ME2113-1-Deflection and Bending Stresses in Beams

ME2113-1 DEFLECTION AND BENDING STRESSES IN BEAMS (EA-02-21)

SEMESTER 3 2011/2012

NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MECHANICAL ENGINEERING

ME2113‐1 Deflection and Bending Stresses in Beams

2011



Contents Objective ................................................................................................................................................................ 2  Sample Calculations .............................................................................................................................................. 2  Calculation of Young's Modulus ........................................................................................................................ 2  Calculation of Poisson's Ratio ............................................................................................................................ 2  Slope of the Graph .............................................................................................................................................. 3  Bend Moment Magnitude at Strain gauge location ............................................................................................ 3  Theoretical Magnitude of Longitudinal Stresses ................................................................................................ 3  Experimental Magnitude of Longitudinal Stresses ............................................................................................. 3  Results.................................................................................................................................................................... 4  Table 1 ................................................................................................................................................................ 4  Table 2 ................................................................................................................................................................ 4  Graph 1 ............................................................................................................................................................... 5  Graph 2 ............................................................................................................................................................... 6  Graph 3 ............................................................................................................................................................... 7  Graph 4 ............................................................................................................................................................... 8  Handgrip Force ................................................................................................................................................... 9  Discussion .............................................................................................................................................................. 9  Conclusion ........................................................................................................................................................... 10 





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ME2113‐1 Deflection and Bending Stresses in Beams

2011



Objective The objective of this experiment to correlate beam theory with an actual demonstration of loading a beam differently and examining the resulting stress and final deflection of the beam as a result of the loads. From the measurement of stresses and deflection values, students are supposed to calculate the resulting Young's Modulus and Poisson's ratio of the beam material. Both the magnitude and signs of the strains and stress in the beam are investigated about their correlation with one another with the use of beam theory.

Sample Calculations Calculation of Young's Modulus Linear Least Squares Fits of Load P against vertical deflection ν  5.856467896  0.028426 

m 

c 

σm 

0.03309476 

0.053828 

σc 

2

0.999872282 

0.057973 

σy 

r 

P(g) 

(‐)VL(mm) 

P(N) 

250 

2.4525 

L (where load P  is applied)  b  h 

L  0.37 

U  0.45 

A  0.41 

250  mm  25.6  mm  6.06  mm  474.7627008  mm4 

Izz 

Young' s Modulus, E   

P L3  VL 3I zz

 250 3 2.4525  0.41 3474.7627

 65621.67912 MPa  65.62167912 GPa

Calculation of Poisson's Ratio εzz1   ‐1.5E‐05 

εxx1  

εzz2  

εxx2  

4.45E‐05  0.000008  ‐2.3E‐05

Poisson' s Ratio,    -

 zz  xx

- 1.5E - 05 0.000008 or  4.45E - 05 - 2.3E - 05

 0.337079

or

0.347826

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ME2113‐1 Deflection and Bending Stresses in Beams

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Slope of the Graph P (N)  2.4525 

εxx1  4.45E‐05 

9.81 

0.000183 

Slope of the Graph 

P1  P2 ( xx1 )1  ( xx1 ) 2

9.81  ( 4.452E.4525  05 )  0.000183

 53315.21739N

Bend Moment Magnitude at Strain gauge location P(N) 

d1 

L (mm) 

2.4525 

50 

250 

M xz   P(L - x)   2.4525(250 - 50)  -0.4905 Nm

Theoretical Magnitude of Longitudinal Stresses Mxz(Nm) at x = 50mm  b  h  Izz 

 xx(y-h/2)  

0.4905  25.6  6.06 

Nm  mm  mm 

474.7627  mm4

M xz h ( ) I zz 2

0.4905  6.06     1000   474 .7627  2 

 3.13043758 MPa

Experimental Magnitude of Longitudinal Stresses E  εxx1  

65.62167912  0.0000445 

 xx  E xx  65.62167912  0.0000445  2.920164721 MPa



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ME2113‐1 Deflection and Bending Stresses in Beams

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Results Table 1 P(g)  250  500  750  1000  1250  1500 

P(N)  2.4525  4.905  7.3575  9.81  12.2625  14.715 

Axial Strain 

(‐)VL(mm)  L  0.37  0.79  1.25  1.67  2.06  2.51 

U  0.45  0.87  1.27  1.69  2.09  2.51 

A  0.41  0.83  1.26  1.68  2.075  2.51 

L  42  86  137  182  226  272 

εxx1(μ) +  U  A  47  44.5  91  88.5  136  136.5  183  182.5  225  225.5  272  272 

L  22  46  67  91  113  134 

εxx2(μ) ‐  U  24  48  69  91  113  134 

A  23  47  68  91  113  134 

L  14  29  43  59  72  88 

Transverse Strain  εzz1(μ) ‐  εzz2(μ) +  U  A  L  U  16  15  8  8  30  29.5  16  16  44  43.5  22  22  59  59  29  29  72  72  36  36  88  88  43  43 

A  8  16  22  29  36  43 

Table 2 P(g) 

P(N) 

250  2.4525 500  4.905 750  7.3575 1000  9.81 1250  12.2625 1500  14.715

Mxz(Nm)  at x =  50mm  0.4905  0.981  1.4715  1.962  2.4525  2.943 

Mxz(Nm)  at x =  150mm  0.24525 0.4905 0.73575 0.981 1.22625 1.4715

Theoretical  σxx1 (Mpa)  at x =  50mm  3.13043758 6.26087516 9.39131274 12.5217503 15.6521879 18.7826255

σxx2 (Mpa)  at x =  150mm  1.56521879 3.13043758 4.69565637 6.26087516 7.82609395 9.39131274

Experimental  σxx1 (Mpa)  at x =  50mm  2.92016472  5.8075186  8.9573592  11.9759564  14.7976886  17.8490967 

σxx2 (Mpa)  at x =  150mm  1.50929862 3.08421892 4.46227418 5.9715728 7.41524974 8.793305

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ME2113‐1 Deflection and Bending Stresses in Beams

2011



Graph 1

Graph of Load P against vertical deflection ν 16 y = 5.8565x + 0.0284

14 12

Load P (N)

10 8

Series1 Linear (Series1)

6 4 2 0 0

0.5

1

1.5

2

2.5

3

Vertical Deflection, ν (mm)





m 

Linear Least Squares Fits of Load P against vertical deflection ν  5.856467896  0.028426 

c 

σm 

0.03309476 

0.053828 

σc 

2

0.999872282 

0.057973 

σy 

r 

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ME2113‐1 Deflection and Bending Stresses in Beams

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Graph 2

Graph of εzz1 against εxx1  & εzz2 against εxx2 0.00006 0.00004 0.00002 y = ‐0.3119x + 9E‐07 0 ‐0.00015

‐0.0001

‐0.00005

εzz

‐0.0002

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

εzz1  against εxx1   εzz2  against εxx2 

‐0.00002

Linear (εzz1  against εxx1  ) ‐0.00004

Linear (εzz2  against εxx2 )

‐0.00006 ‐0.00008 y = ‐0.3186x ‐ 8E‐07 ‐0.0001

εxx



Linear Least Squares Fits of εzz1  against εxx1    m  ‐0.318573  ‐7.5254E‐07  c 

Linear Least Squares Fits of εzz2  against εxx2    m  ‐0.311861  9.25705E‐07  c 

σm 

0.003288 

5.79899E‐07 

σc

σm 

0.003421 

3.00697E‐07 

σc

r2 

0.999574 

6.26763E‐07 

σy

r2 

0.999519 

3.17307E‐07 

σy

ν1 = 

0.318573 

ν2 = 

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ME2113‐1 Deflection and Bending Stresses in Beams

2011



Graph 3

Graph of P against εxx1  16 y = 53824x + 0.0661 14 12

Load P(N)

10 8

Series1 Linear (Series1)

6 4 2 0 0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

εxx1



Linear Least Squares Fits of P  against εxx1   m  53823.94  0.066111  c  σm 

358.434 

0.063208 

σc

r2 

0.999823 

0.068316 

σy 7|P a g e

ME2113‐1 Deflection and Bending Stresses in Beams

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Graph 4

Graph of Maximum Longitudinal Stress xx against location along a beam x 25 Maximum Longitudinal Stress xx(MPa)

Theoretical 1 20

Theoretical 2

15

Theoretical 3

10

Theoretical 4 Theoretical 5

5

Theoretical 6

0 ‐5

0

50

100

150

200

300

350

Experimental 1 Experimental 2 Experimental 3

‐10

Experimental 4

‐15 ‐20

250

Experimental 5 Location along the beam (mm)

Experimental 6



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ME2113‐1 Deflection and Bending Stresses in Beams

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Handgrip Force εxx1hand=  m=  c= 

1.18E‐03  53823.9  0.06611 

P  m xx1hand  c  ( 53823.9  1.18E - 03)  0.06611  6.35 E  1 N

Discussion 1.

Comment on the signs of the strains (xx1, zz1, xx2 and zz2) with respect to the location and orientation of the strain gauges and how the beam is loaded.

Sign Conventions +ve - Tensile Stress -ve - Compressive Stress +ve Stresses  xx1 & zz2 - ve Stresses  xx2 & zz1 xx1 & zz1  Top part of the beam xx2 & zz2  Bottom part of the beam

Hence, xx1 & zz2experiences tensile stress as a result of loading while xx2 & zz1 experiences compressive stress due to loading. As strains on the longitudinal axis can be visually observed experiencing the tensile stresses on the top and compressive stresses at the bottom, strains in zz directions could be characterized by formulas zz1= -vxx1 & zz2=- vxx2 as the resultant stresses due to loading. (This is due to relative small magnitude of experienced stresses (about 1/3 strain) as compared to strains in the xx direction.) 2.

With reference to Graph 4, comment on the slopes of the six theoretical lines and also on how stress varies with beam location. As applied force P/longitudinal stress xx1 ↑  gradient ↑ As x increases, σxx1 tends to 0 independent of Load P at x=250mm. On the other hand, as x increases σxx1 increases. This means the fixed end of the beam experience the highest longitudinal stress while the free end experience no stresses theoretically.

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2011



Comment on the accuracy of your handgrip force. The gripper force is assumed by calculations to be exactly perpendicular to the bottom surface of the beam  May not be true Handgrip Force is postulated from data derive experimentally  experimental errors However, as the error is not significant which can be shown by comparing the theoretical and experimental results, we can conclude that the handgrip force evaluated is accurate.

Conclusion In conclusion, the application of beam theory has been studied and examined in this experiment. Young’s Modulus and Poisson’s Ratio of the material used in this experiment are 65.6GPa and 0.34 respectively. Lastly, the magnitudes and signs of the strains and stresses at two locations along the cantilever beam has been investigated.

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