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1. Arches and Cables for 4th semester B.E, KLEIT, Hubballi - 580027 Prof. Sandesh S. Huggi

2. Contents Introduction Types of Arches Introduction to Three Hinged Arches Different Types Derivation

3. Introduction: Beams transfer the applied load to end by bending and Shear action. In this process either one or two points at particular section is subjected to maximum stress. The material in most of portion is under stressed and hence is under utilized. The horizontal distance from one support to another is called span . For large spans like bridges, arches are provided in stead of beams. Arches are curved beams that transfer loads to their planes . Arches transfer loads to abutments at springing points. Hinges may be provided at springing points. The top most point is called crown . The height of the crown above support level is called rise.

4. Typical Arch Bridge Due to curved nature of arches, they give rise to horizontal forces. Abutments are designed for horizontal forces also. Any section in the arch is subjected to normal thrust, radial shear and bending moment . Loads in the arches partly transferred by axial compression and partly by flexure Typical Arch Bridge

5. Different types of Arches There are mainly three types of arches that are commonly used in practice: Three hinged arch, Two-hinged arch Fixed-fixed arch or Hingeless arch

6. Three-hinged arch is statically determinate structure and its reactions / internal forces are evaluated by static equations of equilibrium. Two-hinged arch and fixed-fixed arch are statically indeterminate structures. The indeterminate reactions are determined by the method of least work or by the flexibility matrix method.

7. Three Hinged Arch Bridges Two Hinged Arch Bridges

8. Three Hinged Arches: Three hinged arches is statically determinate structure and its reaction and internal forces are evaluated by static equations of equilibrium . Arches transfer loads to abutments at springing points . The horizontal distance between one support to another support is called span . The Top most point of the arch is called crown . The height of the crown above the support level is called rise . Figure 1:Three Hinged Arch

9. Types of Three Hinged Arch: There are two types of three-hinged arches according to the shape of the structure . Circular three-hinged arch Parabolic three-hinged arch

10. Circular three hinged arch From the property of a circle, the radius ‘R’ of the circular arch of span ‘L’ and rise ‘h’ Taking ‘A’ as origin at support, the coordinates of any point ‘D’ on the arch is:

11. Parabolic three-hinged arch In the case of a parabolic arch, taking the springing point as the origin its equation If the crown is taken as the origin, the equation of the parabolic curve = = a   Parabolic arch with origin at ‘A’

12. If Springing points are at same level: Parabolic arch with origin at ‘Crown’

13. Let h 1 and h 2 is the depth of the abutments from the crown and let ‘L’ is the span. If Springing points are at different level:

14. Cables Introduction: Cables are flexible structures that support the applied transverse loads by the tensile resistance developed in its members . Cables are used in suspension bridges, tension leg offshore platforms, transmission lines, and several other engineering applications . The distinguishing feature of a cable is its ability to take different shapes when subjected to different types of loadings.  Under a uniform load, a cable takes the shape of a curve, while under a concentrated load, it takes the form of several linear segments between the load’s points of application.

15. Stayed Cable Bridge over Neva River, St. Peetersburg

16. Suspension Bridge

17. Cables are used as temporarily guys during the erection and as permanent guys for supporting masts and towers . Cables are used in the suspension bridges. A suspension bridge consists of two cables with the number of suspenders ( hangers ) which support the roadway . Figure 1 shows a typical suspension bridges in which the cable is supported over towers:

18. To reduce the bending moment in the towers anchor cables are provided . The central sag or dip of the cable varies from 1/10 th to 1/15 th of span. The cables will be having either guided pulley support or roller pulley support as shown in Figure 2 below. In case of pedestrian suspension bridges, suspenders support the roadway directly.

19. For heavy traffic, large spans stiffening girders are provided to support the roadway as shown in Figure 3. Laksman Jhula ( Rishikesh ) and Howrah bridge ( Kolkatta ) are popular example of suspension bridges. Since , the number of suspenders are very large, the load on the cable may be taken as uniformly distributed load.

20. Equilibrium of Cables: A cable is a flexible structure which can not resist Bending Moment. In deflected shape of cable, the bending moment at any point of cable is zero which is achieved by developing horizontal thrust at the support. Consider the cable shown in Figure 4, which is subjected to various loads. Let the horizontal force developed at support is H. Let the vertical reactions at support A and B is VA and VB respectively

21. At section X-X, let the deflection be ‘y ’. Moment at section x-x = Mx = VA x – W1 (x-a1 ) – W2 (x-a2 ) – H y Since the cable is flexible, Mx = 0. Therefore , H y = VA x – W1 (x-a1 ) – W2 (x-a2 ). H y = Beam Moment. The loaded cable can be analyzed by using above equation at any segment of cable.