mechanic of solid report

1. BENDING STRESS IN A BEAMS
 NAME : P.NIROJAN
 Ad.No : 0041
 GROUP : CIVIL 2nd
YEAR
 DATE OF PERFORMANCE : 11 / 12 / 2015
 DATE OF SUBMISSION : 14 / 12 / 2015

2.  EXPERIMENT NO :- 04
 DATE :- 11 – December - 2015
 TITLE :- Bending stress in a beams
 AIM :- To examine how bending moment varies with an increasing point load
 APPARATUS :-
 DigitalForceDisplay
 Loadingframe
 Digitalstraindisplay
 Beam
 Steelstructure
 Straingauges
 STR8Aloadcell
 EXPERIMENTAL SETUP :-

3.  INTRODUCTION :-
 STRESS: resolve the force ΔF in normal and tangential
direction of the acting area as Fig. The intensity of the force
or force per unit area acting normally to section A is called
Normal Stress, σ (sigma), and it is expressed as:
𝜎 = lim
∆𝐴→0
∆𝐹𝑛
∆𝐴
 STRAIN: Let’s take the arbitrarily
shaped body in Fig as an example.
Consider the infinitesimal line segment
AB that is contained within the
undeformed body as shown in Fig. The line AB lies along the n-axis and has an original
length of ΔS. After deformation, points A and B are displaced to A’ and B’ and in
general the line becomes a curve having a length ΔS’ the change in length of the line
is therefore ΔS-ΔS’. We consequently define the generalized strain mathematically as:
𝜀 = lim
𝐵→𝐴
∆𝑆′
− ∆𝑆
∆𝑆
( 𝐵 → 𝐴 𝑎𝑙𝑜𝑛𝑔 𝑛)
Unit of Strain - From Equation, we can notice that the normal strain is a dimensionless
quantity since it is a ratio of two lengths. Although this is the case, it is common in
practice to state it in terms of a ratio of length units.
i.e. meters per meter (m/m)
Usually, for most engineering applications 𝜀 is very small, so measurements of strain
are in micrometers per meter (μm/m) or (μ/m). Sometimes for experiment work,
strain is expressed as a percent,
e.g. 0.001m/m = 0.1%.
A normal strain of 480μm for a one-meter length is said:
𝜀 = 480×10-6
= 480(μm/m) = 0.0480% = 480μ (micros) = 480μs (micro strain)
a) Before deformed b) After deformed

4.  STRESS-STRAIN RELATIONSHIP, HOOKE'S LAW
The stress-strain linear relationship mathematically represented by Equation:
𝜎 = 𝐸 𝜀
Where: E is terms as the Modulus of Elasticity or Young's Modulus with units of N/m2
or Pa.For most of engineering metal material, GPa is used
 THEORY :-
Bending Moment Equation:
𝜎
𝑦
=
𝑀
𝐼
=
𝐸
𝑅
Where:
 M = Bending moment (Nm)
 𝜎 = Stress (Nm-2
)
 𝑦 = Distance from the neutral axis (m)
 𝐼 = Second Moment of Area or Moment
of Inertia of the beam about the Neutral Axis (m4
)
 PROCEDURES :
1) Check the Digital Force Display meter reads zero with no load.
2) Ensure the beam and load cell are properly aligned
3) Turn the thumbwheel on the load cell to apply a positive (down-ward) preload
to the beam of about 100N
4) Take the 9 zero strain readings by choosing the number with the selector switch.
5) Fill the table 1 with the zero force values
6) Increase the load to 100 N and note all 9 of the strain readings
7) Repeat the procedure in 100 N increments to 500 N
8) Finally; gradually release the load and preload.
Beam before and after a positive bending moment is applied
Beam deformed by positive bending moment

5.  DATA ANALYSIS : [ ALL DIMENSION ARE IN mm ]
1) Area, A1 = 38.1 x 6.4 = 243.84 mm2
Y1 = 3.2 + 31.7 = 34.9 mm
2) Area, A2 = 31.7 x 6.4 = 202.88 mm2
Y2 = 15.85 mm
Y – Distance of center of gravity of section form
the bottom of end
(A1 x Y1) + (A2 + Y2) = (A1+A2) Y
( 243.84 x 34.9 ) + ( 202.88 x 15.85 ) = ( 202.88 + 243.84 ) x Y
8510.016 + 3215.648 = 446.72 x Y
11725.664 = 446.72 x Y
Y = 26.248 mm
 Moment of Inertia Ixx: [
𝑏ℎ3
12
] + [𝐴𝑟𝑒𝑎 × (𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑌).2 ]
= [
38.1 × 6.43
12
] + [243.84 × (34.9 − 26.248).2 ] + [
6.4 × 31.73
12
] + [202.88 × (26.248 − 15.85).2 ]
= (832.307 + 18253.156) + (16.989.34 + 21935.062)
= 19285.463 + 38924.402
= 58009.865 mm4
= 58.01 x 10-9
m4
 BEAM NOMINAL DIMENSIONS AND STRAIN GAUGE POSITIONS
x x
All dimensions are in mm

6.  OBSERVATION : Strain reading unit – 𝜇 (Table 1)
Gauge
number
Load (N)
0 100 200 300 400 500
1  54  155  205  349  441  536
2  418  485  537  608  672  738
3  450  514  525  650  667  753
4  875  882  901  918  930  947
5  867  873  911  926  906  921
6  910  900  890  851  847  781
7  920  888  860  833  811  783
8  1079  1080  1082  1082  1083  1084
9  1079  1080  1082  1082  1082  1074
 CALCULATION :
 Moment (M) =
E × ε × I
y
Example 1: Gauge no:1, Zero N reading,
Moment (M) =
69 × 109 × −54 × 10−6 × 58.01 ×10−9
26.248 × 10−3 Nm
=
69 × −54 × 58.01
26248
Nm
= - 8.235 Nm
Data’s : Out Steel beam material is Aluminum ,
Typically, E = 69 GPa = 69 x 109
Nm-2

7.  RESULT :
Gauge
number
Bending moment (Nm)
0 17.5 35 52.5 70 87.5
1  8.235  23.637  31.261  53.220  67.250  81.737
2  63.743  73.960  81.890  92.717  102.477  112.541
3  68.623  78.382  80.060  99.122  101.714  114.829
4  133.433  134.501  137.398  139.990  141.820  144.413
5  132.213  133.128  138.923  141.210  138.161  140.448
6  138.770  137.246  135.721  129.773  129.163  119.099
7  140.295  135.416  131.146  127.028  123.673  119.404
8  164.542  164.695  165.000  165.000  165.152  165.305
9  164.542  164.695  165.000  165.000  165.000  163.780
Polt the graph strain against bending moment for all 9 gauges
𝑀 = (
𝐸 × 𝐼
𝑦
) × 𝜀
“X” axis – Strain
“Y” axis – Bending moment
Y m X=

8. -54, -8.235
-155, -23.637
-205, -31.261-349, -53.22
-441, -67.25
-536, -81.737
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 1 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

9. -418, -63.743
-485, -73.96
-537, -81.89
-608, -92.717
-672, -102.477
-738, -112.541
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 2 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

10. Force Verse Bending Moment for Gauge Number 4
BENDINGMOMENT
-450, -68.623-514, -78.382
-525, -80.06
-650, -99.122
-667, -101.714
-753, -114.829
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 3 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

11. -875, -133.433
-882, -134.501
-901, -137.398
-918, -139.99
-930, -141.82
-947, -144.413
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 4 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

12. -867, -132.213
-873, -133.128
-911, -138.923
-926, -141.21
-906, -138.161
-921, -140.448
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 5 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

13. -910, -138.77
-900, -137.246
-890, -135.721
-851, -129.773
-847, -129.163
-781, -119.009
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 6 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

14. -920, -140.295
-888, -135.416
-860, -131.146
-833, -127.078
-811, -123.673
-783, -119.404
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 7 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

15. -1079, -164.542
-1080, -164.695
-1082, -165
-1082, -165
-1084, -165.305
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 8 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

16. -1079, -164.542
-1080, -164.695
-1082, -165
-1082, -165
-1082, -165
-1074, -163.78
0-200-400-600-800-1000
Strain Verse Bending Moment for Gauge Number 9 (-Ve) X
(-Ve) Y
0
-20
-40
-60
-80
-100
-120
-140
-160
-180
-200

17.  DISCUSSION :
Our comparing the prediction accuracy for strain,there is a relatively small range of
accuracy.similar to moment predictions,the experimental behavior of the aluminum beam
most closely resembles the predicted strain values throughout the course of loading.
Discrepancies between the predicted moment and strain behaviors and the experimentally
determined values arise from both systematic and precision errors that occurred
throughout the experiment.though effect to limit both types of errors were taken,they have
a considerable impact on the accuracy of the predicted beam behaviour regardless.possible
systematic or bias error come from the experimental apparatus equipment used.
On inspection of the percentage error, it is evident that the experimental strain results
correlate more closely with the theoretical behavior predicted. As previously stated, the
assumptions the experiment violated change the actual boundary conditions of the beam
under which the predictions were derived.
 CONCLUSION :
In this experiment, the bending moment of aluminum beam specimens were analyzed
through the collection of strain values over the course of identical loading patterns.
theoretical moment and strain behaviors were calculated for each specific specimen. With
strong efforts to model the loading condition and boundary condition assumptions used in
the derived theoretical behavior models in the experimental apparatus, the experimental
data was compiled and graphed