C Section M312-Report1

8/2/2019 C Section M312-Report1

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Project P3.8 Ultimate Load of a Beam Under Pure Bending

ABSTRACT

Bending of beams is a frequently encountered loading situation in practice. A slender member

subject to traverse loads is termed as a beam under bending. At any cross-section, the traverseloads generate shear and bending moment to maintain equilibrium. One of the common principles

used to determine the loading capacity of a structure is the first yield criterion which assumes thatthe maximum load is reached when the stress in the extreme fabric reaches yield stress. However,

the design based on this rule is not economical for a beam carrying static load, and a substantialreserve of the strength is disregarded. In order to make use of the material strength fully, we must

explore possibilities of loading the beam into the plastic region.

The objectives of the experiment are to verify the shape factor and the limit load for beam ofrectangular cross-section under pure bending, determine experimentally and theoretically the

shape factor and (plastic) limit load of a simply supported beam and study the mode of failure fora thin-walled(1mm thickness) C-section and to propose and experimentally verify an appropriate

reinforcement scheme to prevent or reduce buckling.

This project focuses on the analysis of stresses in beam bending, and was carried out by loadingvarious beams of different cross-sections. It is observed that the experimental ultimate load is

lower than that of the theoretical load, for each of the cross-sections tested. One of the possiblereasons for this is that failure in each of these beams occurs by other modes rather than the pure

bending process stipulated.

The reinforcement of the thin-walled C-channel was carried out on the thin wall C channel was toreinforce the sides of the C Channel using strips bent into s-shape and using adhesive bonding.

From the data collected, it was found that warpage of the sides will not occur when the side of the

C channel is thicker. Hence, part of the reinforcement strips will result in an increase in thicknessof the walls. The middle portion of the S-strip (part b) is used with the intention of increasing the

strength of the C-Channel, without a significant corresponding increase in weight


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1. INTRODUCTION

Bending of beams is a frequently encountered loading situation in practice. A slender member

subject to traverse loads is termed as a beam under bending. At any cross-section, the traverseloads generate shear and bending moment to maintain equilibrium. The bending causes a change

in curvature of the beam and induces tensile and compressive stresses in the cross-section of thebeam. Maximum stresses are achieved in layers furthest from the neutral axis, the layer at which

strain is zero.

1.1BackgroundOne of the common principles used to determine the loading capacity of a structure is the firstyield criterion which assumes that the maximum load is reached when the stress in the extreme

fabric reaches yield stress. While this criterion is easy to apply and safe to use, the design based

on this rule is not economical for a beam carrying static load. According to the elastic flexureformula, the stress in a beam is proportional to the distance from the neutral axis. When a beam ismade to carry a moment causing the extreme fabric to yield, the material below this layer will still

be elastic and is capable to carrying further load. Therefore, if the first yield criterion is applied, asubstantial reserve of the strength is disregarded. In order to make use of the material strength

fully, we must explore possibilities of loading the beam into the plastic region.

1.2ObjectivesThe objectives of the experiment are:

1. To verify the shape factor and the limit load for beam of rectangular cross-section under purebending.

2. To determine experimentally and theoretically the shape factor and (plastic) limit load of asimply supported beam.

3. To study the mode of failure for a thin-walled(1mm thickness) C-section and to propose andexperimentally verify an appropriate reinforcement scheme to prevent or reduce buckling.

1.3ScopeThis project focuses on the analysis of stresses in beam bending, and was carried out by loadingvarious beams, namely 2 C-channels (thick and thin), a box beam and 2 rectangular cross-section

beams. Theoretical analysis and experiments were carried out to determine the strength andstiffness of the beams, which were simply supported at 2 ends with a central load. A C-beam was

chosen and a reinforcement scheme was done to prevent buckling and to increase the strength ofthe beam.


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1.4Equipmenta. ENERPAC hand-pump base loading systemb. Linear Voltage Displacement Transducer (LVDT)c. YOKOGAWA 3025 X-Y Recorder

Figure 1. Experimental Setup.

Figure 2. YOKOGAWA 3025 X-Y Recorder.


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2. THEORY

This project explains the theory of ultimate load applied to a simply supported beam based on theconcept of ultimate load for beam of rectangular cross-section under pure bending. Pure bending

refers to flexure of a beam under a constant bending moment. Therefore pure bending occurs onlyin regions of a beam where the shear force is zero. In the simplified engineering theory of bending,

we make the following assumptions:

1. The beams are assumed to internally statically indeterminate.2. The strains caused by the deformations have a relationship with stresses.3. When Mp is reached, a plastic hinge is formed.The shape factor gives a very good estimate as to how much the yield moment My, could be

exceeded before the ultimate plastic capacity is reached. Basically, shape factor can be calculatedby this simple equation:

Shape factor = M p/ M y

Sc

IM y

y

y

==

where My = yield moment,

y = yielding stress I = moment of inertia c = distance from the neutral axis to load

S = modulus of the cross section

For a beam that experiences plastic yielding, the above equation will be altered to:

Mp= yZ

in which

2

)( 21 yyAZ+

=

A = area of cross section, y1 and y2 = distance of loads to neutral axis.


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3. PROCEDUREThe procedure below is used to investigate the ultimate load of a beam subjected to:(i) pure bending (four point bending)(ii) a simply supported beam

Pure Bending

1. Measure the required dimensions of the beam for the calculations of moment of first yield

and ultimate moment.

2. Place the beam on the roller supports of the test rig.

y1

y2

A

z

y

y

y

Load

Load

y2

y1

Figure 3. Theory of ultimate load.


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Figure 4.

3. Place and centralize the upper rig onto the beam.

4. For C cross-section beams, insert the provided supports before placing the upper rig.

5. Bring the jet into position with the upper rig by using the hand-brake.

6. Set the Liner Voltage Displacement Transducer (LVDT) perpendicular to the stopper.

7. Use the resistor to calibrate the Y-axis by shorting from the amplifier. Note that one

deflection in the Y-axis is equivalent to 1 kN.

8. Calibrate the X-axis to ensure that the plot stays within the range.

9. Ensure that the chart is set to hold and the pen is set to down before proceeding with theexperiment.

10. Pump the hand-pump continuously to apply force onto the beam.

11. Stop the experiment once the graph plot maintains a horizontal straight line.

12. Release the control valve to reset the jet to its original position.

13. Remove the LVDT and the upper rig.

Simply supported beam

Fix a wedge to the jet to stimulate a point load on the beam. Bring the wedge within closeproximity before placing a piece of support between the And place a support between the wedge

and the beam. Repeat steps 6 to 13.

4. RESULTS

5. Rectangular Cross-sectional areaThe parameters are: b=51mm h=13.1mm

Moment of Inertia,

443

10681.612

mbh

I==

Given that the yield stress for aluminium is MPay 300=


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Figure 5.

Figure 6.

Nmh

bhM

I

My

ydyd 6.4372

12

3

==

=

To find Mult, we note that the distance between the resultant forces = h/2

Nmhbh

M ydult 4.65622==

Shape factor = 5.16.437

4.656==

yd

ult

M

M

Mult = Pa/2

Hence, Pult = 6564N

6. Thin-walled C Channelx = centriod of section = 35.9mmy = centriod of section

= ( Ay )/A

= (2 25.3 0.8 25.3/2 + 76.2 1 24.8) / (2 25.3 0.8 +76.2 1)

= 20.58 mm

I total = (Io + Ad2)

= 23

23

22.412.7612

12.76)65.1258.20(8.03.25

12

3.258.02 +

+

+

= 6.068 10-9 m4

max = yd at y = 20.58mm

Myd = yd I / y= 300 106 6.068 10-9 / 20.58 10-3

= 88.5 Nm

For the ultimate moment, we have to locate the position of the neutral axis as the axis is notsymmetrical about the centroid.


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Figure 7.

Figure 8.

Figure 9.

2 24.3 0.8 + 77.8 (y-24.3) = (25.3 y) 77.8y = 24.55mm

Consider the area below the neutral axis

T1 = (0.25 10-3

) (77.8 10-3

) (300 106

)= 5835N acting at 0.125mm from N.A

T2 + T3 = 2 (24.3 10-3

) (0.8 10-3) (300 106)= 11664 N acting at 12.4mm from N.A

Take Moments about C

Mult = 5835 (0.125 + 0.75/2) 10-3

+ 11664 (12.4 + 0.75/2) 10-3

=151.9251 Nm

Shape Factor = 72.15.88

9251.151 =

0.5 P = Mult / 0.2P = (151.9251 / 0.2 ) 2

= 1519.251 N

7. Thick-walled C-Channelx = centriod of section = 35.9mm

y = centriod of section = 19.12 mm

Myd= 297.91 Nm

Mult =533.66 Nm


66.533=

P = 5336.6 N

4.4 Hollow Rectangular Cross-sectional area

The parameters are: b=76.0mm, h=25.2mm

Nmh

IM

yd

yd 43.6312/

==


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Figure 10.

Mult = 743.7 Nm


7.743=

P = 7437 N

On the graph plot, the experimental ultimate load is defined as the maximum point on the graph,and the load of first yield is defined on the maximum point on the graph that lies on the tangent

line drawn from the origin. The Experimental Shape Factor is thus defined as the division of the 2values.

The results are summarized in Table 1 below:

Table 1. Summary of results obtained.

X-section Theoretical

Ultimate

Load (N)

Experimental

Ultimate Load

(N)

Experimental

Load of First

Yield (N)

Theoretical

Shape

Factor

Experimental

Shape Factor

1 Solid

cross-section

6564 4722 3888.9 1.5 1.21

2 Thin C-channel

1519.251 1379.7 1207.55 1.72 1.14

3 Thick C-channel

5336.6 4216.9 2452.3 1.79 1.72

4 Hollow

rectanglecross-

section

7437.045 2200 1764.7 1.178 1.25


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Figure 11.

4.5 Thin-walled C-Channel Subjected to Point Loading

When a concentrated force P is applied at the middle of a simply supported prismatic beam, the

shape of the bending moment is the same regardless of the load magnitude. For any value of P,

the maximum moment M = PL/4, and is M Myp, the beam behaves elastically. When themoment is at Myp, the force at yield , Pyp=4Myp/L.

When Myp is exceeded, contained yielding of the beam commences and continues until the plastic

moment Mp is reached. The curvature diagram prior to reaching Mp at the middle of the beam isshown above. The curvature at the middle of the beam becomes very large as it rapidly

approaches Mp and continues to grow without bound.

By setting the plastic moment Mp equal to PL/4, we have:Pult = 4Mp/L

Pult = (Mp/Myp) Pyp = kPyp,

where the difference between the 2 forces depends only on k, the shape factor.

Given that MPay 300= ,

Diagram for simply supported beam

Bending moment diagram for Aluminium Curvature diagram of Aluminium

Diagram of a plastic hinge

Plastic hinge


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Nmy

IM

I

My

ydyd 9.1432/103.25

10068.610300

3

96 =

==

=

Pyp=4 Myp/ L =4 143.9 / 0.73 = 788.52N

From the above calculations, the shape factor for the thin-walled C-channel is 1.72. Hence, Pult =1356.3N.

The experimental result shows that Pult = 750N. The reason for such a discrepancy is because the

point load causes a hinge to be formed at the midpoint of the thin-walled C channel. As such, theLinear Voltage Displacement Transducer (LVDT) is unable to register further increases in

displacement.

5 DISCUSSION

5.1 Analysis of Results

Table 1 shows the values of the theoretical shape factors, as well as the theoretical and

experimental ultimate loads experienced by beams of various cross-sections. It is observed thatthe experimental ultimate load is lower than that of the theoretical load, for each of the cross-

sections tested.

One of the possible reasons for this is that failure in each of these beams occurs by other modesrather than the pure bending process stipulated. In the case of the thin C-channel, buckling and

warpage of the channel occurred shortly after the application of the load force. This resulted in thematerial failing, and hence the ultimate experimental load attained was lower than the theoretical

calculated value. In the case of the beam with the rectangular hollow cross-section, it wasobserved that a region of bending occurs in the region where the roller support was placed. The

deformation in that region caused the material to fail.

5.2 Suggested Reinforcement

Various types of reinforcement schemes have been suggested. One of the suggestions is to use

riveting joints. Theoretically, when the distance between rivets is small, the resultant stressconcentration ratio is relatively small as the rivet can be taken to be over a large area. However,

introducing rivets will result in stress concentration at the riveting points and this may result inmaterial failure.


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Thus the modification that was carried out on the thin wall C channel was to reinforce the sides of

the C Channel using strips bent into s-shape and using adhesive bonding.

From the data collected, it was found that warpage of thesides will not occur when the side of the C channel is thicker.

Hence, part of the reinforcement strips will result in anincrease in thickness of the walls (see A and C). The middle

portion of the S-strip (part B) is used with the intention ofincreasing the strength of the C-Channel, without a

significant corresponding increase in weight.

Observations

The weights of the C channels are as follows:

Table 2. Comparison of the weights of various C-channels.

Weight

(g)

Wt as a

percentage of

thin wall C-

channel

Experimental

Load of First

Yield

Experimental

Ultimate Load

Experimental

Shape Factor

Thin wall C-

channel (purebending)

284.6 100 1207.55 1379.7 1.14

Thick-walledC-Channel

(pure bending)

790 278 2452.3 4216.9 1.72

Thin wall C-Channel with

reinforcement(pure bending)

450.9 158 1148.5 1505 1.31

Thin wall C-channel (3-

point bending)

284.6 100 657.5 767.1 1.167

The results show that the experimental ultimate load for the C-Channel with reinforcement is1505N which is an increase over the 1379.7N obtaining a thin wall C-channel without

reinforcement. However, the load required for the thin-walled C Channel to reach first yield hasdecreased, hence resulting in a higher shape factor.

Figure 12. Schematic view of

suggested reinforcement.


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From the experimental shape factor that is taken, it is observed that there is an increase in loadingand the C-Channel can withstand a much higher load before it yields.

As the introduction of force, a cracking sound was heard. This is the result of the failure of the

adhesive bond. As the applied force is concentrated at the mid-span, therefore the adhesivebonding will fail at that point. The mode of failure is similar to that of a thin wall C-Channel under

a 3 pt loading, by buckling. One of the possible reason, is the reinforcement at the midpoint hasalready been weakened substantially. Hence the warpage of the wall has been prevented at the

expense of the strength of the channel, which results in this mode of failure.

Comparing the experimental shape factors that are obtained (see Table 2), the inclusion ofreinforcements has resulted in an increase in the experimental shape factor of the thin-walled C

Channel. This is the result of an increase in the ultimate load that the C Channel can now take.The mode of failure for this thin-walled C Channel with reinforcement has also been altered to that

of a 3-point bending, as evident in the hinge that was observed when the beam failed..

One of the conclusions that can be drawn is that with reinforcement, the shape factor of the thin-

walled C Channel has been increased to 1.31, which is, in comparison, closest to that of arectangular channel. Although this value is slightly lower, it is actually more cost-effective to use

reinforcement like the one that was designed. This is because the material usage (and hence theaddition in weight) in a thin-walled C Channel with reinforcement is much lower as compared to

that for a beam with a rectangular cross-sectional are.

6 CONCLUSIONSBending of beams is a frequently encountered situation in practice. The objective of thisexperiment is to study the ultimate load of a beam under bending. For the experiment, specimens

Figure 13. Formation of a hinge at failure.


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are made of aluminum, which is a nonferrous metal; its properties differ from ferrous metals,

which is iron based. As most high stressed structures are made of ferrous metals, we need tocorrelate our experiment results and observations.

Judging from the results, the solid cross section beam is the strongest and the hollow cross section

beam is the weakest, which verifies the theoretical calculations. Since the proposed reinforced Cchannel has no theory to support, we need to present two hypothesis to explain the results and

observations.

Hypothesis I:

The proposed reinforced C channel is assumed to be a thick wall beam. The warpage of the wall

has been prevented and the specimen failure was to expectation of the thick wall beam. However,the strength of the beam did not increase much, which led to many arguments. One of the reasons

could be that adhesive bonding is not suitable for aluminum, as the joint strength is also

determined by the strength of attachment between adhesive and adherend. And most importantly,based on the thickness ratio of the walls and the strength to weight ratio, the proposed reinforcedC channel did not perform too badly. Further improvements could be made.

Hypothesis II:

The proposed reinforced C channel is assumed to be a solid rectangular beam. As mention above,the strength of the beam did not increase much. In addition, the mode of failure is not what

expected of a solid rectangular beam. The only argument we could put is that the strength toweight ration between the proposed reinforced C channel is very big.

In considering the best structure we would have to take into account the respective strength of the

beams relative to that of their weights. The best combination would be to be a design that is ableto provide high strength and it should be lightweight which we had aimed to achieve.

7 RECOMMENDATIONS

In the process of the experiments, the following recommendations have been made to furtherimprove on the test results:

(i) It was found that loose connections at the ends of the transducer could result influctuations, and hence inaccuracies, in the calibrations. Hence, as a precautionarymeasure, the connections should be checked and tightened, whenever necessary.

(ii) Inclusion of proper supports to hold the adhesion in place. In the reinforcement of thethin-walled C-channel, the adhesion formed between the channel and the S-strips may have


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been stronger if proper supports can be given to secure the strips firmly to the walls of the

C-channel. The availability of such supports may give rise to a higher resultant strength.

8 ACKNOWLEDGEMENTSThe authors wish to express their appreciation to their supervisor, Associate Professor Anand

Asundi, and Mr Tan of the Strength of Materials Lab for their assistance rendered in the project.

9 REFERENCES1. E.P Popov, Engineering Mechanics of Solids, 2nd ed. Chapter 20, Prentice Hall (Singapore),

1999.

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C Section M312-Report1