DEFLECTION OF BEAMSIntroduction
Any kind of modern day construction makes use of a component called structural beam, or beam. A beam helps in adding strength of any design or structure. Generally manufactured using steel, wood, or concrete, it is commonly used for spanning any open element of a structure. In any design work based on iron, steel I beam is the most commonly used one. On the other hand, home-based constructions generally make use of wooden beams. When steel I beam is encased within concrete, it forms a concrete beam and is typically used in the construction of heavy structures, such as bridges, buildings, etc. Some other less commonly used beam styles are manufactured using angle iron, C channel, or round steel pipe.
A structural beam is generally used across huge and open components, such as huge windows and doorways, or connecting the openings of one room to another in a house. Generally, floors above a basement or crawlers are also divided using structural beams. They act as strong components in the structures, which prevent sagging or bouncing of the finished floors. These type of beams are generally made of steel I beam, but with modern technology advancing, wooden beams are also gaining popularity.
The most common shape for the beams, both wooden and steel, is the I shape. When used in horizontal applications, it provides superior strength against bending. However, it can also be used in vertical structures, such as pillars. It is generally installed in a flat level position. However, some beams, which are used in the construction of bridges or the road reinforcing, are typically installed in the pre-stressed forms as well.
A pre-stressed beam allows the beam to provide improved strength for supporting heavier loads. Based on the beam design, a round steel pipe has the ability to provide better strength, as compared to the equivalent beam made of angle iron or C channel, but will be beaten by a properly placed I beam.
An I beam is typically preferred in situations where only the vertical load is pushing down on the beam. The applications where the back and forth forces accompany the downward force, an I beam can give way to the shifting of building materials.
Types of beams
Beams are the components which have the ability to transfer the loads horizontally, along the length of the component, and the loads are usually the vertical forces. Different types of beams are:
Simple beam: or simply supported beam, is supported at both the ends of the beam only. If a downward pressure is applied to the beam, the bending of the beam would occur at the middle.
Continuously supported beam: These beams are supported by multiple support points, usually three or more. The deflection exhibited by these types of beam is lesser than the simple beams, because of the cancellation effect of the positive and the negative bending experienced by the beam. Generally, a continuous beam span shows upto 20% more efficiency than a span of a simple beam, because it is able to span longer distances.
The downward bending occurs in between the first and the middle support points, and another between the middle and the last support points. The middle of the beam exhibits an upward bending.
Cantilever beams: These beams overhang on their supports. The cantilever area of the beam is unable of supporting the loads same as the back span, i.e. the section of the beam between the supports. Typically, the amount allowed for cantilever section in a beam is 30% of the back span. For example, a 45 m beam can have a cantilever section of maximum 15 m.
This beam bends only between the supports.
Deflection of beams
The deformation caused in a beam because of the load is measured in terms of deflection as compared to its original position, when it was unloaded. It is measured from a neutral surface of the beam to the neutral surface of the deformed beam. The configuration, which a deformed beam assumes is known as the elastic curve of the beam.
Methods to determine beam deflections
Deflection in beams can be determined using numerous methods. Some of these methods are:
Double-integration method
Area-moment method
Strain-energy method
Conjugate-beam method
Method of superposition
Out of these methods, the double integration method and the area-moment method are the most common ones that are used to determine the beam deflections.
Young’s Modulus
A modulus is a number, which is used to represent a physical property of a material. Young’s modulus is the modulus used to measure the elasticity, meaning, it indicates how easy or difficult it is to deform or stretch a material.
When a material is stretched, the stress experienced by it is directly proportional to the strain applied on it, provided the stretch is not beyond the proportionality limit.
Here, stress can be defined as the force per unit area.
σ = F / A
where,
σ = stress (N/m2)
F = Force (N)
A = Area of object (m2)
Types of Stress:
Tensile stress: The stress which, when applied on an elastic material, stretches or lengthens it. This stress acts normal to the area stressed.
Compressive stress: The stress, which shortens or compresses the material. Like tensile stress, this stress also acts normal to the area stressed.
Shearing stress: This type of stress has the ability to shear the material. This stress acts on the plane of the stressed area, and thus, is normal to the compressive or tensile stress.
Strain can be defined as the ratio of deformation over the initial length of the material.
It can be calculated by the change caused in the dimensions of an object divided by the original dimensions. It is expressed as below:
ε = dL / L
where,
ε = strain (m/m)
dL = stretch / elongation or compression (m)
L = original length of the object
Below is the graph gradient, which gives us the value of Young’s modulus for a particular material:
Young’s modulus = (Tensile Stress) / (Tensile Strain)
E = (F/A) / (ΔL/L)
E = FL / AΔL
Where, E = Young’s modulus (in Pascal)
F = Force (in Newton)
L = original length (in metre)
A = Area (in square metre)
ΔL = Change in length
Young’s modulus is also known as Tensile Modulus, and is used to measure the stiffness of an elastic material. It describes the elastic properties of materials when they are stretched or compressed, such as wires, columns, or rods. Thus, Young’s modulus can also be defined as the ratio of stress along strain.
The modulus number can be used to represent the elongation/compression of a material, provided the stress applied is lesser than the strength of the material.
Elasticity
Elasticity is the property of a material to restore back to its original shape after it is distorted on the application of external or internal load. A very good daily example of an elastic object is a spring. When it stretches on the application of external load, it internally exerts an internal force, which restores it back to its original form/length. This internal force for restoration is directly proportional to the stretch exhibited to the spring, and can be expressed by Hooke’s law.
Hooke’s law
One basic yet important property of an elastic object is that it takes double the amount of force to stretch a spring to double the distance. This linear dependence of the displacement exhibited on the application of stretching force is called Hooke’s law:
Fs = -k dL
Where
Fs = Force in the spring (N)
k = Spring constant (N/m)
dL = elongation/stretch of the spring
Yield Strength
Also known as yield point, yield strength can be defined as the amount of stress that a material can bear and undergo before it moves from elastic deformation to the plastic deformation, i.e. to a point from where it can not restore back to its original shape/length.
Ultimate Tensile Strength
The Ultimate Tensile Strength, or UTS, of a material is the maximum stress at which the material reaches its breaking point, or it actually breaks. Due to the break point, it suddenly releases the elastic energy stored within the material.
Procedure of the Experiment
The experiment was carried out using the following instructions:
Measured the length, width and height for each beam 3 times. Calculated the average to be used in the calculations.
Marked the centre of the beam on each side at a distance of 600 mm.
Measured the height of the central point on the deflection measuring device.
Applied the load in the increments as shown in the table below:
Entered the results in the results table and completed the calculations
Plotted a graph of Load (N) against deflection (mm) and calculated the gradient.
Repeated the above steps for aluminium and timber beams as well.
Data and Results
The load for steel beam increments at 10 N. Lower the deflection, more rigid the material. The deflection for steel per 10 N load increases at 0.3 mm on an average. This indicates the high rigidity of the material
When the load is completely removed, the steel comes back to its initial position with the minimum deflection of 0.01 mm.
The load for aluminium beam increments at 5 N. The deflection for aluminium per 5 N load increases at 0.6 mm on an average. This indicates that aluminium exhibits the rigidity of medium level.
When the load is completely removed, aluminium comes back to its initial position with a low deflection of 0.02 mm.
When compared to steel, we can see that at lesser load increments, aluminium deflects more than steel.
The load for timber beam increments at 1 N. The deflection for timber per 1 N load increases at 1 mm on an average. This indicates very low rigidity of the material, compared to steel and aluminium.
When the load is completely removed, the timber comes back to its initial position with a high deflection of 0.26 mm. This shows that it’s plastic in nature.
When compared to steel and aluminium, we can see that at extremely less load increments, timber deflects more than aluminium and steel.
Conclusion
Young’s modulus, or the modulus of elasticity, is a number, which is used to measure the elasticity of the material. Elasticity can be defined as the ability of a material to retain back its original form after being stretched or deformed. The change in original shape of a material caused due to the application of an external or internal load is called deflection.
As we can see from the experimentation results, the steel exhibited a very low deflection of 0.01 mm even on the application of extremely high load increments of 10 N, when compared to the load increment applied on timber, i.e. 1N. Timber, on the other hand, at low increments of 1 N, showed a high deflection of 0.26 mm.
This shows that the steel has a high ability of retaining back its original form on the removal of external load, while timber, once deformed, cannot come back to its original shape. As a result, we can confidently say that lower the deflection, higher the rigidity of the material.
It is also known that a stiff material requires a high amount of force to be deformed. Hence, we can say that Young’s modulus measures the stiffness of a solid material. From the results, a high value for Young’s modulus clearly points towards the high rigidity of steel (since the number is supposed to be high for the linear rigid materials. Higher the value of Young’s modulus, higher the rigidity.
Steel > Aluminium > Timber
Bibliography
Fleming, Robins. The Deflection of Beams. American Institute of Steel Construction, 1941. Print.
Megson, T.H.G. Structural and Stress Analysis. Elsevier Botterworth-Heinemann, 2005. Print.
Wolfenden, Alan. Dynamic Elastic Modulus Measurements In Materials. Philadelphia: American Society for Testing and Materials, 1990. Print.
