UNIVERSITY OF EAST LONDON

Stiffness determination of a beam structure uniform section.

CE2203: Analysis and design of structure elements.

Written by: Abdullah Sharriff

Student No: 0945251

Module Leader: Iounnou Constantinou

Content

Aim..............................................................................................Page 3

Apparatus...................................................................................Page 3

Observation................................................................................Page 4

Calculation..................................................................................Page 5

Discussion and interpretation...................................................Page 9

Error.............................................................................................Page 9

Conclusion...................................................................................Page 9

Appendix.....................................................................................Graphs

PART 1:

Aim: the aim of this experiment is to determine the stiffness of a beam structure of uniform section and relationship between moment and rotation

OBSERVATION

    PART ONE  Loading

W Dial Gauge Distance (a)Distance

(b)between

(Kg)Reading

(mm)between load

and Dial Gauge and

    support B (mm) support B (mm)0.000 8.200 150.000 250.0000.500 8.580 150.000 250.0001.000 9.030 150.000 250.0001.500 9.490 150.000 250.0002.000 9.940 150.000 250.0002.500 10.400 150.000 250.0003.000 10.840 150.000 250.0002.500 10.750 150.000 250.0002.000 10.090 150.000 250.0001.500 9.640 150.000 250.000

1.000 9.210 150.000 250.0000.500 8.740 150.000 250.0000.000 8.230 150.000 250.000

Table 1: shows the measurements results

CALCULATION

    PART ONE      Load,

WDisplacemen

t Distance (a) Moment Distance (b) Rotation

(N) δbetween load

and (MB ) between Dial ( θB )

  (mm)support B

(mm)W * a (N– mm)

Gauge and support δ /b

B (mm) 0.00 0.00 150.000 0.000 250.000 0.0004.91 0.38 150.000 736.500 250.000 0.0029.81 0.83 150.000 1471.500 250.000 0.003

14.72 1.29 150.000 2208.000 250.000 0.00519.62 1.74 150.000 2943.000 250.000 0.00724.53 2.20 150.000 3679.500 250.000 0.00929.43 2.64 150.000 4414.500 250.000 0.01124.53 2.55 150.000 3679.500 250.000 0.01019.62 1.89 150.000 2943.000 250.000 0.00814.72 1.44 150.000 2208.000 250.000 0.0069.81 1.01 150.000 1471.500 250.000 0.0044.91 0.54 150.000 736.500 250.000 0.0020.00 0.03 150.000 0.000 250.000 0.000

Table 2: shows the calculation of the experiment

Moment

Rotation

(MB ) ( θB )W * a δ /b

   0.000 0.000736.5 0.0021471.5 0.0032208 0.0052943 0.007

3679.5 0.0094414.5 0.0113679.5 0.0092943 0.0082208 0.006

1471.5 0.004736.5 0.0020.00 0.000

Table 3: this set of readings will be used to determine the slope of a straight line plotting moment (y axis) against rotation (x axis).

PART 2:

Loading, WA Loading, Wd Dial Gauge Dial Gauge

(kg) (kg)Reading 1 (mm)

Reading 2 (mm)

0.000 0.00 8.23 17.180.500 0.26 8.31 17.181.000 0.54 9.60 17.181.500 0.70 9.68 17.182.000 1.00 9.92 17.182.500 1.22 10.11 17.183.000 1.40 10.53 17.182.500 1.19 10.10 17.182.000 0.99 9.84 17.181.500 0.70 9.65 17.181.000 0.50 8.48 17.180.500 0.26 8.28 17.180.000 0.00 8.24 17.18

TABLE 4: shows the measurements results

Load, WA Displacemen Distance Distance(b Moment, Rotation

(N) tAt point 1 δ1(mm)

(a)(mm)

)Between dial Gauge and support B(mm)

MB WA * a(N –mm)

θB

δ 1 / b

0.00 0.00 150.0 250.0 000.000 0.00004.91 0.08 150.0 250.0 736.500 0.00039.81 1.37 150.0 250.0 1471.500 0.0050

14.72 1.45 150.0 250.0 2208.000 0.006019.62 1.69 150.0 250.0 2943.000 0.007024.53 1.88 150.0 250.0 3679.500 0.008029.43 2.30 150.0 250.0 4414.500 0.009224.53 1.87 150.0 250.0 3679.000 0.008019.62 1.61 150.0 250.0 2943.000 0.006414.72 1.42 150.0 250.0 2208.000 0.00609.81 1.25 150.0 250.0 1471.500 0.00504.91 0.05 150.0 250.0 736.500 0.00020.00 0.01 150.0 250.0 000.000 0.0000

TABLE 5: Shows the calculation on the experimental stiffness of the member.

Moment, MB WA * a(N –mm)

Rotation θB

δ1 / b000.000 0.0000736.500 0.00031471.500 0.00502208.000 0.00602943.000 0.00703679.500 0.00804414.500 0.00923679.000 0.00802943.000 0.00642208.000 0.00601471.500 0.0050736.500 0.0002000.000 0.0000

Table 6: this set of readings will be used to determine the slope of a straight line plotting moment (y axis) against rotation (x axis).

Carry over factor calculations.

Loading,WA

(N)

Loading,WD

(N)

Dial GaugeReading2(mm)

Distance(a)(mm)

Distance(d)(mm)

Moment,MB

WA * a(N –mm)

MomentMcWD* d(N – mm)

0.00 0.00 17.18 150.0 170.0 0.00 0.004.91 2.55 17.18 150.0 170.0 736.50 433.509.81 5.30 17.18 150.0 170.0 1471.50 901.00

14.72 6.87 17.18 150.0 170.0 2208.00 1168.0019.62 9.81 17.18 150.0 170.0 2943.00 1667.7024.53 12.00 17.18 150.0 170.0 3679.50 2040.0029.43 13.73 17.18 150.0 170.0 4414.50 2334.1024.53 11.67 17.18 150.0 170.0 3679.50 1984.0019.62 9.71 17.18 150.0 170.0 2943.00 1650.7014.72 6.87 17.18 150.0 170.0 2208.00 1168.009.81 4.91 17.18 150.0 170.0 1471.50 834.704.91 2.55 17.18 150.0 170.0 736.50 433.500.00 0.00 17.18 150.0 170.0 0.00 0.000

TABLE 7: Carry over factor calculations.

Moment,MBWA a(N –mm)

MomentMcWD* d(N – mm)

0.00 0.00736.50 433.50

1471.50 901.002208.00 1168.002943.00 1667.703679.50 2040.004414.50 2334.103679.50 1984.002943.00 1650.702208.00 1168.001471.50 834.70736.50 433.50

0.00 0.000

TABLE 8: this set of readings will be used to determine the slope of a straight line plotting MB (y axis) against MC (x axis).

ERROR:

First time when the result were been recorded team members did not read the fully instruction properly this led to a massive error which did not match with the experimental tolerance this occur due to the fact that while the load was applied to the hanger the dial gauge was not tapped therefore every time the reading was taken few divisions were counterbalance which led to a big difference in tolerance. However the loading and uploading of the weight has played a major part as great care did not taken into consideration therefore while putting the weight of slowly the hand adds weight to it when not letting the weight off accurately.

DISCUSSION:

From research done Hook`s law state that (load is directly proportional to deflection) which gives linear relationship, loading and unloading are same relationship. However this experiment was to prove that the stiffness of the steel bar, nevertheless the stiffness of the bar was calculated by acquire the gradient of the straight line from the graph which was plotted moment (Mb) against rotation (θB). As the gradient was 1000/0.0025 = 400000, this was then checked theoretically by using the formula kBC = 3EI / L. Second moment of area, I = (bd3 / 12) = (25x103 / 12) = 450mm4. Kbc = 3 (200x103) x (450) / 750 = 360000 Nmm/unit rotation. Therefore 400000 – 360000 = 40000N/mm or 40kN/mm. Looking at this value one can tell that this value was over the tolerance allowable for this experiment and the reason were clearly explained on the “error” section. Looking at the three graphs gained in this experiment it clearly shows that this experiment has relationship between the load and the rotation of the beam.

CONCLUSION:

The final conclusion to this experiment shows that the relationship is present between applied load and rotation, therefore the practical and theoretical corroborate with the experiment analysis. Few errors were also gained due to different type of reasons including, carelessness and old equipment used during the laboratory. If chance was given to do the experiment again, taking more time is really vital and read the information given and understand it well before undertaking the experiment. Overall this experiment is very good and can be used and applied in real work.