Stress Analysis Beam Bending

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UCL MECHANICAL ENGINEERING MECH2005 - Stress Analysis

LABORATORY REPORT

Lecturer: G BURRIESCI

SA2 Laboratory - Beam Failure

through Plastic Hinge Formation

STUDENTS DETAILS

Ahmed Mahmood MEng (Dr. Ben Hanson)

Honor Brannelly MEng (Dr. Rebecca Shipley)

Douglas Stridsberg MEng (Prof. Ventikos)

Yuh-Chih Chen BEng (Prof. Bucknall)

Jia Shen Lim BEng (Dr. Jayasinghe)

Monzer Filipp Shebbo BEng (Dr. Torii)

George Harker-Smith BEng (Mr. Selfridge)

LABORATORY DETAILS

Lab Group: 7

Date of Lab: 21/02/14

Date Due: 14/03/14



SA2: Beam Failure through Plastic Hinge Formation page 2 of 16

CONTENTS

1. INTRODUCTION ............................................................................................................................. 3

2. MATERIALS AND METHODS ..................................................................................................... 3

2.1 Materials ........................................................................................................................................... 3

2.2 Methods ............................................................................................................................................ 4

Test A .................................................................................................................................................. 4

Test B .................................................................................................................................................. 4

3. RESULTS AND CALCULATIONS ............................................................................................... 5

Test A ...................................................................................................................................................... 5

Test B ...................................................................................................................................................... 6

4. DISCUSSION ...................................................................................................................................... 7

Test A ...................................................................................................................................................... 7

Test B ...................................................................................................................................................... 8

Error discussion ...................................................................................................................................... 9

5. CONCLUSION ................................................................................................................................. 10

6. REFERENCES ................................................................................................................................. 10

APPENDICES ......................................................................................................................................... 11

Tabulated Data .................................................................................................................................... 11

Test A ................................................................................................................................................ 11

Test B ................................................................................................................................................ 11

Relevant Equation Derivations ......................................................................................................... 13

Test A ................................................................................................................................................ 13

Test B ................................................................................................................................................ 14




1. INTRODUCTION

This experiment involves investigating the formation of plastic hinges as a beam is subjectedto loading up until the point where it fails by bending. This is done by observing therelationship between the applied load and the corresponding vertical deflection of the beam.

It is important to do so in order to be able to predict the plastic collapse of a staticallydeterminate beam. Initially, the deflection increases gradually for a given load. However, afterthe plastic hinge formation, the deflection increases significantly over a short time periodbefore collapse.

Theoretically, the cross-section of the beam is expected to behave plastically up until theplastic moment, meaning the load applied should have a proportional linear relationship tothe deflection measured. When the plastic hinge forms, the effect is equivalent to theintroduction of a pin joint with a concentrated bending moment, Mp. Collapse will only occuronce the sufficient number of hinges required to reduce the structure to a mechanism haveformed.

2.

MATERIALS AND METHODS

Figure 1

: Schematic diagram of the steel beam

2.1 Materials

1. Loading frame is provided to simply support the steel bar at three points A, C, and E.Points B, D, and F are the loading points on the beam. (a=101.6mm)

2. Steel bars of approximately 360 mm (test A) and 660 mm (test B)

3. Long stroke dial gauges, which is attached to the loading points are used to measurethe deflection.

4. Scale pan to put the load on.5. Variations of weights (1lb, 2lb, 5lb, 10lb)6. Small-notched Perspex blocks fixed to the beam in order to provide a seating point for

the loading pan sharp edges.7. Micrometer8. Steel rule




2.2 Methods

Test A

The 360mm steel bar was used as a simply supported beam across support C and E with its

mid-span at the center, F. (F is 152.4mm away from C and E). The load was then appliedincrementally with 2lb each time at position F on the Perspex block. Before loading, the longstroke dial gauge was zeroed. From 20lb onwards, the incremental weight was reduced to 1lbbecause the steel bar was approaching yielding point and collapse mode. The load wasincreased until the beam collapsed. Note that the load was added gently to the scale pan eachtime so that no additional force was exerted. The data collected were then plotted into a loaddeflection curve, which indicates the load at which first yield and total collapse occurs. In

addition, stress, yield moment, plastic moment and the beam’s stiffness were determinedusing information from the graph plotted.

Test B

The longer steel bar (660mm) was used across support A, C and E. The loading points wereat B and D, which are both 101.6mm apart from the center support, C. Again, before loading,the long stroke dial gauge was zeroed. The incremental weight was 2lb each time but forevery 20lb, all the 2lb weights were replaced by a single 20lb weight. This was followed withzeroing the long stroke dial gauge. The incremental weight was reduced to 1lb when the steelbar had reached its yielding point. Note that two sets of data were collected for each loadingpoint. The load was increased until the beam collapsed. The graph of load against averagedeflection was plotted and the order of plastic hinge was noted from the graph.




3. RESULTS AND CALCULATIONS

Test A

Figure 2

Plot produced using table 1 (in the appendix). Graph of load against mid-spandeflection.

Rearranging equation 1 (from the appendix), to make Young's modulus, , the subject:

Where:

Rearranging equation 2, to make the plastic moment, , the subject:

Where:

Similarly, the yield moment, , was found by rearranging equation 3:

y = 10087x + 9.1505

0

20

40

60

80

100

120

140

160

180

0 0.005 0.01 0.015 0.02

L o a d

[ N ]

Deflection [m]

Graph of Load against Deflection for Test A

Elastic Region

Plastic Hinge

Yielding Region

(0.0181, 156)

(0.0093, 103)




Where:

From these two values, the shape factor, , could be found, using equation 4:

Using equation 5, the yield stress of the steel specimen can be found:

Where:

Test B

Figure 3

Plot produced using table 2 (in the appendix). Graph of total load against averagedeflection.

0

100

200

300

400

500

600

700

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

T o t a l L o a d

[ N ]

Average Deflection [m]

Graph of Total Load against Average Deflection

for Test B

Elastic Region

Plastic Hinge 1

Plastic Hinges 2 & 3

418

514

0.0048 0.0085

580

0.0134




Equation 10 was used to calculate the plastic moment of the beam, based on the figure above:

Where:

Similarly, equation 11 was rearranged to make the plastic moment, , the subject:

Where:

4. DISCUSSION

Test A

In order to be able to compare the theoretical values and the experimental values of the

experiment the theoretical mild steel yield strength was found to be [1].

The equation for calculating the theoretical plastic moment M

p

was:

Therefore, the theoretical plastic moment was found to be M

p

= 12.04 Nm.

The percentage error between the theoretical and experimental plastic moment values wascalculated as such:

Therefore,

The equation for calculating the theoretical yielding moment M

y

was:

Therefore, the theoretical yielding moment was found to be M

y

= 8.024 Nm.




The percentage error between the theoretical and experimental yielding moment values wascalculated the same way as for the plastic moment as shown below:

The percentage error present when comparing the experimental value to the theoreticalvalues for plastic moment and yielding moment. The small percentage error indicates that theexperiment was accurate and the behaviour of the beam under the loading can be successfullypredicted by theoretical calculations with a slight inaccuracy.

In the results section the yielding strength σ was calculated twice for the yielding moment M

y

and plastic moment M

p

which was then compared to the theoretical value of yielding strength

of the beam used. By using the same procedure previously to obtain the percentage errorbetween the theoretical and the experimental values for the yielding strength was found to be2.22 and 1.13 calculated for yielding moment and plastic moment, respectively.

The inaccuracies and errors are still present in the experiment due to various factors.

The slight percentage error might have aroused from the procedure of the experiment and theapparatus used. For example the dead weights had hollow stripe in order to place them onscale pan which results in causing an offset. The weights might have been placed quicklyopposed to putting them carefully and gently. It was impossible to perform the experiment inorder to be ideal since the point where the load was supposed to be concentrated is locallydistributed therefore the circular bend after failure. In ideal conditions there is no increase inlength of the beam and the beam does not slide from its supports, while in the experimentalcase the beams was slightly elongated on both sides of the point of load application.

In addition to the listed errors in the previous paragraph errors that might have been presentwere the readings of the gauges by done by students which can be inaccurate which is ahuman error. The material itself might have not been homogenous.

Test B

As seen in the graph of total load against average deflection for Test B, there are three

separately identifiable regions of data — an elastic region, a region post first plastic hingeformation and a last region post second and third plastic hinge formation. The three regions

display a linear relationship between deflection and load but have decreasing slopes — suggesting that the rate of deflection accelerated as the hinges formed and the load wasincreasing. This was indeed what happened during the experiment: as the second and third

plastic hinges formed, the beam started deflecting by itself after additional load had beenplaced on it. This made measuring the deflection difficult, as will be discussed below.

The test scenario in Part B can be approximated to a beam collapse, i.e. when three differentplastic hinges are formed. The bending moment required to form a plastic hinge in this case istwice as large as in the case of one plastic hinge forming. The experimentally derived valuefor M

p

shows this clearly.




Error discussion

The inaccuracies and errors are still present in the experiment due to various factors.

The slight percentage error might have aroused from the procedure of the experiment and the

apparatus used. For example the dead weights had hollow stripe in order to place them onscale pan which results in causing an offset. The weights might have been placed quicklyopposed to putting them carefully and gently. In ideal conditions there is no increase inlength of the beam and the beam does not slide from its supports, while in the experimentalcase the beams was slightly elongated on both sides of the point of load application.

A problem that was present in both Test A and Test B was the fact that the Perspex blocksused to fix the load in fact made it act as a slightly distributed load (across the bottomsurface of the block, roughly 1 cm). In our experimentally derived equations we have assumedthe loads to be concentrated but the aforementioned fact meant that they weren't quiteconcentrated and this naturally rendered these equations slightly inaccurate. Regardless,plastic hinges were in all cases observed, despite the fact that the loads were not

concentrated. The crackling layer of anti-corrosive paint on the beam at the points of hingingwere part of the proof of this.

Another major issue that plagued the measurements of deflection, in particular for Test B,was the fact that the rate of deflection started accelerating as the load was increased andhinges started forming. After the first hinge formation, but even more so after the second andthird ones formed, the beam started deflecting by itself at a greatly accelerating pace. Thismade measuring the deflection in a precise manner virtually impossible. It seems not to havehad a major impact on our results, however, as shown by the good fit of the data to the lineof best fit drawn in the graph for Test B.

In addition to the listed errors in the previous paragraph errors that might have been presentwere the readings of the gauges by done by students which can be inaccurate which is a

human error. The material itself might have not been homogenous.




5. CONCLUSION

In conclusion we can see that the correlation between our experimental results and thetheoretically expected performance of the beam was strong with the errors in calculationsranging in the 1-2% region. Considering the somewhat simple apparatus used in thisexperiment plus the strong reliance on human reliability, this seems like a sensible accuracy

and reliability for our experiments. Hence we have been able to observe the effects plastichinges have on loading systems especially when being loaded to fail. With Test B’s longer

beam and three plastic hinges being able to survive almost exactly double the load as test A’ssingle plastic hinge system.

From the errors discussed above, several improvements/solutions could potentially beoffered and suggested. A digital readout for the long stroke dial gauges for instance wouldeliminate the human error generated from reading oscillating analogue gauges, as well aspotentially eliminating the errors induced into the system from basic parallax and re-zeroingerrors. The scale pans the dead weights sit on could potentially be made thinner, so as toenable the load to act more like a point force and less of the distributed force it has, hopefullymaking the system more theoretical. This is obvious from the smooth curves/bends produced

around the load points were we would otherwise theoretically expect sharper edges. Lastlyanother improvement could be made to the loading system, which is inherently reliant on thesteady hand of the individual loading the deadweight slowly and in a perfectly horizontalmanner. A potential replacement could be the use of something along the lines of a tensionmachine, to ensure a steady and balanced loading of the beam.

Finally we are in the position to conclude with some degree of certainty that thisexperiment was successful in completing its aims. As the experiment allowed us to see thetransition from theory reality, this was made most evident when the beams reached theircritical loads, and observe directly as the beams ability to resist deformation rapidlydeteriorated. This will go on to further inform us of the nature of beams deforming in realityand the importance this carries when designing load bearing devices.

6. REFERENCES

1. Eagle Steel. Carbon Steel Grades .http://www.eaglesteel.com/download/techdocs/Carbon_Steel_Grades.pdf (accessed04/03/2014).




APPENDICES

Tabulated Data

Test A

Mass Load Deflection

[lb] [kg] [N][0.001

inches][m]

0 0 8.900 0 0.00000

2 0.907 17.799 33.5 0.00085

4 1.814 26.699 68.5 0.00174

6 2.722 35.598 103 0.00262

8 3.629 44.498 138 0.00351

10 4.536 53.397 172 0.00437

12 5.443 62.297 207 0.00526

14 6.350 71.196 242 0.00615

16 7.257 80.096 276.5 0.00702

18 8.165 88.995 312 0.00792

20 9.072 97.895 348 0.00884

21 9.525 102.344 363 0.00922

22 9.979 106.794 381 0.00968

23 10.433 111.244 400 0.01016

24 10.886 115.694 419 0.01064

25 11.340 120.143 441 0.01120

26 11.793 124.593 461.5 0.01172

27 12.247 129.043 480 0.01219

28 12.701 133.493 510 0.0129529 13.154 137.942 540 0.01372

30 13.608 142.392 572 0.01453

31 14.061 146.842 609 0.01547

32 14.515 151.292 662 0.01681

33 14.969 155.741 714 0.01814

Table 1. Data collected from Test A. Yielding point highlighted in blue.

Test B

Mass per

Pan Total Load Deflection at B Deflection at D

Average

Deflection

[lb] [kg] [N][0.001inches]

[m][0.001inches]

[m] [m]

2 0.907 35.599 7.5 0.00019 8 0.00020 0.00020

4 1.814 53.398 15 0.00038 17 0.00043 0.00041

6 2.722 71.197 28 0.00071 25 0.00064 0.00067

8 3.629 88.996 32 0.00081 34 0.00086 0.00084




10 4.536 106.795 43 0.00109 42 0.00107 0.00108

12 5.443 124.594 48.5 0.00123 51 0.00130 0.00126

14 6.350 142.393 56.5 0.00144 59.5 0.00151 0.00147

16 7.257 160.192 65 0.00165 68 0.00173 0.00169

18 8.165 177.991 79 0.00201 76.5 0.00194 0.00197

20 9.072 195.790 85 0.00216 84.5 0.00215 0.00215

22 9.979 213.588 88.5 0.00225 92.5 0.00235 0.00230

24 10.886 231.387 96 0.00244 100.5 0.00255 0.00250

26 11.793 249.186 107 0.00272 109 0.00277 0.00274

28 12.701 266.985 116 0.00295 117 0.00297 0.00296

30 13.608 284.784 125 0.00318 125 0.00318 0.00318

32 14.515 302.583 130.5 0.00331 133.5 0.00339 0.00335

34 15.422 320.382 140.5 0.00357 142.5 0.00362 0.00359

36 16.329 338.181 147.5 0.00375 151.5 0.00385 0.00380

38 17.236 355.980 159.5 0.00405 160 0.00406 0.00406

40 18.144 373.779 173 0.00439 171.5 0.00436 0.00438

42 19.051 391.578 185 0.00470 183.5 0.00466 0.00468

44 19.958 409.377 191 0.00485 196.5 0.00499 0.00492

46 20.865 427.176 205 0.00521 208.5 0.00530 0.00525

47 21.319 436.075 218 0.00554 219.5 0.00558 0.00556

48 21.772 444.975 232 0.00589 231.5 0.00588 0.00589

49 22.226 453.874 246 0.00625 251.5 0.00639 0.00632

50 22.680 462.774 262 0.00665 265.5 0.00674 0.00670

51 23.133 471.673 273 0.00693 277.5 0.00705 0.0069952 23.587 480.573 288 0.00732 287.5 0.00730 0.00731

53 24.040 489.472 297 0.00754 297.5 0.00756 0.00755

54 24.494 498.372 310 0.00787 310.5 0.00789 0.00788

55 24.948 507.271 325 0.00826 326.5 0.00829 0.00827

56 25.401 516.171 346 0.00879 339.5 0.00862 0.00871

57 25.855 525.070 388 0.00986 372.5 0.00946 0.00966

58 26.308 533.970 403 0.01024 386.5 0.00982 0.01003

59 26.762 542.869 425 0.01080 403.5 0.01025 0.01052

60 27.216 551.769 448 0.01138 421.5 0.01071 0.01104

61 27.669 560.668 476 0.01209 441.5 0.01121 0.01165

62 28.123 569.567 511 0.01298 475.5 0.01208 0.01253

63 28.576 578.467 550 0.01397 509.5 0.01294 0.01346

Table 2. Data collected from Test B. Yielding points highlighted.




Relevant Equation Derivations

Test A

Figure 4

Free Body Diagram for Test A.

Using Macaulay's method, the bending moment is given by:

Where is the reaction force at point C, is the applied load and is the beam length.

From a vertical force balance, it is given that:

Since the applied load acts at the centre of the two reaction forces, and the beam issymmetric:

Integrating this expression gives: Integrating again gives:

Using the boundary conditions: It can be found that:

Since deflection at the centre of the beam was recorded, the deflection can be calculated as:




The mid-span is the position of maximum deflection as well as maximum bending:

The plastic hinge begins to form when:

The shape factor is defined as the ratio of the plastic moment to the yield moment:

It is also possible to calculate the yield stress of the specimen:

Test B

Figure 5

Free Body Diagram for Test B.

From a vertical force balance, it is given that:

Where:

(1)

(2)

(3)

(4)

(5)

(6)

(7)




Taking moments about A:

With two equations and three unknowns, the system is statically indeterminate.

Applying Macaulay's method gives the following: Hence:

And:

Using the boundary conditions:

It can be found that:

Additionally:

And so substituting and simplifying yields:

Hence:

Combining equations 6, 7 and 8 gives the following:

Substituting back into the moment equation:

(8)

(9)




Hence:

The first plastic hinge forms at the centre of the beam (i.e. at point C), since this is themaximum:

By sectioning at the centre of the beam, then calculating support reactions, for the case when

(i.e. when the first hinge has formed), and applying Macaulay's method, to find thebending moment at points B and D, it is given that, for collapse:

(10)

(11)

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Stress Analysis Beam Bending