The aim of this experiment is to verify Clerk-Maxwell’s Reciprocal Theorem and to determine the Young’s modulus of a beam using the apparatus provided. The theorem states that in a linearly elastic structure, the deflection at any point A due to a load applied at some other point B will be equal to the deflection at point B when the same load is applied at point A.
The Clerk-Maxwell’s Reciprocal Theorem apparatus consists of a rigid frame and a light beam with simple end supports in the form of a hinge at one end and a roller at the other end. The beam is loaded using a tire rod assembly with weights, and a traveling pedestal is used to support the dial gauge for measuring deflections.
To perform the experiment, the beam is placed correctly over the supports, and the dial gauge is mounted on the pedestal and placed under the beam exactly at mid-span, adjusted to read zero on the scale. The tire rod assembly is hung exactly at quarter-span, and the dial gauge reading is noted. The tire weights are then placed one by one, and the dial gauge reading is noted every time a weight is added or removed. The observations are continued while unloading the beam, and the process is repeated after interchanging the positions of the dial gauge and the tire rod assembly.
The observations are recorded in a tabular form and a graph is plotted with deflection on the X-axis and load on the Y-axis for both cases. The deflection at the quarter point due to the load at the center is calculated using the data obtained from the experiment. The Young’s modulus of the beam is also determined using the same data.
In structural engineering, the Clerk-Maxwell reciprocal theorem is used to relate the effects of a load on different points of a structure. The theorem states that the deflection of a beam caused by a load is equal to the deflection caused by the same load acting on a second beam, while the first beam is subjected to a unit load at the point where the second beam was originally loaded.
To find the Young’s modulus of the material of the beam, one can use the formula that relates deflection, load, span, moment of inertia, and Young’s modulus. The deflection under different loads in case (1) can be compared with those in case (2), and they will be found to be the same, thereby verifying the theorem. Alternatively, one can superpose the plot of load vs deflection of case (1) with that of case (2), and they will coincide, also verifying the theorem.
The Clerk-Maxwell apparatus used for this purpose typically consists of a beam with two types of supports: a hinge support at one end and a roller support at the other end. The depth of the beam in the apparatus is less than its width to ensure that the beam deflects only in one plane.