mechanical engineering

Running Head: PRINCIPAL STRESSES AND STRAINS 1

PRINCIPAL STRESSES AND STRAIN 13

Principal Stresses and Strains

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PRINCIPAL STRESSES AND STRAINS

Introduction

Mechanical properties of materials are very important in engineering. These are the physical properties that materials portray when forces are applied on the materials. Examples of mechanical properties include; elasticity or stiffness; that measures the elastic deformation of a material under stress which can be recovered after the stress is removed, ultimate tensile strength; the point beyond which application of more stress causes a body to fracture, fatigue limit, toughness; this is the amount of energy needed to fracture a unit volume of a material, ductility; the quantity of plastic deformation that a body can withstand before fracture, and hardness; which the ability of a body to resist deformation from another body, (Steven, 2012).

Maximum stress and strains are main fundamentals properties for design of material that will undergo loading resulting in bending, torsion and shearing. To determine principal stresses and strains, experiments are done on material and strains are measured. These can then be calculated to determine principal stresses and strains.

Abstract

Principal stresses and strains can be obtained by measuring strains using triaxial rosette gage. A loaded beam cantilever is attached on the gage and load is applied to obtain strains.

The experiment was performed on strain measurement in determining principal stresses and strains. A hollow steel pipe was mounted on the strain gage sand the load applied sequentially at offset point.

Three forces were experienced: bending, torsion and shearing. The gages recorded different strains along their axes. The three strains along the different axes on the same point are to be measured in determining the principal strains and stresses with the strain gages. Though the stress field on the surface of a symmetrically loaded cantilever beam is axial except nigh the clamped end and the loading point, the stress at any point varies with angle around that point. The strain field in this case biaxial, it varies similarly due to the Poison stress, (Fung, 2013).

The three axes along which strains are to be measured can be arbitrarily oriented about the point of interest. Using outline formulas, measured strains can be calculated to obtain principal stresses and strains as well as shearing stress.

Equipment

· The triaxle strain gage rosette

· Vishay flexor cantilever flexure frame

· Hollow steel pipe

· Support bracket

· A table with two C clamps

· Horizontal Pipe

· Thumb screws

· Weight Hanger

· A load

· Quarter bridge completion blocks.

Theory

It is facts that when a material is subjected to forces it transformed in reaction to those forces. The extent of transformation depends on geometry of material.

When an object is subjected to a force, there will be appearance of stresses in that object. Stress is a force acting on body per unit area. If stress is enough to overcome object strength, it causes strain. Strain is the reactive change in shape and sizes when an object is subjected to external forces. When an object is rotated, it affects stress. This gives principle strains, planes, angles and stresses. Measuring strain on machine components or structural elements is crucial in real life issues in understanding the actual stress states on the materials. Strain gages are essential in measuring the strains accurately and conveniently, (Ugural, 2009).

Strain gages are made of; thin electrical wires glued parallel to the direction where measuring is to be taken. The elongation or shortening of a body causes change in the electrical resistance of the strain gage. When the current passing through the gage is measured, strain is accurately determined for the loading condition, (Popov, 2010).

When cantilever beam is subjected to loading it undergoes strains and stress. A plane stress condition on a free surface machine component, the X-Y plane is assumed to be the plane on which the plane stress occurs. The condition therefore holds uniform isotropic material that is in line with Hooke`s law;

Where E is the elastic modulus and is constant for particular material. When a steel pipe is mounted on strain gage rosette at one point of beam, it measures normal strains along its axes.

Consider 3 normal strain measured on a point on the surface of the machine component; (), a calculation can be made on two normal and one shear strain for the point on the XY plane, (Ex, Ey and Yxy). The three strain gages arranged forms the strain rosette. The state of strains along x and y direction can be found using equations:

+

Where and are longitudinal and transversal axes, is shear stress and are normal stresses measured by rosette at a point

The shear stress on the principle plane is zero. Principle plane is a plane in which normal stress attains minimum and maximum stresses. Shear stress is given by the following formula

Where is given as

Where M is bending moment, c is distance to neutral axes and I is the moment of inertia.

τ where T is the torque and J is molar polar of inertia.

The change n deformation to its original length constitute a shear strain. Principle shear strain is given by

Where ɛ pane strain given by

Where E is young modulus and v is poison ratio

Methodology

Support bracket and main pipe was mounted to the table with two C – clamps. Horizontal bar was attached to the end of the pipe and the thumb screws was used to adjust the distance between screws to 12 inches. Unloaded weight hanger was placed against the outer screw. Dimensions of the pipe and the loaded bar was measured. The three strain gages was connected to the top of the beam to NI-9237 DAQ using three quarter bridge completion blocks. LabVIEW code which collect string from each gage as different loads are applied were written. The calibration for each gage with the hanger attached but with no weights was run. The gage information was taped to the pipe. Weights was added sing 2 lbf increments to the hanger until 20 lbf is reached while recording strain for each load. The process was then reversed by removing weights and repeated until required measurements were obtained. The experiment was then return to initial state and al the equipment’s was removed

Results

For the bar

Polar moment of inertia I

Shearing stresses on different plane

The torsion shearing stress

Shear strains

From the data

Let, strain 1=

Strain 2=

Strain 3=

Taking the maximum values;

Now finding the shear strain

We use strain transformation to normal strain at 450

And since +

Substituting;

Principle strains

Principle stresses

Predicted strains

From Hooks law

Taking assumption that

Taking assumption, no stress on z plane and since

Since +

0=54.94 cos 45 + (-8.05)

=23.445

Predicted stress

Comparison of stress and strain

Stress/strain	Measured value	Predicted
	172.71	14271.15
	14629.9	14271.15
	-898.74	-536.15

Error Analysis

Sources of Errors

1. Errors due to parallax in making the readings.

2. Errors due to mechanical default of the machines.

The percentage error in shear stress 1 & 2 will be;

a. 14271.15 – 14629.9/ 14629.9 = -0.0245

b. -536.15 –898.74/ -536.15 = -0.6763

The percentage error in the principle shear strain 1 & 2 will be;

a. - 182.7u)/ 182.7u = -0.6147

b. 14271.15 – 172.71/ 172.71 = 81.6307

Conclusion

From the results obtain, it was noticed that there was much difference in measured principal strains as compared to predicted ones. This could have been due as a results of measurement errors in the material used. However, principle stresses showed closed values, meaning the method is preferable and is accurate in obtaining principal stresses.

Recommendations

1. The experiment after making sure that all the instruments are correctly adjusted.

2. The experiment of triaxle rosette gage should be compared to that of the uniaxial rosette gage to determine which is better.

Reference

Fung, Y. C. (2013). Biomechanics: mechanical properties of living tissues. Springer Science & Business Media.

Popov, V. L. (2010). Contact mechanics and friction (pp. 231-253). Berlin: Springer Berlin Heidelberg.

Steven M. (2012). Mechanical Properties, retrieved from; June 27, https://www.warwick.ac.uk/fac/sci/

Ugural, A. C. (2009). Stresses in beams, plates, and shells. CRC press.