An arch is defined as a plane-curved bar or rib supported and loaded in a way that makes it act in direct compression. Arch is one of the oldest and enduring structural elements of traditional architecture and is designed to carry predominantly vertical loads. There are three major types of arches in construction practice namely three hinged, two hinged and hingeless arches.
Two hinged and hingeless arches are statically indeterminate structures which are generally more economical, stiffer and stronger. The former is indeterminate to the first degree whereas the latter is indeterminate to the third degree. A hingeless arch is a very effective element, but does not suit lightweight applications such as transformable structures.
1. Two hinged Arches
In two hinged arches, supports permit the rotation of the arch at the ends under loads, temperature fluctuations, and horizontal support settlements. These make an arch relatively flexible and less prone to developing high bending stresses. Two hinged arches are statically indeterminate to the first degree with four reaction forces and three equilibrium equations.
Analysis of Two Hinged Arches
Since two hinged arches are statically indeterminate to the first degree, it becomes necessary to develop another equation to compute all the reactions and eventually draw shear and bending moment diagram.
The fourth equation is developed considering the of the arch. The unknown redundant reaction Hb is computed by noting that horizontal displacement of hinge B is zero.
The Hb is calculated using the theorem of least work which states that the partial derivative of the strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish. The general form of the equation used to evaluate Hb is as follows:
s: Length of centerline arch
M: Bending moment
E: Young’s modulus of the arch material
I: Moment of inertia of the arch cross-section,
H: Horizontal reaction
N: Axial Compression
A: Cross-sectional area at any coordinate
2. Hingeless Arches
In hingeless arches, also known as fixed-end arches, supports do not allow rotation of any kind and that is why a relative rotation or settlement at support creates significant additional stresses.
Hingeless arches are statically indeterminate to the third degree with three reactions and three equilibrium equations. Two hinged arches are used in common applications and are not suitable for lightweight applications.
Analysis of Hingeless Arches
This type of arch is statically indeterminate with third degree. Therefore, three more equations need to be developed to compute reactions and draw shear and moment diagrams.
There are a number of methods by which fixed-end arches can be analyzed, for instance: least energy method, column analogy method, and elastic center method. Least energy method which is discussed here is used for symmetrical arches with symmetrical loading.
As for unsymmetrical arches with unsymmetrical loading, least energy method can be used but requires extra effort. In this case, column analogy or elastic method can be adopted.
Unknown reactions can be found using strain energy formula. Considering only the strain energy due to axial compression and bending, the strain energy U is expressed as:
M: Bending moment
N: Axial force of the arch rib.
Because the support at A in fig.2 is fixed, equations for , shear, and axial force at that point can be written as follows:
As dimensions of the arch and loading are known, unknown redundant reactions Ma, Ha, Ray can be evaluated using the above three equations.
Due to the fact that the arch and the loading are symmetrical, the shear force at the crown is zero. Therefore, there would be only two unknowns at the crown. Hence, if the internal forces at the crown are taken as redundant, the expression would be simplified to:
s: Length of centerline of the arch
I: Moment of inertia of the cross section
A: Area of the cross section of the arch
If Mo and No are the bending moment and the axial force at any crosssection due to external loading, the bending moment and the axial force at any section is given by:
Equations 8 and 9 then can be further simplified by using equation 10, 11, and 12.