Bending Experiment on steel Stahl -36 and Cyprus limestone

May 20 2015 BENDING TEST

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National Technical University of Athens

School of applied mathematical and

physical sciences

Department of Mechanics

Experimental mechanics of materials

(Jacob Bernoulli 1655 – 1705)

Bending Experiment on steel Stahl -36 and

limestone Cyprus

George Kleitsiotes

Undergraduate student


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The purpose of this experiment is the comparison of Bernoulli’s

theory theoretical prediction of deformations in 3-point and 4-point

bending actual results.

The theory of Bernoulli for bending

Bernoulli’s bending theory is based on the following two basic

assumptions:

1. The transverse sections of the beam remain plain during the

bending deformation

2. the beam behaves as a sum of independent fibers

Based on these assumptions, if the torque exerted on every point

of the beam is given by the function M (x), then the proper stress

at any point is given by the relationship:

sXX = m (x)* y / Izz , with

yy the vertical transverse axis with positive orientation on the

upper base of the beam,

XX, the longitudinal axis which coincides with the theoretical axis

of neutral beam charge (i.e. sXX = 0 for every x)

zz , the horizontal transverse axis

I zz , the second grade surface Momentum of the zz axis.

In case you ignore the scale of the specimen the shear stress

generated during the load, knowing that for any point the tensors

of stress and strain, and through Hooke’s law of deformation

(Eyoung* eXX = sXX) If you call w (x) the arrow bending at any point x,

then the function of elastic line is given by the solution of the

differential equation:

w''(x) = M(x)/(Eyoung*Izz)

Experimental equipment, useful sizes and standards

For the tests on the steel used an INSTRON ® model 1120

(upgraded and modernized) machine, with maximum 250 load kNt ,


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and an electronic device which gives the Volt equivalents of the

load and machine cross displacement as well as the indications of

electrical stain gages.

For testing in the limestone was used one MTS digital machine with

maximum tensile and compression load of 5 kNt as well as the

above electronic device.

And in all 3 we have used mechanical Velometers with accuracy of

10-5 m, for the determination of the actual arrow bending.

Dimensions of test specimens

Stahl – 36 :

l =300 mm

Squares: 30 mm X 30mm

Cyprus limestone:

l = 150 mm

Squares:

41.105 mm X 41105 mm (seen from the average measurement of

dimensions)

Templates

The regulations set out the bending tests are for Deutscher Institut

für Normung the DIN 1050

Feed back

(Stahl – 36)

1 Volt = 6 kNt


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1 Volt = 0.3 cm

(Cyprus Limestone)

1 Volt = 9kNt

1 Volt = 1 cm.

Processing of measurements and experimental

procedure

3 point bending

(Picture 1 layout 3pb)

In the case of three-point bending flexural torque increases linearly

until the point at which the power (blue cylinder – picture 1) gets its

maximum value of (m = Force*lactive /4 to lactive distance of Orange

cylinders like these in picture 1).

So according to Bernoulli’s theory as mentioned above we

can calculate the theoretical bending arrow and to

compare it with the actual deformation. For this reason the

velometers have been placed on the specimen.

Stahl – 36 Test


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(Picture 2 Stahl – 36 during the test)

Note: the Velometers (in positions l/4 & l/2) in Picture 2

have been removed since by the time the photo have been

received the material had already failed

We have added two strain gages on the specimen, one on the

base in order to calculate the Young’s modulus of the material and

one at the theoretical position of neutral axis in order to ascertain

the degree of precision in our calculations.


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(Chart 1 "raw" data charge – front strain gage)

From the above chart, we discover that there is an error in our

calculations as we have incorrectly identified the neutral axis.

Nevertheless, we accept this error is negligible and we move on

to our calculations.


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(Chart 2 Chart trends deformation (strain gauge on base))

Note: Observe possible detachment of the strain gage.

Nevertheless, our measurements are not affected.

Based on the data shown in the diagram above we chose the

data that seem to move in the elastic region (Matlab® Brushing)

and calculate the "important Values” of the specimen.

The modulus of elasticity was calculated through the command

"polifit" (Chart 3)


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(Chart 3)

Based on the above results and the theory we calculate the elastic

line (The script is given at the appendix) as shown in Figure 4

(Chart 4)

As indicated in chart 15 kNt the specimen follows the theory, while

at 30 kNt we have a large margin due the possibility of hardware


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failure. In this case we consider Bernoulli’s theory satisfactory.

(Picture 3 Aftermath)

Note: for the next trials we have followed an identical procedure.

Thus, the sequence will not be analized.


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Cyprus limestone

(Picture 4 Cyprus Limestone during test)

(Chart 5 Raw data)


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(Chart 6 Stress-Strain Diagram)

(Chart 7 Results)


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In the case of limestone, a major failure of the theory is observed.

This may be due to wrong choice of velometers or even the wrong

choice of material, since limestone is extremely porous. As a result

that there are many " stress amplifiers" and the material deforms

at a faster pace. Besides, the theory predicts application only on

homogeneous and isotropic materials.

Stahl - 36 4pb

The experiment of 4-point bending does not differ in comparison

with the process of 3pb. It differs however in calculations as there

is now a large area where that the momentum remains constant as

shown in Picture 5.

(Picture 5 3pb and 4pb comparison)


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Note: since it is very difficult and time-consuming to calculate the

boundary conditions for the equation of elastic four-point line,

received the theoretical values of the arrow bending for particular

locations and worked with the numerical interpolation method for

the determination of the line

(Chart 8 ' raw ' data 4pb)

Note: As no strain gage was placed in the specimen there I no

need to give stress strain chart. The load-displacement diagram

can still be given though.


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(Chart 9 load-displacement diagram)

(Chart 10 results)


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From what we see the deviations are small. So Bernoulli’s theory

is acceptable.

****************************************************************************

(Picture 6 the Biomechanical engineer’s pet)

Comments

1. From that we saw in the first experiment the strain gage is

not set correctly at the location of the neutral axis meaning

that there is an error in procedure.

2. The velomters are analogical something that results in less

precision.

3. An error is still going due to the ignored shear stress.

4. On the bending of the limestone a smaller magnitude

velometer could have been used

5. Punch effect was not taken into account (deformation near

the loading cylinder)


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Appendix

Matlab scripts

% @author : Georgios “Blaze” Kleitsiotes % @date : 11/05/2015 (mm/dd/yyyy) % @version: 1.0.0 % @param : p; the load being applied by the machine at the specimen % : l; the length of the specimen % : e; the young's constant % : I; the moment of inertia % : x; the definition field of the line's function % @description: evaluates the elastic line of a specific specimen under a % 4pb test function b = biegelienie_4pb_evaluate(x,p,l,e,I) xx=[0;l/6;l/3;l/2;(2*l)/3;(5*l)/6;0]; ww = [0;(0.0176*p*(l^3))/(e*I);(0.0309*p*(l^3))/(e*I);(0.036*p*(l^3))/(e*I);(0.0309*p*(l^3))/(e*I);(0.0176*p*(l^3))/(e*I);0]; ww = -ww; pp = polyfit(xx,ww,3); b = polyval(pp,x); end ****************************************************************************

% @author : Georgios “Blaze” Kleitsiotes % @date : 11/05/2015 (mm/dd/yyyy) % @version: 1.0.0 % @param : p; the load being applied by the machine at the specimen % : l; the length of the specimen % : x; the definition field of the line's function % @description: evaluates the bending moment of a specific specimen under a % 4pb test function [M] = Bending_Moment_4pb(l,p,x) if (x<=l/3) M = (p./2).*x; elseif(x<=(2.*l)/3) M = (p.*l)./6; else M = (p.*l)./3 - (p./2).*x; end end ****************************************************************************

% @author : Georgios “Blaze” Kleitsiotes % @date : 05/05/2015 (mm/dd/yyyy) % @versiom: 1.0.0 % @param : Load; the load being applied by the machine at the specimen % : l; the length of the specimen % : x; the position of interest % @description : calculates the bending moment on a specimen with % specified length, for a specified load, at a specified position function [M] = Bending_Moment_3pb(Load,l,x) if x<=l M = Load.*(x/2); else M = (Load.*l)./2 - (Load/2).*x; end end

****************************************************************************


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% @author : Georgios “Blaze” Kleitsiotes % @date : 05/05/2015 (mm/dd/yyyy) % @versiom: 1.0.0 % @param : f; the applied force % : l; the length of the specimen % : x; the position of interest % : E; the young's constant % : I; the moment of inertia % @description : evaluates the elastic line % "Biegelienie is the German word for elastic line" function [w]=Biegelienie(f,E,I,x,l) c=E*I; b=l/2; w = -(((f*b*(l.^2)).*x)./(6*l*c)).*(1-(b/l).^2-(x./l).^2); end ****************************************************************************

Bibliography

1. Πειραματική αντοχή των υλικών – Θεωρία και Εργαστήριο

(Ι. Ν. Πρασιανάκης – Σ. Κ. Κουρκουλής) Εκδόσεις

Συμμετρία

2. Technische Mechanik 2 (D. Gross – W. Hauger – J.

Schröder – W. A. Wall) Εκδόσεις Springer Vieweg

3. Τεχνική Μηχανική 2 (Ι. Βαρδουλάκης) Εκδόσεις Συμμετρία

4. Μηχανική των υλικών (F. P. Beer – E. R. Johnston, Jr- J.

T. DeWolf – D. F. Mazurek) Εκδόσεις Τζιόλα

5. Αριθμητική Ανάλυση (Γ. Σ. Παπαγεωργίου – Χ. Γ.

Τσίτουρας) Εκδόσεις Συμεών

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Bending Experiment on steel Stahl -36 and Cyprus limestone

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