m5l33

m5l33 - Module 5 Cables and Arches Version 2 CE IIT,...

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Module 5 Cables and Arches Version 2 CE IIT, Kharagpur
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Lesson 33 Two-Hinged Arch Version 2 CE IIT, Kharagpur
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Instructional Objectives: After reading this chapter the student will be able to 1. Compute horizontal reaction in two-hinged arch by the method of least work. 2. Write strain energy stored in two-hinged arch during deformation. 3. Analyse two-hinged arch for external loading. 4. Compute reactions developed in two hinged arch due to temperature loading. 33.1 Introduction Mainly three types of arches are used in practice: three-hinged, two-hinged and hingeless arches. In the early part of the nineteenth century, three-hinged arches were commonly used for the long span structures as the analysis of such arches could be done with confidence. However, with the development in structural analysis, for long span structures starting from late nineteenth century engineers adopted two-hinged and hingeless arches. Two-hinged arch is the statically indeterminate structure to degree one. Usually, the horizontal reaction is treated as the redundant and is evaluated by the method of least work. In this lesson, the analysis of two-hinged arches is discussed and few problems are solved to illustrate the procedure for calculating the internal forces. 33.2 Analysis of two-hinged arch A typical two-hinged arch is shown in Fig. 33.1a. In the case of two-hinged arch, we have four unknown reactions, but there are only three equations of equilibrium available. Hence, the degree of statical indeterminacy is one for two- hinged arch. Version 2 CE IIT, Kharagpur
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The fourth equation is written considering deformation of the arch. The unknown redundant reaction is calculated by noting that the horizontal displacement of hinge b H B is zero. In general the horizontal reaction in the two hinged arch is evaluated by straightforward application of the theorem of least work (see module 1, lesson 4), which states that the partial derivative of the strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish. Hence to obtain, horizontal reaction, one must develop an expression for strain energy. Typically, any section of the arch (vide Fig 33.1b) is subjected to shear force V , bending moment M and the axial compression . The strain energy due to bending is calculated from the following expression. N b U = s b ds EI M U 0 2 2 (33.1) The above expression is similar to the one used in the case of straight beams. However, in this case, the integration needs to be evaluated along the curved arch length. In the above equation, is the length of the centerline of the arch, s I is the moment of inertia of the arch cross section, E is the Young’s modulus of the arch material. The strain energy due to shear is small as compared to the strain energy due to bending and is usually neglected in the analysis. In the case of flat arches, the strain energy due to axial compression can be appreciable and is given by, ds AE N U s a = 0 2 2 (33.2) Version 2 CE IIT, Kharagpur
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This note was uploaded on 06/30/2009 for the course CE 358 taught by Professor Trifunac during the Fall '07 term at USC.

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m5l33 - Module 5 Cables and Arches Version 2 CE IIT,...

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