Stayed Bridge

Structural Studies Bridges

Figure 1 The Sunniberg Bridge, Klosters, Switzerland (image credit: Maria Garlock)

Cable Stayed Bridges

WContents 1. Form: The Cable Stayed Bridge 2. Forces: The Cable Stayed Bridge 3. Forces: Axial Force in a Cable 4. Tower Reactions 5. Anchor Reactions Optional Reading

Princeton University Department of Civil and Environmental Engineering Professor Maria E. Garlock Moreyra

Structural Studies Bridges

1. Form: The Cable Stayed Bridge A cable stayed bridge belongs to the family of cable supported bridges, such as suspension bridges, but unlike these, the cables directly connect the deck to the towers (often referred to as “pylons” in cable stayed bridges). The Sunniberg Bridge (Figure 1), close to the village of Klosters, Switzerland, designed by Christian Menn is an example of a cable stayed bridge.

It was completed in 1998 and it has gained worldwide recognition as a symbol for technical excellence.

There are two main cable-stayed systems. One is a fan system (Figure 2, top) where all the cables are connected to a single point on the tower top. The other one is a harp system (Figure 2, below) where all the cables are connected at different heights along the tower (parallel stay cables are used). Aesthetics is the main reason why we use a system or another.

(Pylon)

Figure 2 Cable stayed bridge systems: Top: fan arrangement Below: harp arrangement

2. Forces: Deck, Cables, Towers As with suspension bridges, the dead load of the bridge deck and the live load from the traffic carried by the bridge are the dominant loads in this structure. Wind loads, though important for the analysis and design of the bridge, will be neglected in this study for simplicity. The loads from the deck are transmitted by the cables directly to the towers. This analysis will focus on the reactions at the end of the cables on the tops of the towers and the anchors, and the axial force in the cables.

3. Forces: Axial Force in a Cable The tensile force that develops in every cable in a cable stayed bridge acts along the axis of the cable, and must be resisted by a horizontal and vertical reaction at the towers and the anchors. Unlike suspension bridges, the axial force in a generic cable in a cable stayed bridge is constant from the tower to the deck.

Cable Stayed Bridges — 2

Structural Studies Bridges

Let us start determining the axial force in a generic cable assuming the stay cables anchorages at the deck will generally be spaced equidistantly, l1 (Figure 3), and that each

cable carries a proportion of the gravity loads based on the distance to the adjacent cables.

This distance is often referred to as the tributary length.

We can approximate the gravity loads as a uniformly distributed load, q (lb/ft or k/ft) acting on the deck (Figure 3, detail). Let us define θ as the angle between a generic cable and the deck (Figure 3, detail).

Figure 3 Middle of the main span in a harp design Detail: Tributary length and free-body diagram of the cable attachment

To simplify our analysis, we will use a harp arrangement and analyze the longest cable in the middle of the main span. The axial force can be found by maintaining vertical equilibrium in the free-body diagram of the cable:

Equation 1 Axial force in a cable

It turns out that since we used a harp design with even spacing, each of the cables is attached at the same angle so the axial force will be the same for each cable.

Would we obtain the same results if we use a fan design for the cable stayed bridge? With a fan design all of the cable angles will be different (Figure 4), so applying Equation 1 it turns out that the cable with the smallest angle, θ1, will have the highest axial force.

Figure 4 Middle of the main span in a fan design

Cable Stayed Bridges — 3

Structural Studies Bridges

4. Tower Reactions The behavior of a cable stayed bridge tower differs from that in a suspension bridge tower.

In a suspension bridge, the cable continuously passes over the tower and sits on a “saddle”.

Therefore, the cable of a suspension bridge typically only induces vertical loads on the tower (the horizontal component of the main span cable force is balanced by the horizontal component of the back span cable force).

In a cable stayed bridge, unbalanced loads (meaning loads that are not uniform along the entire bridge) create a horizontal component of force on the tower. The cables connecting the deck to the towers therefore apply horizontal and vertical forces to the towers, which produce compression (vertical reaction of the cable at the tower) and bending (horizontal reaction on the tower) (Figure 5).

The vertical reaction due to one cable Vcable (lbs or k), at a tower is, based on Equation 1 (Figure 5):

Equation 2 Vertical reaction due to the cable at the tower

where q is the uniformly distributed load applied to the bridge (lbs/ft or k/ft) and l1 (ft) is the tributary length, to the distance between anchorages in the deck.

Figure 5 Tower reation due to a cable

The horizontal reaction due to one cable Hcable (lbs or k), at a tower is, taking into account Equation 1 (Figure 5):

Equation 3 Horizontal reaction due to the cable at the tower

where θ is the slope of the cable.

This analysis is for one cable. The total reaction at the tower is the sum of all cables.

Cable Stayed Bridges — 4

Structural Studies Bridges

It turns out that since we used a harp or a fan design with even spacing, the vertical reaction induced in the tower by each of the cables is the same, independent of the slope θ of the cable, while the horizontal reaction increases with decreasing slope of the cable.

Unbalanced horizontal reactions at the tower induce bending. Considering dead and live loads, towers in cable stayed bridges are therefore typically stiff and strong to resist bending due to unbalanced live loads.

Example Fan Arrangement: Millau Viaduct Determine: The axial force in cables A and B (See Figure 6) of the Millau Viaduct (France).

The horizontal and vertical reaction in the tower due to the sum of cables A and B only See Figure 7.

Given: Gravity Load q = 24 k/ft

Tributary Length, l1 = 41 ft

Angle of cable A and B, θA = 46°, θB = 27°

Figure 6 Millau Viaduct Elevation (not to scale)

Figure 7 Millau Viaduct Photo credit: Simon Hart 1

Cable Stayed Bridges — 5

Structural Studies Bridges

Solution: Step 1: Solve for axial forces in cables:

Step 2: Solve for horizontal and vertical reactions in towers (cables A+B only)

Note that to find the total reactions at the tower you need to sum all cables.

5. Anchor Reactions We can find two different ways the cable systems of a cable supported bridge can be anchored at the end supports.

In the earth anchored system (Figure 8) both the vertical and the horizontal components of the cable force are transferred to the anchor block (like in a typical suspension bridge), whereas in the self-anchored system, (Figure 9), the vertical component is taken by the anchor pier and the horizontal one is transferred as compression in the deck.

Figure 8 Earth-anchored system

Cable Stayed Bridges — 6

Structural Studies Bridges

Figure 9 Self-anchored system

In principle, both systems can be applied in cable stayed bridges and in suspension bridges.

Nevertheless, self-anchoring is typically used for cable stayed bridges and earth anchoring for suspension bridges.

To maintain vertical equilibrium, the vertical reaction in the anchor must have the same value as the vertical reaction of the anchored cable, Vanchor = Vcable.

The next section is intended for those who want a more advanced understanding of the relationship between the anchor reactions and foundations. It is optional reading.

Cable Stayed Bridges — 7

Structural Studies Bridges

Optional Reading: Detailed effect on foundations In self-anchored systems the horizontal component of the axial load of the cable increases the compressive forces at the deck, while, in earth anchored systems that horizontal component must be supported by the counterweight where the cable is anchored. The size and dimensions of these counterweights are defined so that they are prevented from overturning, being uplifted or sliding for the highest tensile load transferred by the cable.

In a self-anchored cable stayed bridge, the way the vertical components of the cable is supported is shown in Figure 10, which is not as simplified as Figure 9.

Figure 10 Self-anchored system

The vertical component of the cable axial load, Vcable is balanced by the weight of the pier,

Vpier and the weight of the footing, Vfooting.

To prevent the footing from being uplifted, the weight both of the pier and the footings must be greater than Vcable.

In an earth anchored system of a suspension and cable stayed bridge, the load of the cable,

Ncable, must be supported by a counterweight. As shown in Figure 11, the reaction of the ground, Vground, must be positive, since no tension stress can be transferred through the soil-counterweight interface. Since Vground = Vanchor – Vcable, it means that at least, the weight of the counterweight must be greater that the vertical component of the cable.

Figure 11 Earth-anchored system

Cable Stayed Bridges — 8

Structural Studies Bridges

In addition, Vground will be eccentric (i.e., not in line with) to Vanchor, since it needs to balance

the overturning moment induced by the cable. The counterweight will tend to rotate about the point E. The distance dG between Vground and E can be obtained by taking moments at E:

Since Vground = Vanchor – Vcable, and,

Since Vanchor > Vcable, as we said before, if dG is to be positive (i.e. is the counterweight is not going to overturn)

Additionally, the counterweight cannot slide. The maximum value of the horizontal load that can be developed at the soil-structure equals:

where μ is the friction coefficient. It must be verified that:

In addition to what is discussed here, there are additional ways to ensure the stability of counterweights in suspension and cable-stayed bridges so that footing are not uplifted, such as ground anchorages.

Cable Stayed Bridges — 9

Structural Studies Bridges

w Summary of Terms q L l1 θ Ncable Vcable Hcable Vanchor Hanchor

distributed load acting on the deck

lbs/ft or k/ft

main span length

ft

distance between cables anchorages at the deck

ft

slope of the cables axial force in a cable

lbs or k

vertical reaction at tower due to a cable

lbs or k

horizontal reaction at tower due to a cable

lbs or k

vertical reaction in the anchor

lbs or k

horizontal reaction in the anchor

lbs or k

w Summary of Equations Axial force in a cable

Vertical reaction due to the cable at the tower

Horizontal reaction due to the cable at the tower

Vertical reaction at anchor (self-anchored system)

w Visual Glossary (Pylon)

Notes 1 Image obtained from Flickr from author Simon Hart (https://www.flickr.com/photos/sihart/4065774479) , licensed under the Creative Commons Attribution-ShareAlike 2.0 Generic

Cable Stayed Bridges — 10