-Measure strains (using strain gages) on a beam in bending at fixed locations along the length of the beam, on tensile and compressive fibres (top and bottom surface)
- Obtain the plot of strains versus time (fig. ), as loading takes place, and show that the tensile and compressive fibres at same spanwise location read equal and oppposite strains
- The equal and opposite strains at fixed spanwise location indicates plane bending.
This theory is also based on euler bernoulli beam theory and is used to prove that strains on either end of the neutral axis are same for same loading under a set of assumptions.
Under this theory, the following assumptions are made:
- The beam is initially straight and has a constant cross-section
- The beam is made of homogenous material and the beam has longitudinal plane of symmetry
- Resultant of the applied loads lie in the plane of symmetry
- The geometry of the overall member is such that bending is the primary cause of failure and not buckling
- The stress levels attained during tension and compression do not exceed the elastic limit and it is well within the linear graphical region where Young’s modulus is obeyed
- The plane cross sections remain plane before and after bending
Fig 1 (a) shows the beam initially unstressed. Fig 1 (b) shows the state where the beam is subjected to a constant bending moment along its length as would be obtained by applying couples at each end and the beam bends to form with a radius R.
As a result of this bending, the top fibres of the beam will be subjected to tension and the bottom to compression. This means that there are points between these two where the stress is zero. The locus of all such points is known as the neutral axis.
The Euler Bernoulli equation describes the relationship between the beam’s deflection and the corresponding applied load. To analyse this, the concept of pure bending is applied such that the internal reactions developed on any cross-sections are considered to be by virtue of bending effects alone. The normal and shear force are considered to be zero on any cross section that is perpendicular to the longitudinal axis of the member.
The Elastic Flexural formula:
Consider the cross sections HE and GF in fig 1(a), when the beam is to bend, it is assumed that these sections remain parallel, i.e. H’E’ and G’F’, the final position of the sections are still straight lines, they then subtend some angle, q.
The fibre AB in the material is at a distance y from the neutral axis. As the beam bends, this will stretch to A’B’
Since CD and C’D’ are on the neutral axis and it is assumed that the stress on the neutral axis is zero. Therefore, strain on neutral axis = 0.
Consider an arbitrary cross section of the beam as shown above. The train on a fibre at a distance ‘y’ from the neutral axis is given by,
This is the bending theory equation.
The Experimental setup
The experimental setup that we are using for validation of Euler Bernoulli’s Theory has a cantilever beam with four strain gages attached on its surface. There is a motor on the base which uses a screw mechanism for translating the rotational motion to linear vertical movement. The linear end of the screw mechanism is then linked to the end of the cantilever beam using an S-type Load cell as seen in the below picture. When the motor is activated, the rotational motion pulls or releases the end of the cantilever beam depending on the direction of rotation. The load applied is then fed to the computer along with the respective deflection value from the strain gage. The beam is tested up to a predefined safe limit and then the unloading process takes place.
As per the Euler-Bernoulli’s theorem, a beam of sufficient length, when subjected to external load undergoes deformation and the bending deformation is proportional to the applied load. If the beam is pulled down fixing one end (like in a cantilever beam), then the upper surface of the beam will undergo tension and the lower surface will be experiencing compression. The stress on the top surface will be equal and opposite to the stress experienced in the lower surface of the beam. Since stress is proportional to strain as Young’s modulus for an isotropic material is constant, measuring the strain can help us determine the magnitude of stress in the upper as well as lower surface of the beam. This is accomplished by attaching a total of four strain gages, two at the top and two at the bottom of the beam. They have to be placed at a certain distance from the point of application of load (the free end of the cantilever beam) such that we are able to quantify the strain values obtained at the upper and lower portions of the beam at any 2 specific points. The loading and unloading is done using a motor as discussed above.
Fig 4: View of the Hooke’s law experimental setup
Fig 5: View of the Cantilever beam with strain gage mounted on it
Finally the motor is run and the beam loaded. The respective strain values are plotted against time and the graphs are seen to be symmetrical about the x-axis. The results are then verified and errors (if any) are accounted for.