Principal Stresses And Strains Repaired)

Principal Stresses and Strains

Edward W. Eaves

Objective The purpose of this lab is to calculate the principle stresses from the strains of an aluminum beam, the principal angle, and the percent error in between the measured and theoretical values of stress. Background

Since stress is impossible to compute, a vital relation to help quantify this occurrence is the relation linking stress and strain. Since strain can be measured, it is easy to verify the quantity of strain. Principal stresses are the magnitudes of stress that arise on definite planes inside a solid body. The coordinate system for these definite planes is aligned such that no shear stresses occur along them. For a rectangular aluminum alloy beam, a edge of the beam can be considered as the starting point of this coordinate system.

Figure 1 Theory Principal stresses are able to be determined straightforwardly via aligning strain gages along the primary axis as shown in figure 1. The equations listed below explain the usage of a rectangular rosette, an essential arrangement of strain gages positioned in distinctive orientations in submultiples of π all-around a single point, to find out principal strains.

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Figure 2 (rectangular rosette) For a rectangular rosette, the two principal strains the length of the two-dimensional plane is given by

ε p ,q =

ε1 + ε 3 1 ± 2 2

( ε1 − ε 2 ) 2 + (ε 2 − ε 3 ) 2

and

θ p ,q =

2ε − ε 1 − ε 3 1 tan −1 2 2 ε1 − ε 3

The stress was found by using Hooke's Law. Because the strains calculated are not aligned with the weighted down stress, the universal form of Hooke's Law was used:

σp =

E (ε p + v ⋅ ε q ) 1− v2

σq =

E (ε q + v ⋅ ε p ) 1− v2

and

where E is the modulus of elasticity or Young’s modulus.

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Equipment: 1. Cantilever flexure frame: An apparatus to hold a rectangular beam at one end while applying a load at the other end. 2. Aluminum beam: In this experiment, aluminum alloy was tested. Specific dimensions are (.125 x 1 x 12.5) inches with rectangular strain gage attached. 3. P-3 strain indicator: A device that accurately translates to the output of strain gages into units of µ strain. Procedure 1. Secure aluminum alloy beam in flexure frame so that load can be applied to the unbound end of the beam. 2. Calibrate P-3 strain gage indicator using three-quarter bridge arrangement and inputting gage factors for each gage. 3. After a zero gage readout achieved, 700 gram load, (1.543lbs), should be applied to the end point. 4. The strain reading for each gage should be recorded in the correct order. 5. Remove the load and the readings supposed to return to the initial zero reading. Results The load applied to the beam was 1.54 pounds. The subsequent table lists other identified information contained by this experimental arrangement. Beam Dimensions

Gage Factors

Angles

Sg1 = 2.06

Tp =30

(inches) b=1

3

t = .125

Sg2 = 2.075

L = 10.25

Sg3 = 2.06

Tq = 120

Table 1 The gage readings for the deflection of the beam are as follows: Gage

Reading

1

408

2

564

3

21 Table2

The two principal strains are found to be 613.99 and -184.99 micro-strains. Sample calculation shown below:

εp =

408 + 21 1 + 2 2

( 408 − 564) 2 + ( 564 − 21) 2

= 613.99 µε

Poisson's ratio is defined as the lateral strain divided by the longitudinal strain, v=

εq εp

=

− 184.99 = 0.301 613.99

The angle between Gage 1 and the principal axes,

θp =

1 2 * 564 − 408 − 21 tan −1 = 30.5145 2 408 − 21

The stress down the principal axes is found by using the universal Hooke's Law, as shown:

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σp =

10.4 × 106 613.99 × 10 −6 + 0.301⋅ −184.99 × 10 −6 = 6.385 ksi 2 1 − 0.301

σq =

10.4 × 106 − 184.99 × 10 −6 + 0.301⋅ 613.99 × 10 −6 = −2.05ksi 2 1 − 0.301

(

)

(

)

As a contrast, theoretical value can be obtained by calculating the principal stresses.

σ longitudinal =

6 ⋅ P ⋅ L 6 ⋅ 1.54 ⋅ (10.25) = = 6.06 ksi b⋅t2 1.000 ⋅ 0.1252

Employing the specified equations, εp and εq were found to be 613.99 and -184.99 με correspondingly. This relates to a Poisson's ratio of 0.301. This correlates to a .33% error from the tangible value 0.3. Possible sources for this error could be operator error with weights or P-3 strain indicator. The angle at which the gage was positioned was computed to be 30.5145º. Employing Hooke's Law, the strain was determined to be 6.385 and -2.05 ksi respectively. The longitudinal stress shows evidence of a 5.36% error since the theoretical value is 6.06 ksi. Conclusions The findings in this experiment exhibit prohibitive reliability when judge against the theoretical values. The rectangular rosette has demonstrated in this experiment as being an authoritative tool for determining stress along a two dimensional plane. The only real question left unanswered is the reason for a -2.05 ksi stress finding on the principle axis when the measured amount should probably be zero. The error in the calculations for this calculation was found.

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Appendix

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