ME2113-1
DEFLECTION AND BENDING STRESSES IN BEAMS
(EA-02-21)
SEMESTER 3
2011/2012
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MECHANICAL ENGINEERING
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
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 m 5.856467896 0.028426 c σm 0.03309476 0.053828 σc r2 0.999872282 0.057973 σy P(g) P(N) (‐)VL(mm) L U A 250 2.4525 0.37 0.45 0.41 L (where load P is applied) 250 mm b 25.6 mm h 6.06 mm Izz 474.7627008 mm4 GPa
2
65.6216791
MPa
2
65621.6791
3I
V
P
E
Modulus,
s
Young'
7627 . 474 3 250 0.41 4525 . 2 zz 3 L 3
L
Calculation of Poisson's Ratio
εzz1 εxx1 εzz2 εxx2‐1.5E‐05 4.45E‐05 0.000008 ‐2.3E‐05
0.347826
or
0.337079
05
-2.3E
-0.000008
-or
05
-4.45E
05
-1.5E
Ratio,
s
Poisson'
zz
xx
Slope of the Graph
P (N) εxx1 2.4525 4.45E‐05 9.81 0.0001839N
53315.2173
)
(
)
(
P
Graph
the
of
Slope
0.000183 ) 05 45 . 4 ( 81 . 9 4525 . 2 2 xx1 1 xx1 2 1
EP
Bend Moment Magnitude at Strain gauge location
P(N) d1 L (mm) 2.4525 50 250Nm
-0.4905
x)
-P(L
M
50) -2.4525(250 xz
Theoretical Magnitude of Longitudinal Stresses
Mxz(Nm) at x = 50mm 0.4905 Nm b 25.6 mm h 6.06 mm Izz 474.7627 mm4MPa
3.13043758
)
2
(
I
M
1000 2 06 . 6 7627 . 474 0.4905 zz xz -h/2) xx(y
h
Experimental Magnitude of Longitudinal Stresses
E 65.62167912 εxx1 0.0000445MPa
1
2.92016472
0.0000445
2
65.6216791
xx
E
xx
Results
Table 1
Table 2
P(g) P(N) Mxz(Nm) at x = 50mm Mxz(Nm) at x = 150mm Theoretical Experimental σxx1 (Mpa) at x = 50mm σxx2 (Mpa) at x = 150mm σxx1 (Mpa) at x = 50mm σxx2 (Mpa) at x = 150mm 250 2.4525 0.4905 0.24525 3.13043758 1.56521879 2.92016472 1.50929862 500 4.905 0.981 0.4905 6.26087516 3.13043758 5.8075186 3.08421892 750 7.3575 1.4715 0.73575 9.39131274 4.69565637 8.9573592 4.46227418 1000 9.81 1.962 0.981 12.5217503 6.26087516 11.9759564 5.9715728 1250 12.2625 2.4525 1.22625 15.6521879 7.82609395 14.7976886 7.41524974 1500 14.715 2.943 1.4715 18.7826255 9.39131274 17.8490967 8.793305P(g) P(N) (‐)VL(mm) Axial Strain Transverse Strain εxx1(μ) + εxx2(μ) ‐ εzz1(μ) ‐ εzz2(μ) +
L U A L U A L U A L U A L U A
250 2.4525 0.37 0.45 0.41 42 47 44.5 22 24 23 14 16 15 8 8 8 500 4.905 0.79 0.87 0.83 86 91 88.5 46 48 47 29 30 29.5 16 16 16 750 7.3575 1.25 1.27 1.26 137 136 136.5 67 69 68 43 44 43.5 22 22 22 1000 9.81 1.67 1.69 1.68 182 183 182.5 91 91 91 59 59 59 29 29 29 1250 12.2625 2.06 2.09 2.075 226 225 225.5 113 113 113 72 72 72 36 36 36 1500 14.715 2.51 2.51 2.51 272 272 272 134 134 134 88 88 88 43 43 43
Graph 1
Linear Least Squares Fits of Load P against vertical deflection ν m 5.856467896 0.028426 c σm 0.03309476 0.053828 σc r2 0.999872282 0.057973 σy y = 5.8565x + 0.0284 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 2.5 3 Load P (N) Vertical Deflection, ν (mm)
Graph of Load P against vertical deflection ν
Series1 Linear (Series1)Graph 2
Linear Least Squares Fits of εzz1 against εxx1 Linear Least Squares Fits of εzz2 against εxx2
m ‐0.318573 ‐7.5254E‐07 c 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 = 0.311861 y = ‐0.3186x ‐ 8E‐07 y = ‐0.3119x + 9E‐07 ‐0.0001 ‐0.00008 ‐0.00006 ‐0.00004 ‐0.00002 0 0.00002 0.00004 0.00006 ‐0.0002 ‐0.00015 ‐0.0001 ‐0.00005 0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 εzz εxx
Graph of ε
zz1
against ε
xx1
& ε
zz2
against ε
xx2
εzz1 against εxx1 εzz2 against εxx2
Linear (εzz1 against εxx1 ) Linear (εzz2 against εxx2 )