Arch Analysis

Assumptions and limitations adopted while analyzing an arch

1.     The cross section of the arch is assumed to be very small compared to its length.

2.    Action of torsion or twist is neglected, since the load act in the transverse direction of longitudinal axis.

3.     Self-weight of the arch is neglected.

4. The material of arch is isotropic and homogeneous with a constant Modulus of elasticity throughout.

5.     The resultant moment of bending stress is equal to the external moment along entire length of beam.

6.     The neutral axis neither undergo stresses nor change in length.

7.     Deflections are considered as very small compared to the length of the arch.

8.     In case of circular arch, the deflected shape follows a circular arc whose radius of curvature is large compared to its other dimensions.

Expression for radius of curvature of a circular arch

Consider the above figure Let R = Radius of the arch, L = Span of the Arch, r = Rise of the Arch, x and y are the co- ordinates of the point P from Origin O.

From Triangle OEP,

OP² = OE² + PE²

R² = (OC – EC)² + x²

R² = (R – (r-y))² + x²

R² = (R – r + y)² + x²

From the figure, x = OP sin θ = R sin θ

Similarly, y = OE – OD

                 

y = R Cos θ – R Cos α

We Know that in a segment of a circle, (2R – r) r = L²/4

Therefore, 2R = (L²/4r) + r

Hence, R= (L²/8r) + (r/2)

Expression for rise of an arch in a parabolic arch

Consider the above figure Let AB = L = Span of the Arch, CD = r = Rise of the Arch, 

x and y are the co- ordinates of the point P from Origin O

The general Equation of Parabola is given by

y=K x(L-x)

Where K is a Constant

At x = L/2, y = r

Substitute the above values in the general equation

r = K(L/2) (L – (L/2))

Therefore, K = 4r/L²

Substitute the value of K in the general equation

y = 4rx (L- x)/L²

Slope of the arch is obtained by differentiating the above equation wrt x

Tan θ = dy/dx = 4r(L-2x) / L²