Plastic Analysis Full Report

1.0

OBJECTIVE

1.1

To find the form factor

1.2

To investigate the load deflection relationship for a beam to the point of plastic

collapse.

2.0

LEARNING OUTCOME

2.1

Able to apply the structural knowledge in practical application.

2.2

To improve the technical efficiency through the laboratory work.

2.3

Able to communicate effectively in team work.

2.4

Able to recognize the problem, solving and getting the solution through experimental work

3.0

INTRODUCTION During the design process for beams it would not be unreasonable for one to

assume that no part of the beam should experience a stress greater than that allowable for the working material. However, it can be found that a beam will withstand much larger forces before collapse than simple elastic theory predict. In the analysis, as the stress increased further the plasticity spread inwards until an entire cross section of structure has yield point. At the yield point, the steel attain its maximum possible moment capacity called the plastic moment, Mp. The development of the hinge caused a redistribution enables the structure to carry more loads after first hinge has formed. The second plastic hinge forms at the next most critical stage. On further increase in stress, the bending moments at the section of the two plastic hinges remain constant at their plastic moments and it keep increasing until the third plastic hinge forms. The process of the formation of successive plastic hinges continues until collapse of structure.

1

The purpose of plastic analysis is to determine the collapse load or ultimate load. Plastic analysis considers the behavior of structure in plastic limit before the structure collapse. 4.0

THEORY

When a beam is bent around the neutral axis, the stress through the beam section varies with the distance from the neutral axis, form the greatest at the extreme fibers (y= maximum) to zero at the neutral axis (y = 0)

If the beam is subjected to an increasing bending moment, the stress will build up through the section to a maximum at the extreme fibers. This means that although the outer parts of the beam may well have yielded and are behaving plastically, the inner parts may still be behaving elastically and resisting load. If the bending moment continues to increase, the plastic portion will move further into the beam leaving a smaller elastic core. This called the partially plastic condition. The beam will continue to resist the bending moment although with an increasing rate of deflection as the plastic portion moves further toward the neutral axis. (See diagram below)

Eventually the elastic portion will far enough into the beam and the beam will be “fully” plastic. It will form a plastic hinge and be unable to resist any further bending moment (shown below)

The ratio of the “fully’ plastic bending to the “just” plastic moment is call FORM FACTOR. The form factor is entirely dependent on the shape of the beam and not on the size, material or fixing condition.

For cantilever beam

For the simply supported beam

Experiment from factor = Mp / My where the text book value is 1.5

50

PROCEDURE

Experiment (Simply Supported Beam)

Figure 1 : Specimen beam 1. The specimen beam were taken and the cross section were measured using the

steel ruler, and then the second moment of area for the specimen were calculated.

Figure 2 : Ruler to measure the specimen

Figure 3 : Placing the specimen beam

2. The clamp plates were ensure removed and the specimen beam placed were

across the chucks of the unit. 3. The roller mechanism were push outwards to its stop. 4. The pin were put through the load cell fork and winded the load cell down until

the pin just touches the specimen beam zero both the load cell and the indicator.

Figure 4 : Winding the load 5. Wind the load cell down to cause a measured deflection of 3mm and the reading of the force required. 6. Continue to wind the load cell down in 3mm step until there is no or very little

increase in load for each increment of deflection.

6.0

APPARATUS

Sketch the apparatus used un this experiment and named the essential components

Figure 5 : The Sketch of the apparatus

7.0

RESULT

DEFLECTION (mm)

FORCE (N)

0

0

3

52

6

88

9

109

12

129

15

150

18

161

21

167

24

187

27

186

Table 1 : Result for experiment ( Simply Supported Beam)

8.0

DISCUSSION

1. Plot the graph Force versus Deflection and from your result comment on the shape of the resulting plot.

Force vs Deflection

Force (

200 150 100

force (

50 0 0

5

10

15

20

25

30

Deflection (mm)

Figure 6 : The graph force versus deflection

From the Force versus Deflection above, it is show that the graph is increased steadily. We also can see that when the value of force increase, the value of deflection also increase due to the increasing of the force. When the value of deflection is 27mm the value of the force is 186N decrease from the value before that which is when the deflection is 24mm, the value of force is 187N. From the graph, we also can know that when the value of deflection is decrease the value of force also decrease. According to the graph that have been plotted, the maximum load is 186N when the deflection reaches at 27mm. If we continue to do the experiment with high load, the beam will be achieved the collapse load. Generally, if the deflection increased further, it can cause more forces on the beam.

2. From Table 1, note the collapse load, and using the bending moment diagram calculates the plastic moment (Mp).

M=

Y1 - Y2 X1- X2 150 – 88 15 – 6

=

62 9

= 6.89 N/mm

The maximum deflection is 27mm, when the force reaches to 186 N.

Mp = wL 4 = 186 (750) 4 =

34.875 x 103 Nmm

Mp + Mp = PL 2

4

3Mp = PL 2

4

P =

6Mp

L

=

6 (34.875 x 103) 750

=

279 N (collapse load)

3. Using yield stress of 325 Mpa* Calculate the bending moment (My) to just cause yielding of the extreme fibers My = σyI y σy = =

4mm

325 Mpa 325 N/mm2

= bd 3

I

1 = 8.0 mm ( 8.0 mm ) 3 12 = 341.33mm4

My =

(325 N/mm2) (341.33mm4) 4mm

= 27733.06 Nmm

=

27.73 Nm

4. Calculate the form factor (Mp/My) . Compare to the text book value

Form Factor = Mp My

=

34.875 x 103 Nmm 27733.06 Nmm

=

1.258

Compared to the text book value, For the rectangular section, S = bh4/4 = 1.5 bh2/6

The form factor is 1.258 < 1.5.

5. Discuss the advantages of considering the extra available strength due to the plastic beam theory when designing structures.

The advantages of considering the extra strength when designing the structures are:

iii)

i)

To determine the collapse load or the ultimate load.

ii)

It is give the additional safety for structure.

It is reduce the risk of failure due to the additional load or calculation error. 12

iv)

To prevent the structure from collapse.

v)

To increase the stability of the structure.

9.0

CONCLUSION

By doing this experiment, we will be able to know about the relationship of load deflection to the plastic collect which we can investigate the point of the beam may collapse. We also can assumed that in a plastic analysis the resulting of the strain distribution is linear about the neutral axis and the resulting of the stress distribution is nonlinear and is dependent on the beam’s material. Note the deflections necessary to develop the stresses indicated in a plastic analysis are generally excessive, frequently to the point of incompatibility with the 13

function of the structure. The large deflections and stiffness changes usually associated with plastic analysis can significantly change the internal load distribution, particularly in statically indeterminate beams. We also can know about the advantages of considering the extra strength when designing the structures which are to determine the collapse load or the ultimate load. It also can give the additional safety for structure and it also reduce the risk of failure due to the additional load or calculation error. Beside that it can prevent the structure from being collapse and it is also to increase the stability of the structure. Based on the result, we can conclude that the experimental value is a bit different compare to the theoretical value. It is because the error that occur during the experiment time happened such as the condition of the beam, wind which make it difficult to get an exact data and also the material that used in this experiment such as the specimen beam. We must use the new specimen when doing this experiment not the second-hand specimen. When we used the second-hand specimen, it will effect our finding and result.

REFERENCES

1.

Department of Structure and Material Engineering (2008), Structural Analysis.

2.

Ferdinand P. Beer, E. Russell Johnston Jr, John T. Dewolf, (2006), Mechanics of Materials (Fourth Edition in SI Units) by SEARCH ENGINES: 3.

http://en.wikipedia.org/wiki/Plastic_bending

4.

R.C Hibbeler, (2006), Structural Analysis, (Sixth Edition in Unit SI), PEARSON Education.