A brief introduction to analyze Fixed Arches

Hello,

Fixed arches are used as bridge arches or as tunnel roofs, where the span is short. Fixed arches restrict the horizontal, vertical or rotational movement at its two ends, so a fixed arch is an indeterminate structure.

As they have total of 6 supporting reactions, 3 at each end, they are indeterminate to a degree of 3. This can be  numerically justified as below:

Total numbers of support reactions:
R=3+3
Total numbers of equations of static equilibrium = r=3

So Indeterminacy,
E = R-r = 6-3 = 3
So, as to analyze a fixed arch we need a minimum of three extra equations, which can be derived from compatibility of the structure.

Here we will use the following compatibility conditions:
The displacement of the joint A with respect to joint B are zero in
(a) x- direction
(b) y-direction
(c) rotational movement (Angular)

These conditions can be used to form three equations.
Use the Castigliano's first theorem, which says that partial derivative of total strain energy of the structure with respect to the load gives us the displacement along the direction of the load.
So you can form the following three equations using this theorem:
(1) Partial derivative of total strain energy with respect to Ha is zero
(2) Partial derivative of total strain energy with respect to Ra is zero.
(3) Partial derivative of total strain energy with respect to Ma is zero.

These three equations along with the other three equations of static equilibrium makes total of six equations.
When they are solved simultaneously, you will get the six unknowns which are Ra, Ha, Ma, and Rb, Hb and Mb.

That is all for now, please write in to make this article better.
Thanks