Structure Analysis - Arches and Cables Complete Study Notes

INTRODUCTION

Beams generally transfer the applied load to end supports by bending and shear action but arches transfers load to abutments at spring points. The topmost point is called the crown which sometimes has a hinge. The height of the crown above the support level is known as rise. An arch is generally economical for a larger span compared to a simply supported beam.

Design Forces in Arches:

An arch will be subjected to three forces.

(a) Bending Moment

(b) Normal Thrust

(c) Radial shear force

1. TYPES OF ARCHES

There are three types of arches depending upon the number of hinges provided.

(i) Three hinged arch (Determinate)

(ii) Two hinged arch (Indeterminate to 1 degree)

(iii) Fixed arch (Indeterminate to 3 degree)

2.1.  Three hinged Arch

The three hinged arches are statically determinate structure as equations of equilibrium alone are sufficient to find all the unknown quantities.

Circular Arch:

From the properly of a circle the radius r of the circular arch of span
L and rise h may be found as

Note: If a two hinged or three hinged arch is subjected to UDL throughout its length, then BM and R_s are zero everywhere. The cross section is subjected to N_T (Normal thrust) only.

(iv) Temperature effects in 3-hinged arches

Case 1: When the arch is subjected to temperature rise alone

Since, three hinged arch is a statically determinate structure due to temperature variation or due to settlement of support, no stress are developed anywhere in the whole structure.

Although, there is no stress developed in the structure, the crown will rise by the value of Δh due to increase in temperature to accommodate the free expansion of the arch.

Here,

75 x 4 - 100 x 2 - H x 4 =0

H= 25 kN

Eddy’s Theorem: This theorem states that,” The Bending moment at any section of an arch is directly proportional to the vertical intercept between the actual arch and linear arch (Theoretical arch)”. Where linear arch is the shape of funicular polygon due to the loading.

2.2.   Two hinged arches

A two hinged arch is an indeterminate arch. The horizontal thrust is determined using Castigliano’s theorem of least energy.

Where,

αtl = due to increase in temperature

l/AE due to rib shortening

K = yielding of support/unit horizontal thrust.

In a two-hinged parabolic arch as the temperature increase, horizontal thrust increases. If the effect of rib shortening and yielding of support are considered then horizontal thrust decreases.

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