MEEG 215: Mechanics of Solids – Fall 2008 · MEEG 215: Mechanics of Solids – Fall 2008 ... to write the lab report. ... Construct the quantitative shear force and bending moment

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MEEG 215: Mechanics of Solids – Fall 2008

Laboratory #4: Bending and Stress Transformation

Pre-lab Questionnaire: to be handed in on entering the lab.

1. Explain the term “principal strains”. 2. What is a “strain-rosette” and how does it work? 3. Other than as the ‘skeleton’ of many buildings, give three examples of where I-beams or similar sections may be commonly used. 4. Why might you want to perform a strain (or stress) transformation?

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MEEG 215: Mechanics of Solids – Fall 2008

Laboratory #4: Bending and Stress Transformation

Introduction Mechanical components are frequently subjected to a variety of external shear and normal stresses. Nevertheless, it is always possible to define an element, in a highly specific orientation, such that the only resultant stresses are normal stresses: i.e., despite the fact that shear and/or normal stresses are applied externally, the body behaves as though subjected only to normal stresses. These stresses are then called principal stresses and the planes on which they act are called principal planes. Correspondingly, the element can be oriented such that the strains imposed on it can all be represented by normal strains, with no shear strains acting. The strains are now called principal strains and they act on planes whose normals coincide, for isotropic materials, with the principal planes. Broadly speaking, the failure behavior of a brittle material is determined by the maximum normal stress and that of a ductile material by the maximum shear stress: thus, it is often necessary to determine the orientation of the principal planes, principal stresses and principal strains. In this lab you will determine the principal strains and stresses as well as the stress distribution over the cross section of an aluminum I-beam subjected to three-point bending. You will do this by investigating the stresses and strains measured with multiple strain gage rosettes placed on the beam. You will then transform the strain data – recorded in the directions of the three gages – into the strains along the principal axes.

The web site <http://www.vishay.com/docs/11065/tn515.pdf> (Vishay Intertechnology, Inc) describes the use of strain gage rosettes. There are some slight differences in the naming conventions used in this site and those used in our class. You will need to read and understand this information to be able to write the lab report. You should also consult Section 10.5 of the Hibbeler text, “Mechanics of Materials”. Materials and Methods

An aluminum (6061-T6) I-beam, as illustrated above, will be loaded in bending in a materials testing system machine (Instron Corp. [http://www.dynatup.com/index.asp] and updated by Instrumet). The

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beam is 483 mm long, with a height of 76.2 mm and width of 63.5 mm. The web thickness is 8.9 mm and the flange height averages 3.8 mm. The I-beam will be loaded in three-point bending supported by two knife-edges 432 mm apart and is loaded transversely at the center between these two supports. During testing, the applied load and the displacement at the loading head will be measured by the Testworks (MTS Inc) software.

Strain gage rosettes have been mounted on the I-beam at five locations through the height, 349.3 mm from one end of the beam. Three of the gages are rosettes (gage numbers 1, 2 and 3) mounted on the web of the beam as shown here, with gage 1 on the neutral axis and gages 2 and 3 located 19.1 mm above and below the neutral axis, respectively. Gages 4 and 5 are unidirectional and are mounted on the upper and lower flanges and oriented to measure the axial strain at those locations. The outputs of the individual gages are run through the strain gage amplifier and, by using the correct gage factor, converted directly to strain. The strain gage rosettes consist of three individual gages oriented 45° from each other. They are mounted on the web of the beam to measure the strain in the axial direction and at ±45° from the axial direction. Note that, although there is an obvious long axis to the I-beam, this does not necessarily correspond to a principal axis as defined above. Apply load to the beam in at least 4 incremental loads with a final load not exceeding 300 kgf.

Experimental At each loading/displacement increment, record the load, displacement and strain levels from each of the strain gages. When you record the strain data, the data logger takes 3 readings in rapid succession: report just the average of these readings as your measured strain level. Present your data in a clearly laid-out and labeled table. Treatment of Results 1) Use the strain transformation equations to convert the strains at strain gage rosette 1 into εx, εy and

!

" xy

!

"1(#45),"

2(0),"

3(+45)$ "x ,"y , %xy[ ] by solving these three equations (c.f. Hibbeler Eqns. 10-

16) for the three unknowns (θ1, θ2, θ3 are the known orientations of the strain gages) at each load.

!

"1

= "x cos2#

1+ "y sin

2#1+ $ xy sin#1 cos#1

!

"2

= "x cos2#

2+ "y sin

2#2

+ $ xy sin#2 cos#2

!

"3

= "x cos2#

3+ "y sin

2#3

+ $ xy sin#3 cos#3

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Similarly, repeat the procedure for strain gage rosettes 2 and 3 at each load. Tabulate the results. For one of the strain gage rosettes, plot each of the strains (εx, εy, γxy) as a function of load and generate a least squares fit to the data for each. Report the equation of the fitted line. 2) Pick a convenient load from the least squares line generated in 1) above and calculate the principal strains

!

"P,"Q for rosettes 1, 2, and 3. Tabulate the results.

!

"P ,Q

="1+ "

3

2±1

2"1#"

2( )2

+ "2#"

3( )2

3) Also calculate the angle that the principal strain directions make with the x,y axes (θ).

!

" =1

2tan

#1 $1 # 2$2 + $3

$1#$

3

%

& '

(

) *

4) Calculate the maximum shear strain using,

!

"max

= #P$#

Q . 5) Using εx, εy, & γxy, plot Mohr’s circle of strain at rosettes 1, 2, and 3 (i.e. 3 separate circles) for your selected load. Compare the principal strains calculated in 2) above with the values you obtain graphically here from the Mohr’s circle diagrams. 6) Calculate the principal stresses from the principal strains

!

"P,"Q#$

P,$

Q[ ] at rosettes 1, 2, and 3 at each of the loading increments used above. Take E = 210 GPa, ν = 0.35.

!

"P

=E

1#$ 2%P

+ $%Q( ) ;

!

"Q

=E

1#$ 2%Q

+ $%P( )

7) Plot the principal stresses for rosettes 1, 2 and 3 at your selected load. 8) Calculate the maximum shear stress at rosettes 1, 2, and 3 using,

!

"max

=#P$#

Q . 9) Plot Mohr’s circle of stress for rosettes 1, 2, and 3 (i.e. 3 separate circles) at your selected load using the principal stresses calculated in 7) above. Then, use these Mohr’s circle diagrams to find σx, σy, and τxy graphically for each rosette and compare this value with the measured value. Discussion Discuss the results of your calculations above. Note, for example, whether the values measured from your Mohr’s circle constructions coincide with those measured experimentally. Discuss the utility of the Mohr’s circle construction. Then: 1. Construct the quantitative shear force and bending moment diagrams for any two of the applied loads you used in this experiment. Clearly label the values of shear force and bending moment at the gage location.

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2. Calculate and plot the theoretical shear stress distribution across the cross-section of the I-Beam (follow the example in Hibbeler). Find the theoretical shear stress at the location of the rosettes and compare with the shear stress values from the Mohr’s circle. 3. The present experiment used 3-point bending but another common test method uses 4-point bending (which is also often referred to as “pure bending”). In 4-point bending each end is supported and the loading is applied at l/4 and 3l/4. Construct schematic shear force and bending moment diagrams as a function of the applied load P for this configuration and explain how this compares with and/or differs from the diagrams you drew in 1 above. 4. Describe how the stresses along the axis of the beam in the top and bottom flanges and in the web differ (for any load you choose). Based on these observations, and the observed fact that there is no resultant axial force on the beam, can you suggest a possible relationship between the stresses at equal distances above and below the center of the beam? Conclusions Summarize the principal lessons demonstrated and learned from this exercise.

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TEST DATA

MEEG215 Lab 4

Least squares fit for selected load value

Load Displace. Data Logger Channel Gage # 0 2B (ε3) 1 2A (ε2) 2 2C (ε1) 3 1B (ε3) 4 1A (ε2) 5 1C (ε1) 6 3B (ε3) 7 3A (ε2) 8 3C (ε1) 9 5 (bottom) 10 4 (top)

NOTE: in order to avoid dealing with strains in terms of small numbers with a large number of decimal points (e.g. ε = 0.00022546) you may find it clearer and more convenient to express these numbers as microstrain, i.e. strain x 106: the strain here would become a microstrain of 225.5. This avoids awkward nomenclatures such as “e-4”, “10^5”, etc. Just remember to state clearly the units you are using.