Variation of Deflection of a Simply Supported Beam with Load, Beam Thickness and Material

Table of Contents

Abstract

Introduction

Theory:

Stress ( $\mathit{\sigma }$

):

Strain ( $\mathit{ϵ}$

):

Young’s Modulus ( $\mathit{E}$

):

Second Moment of Area of a Beam ( $\mathit{I}$

):

Relation between Stiffness, Thickness and Load

Pure Bending

Apparatus:

Procedure:

Results:

For Steel and Brass beams:

For 6.4 mm Aluminium beam:

Result Analysis:

Graph 1

Graph 2

Relation between Stiffness & Thickness:

Graph 3

Comparing Theoretical & Practical Values of ‘k’

Material Properties:

Graph for Steel & Brass:

Graph 4

Conclusion

References

Abstract

Bending of beams under load is a common phenomenon observed in daily life situations and is a very important factor to be considered while designing a structure or a component.

The objectives of this experiment are to prove that – stiffness and the beam thickness are proportional, the relationship between modulus of elasticity, stiffness and the beam dimensions. Finally, to show that beams with different materials and thickness have different stiffness values. There are many experimental methods for this and this experiment is done with simply supported beam arrangement.

This experiment mainly focuses on the deflections shown by beams with different materials and thicknesses. By plotting graphs for deflection (z) and load (W), it is observed that the graphs are straight lines for all the beams tested. This proves that the beams have deformed in their linear, elastic region and that ‘z’ is proportional to ‘W’.

Also, for same thickness of beams with different materials (Steel, Brass and Aluminium), it is observed that the steel beam has the highest stiffness value followed by brass and then aluminium. This confirms that the material property (Young’s Modulus) can determine the stiffness.

Introduction

Bending in beams is a fundamental characteristic which must be considered while designing any structure or a component. This phenomenon is observed in daily life situations and are caused by stresses generated due to loads applied. These stresses in-turn cause distortion in the length of the beam. If the beam bends in its elastic region, there is no permanent deformation in its length i.e., the beam gets back to its original length when the load is removed. Along with stresses and strains, the material property also plays a key role in determining the stiffness of a beam.

Theory:

Stress ( $\mathit{\sigma }$

):

It is the force applied to a component over a specific area and is given by,

$\mathit{Stress\; \sigma }=\frac{F}{A}$

Strain ( $ϵ$

):
It is the force applied to a component over a specific area and is given by,

$\mathit{Strain\; ϵ}=\frac{\mathit{\delta l}}{l}$

Young’s Modulus ( $E$

):
It is the measure of stiffness of a material. (A material with higher stiffness has the higher value of Young’s Modulus).

$E=\frac{\sigma }{ϵ}$

If a graph is plotted between Stress and Strain, the gradient gives the Young’s Modulus.

Second Moment of Area of a Beam ( $I$

):
For a rectangular cross-sectional beam, the second moment of area is given by,

$I=\frac{b{d}^3}{12}$

If a graph is plotted between Stress and Strain, the gradient gives the Young’s Modulus.

Relation between Stiffness, Thickness and Load

A beam with high thickness deflects less for a given load (W) than a less stiff beam. Stiffness depends on the material and dimensions of the beam. Stiffness (S) is the ratio of applied load to the deflection (z),

$S=\frac{W}{z} \left(N/m\right)$

Stiffness is proportional to thickness cubed, so the ratio of the stiffness to the thickness cubed is constant,

$\frac{S}{d^3}=\mathit{Constant}$

Pure Bending

When a beam is loaded such that it bends only in the plane of applied moment, the stress distribution and curvature of the beam are related by,

$\frac{M}{I}=\frac{\sigma }{y}=\frac{E}{R}$

Also, deflection of a beam subjected to a point load can often be expressed in the form of,

$\frac{z/W}{1/E}=\frac{k{l}^3}{I}$

Apparatus:

The main component of the beam apparatus is its steel frame which holds the beam, load cell supports, moving digital deflection indicators and the cantilever support. This whole setup sits on a level bench.

The digital deflection indicators measure the deflection of the beam at any point. The cantilever support holds the beam at one end. Load cells measure distance moved but have a calibrated support spring so that each 10N of downward force moves the indicator by 1 mm. They can act as reaction force indicators when not locked and simple beam supports when locked.

A weight hanger holds the weights to load the beam at any desired point on the beam. There is a graduated scale at the top of the apparatus which helps in applying loads at repeatable distances. Storage hooks helps in the storage of unused beams.

Procedure:

1. Draw two blank result tables to record the deflections of the beams for different loads and thicknesses. One table is for the steel and brass beams, the other for aluminium beam because the aluminium beam quickly bends out of the range of deflection indicator with large weights. So, smaller weight divisions must be used in a separate table.
2. Measure the length, thickness and width of the beam and mark it at mid-span of its length using a pencil.
3. Choose a suitable reading close to the centre of the graduated scale of the apparatus to match with the pencil mark on the beam.
4. Set up the beam and two load cells. Make sure that the two load cells are equidistant from the centre of the beam and their locking pins are fitted.
5. The centre mark on the beam must be directly under the scale reading chosen in
   step 3.
6. The beam will now have an overhang on both the ends.
7. Hang the weight hanger at the centre of the beam.
8. A digital indicator is now placed on the upper cross member such that its contact rests directly above the weight hanger and check whether the stem is vertical and there is enough travel downwards.
9. Zero the indicator and start applying loads to the weight hanger in increments of 5N for steel and brass beams and 2N for aluminium beam.
10. For each load and thickness of the beam, take the readings of the deflection in the respective tables.

Results:

The deflections recorded for different beam thicknesses and loads are tabulated below.

For Steel and Brass beams:

For 6.4 mm Aluminium beam:

Result Analysis:

To find the stiffness of each beam, plot a graph taking deflection (z) on Y-axis and Load (W) on X-axis and find the gradient of each graph. This gradient is equal to the inverse of the stiffness (1/S). The gradients of all the graphs are recorded in a table and their respective stiffnesses are calculated.

Graph 1

Graph 2

Relation between Stiffness & Thickness:

$\mathit{Stiffness }\left(S\right)=\frac{W}{z}\mathit{ N}/m$

This graph is plotted by taking the stiffness value at constant load of 10N for different thicknesses. (6.4 mm, 4.8 mm, 3.2 mm)

Graph 3

Comparing Theoretical & Practical Values of ‘k’

Material Properties:

Graph for Steel & Brass:

Graph 4

Values obtained by Gradient of the Graph 4:

We know that Second Moment of Area for a rectangular cross-sectional beam,

$I=\frac{b{d}^3}{12}=\frac{19*{6.4}^3}{12}= 4980.74\mathit{ m}{m}^4$

$\frac{1}{S}=\frac{z/W}{1/E}=\frac{k{l}^3}{I}$

Also, $z=\frac{W{l}^3}{48\mathit{EI}}$

for load acting at the centre of a simply supported beam.

Therefore, the theoretical value of ‘k’ is $\frac{1}{48}=0.021$

Conclusion

• The graphs for all five beams are straight lines which confirms that the deflection is proportional to load and that they are deformed in their linear, elastic region.
• Among Steel, Brass and Aluminium, steel beam has the highest value of stiffness followed by Brass and Aluminium. This proves that Stiffness can be determined by the material and its Young’s Modulus.
• The beam with highest thickness (6.4 mm) has the highest value of stiffness followed by
  4.8 mm beam and 3.2 mm beam. This shows that the Stiffness and thickness of the beam are related and the linearity of the graph of S Versus d³ proved the equation,

$\frac{\mathit{Stiffness}}{{\mathit{Thickness}}^3}=\mathit{Constant}$

References

     SM1004 User Guide

     Mechanics of Materials, SI Edition. Textbook by Dr. James Gere and Barry J. Goodno

     Images from www.images.google.com