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Beam Theory I
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Objective

1.To plot the experimental flexural stress against the applied moment (load cell force times lever arm) and verify graph is a straight line and to compare the plot obtained to the model plot shown in figure.

2. To calculate the slope of the graph and verify it to be the geometric parameter (y/I), where 'y' is max fiber distance from neutral axis and 'I' is the second area moment of Inertia 

Introduction

The Euler Bernoulli’s theory also called classical beam theory (beam theory 1) is a simplification of the linear theory of elasticity which provides a means for calculating the load carrying and deflection characteristics of beams. This theory covers the case for small deflections of a beam that is subjected to lateral loads alone. This theory is based on the flexure formula.

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

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.See 

figure above. 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.

i.e., F=0,

The Elastic Flexural formula:

Consider the cross sections of HE and GF of the beam, when the beam is to bend, it is assumed that these sections remain parallel, the final position of the sections are still straight lines, they then subtend some angle, θ.

 

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’

Figure 2 Beam under bending

 

Figure 3 Cross sectional view

 

The Experimental setup

The experimental setup that we are using for Flexure law has a cantilever beam with 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 (fig. 2 and 3). 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.

Fig 2: View of the Flexure law experimental setup

 

Fig 3: View of the Cantilever beam with strain gage mounted on it.

The Load value is obtained independently from a load cell. This load is multiplied by the lever or moment arm to the respective strain gage and converted to Moment (N.m). The Moment is then plotted on the x-axis as the independent variable

sigma = frac{My}{I}

The Strain, on the other hand, is obtained independently using the affixed Strain Gages. The strain (microstrains) is then converted to the flexural stress by:

sigma = Eepsilon

Finally the Stress vs. Moment graph is plotted. The value of (Y/I) for the cantilever beam made of mild steel is obtained as the slope of Bending Stress-Moment graph, and should remain constant for the beam, unless there is significant deterioration with time. The results are then verified and errors (if any) are accounted for.

Cite this Simulator:

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