Exp 10.docx

Experiment no. 10 1. Objective: To determine the Load Point Deflection of a Simply Supported Beam subjected to an Eccentric Load.

2.Apparatus:      

Deflection of beam apparatus Hangers Weights Meter rod Dial indicator Vernier Calipers

3.Introduction: Deflection of beam apparatus contains a metal beam and fixed supports upon which the beam is supported for this experiment. With the help of clamps arrangement at ends it can be made fixed type of beam. In the simplest of situations, the beam is taken to have a rectangular cross-section and the loads and supporting reactions act in the vertical plane containing the longitudinal axis. The loads and the reactions at the supports are considered external forces and they must be in equilibrium for the entire beam to be in equilibrium. To study the strength of the beam, it is necessary to know how these external forces affect it.

4.Theory: 4.1 Beam: A beam is a structural element that primarily resists loads applied laterally to the beam's axis. The loads applied to the beam result in reaction forces at the beam's support points. The total effect of all the forces acting on the beam is to produce shear forces and bending moments within the beam. Beams are characterized by their manner of support, profile, length, and their material. Beams are traditionally descriptions of building or civil engineering structural elements.

4.1.1 Simply supported beam: A simply supported beam is a type of beam that has pinned support at one end and roller support at the other end. Depending on the load applied, it undergoes shearing and bending. It is the one of the simplest structural elements in existence.

Figure 1 Simply Supported Beam

4.2 Deflection in Beams: In all practical engineering applications, when we use the different components, normally we have to operate them within the certain limits i.e. the constraints are placed on the performance and behavior of the components. For instance, we say that the particular component is supposed to operate within this value of stress and the deflection of the component should not exceed beyond a particular value. In some problems the maximum stress however, may not be a strict or severe condition but there may be the deflection which is the more rigid condition under operation. It is obvious therefore to study the methods by which we can predict the deflection of members under lateral loads or transverse loads, since it is this form of loading which will generally produce the greatest deflection of beams.

4.3 Construction: The slope of the fixed beam is zero at the ends and a couple will have to be applied at each end to make the slope there have this value. The applied couples will be of opposite sign to that of bending moment due to loading. Consider a beam AB of length L fixed at A and B and carrying a load W. The theoretical deflection (yc) is given by;

yc = Where

𝐖𝐚𝟐 𝐛𝟐 𝟑𝐄𝐈𝐋

E= Modulus of elasticity for the material of beam I =Moment of Inertia of the beam

4.4 Eccentric load: A load imposed on a structural member at some point other than the centroid of the section. An eccentric load is a load which resultant is applied outside of the centroid of the element (column, beam or foundation) where it is applied. The bigger the distance to the centroid the bigger the eccentricity. As a result, in addition to the compression or bending that caused by that load, there will also be a moment due to the eccentricity that will need to be considered in the design of the element.

Figure 2 Eccentric Load

4.4.1 Examples of Eccentric load: In structural engineering you must know where the loads occur. Nearly all loading on beams is eccentric. To identify the eccentricity of the load you must know the line of action of the load, and know the location of the beam centerline.  If a Roof beam supports joists on both sides, but the joist spans are unequal, the load on the beam is eccentric.  Brick loads on exterior walls seldom bear on the center line of the beam, and thus are eccentric.

 Nearly all wind loads on wind beams are eccentric (i.e. not equal on both sides of the beam). It is the same with columns. A column with not be loaded eccentrically when: it is an interior column with equal spans on either pair of sides:  All corner columns are loaded eccentrically.  All exterior wall columns are loaded eccentrically.  Any columns near a floor opening (such as stairs or elevator shaft) are eccentrically loaded.

4.5 Modulus of Elasticity: The modulus of elasticity E is a material property, that describes its stiffness and is therefore one of the most important properties of solid materials. When a material deforms elastically, the amount of deformation likewise depends on the size of the material, but the strain for a given stress is always the same and the two are related by Hooke´s Law:

𝝈 = 𝑬𝜺 where σ is stress [MPa], E modulus of elasticity [MPa], ε strain. From the Hook’s law the modulus of elasticity is defined as the ratio of the stress to the strain:

𝐄=

𝛔 𝛆

4.6 Moment of inertia of beam: The Area Moment of Inertia of a beams cross-sectional area measures the beams ability to resist bending. The larger the Moment of Inertia the less the beam will bend. The moment of inertia is a geometrical property of a beam and depends on a reference axis. The smallest Moment of Inertia about any axis passes through the centroid. The following are the mathematical equations to calculate the Moment of Inertia. y is the distance from the xaxis to an infinitesimal area dA and x is the distance from the y-axis to an infinitesimal area dA.

Figure 3 Moment of Inertia

𝑰𝑿 = ∫ 𝒚𝟐 𝒅𝑨 𝑰𝒚 = ∫ 𝒙𝟐 𝒅𝑨

5. Procedure: i. ii. iii. iv. v. vi.

Set the deflection of beam apparatus on horizontal surface. Set the dial indicator at zero. Apply a load of 0.5 lb and measure the deflection using dial indicator. Keep on increasing the load. Take at least five readings of increasing value of load and then take readings on unloading. Measure the length, width and height of the beam using meter rod and Vernier caliper and calculate the theoretical value of Deflection using the given formula. vii. Compare the experimental and theoretical values of deflection.

6. Observations and Calculations: First, perform the experiment using brass beam.

6.1 For Brass:       

Least Count of dial indicator Least Count of Vernier calipers Effective length of beam (L) Breadth of beam Height of beam Moment of inertia of beam (I = bh3/12) Modulus of elasticity of beam

No. of Obs.

Effective Load-W (lbs)

1 2 3 4 5

0.5 1 1.5 2 2.5

= 0.001 inch = 0.05 mm = 30 inch = 1.053 inch = 0.2573 inch = 1.495 ×10-3 in4 = 15 ×106 psi

Actual Central Deflection-𝜹𝒄 (in.) Loading 0.011 0.024 0.038 0.051 0.065

Unloading 0.012 0.025 0.039 0.052 0.065

Average 0.0115 0.0245 0.0385 0.0515 0.065

Average value of error = 10.4%. Now, perform the same experiment using steel beam.

6.2 For Steel:       

Least Count of dial indicator Least Count of Vernier calipers Effective length of beam (L) Breadth of beam Height of beam Moment of inertia of beam (I = bh3/12) Modulus of elasticity of beam

= 0.001 inch = 0.05 mm = 30 inch = 1.053 inch = 0.1768 inch = 0.485 ×10-3 in4 = 29 ×106 psi

Theoretical Deflection 𝜹𝒕𝒉 = Wa2 b2/3EIL (in)

%age Error

0.00193 0.00368 0.00580 0.00770 0.00966

5.6 7.2 12.6 12.7 13.8

No. of Obs.

Effective Load-W (lbs)

1 2 3 4 5

0.5 1 1.5 2 2.5

Actual Central Deflection-𝜹𝒄 (in.) Loading 0.018 0.036 0.056 0.075 0.094

Unloading 0.019 0.038 0.057 0.076 0.094

Average 0.0185 0.037 0.0575 0.0755 0.094

Theoretical Deflection 𝜹𝒕𝒉 = Wa2 b2/3EIL (in)

%age Error

0.0182 0.0360 0.0546 0.0729 0.0910

1.59 2.7 3.5 3.6 3.3

Average value of error = 2.3%

8. Comments:  The errors may be due to poor calibration, or may be due to the negligence of the performer.  The error may be due to the improper standard of applied weights.  Area of cross-section may not be uniform throughout the wire.

9. References:  [1] http://www.mechanicalbooster.com/2016/09/types-of-beams.html

 [2] https://www.slideshare.net/shamjithkeyem/module4-plastic-theory-rajesh-sir  [3] https://www.eboss.co.nz/library/steltech/custom-tapered-beams