Lab Bending In Beam.docx

Abstract If a beam is supported at two points, and a load is applied anywhere on the beam, the resulting deformation can be mathematically estimated. However, if it is due to improper experimental setup, the actual results can vary substantially when compared against the theoretical values. One of the most common principles used to determine the loading capacity of a structure is the first yield criterion which assumes the maximum load is reached when the stress in the extreme fabric reaches yield stress. All the columns experience very high shear force and we must design them all for lateral forces. Support reactions are really important to understand of the shear force in a beam, as the maximum shear force is near the supports. In order to make use of the material strength fully, we must explore possibilities of loading the beam into plastic region.

Introduction Bending is a flexible process where many different shapes can be produced. Standard die sets are used to produce a wide variety of shapes. The material is placed on to the die, and positioned in place with stops and gages. It is held in place with hold-downs. The upper part of the press, the ram with the appropriately shaped punched descends and usually forms the v-shaped bend. When the real life or the experiment the bending characterizes the behaviour of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. In this experiment , the deflection of the beam when load is applied to the specimen at a measure point along the specimen and from that elastic modulus can be calculated by the given data. Bending is done using press Brakes and Programmable back gages, and multiple die sets is available currently in making a very economical process. In the experiment conducted, the specimen is subjected to pure bending. This means that the shear force is zero, and that no torsional or axial loads are involved. The material obeys Hooke’s Law that it is linearly elastic and will not deform plastically.

Theory

A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. When a beam is loaded by a force F or moments M, the initially straight axis is deformed into a curve. Bending deformation of a straight member: 1. Longitudinal axis x ( within neutral surface ) does not experience any change in length 2. All cross sections of the beam remain plane and perpendicular to longitudinal axis during the deformation 3. Any deformation of the cross-section within its own plane will be neglected If the beam is uniform in section and properties, long in relation to its depth and nowhere stressed beyond the elastic limit, the deflection , and the angle of rotation, , can be calculated using elastic beam theory. The basic equation at point along the beam is 𝑑2

𝐸𝐼 𝑑𝑥 2 = M(x) When a beam with a straight longitudinal axis is loaded by lateral forces, the axis is deformed into a curve, called the deflection curve of the beam. Deflection is the displacement in the y-direction of any point on the axis of the beam. Besides, deflections are essential for example in the analysis of statically indeterminate structures and in dynamic analysis, as when investigating the vibration of aircraft or response of buildings to earthquakes. Deflection also are sometimes calculated in order to verify that they are within tolerable limits.

When y is the lateral deflection and M(x) is the bending moment at the point x on the beam, E is Young's modulus and I is the second moment of area. M is constant value 𝑀 𝐼

1

1

=E[𝑅-𝑅 ] 0

Where Ro is the radius of curvature before applying the moment and r the radius after it is applied. Deflections and rotations are found by integrating these equations along the beam. Equations for the deflection, , and end slope, , of beams, for various common modes of loading 𝐹 𝑠

=

8𝐸𝐼 𝐿3

Normal stress at intermediate distance y can be determined from 𝜎=-

𝑀𝑦 𝐼

Stress is negative as it acts in the negative direction (compression) Apparatus Mild steel, aluminium and brass beam, the cantilever beam setup, dial calipers, weights to be hung from the end of the beam, tape measure. Procedure 1. The length (L) from the wall to the end of the beam is recorded and measured. 2. The length (x1 and x2) from the wall to the center of the dial calipers is measured and recorded. 3. The weights are hung on the weight-hanger starting from the lowest 2N, and then increase its increment by 2N. 4. The deflection is measured and the weight is recorded at every increment.

Result Load (N) 0 2 4 6 8 10 12 14 16

Deflection (mm) Mild Steel Brass Aluminium 0 0 0 26 25 22 51 42 42 77 59 62 102 77 82 128 94 102 154 110 122 179 128 142 204 144 162 Table 1.0: Table of load and deflection of mild steel, brass and aluminium

Load vs Deflection 250

Deflection

200 150

Mild steel Brass

100

Aluminium 50 0 0

2

4

6

8

10

12

14

Load

Graph 1.0: Graph of load vs deflection

16

Use this equation to find value modulus of elasticity, 𝐸𝑒𝑥𝑝 :

Eexp =

ØL2 x I8

To find percentage of error: 𝑇ℎ𝑒𝑜𝑟𝑦−𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑇ℎ𝑒𝑜𝑟𝑦

x 100

The value of modulus of elasticity is can be defined in standard value and compare to experiment value that can find from the equation.

Material Eexp

Aluminium 7 GPa

Brass 16 GPa

Mild steel 23 GPa

𝐸𝑡ℎ𝑒𝑜𝑟𝑦

69 GPa

125 GPa

200 GPa

Error

89%

87%

88%

Ultimate tensile strength

110 Mpa

250 MPa

400 MPa

Table 2.0

References 1. http://www.engineersedge.com/strength_of_materials.html 2. http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/mechanics-andstrength-of-materials. 3. http://mechanicalc.com/reference/beam-analysis 4. http://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html 5. http://www.strucalc.com/engineering-resources/normal-stress-bending-stress-shear-stress/