benefit can be expected. 

• Equivalent Truss Elements: 

With the sag increases, the cable stiffness decreases. We can consider these effects as reduced area A_effect. Here, w is the cable weight per unit length and T is the cable tension force. Since T is not constant, this method can make the analysis nonlinear. However, if we can assume T as constant, the calculation can be a lot easier. 

There are pros and cons for each method: 

Catenary elements give the most exact solutions, but iterative analysis is essential, and the rule of superposition is not valid. This means we cannot use influence line analysis, and the calculation would be very voluminous and not practical in many cases.

Nonlinear truss elements have been used before the age of catenary elements (which was not that long ago), but seem not to have much of benefit over other methods. If we need to check the displacements within a single cable element, the linearized truss elements may be a good option, but we have to calculate each nodal coordinate and internal force. The equivalent truss element has been used for a long time and still gives very reasonable results with minimum efforts. 

The second nonlinearity is from P-delta effects.

In the cable-stayed bridges, the girder, just like the tower, resists axial compression. Therefore, we need to consider the P-delta effects for girder design. To see how P-delta analysis is performed in midas Civil, check out the previous tip on geometric nonlinearity explained. 

The third nonlinearity is also part of the P-delta effects. 

In the cable-stayed bridges, the loading between cable anchoring points are transformed into equivalent nodal point loads at the nodes. This converting equation for a beam with axial force is different from that of the conventional beam elements. Strictly speaking, we have to calculate the equivalent loads using more complex equations, which is also nonlinear. However, it is known that conventional methods give reasonable results in most cases. One of the reasons being that the beam element length is restricted to relatively short due to cables.  

What is extradosed bridges? 

The strands that are used for the prestressed concrete and for the cable-stayed bridge are just the same. The only difference is the test item and the test itself is more severe for the cable-stayed bridges. This means the manufacture never makes anything less qualified for the prestressed concrete strands, but the test items are different for different applications.

This means that the allowable stresses are different for each case: 0.6 fpu for the prestressed concrete strands, and 0.45 fpu (or 0.40 fpu according to specifications) for the cable-stayed bridge strands. 

It is understood that this 15% - 20% difference mainly comes from the stress variations. For the prestressed concrete bridge strands, the stress variation is relatively small, and it is rather large for cable-stayed bridges. 

Figure 4. Allowable stress vs. stress variation (Extradosed Bridges in Japan, Kasuga, 2006)

From this, Kasuga proposed to use 0.60 fpu if he stress variation is less than 14.5 ksi, 0.40 fpu is the stress variation is larger than 19 ksi, and interpolate between the two limits for prefabricated wires. 

Extradoses bridges are something between prestressed concrete bridges and long-span cable-stayed bridges. We can increase the allowable stress according to the stress variation in the cables and finally the design would be more economical. 

Figure 5. Minimum cost tower height for 500 ft span railway bridges. 

Figure 5 is preliminary cost comparison to find the most economical tower height for a three span extradosed railway bridge (280 ft + 500 ft + 280 ft). Figure 5 shows tower height 50 ft is the most economical and the optimal H/L ratio of 50/100 is quite low comparing to conventional cable-stayed bridges.