Lab 3 - Unsymmetrical Bending of a Cantil

Laboratory Sheet 1

UNIVERSITI TEKNIKAL MALAYSIA MELAKA SOLID MECHANICS 3 UNSYMMETRICAL BENDING OF A CANTILEVER

No Dokumen:

No Isu./Tarikh

SB/MMSB3/T1/BMCS4353/4 1/15-2-2006 No Semakan/Tarikh 0

Jum Mukasurat:

4

OBJECTIVE

1. To construct a Mohr’s Circle of deflections. 2. To establish values of the principle second moments mom ents of area from the Mohr’s Circle. 3. To compare the principle second moments to theoretical values. THEORY

Figure 1 shows a cantilever beam with a load applied at the free end. The beam has two principle axes, x and y, about which pure bending can take place. The principle axes pass through the centroid of the section but do not necessarily coincide with the arbitrary geometrical axes of the section.

Figure 1: Cantilever schematic and notation

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If applying a moment about one of the principle axis then the beam will deflect in that direction only and the simple bending formula can predict the deflection. However, if the moment is at an angle to either of the axes then the beam will bend about both of the axes. The free end deflection will have two components - one in the direction of pull (U) and at right angles (V). If we were interested in predicting the magnitude of the deflections, we would need to resolve the moment into components acting about the principle axes. This leads to the following formulae (standard texts outline the derivation):

U  =

3 FL ⎡⎛  ⎜1

6 E 

3

V  =

FL

6 E 

⎢⎜ ⎢⎣⎝  I  x

+

⎛   ⎞⎤ ⎟ + cos 2θ ⎜ 1 − 1 ⎟⎥ ⎜  I  I  ⎟⎥  I  y  ⎠⎟  x  ⎠ ⎦ ⎝   y 1  ⎞

⎡1

sin 2θ ⎢

⎢⎣ I  x

−

⎤ ⎥ I  y ⎥⎦ 1

(1)

(2)

Where: L E U V  Θ

I x, Iy 

= Effective length of the specimen (m) = Young’s modulus (GNm-2) = Deflection in the direction of pull (m) = Deflection at right angles to the pull (m) = Angle of pull (°) = Principle second moments of area (m 4)

A good way of reading off the cantilever deflections for various angles is to construct a Mohr's Circle. The Mohr's Circle is an excellent graphical method, which you may have come across before in stress analysis. To construct a Mohr's Circle of deflections you would need to know at least the principle second moments of area ( IX  and Iy ). In this experiment we will use the Mohr's Circle in reverse to establish values of the principle second moments of area from deflections measured off the equipment in each direction (U and V). We can then compare the principle second moments to theoretical values.

Figure 2: Mohr’s circle of deflections to find the principle second moments of area

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SB/MMSB1/T1/BMCS1323/4 EQUIPMENT

1.

The unsymmetrical bending experiment in the structures frame is shown in Figure 3.

Figure 3: Unsymmetrical bending experiment in the structures frame

PROCEDURE

1. Loosen the two rearward facing thumbscrews on the indicator bosses, turn the indicators inward to contact the inner two datum pegs and lock off the thumbscrews. This sets the 90° angle between the two indicators. 2. Select one of the three specimens and fit it into the bottom chuck referring to Figure 5 for correct positions. 3. Fit the top of the specimen into the top chuck in the same relative position, ensure that the specimens are set squarely and all of the screws are tight. 4. Fit the extension piece to the bottom chuck, hook the cord onto the groove and pass it over sliding the pulley. 5. Undo the top chuck hand wheel and rotate the specimen so it is orientated as per Figure 5. When you feel the chuck 'click' into the correct position tighten the hand wheel.

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6. Ensure that the indicators have about 10-11 mm forward and 2 - 3 mm backward travel in this position; if not loosen the indicator top screw and slide the indicator to the correct position, retighten the screw. 7. Tap the frame sharply to reduce the effect of friction and zero the indicators. Apply loads in 100 g increments, up to a maximum of 500 g on the end of the cord. 8. Ensure the cord remains parallel to the lines on the plate below. 9. Tap the frame sharply after adding each load. Record the resulting deflections (left and right) in Table 1 under the 'Head angle: 0°' title. 10. Undo the top chuck hand wheel, rotate the specimen clockwise 22.5° (i.e. to the next location) and tighten. 11. Adjust the indicators, if required, and then repeat the loading procedure, recording the results under 'Head angle: 22.5°' in Table 1. 12. Continue taking results until the specimen has rotated 180°. 13. Repeat step 2 – 12 for the other two specimens.

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