Chapter 4- Arches - Lecture notes 1 PDF

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Arches notes...

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4 Arches The arch is one of the oldest structures. The Romans developed the semi-circular true masonry arch, which they used extensively in both bridges and aqueducts. Quite a few of the early Indian railway and highway bridges use masonry arches. They were constructed with brick or stone masonry with lime or cement mortar as the binding material. Arches are also used in buildings to carry loads over doorways, windows etc., as well as to add an aesthetic touch to the building.

4.1 Types of arches Arches may be classified, of course, based on the materials of which they are built; steel and reinforced concrete are the most common of all materials. From the point of view of structural behaviours, arches are conveniently classified as three-hinged, two-hinged and hingeless (also known as fixed) arches. Based on form, arches may be further classified as parabolic, circular, elliptical, etc. A number of arch forms are indicated in Figure 2.6 which vary in the manner they are supported and in the structural arrangement of the arch ribs. Open web arch ribs, though they resemble trusses, are considered as arches because of the manner in which the loads are transmitted. Of the three types of arches, only three-hinged arches are statically determinate and hence included in this section.

Figure 4-1.a) Three-hinged arch, (b) Two-hinged arch, (c) Fixed arch, (d) Tied Arch, (e) Two-hinged crescent arch, (f) Two-hinged spandrel braced arch

4.1.1 Three hinged arches If an arch contains three hinges such that two hinges are at the supports and the third one anywhere within the span, it is called a three-hinged arch (Figure 2.7). This type of arch is statically determinate wherein reactions, horizontal thrust and all internal structural actions can be easily determined by using the laws of equilibrium and statics. If the third hinge is provided at the highest point, it is called crown of the arch.

Figure 4-2.Three hinged arch 4.1.2 Analysis of three-hinged arches To determine the reactions at the supports, the arch is disassembled. In order to determine the reactions at the supports, the arch is disassembled and the free-body diagram of each member. Here there are six unknowns for which six equations of equilibrium are available. One method of solving this problem is to apply the moment equilibrium equations about points A and B. Simultaneous solution will yield the reactions Cx and Cy. The support reactions are then determined from the force equations of equilibrium.

Figure 4-3. Free body diagram

Once all support reactions obtained, the internal normal force (Nd), shear (Sd), and bending moment (Md) at any point along the arch can be found using the method of sections (Figure 4.4). Here, of course, the section should be taken perpendicular to the axis of the arch at the point considered. Vd Hd

Ha

Ray

Figure 4-4. Internal forces

Radial shear and normal force are given by: S d H a sin Ray cos  N d H a cos Ray sin      

4.1.3 Three hinged parabolic arches Consider the arch shown in Figure 4.5 Let’s refer to its height as h and use L to refer to the horizontal distance between the two supports.

Figure 4-5

Le P be at any point along the parabolic arch with coordinates (x,y), then;

4.2 Example 4.1 (to be done in class) A UDL of 4kN/m covers left half span of the 3-hinged parabolic arch of span 36m and central rise 8m. Determine the horizontal thrust also find (i) BM (ii) Shear force (iii) Normal thrust (iv)Radial shear at the loaded quarter-point from A

A three-hinged parabolic arch is loaded as shown in the figure below. Calculate the location and magnitude of maximum bending moment in the arch.

10KN/m

40KN

y

Ha

A

Ray

C

B

x

10m 40m

Hb

Rby

4.2.1 Circular/Segmental arches A segmental arch is a part of a circular curve. For such arches y 

4hx (l  x) is not applicable l2

since the equation is applicable only for parabolic arches. Similarly, the equation for the circular arch will be different.

4.3 Example 4.2 (to be done in class) For the circular arches shown below, Find: (i)

Support reactions

(ii)

Radial shear, bending moments and axial thrust at 10 m away from A...