Sunday, May 26, 2013

Two Hinged and three hinged arches




Arches are the structures, which look somewhat different from the columns and beam. They have the curved shape, of an arch, which can be circular or parabolic.  In Civil Engineering, you have to study the analysis of the arches.

In engineering terms, there are three types of arches,

  1. Two hinged arches
  2. Three hinged arches.
  3. Fixed Arches

Three hinged arches are the determinate structures, because there are four unknown support reactions, and again there are four numbers of equations of equilibrium, to get the values of these unknowns.

Three hinged arch:

See above in fig.2, there are three hinges in the arch, A, B and C. Generally there are three numbers of equilibrium equation, but the fourth equation is derived from the fact the algebraic sum of all the moments at the hing C is 0.  So there are four numbers of equilibrium equations, and we can determine all the four support reactions, Va, Vb, Ha, and Hb.

Two hinged arch:

In Fig.1 there are two hinges A and B, and there are four support reactions. There are only three numbers of equations of equilibrium, so two hinged arches are indeterminate to the degree equal to 1.

 If we have to find out all the four unknown reactions of the two hinged arch, then, we need one more equilibrium equation. So the indeterminacy of a two hinged arch is equal to 1. 

We can easily find out the Va and Vb, by taking algebraic sum of all the moments about A or B equal to 0.  To find out the horizontal reactions Ha and Hb, many books advise to use the Castigliano's first theorem.

The relative displacement of the either hinge with respect to other is zero, so the partial derivative of the strain energy of the beam with respect to the horizontal reaction will be zero.
 So first we have to find the equation of the strain energy of the whole beam, and then partially differentiate it w.r.t. to the horizontal reactions, and then equate it to zero. 

It becomes the fourth equation, and we can get the value of the horizontal reaction. Now as all the support reactions are found, we can easily plot the bending moment diagram, for the arch. 
Now at any cross of the arch the vertical and the horizontal forces can be resolved along two directions, one is tangent to the cross sectional surface of the arch and another is normal to the cross sectional surface of the arch. It gives rise to another two terms:

  1. Radial Shear
  2. Normal thrust. 

Radial Shear and Normal Thrust

Due to the curved shape of the arch,  unlike shear force and axial thrust in a straight  beam, the direction of the normal thrust and the radial shear change constantly from one end of the arch to the other.
To find out the magnitude of the radial shear and the normal thrust, first we have to find out the direction of these two.

 To find out the direction, we simply find out the slope of the arch(angle w.r.t. horizontal), at that section which, can be found by writing the equation of the circle or the parabola in terms of y and x co-ordinates, depending upon the shape of the arch and  then differentiating the y w.r.t. x gives us the slope of the equation.

Putting the corresponding value of the x at the given cross section, gives us the slope at the given cross section. After finding the angle, it becomes easy to resolve the forces into two directions, of the radial shear and the normal thrust. In the figure shown below, you have to resolve the forces at the section along the R and N, to find out the Radial Shear and the Normal Thrust.

Rib Shortening: 

As we can see that there is a normal thrusting force, which applies the compressing action to the rib/arch. If the magnitude is high it may result in the change in the length of the arch, due to corresponding strain, which can be found out by the Hooke's law. Final result will be the shortening of the arch. This effect is known as the Rib shortening. 

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