b theory of a statically indeterminate bending beam

8/21/2019 b Theory of a Statically Indeterminate Bending Beam

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Universitas Indonesia

SOLID MECHANICS PRATICAL REPORT

MODULE B

THEORY OF A STATICALLY INDETERMINATE

BENDING BEAM

GROUP PIAdam Yuta Prayoga ( 1206292370 )

Asti Diar Safitri ( 1206292414 )

Bimasena Heribowo ( 1206292351 )Christopher Kevinly ( 1206223846 )

Nathan Djumali ( 1206292420 )

Wednesson Lawijaya ( 1206230593 )

Date of experiment : 17042014Experiment Assistant : Aulia Rizky Tansir

Approval date :

Score :

Assistant Signature :

SOLID MECHANICS LABORATORY

CIVIL ENGINEERING DEPARTMENT

FACULTY OF ENGINEERING

UNIVERSITY OF INDONESIA

DEPOK 2014


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A. THEORY OF STATICALLY INDETERMINATE BENDING BEAM

I. PURPOSE

1.

Check the accuracy of the simple bending theory by comparing the value

of E (modulus of elasticity) were obtained from existing experiments with

E literature to concentrated loads and moment loads on the structure

statically indeterminate structure.

2. Check the accuracy of the use of Theorem moment by finding the value of

k (constant) for load moment at midspan on statically indeterminate

structure

II. THEORY

The deflections and rotation angle of the beam or cantilever can be

analyzed by several theories. As an example:

- Unit load Method

- Area of moment or conjugate beam Method

-

Integration Method

For further information, there has been adviced reading the further

structural analysis from C.K. Wang.

III. APARATUS

1HST. 601 Cantilever End and Fixed End

1HST. 602 Cantilever End and roller

1HST. 603 Complete moment used

2HST. 604 Double Pulley

2HST. 605 Wire Collection

3HST. 606 Hanger Clamp

2HST. 607 Hanger Connector

2HST. 608 Big Hangers

7HST. 609 Small Hangers

1HST. 610 Hanger Balancer

1HST. 611 Adjustable Cantilever


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1HST. 6m Watch Gauges

1HST. 6c Metal

1HST. 6d Perspective Beam Test

Figure B.1props structure with upward loadand load moment.

Figure B.1 shows the application of a concentrated load to the top (upward load)

and the load torque on statically indeterminate structure. Many variations can be

made as shown round the corner and deflection on the placement, hanging load or

load evenly split, the theory of reciprocity, and others.

IV. Experiment 1. Expense Focused Amid Placement Landscape With

Pinch-Joints

1/2L 1/2L

HST.HST.

HST.

HST.

L/2 L/2

P

x

C

D

Figure B.2Experiment 1 Condition


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A. How it Works

1.

Setting spans two buffers at 0.9 m thick iron rod and insert the tip of the

tool. Measure the dimensions of the steel plate and the distance x

2. Locking arm drive at point C to produce a state-flops placement and attract

key mover arm at point D to produce a joint placement conditions

3.

Putting hanger and clamp load at midspan and prepare timepiece gauge to

measure the deflection at the concentrated load. Check that the load on the

buffer, free to rotate in the direction of beam deflection.

4.

Adding loads one by one from 2 N to 10 N (load variation can be

determined). Noting watches gauge readings (A and B)

B. Observations and Data Processing

Determining the value of Modulus of Elasticity / Young's Modulus (E) of the lab

results from the deflection formula theoretically calculated by equation

(7PL3/768EI) and from the formula rounds the corner at point D is calculated

according to equation (0.03125 PL

2

/EI), then the results were compared withELiteratur -Steel = 200.000MPa. Round the corner at point D at trial that (DGI readings

on D) / x.

C. Data Processing

From the results of the lab experiments Certain Static Theory of bending beam

with concentrated load at the middle span Pinch-Pinch placement, obtained some

data from a dial readings with each load 2N, 4N, 6N, 8N, and 10N. Dial readings

are then inserted in the formula to get the value of E that magnitude will be

compared with ELiteratur-Steel = 200.000MPa, then proceed with the calculation of

the relative error.


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General Formula:

=Dial AP

Dial D =

P X

37PL

E = Akibat 768I

20.03125PL

E = Akibat I

n 2

Ei-Ei=1

E =EPraktik n

37PL

=768EI

20.03125PL

=EI

E -ETeori Praktik

K =E E

Teori

E -ETeori Praktik

K =E E

Teori

T

T

Experiment 1 Practical Calculation Result

A. Calculation of the value of E from (Rotation Angle)

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4

E Steel = 200000 N/mm2


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No P (N) Dial B (mm) E Due E Average (Ei-)

1 2 0.55 0.0055 363128.2829 277299.7 85828.55

2 4 1.19 0.0119 335664.7993 277299.7 58365.07

3 6 2.71 0.0271 221092.866 277299.7 56206.87

4 8 3.22 0.0322 248100.0691 277299.7 29199.66

5 10 4.57 0.0457 218512.6429 277299.7 58787.09

1386498.66 288387.2

E Average = 277299.7

( (Ei-)2

) / n-1 = 72096.81

E Maximum = 349396.5 MPa

E Minimum = 205202.9 MPa

Therefore, Its obtained that E1= 349396.5 MPa and E2= 205202.9 Mpa

B. Calculation of the value of E from (Deflection)

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4



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No P (N) Dial A (mm) (mm) E Due E Average (Ei-)

1 2 1.43 1.43 366619.901 352826 13793.91

2 4 3.06 3.06 342657.816 352826 10168.18

3 6 4.71 4.71 333927.6805 352826 18898.31

4 8 5.98 5.98 350679.9053 352826 2146.09

5 10 7.08 7.08 370244.674 352826 17418.68

1764129.977 62425.17

E Average = 352826

( (Ei-)2

) / n-1 = 31212.58

E Maximum = 384038.6 MPa


Therefore, Its obtained that E1= 384038.6 MPa and E2= 321613.4 MPa

C. Theoritical Calculation

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

IBeam

= 25.34792 mm4



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No P (N) (mm) E Due E Due

1 2 0.009986028 2.621332292 200000 200000

2 4 0.019972056 5.242664585 200000 200000

3 6 0.029958083 7.863996877 200000 200000

4 8 0.039944111 10.48532917 200000 200000

5 10 0.049930139 13.10666146 200000 200000

D. Relatives Error Calculation (RE%)

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4


No P (N) Theory Practical Relatives Error Theory (mm) Practical Relatives Error

1 2 0.009986028 0.0055 44.92304527 2.621332 1.43 45.44759

2 4 0.019972056 0.0119 40.41674897 5.242665 3.06 41.63273

3 6 0.029958083 0.0271 9.540274348 7.863997 4.71 40.10679

4 8 0.039944111 0.0322 19.38736626 10.48533 5.98 42.967935 10 0.049930139 0.0457 8.472115226 13.10666 7.08 45.98167


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EAverage = (352826+ 277299.7) /2

= 315062.8637

Relatives Error Averagefor E = | |

= = 57.53%

Relatives Error Averagefor = 24.54791001 %

Relatives Error Averagefor = 43.22734%

V. Experiment 2. Centralized Load Central In Landscape With Fixed -

Fixed Placement

A. How it Works

1. Setting spans two buffers at 0.9 m thick iron rod and insert the tip of the

tool. Measure the dimensions of the steel plate

2.

Locking arm drive at point D to produce a condition of placement flops


L/2L/2C D

A

P


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3.

Putting hanger and clamp load at midspan and prepare timepiece gauge to

measure the deflection at the concentrated load. Check that the load on the

buffer, free to rotate towards the beam deflection.

4.

Adding loads one by one from 2N to 10 N (load variation can be

determined). Record gauge readings watches (A)


Determining the value of Modulus of Elasticity / Young's Modulus (E) of the lab

results from the deflection formula theoretically calculated by equation

(PL3/192EI) then the results were compared with ELiteratur-Steel = 200.000MPa. then

proceed to determine the value of the average elastic modulus obtained from

experimental data processing 1 and 2 and calculate the error literature.

C. Data Processing


with concentrated load at the middle span Pinch-Pinch placement, obtained some

data from a dial readings with each load 2N, 4N, 6N, 8N, and 10N. Dial readings

are then inserted in the formula to get the value of E that magnitude will be

compared with ELiteratur-Steel = 200.000MPa, then proceed with the calculation of

the relative error.


A. Calculation of the value of E from (Bending)

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4



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No P (N) Dial A (mm) (mm) E Due E Average (Ei-)

1 2 0.7 0.7 427972.6 402914.8 25057.87

2 4 1.456 1.456 411512.1 402914.8 8597.381

3 6 2.24 2.24 401224.3 402914.8 1690.422

4 8 3.09 3.09 387806.9 402914.8 15107.85

5 10 3.88 3.88 386057.8 402914.8 16856.98

2014574 67310.5

E Average = 402914.8

( (Ei-)2

) / n-1 = 33655.25

E Maximum = 436570 MPa


Therefore, Its obtained that E1= 436570 MPa and E2= 369259.5 Mpa

B. Theoritical Calculation

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4

ESteel

= 200000 N/mm2

No P (N) (mm) E Due

1 2 1.497904 200000

2 4 2.995808 200000

3 6 4.493713 200000

4 8 5.991617 200000

5 10 7.489521 200000


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C. Relatives Error Calculation (RE%)

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4


No P (N) Theo (mm) Prac (mm) Relative Error

1 2 1.497904 0.7 53.26804

2 4 2.995808 1.456 51.39876

3 6 4.493713 2.24 50.15257

4 8 5.991617 3.09 48.42794

5 10 7.489521 3.88 48.19428

EAverage = 50.28832

Relatives Error Averagefor E = | |

= = 101.4574%

Relatives Error Averagefor = 50.28832 %


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VI. Experiment 3. Moment On Load Central Placement Landscape With

Pinch-Joints


A. How it Works

1. Setting spans two buffers at 0.9 m thick iron rod and insert the tip of the

tool . Measure the dimensions of the steel plate.

2. Locking arm drive at point C to produce a state -flops placement and

attract key mover arm at point D to produce a joint placement conditions.

3.

Putting the burden on both the load chain ( thus forming the coupling

moment).

4. Adding loads one by one from 5N to 25N ( load variation can be

determined ). Noted timepiece measuring readout ( A and D )


Determining the value of constants kMiddleand krightof lab results of Middleand

Rightby the general equation = ML / ( KEI ) , followed by calculation of the

relative experimental error . Round the corner at point D at trial that ( DGI

readings on D ) / x .

C. Data Processing


with moment load at the middle span perletakkan Pinch - Joints , obtained some

of the data such as the reading of a dial and dial D by loading each 2N , 4N , 6N ,

L/2 L/2

M

CD

x

A


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8N , and 10N . Dial readings are then inserted in the formula to get the value of

constants which will be compared with the magnitude of K literature followed by

a calculation of the relative error .

General Formula:

DialA -Dial A1 2

=A X

Dial D =

D X

LM

EIK =

A A

LM

EIK =

D D

n 2Ki-K

i=1K =Rltf

K


A. Calculation of the value of K from Middle (Rotation Angle A)

L = 900 mmX = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4


L/EI = 0.00017753 N/mm


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No P (N) Moment's Arm M (Nmm) Dial A (mm) A KA KA Average (Ki-)

1 2 80 160 0.19 0.0019 14.94984 14.25891 0.690937

2 4 80 320 0.41 0.0041 13.85595 14.25891 0.402954

3 6 80 480 0.61 0.0061 13.96953 14.25891 0.28938

4 8 80 640 0.81 0.0081 14.02701 14.25891 0.231892

5 10 80 800 0.98 0.0098 14.49219 14.25891 0.233289

71.29453 0.369691

K Average = 67.40480691

( (Ei-)2

) / n-1 = 26.57295073

K Maximum = 93.97775764

K Minimum = 40.83185618

Therefore, Its obtained that K114.44375 and K2= 14.07406

B. Calculation of the value of K from Right (Rotation Angle D)

No P (N) Moment's Arm M (Nmm) Dial D (mm) D KD KD Average (Ki-)

1 2 80 160 0.43 0.0043 6.605744 5.582061 1.023684

2 4 80 320 1 0.01 5.68094 5.582061 0.098879

3 6 80 480 1.5 0.015 5.68094 5.582061 0.098879

4 8 80 640 2.08 0.0208 5.462443 5.582061 0.119618

5 10 80 800 3.17 0.0317 4.480237 5.582061 1.101824

27.9103 2.442885

K Average = 5.582061

( (Ei-)2

) / n-1 = 1.221442


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K Maximum = 6.803503

K Minimum = 4.360618

Therefore, Its obtained that K16.803503 and K2= 4.360618

C. Relatives error Value of K at Middle (Rotation Angle A)

Relatives Error = (14.25891 - 0.184845/14.25891) x 100% = 98.70365033 %

D. Relatives error Value of K at Right (Rotation Angle D)

Relatives Error = (5.582061 - 1.221442/5.582061) x 100% = 78.118433%

VII. Experiment 4. Load Moment In Central Landscape With Fixed Ends

Placement.

A. How it Works

1. Preparing spans two buffers at 0.9 m thick iron rod and insert the tip of the

tool. Measure the dimensions of the steel plate.

2. Lock the drive arm at point C to produce a condition of placement flops.

3.

Put a strain on both the load chain (thus forming the coupling moment).

4. Adding loads one by one from 2N to 10N (load variation can be

determined). Noted timepiece measuring readout (A)

Figure B.5Experimet 4 Condition

M

C DL/2 L/2

A


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Determining the value of constants kMiddleand krightof lab results of Middleand

Rightby the general equation = ML / ( KEI ) , followed by calculation of the

relative experimental error . Round the corner at point D at trial that ( DGI

readings on D ) / x .

C. Data Processing


with moment load at the middle span perletakkan Pinch - Joints , obtained some

of the data such as the reading of a dial and dial D by loading each 2N , 4N , 6N ,

8N , and 10N . Dial readings are then inserted in the formula to get the value of

constants which will be compared with the magnitude of K literature followed by

a calculation of the relative error .

General Formula:

DialA -DialA1 2

=A

XL

MEI

K =A

A

n 2Ki-K

i=1K =

Rltf K


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A. Calculation of the value of K from Middle (Rotation Angle A)

L = 900 mm

X = 100 mm

BBeam = 25 mm

H Beam = 2.3 mm

I Beam = 25.34792 mm4


L/EI = 0.00017753 N/mm

No P (N) Moment's Arm M (Nmm) Dial A (mm) A KA KA Average (Ki-)2

1 2 80 160 0.07 0.0007 40.57814 14.25891 26.31924

2 4 80 320 0.04 0.0004 142.0235 14.25891 127.7646

3 6 80 480 0.14 0.0014 60.86722 14.25891 46.60831

4 8 80 640 0.21 0.0021 54.10419 14.25891 39.84529

5 10 80 800 0.36 0.0036 39.45097 14.25891 25.19207

337.024 53.1459

K Average = 14.25891

( (Ei-)2) / n-1 = 26.57295

K Maximum = 40.83186

K Minimum = -12.314

Therefore, Its obtained that K1= 40.83186 and K2= -12.314


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C. Relatives error Value of K at Middle (Rotation Angle A)

Relatives Error = (67.40480691 - 26.57295073/67.40480691) x 100% =

60.57706869%

VIII. Pratical Analysis

A. Experimental Analysis

Experiments on the theory of beam deflection in statically indeterminate aims to

determine the accuracy of the deflection theory by finding the value of the

modulus of elasticity (E). E values were calculated from lab results will then be

compared with standard E value is 200,000 MPa. Besides this lab also conducted

to determine the constant rotation angle of rotation angles that occur in the joint

placement and at midspan moment due to the coupling rod that works.

At the first experiment, external force is applied on the mid span of the beam, this

beam is placed in structurally indeterminate support reaction which is fixed ends

and fixedpin ends. There is adjustments on the laboratory equipment, there is a

locker at the support reation, therefore during the experiments of fixed ends,

locker/ key are locked but during fixed pin support reation we need to unlock

one of the supports can fuction as pin support.

During the second experiment at the fixed ends conditions, dial gauge is set 0

and variations of load is applied at the middle of the beam. Load variations consist

of 2N , 4N, 6N, 8N and 10N. Only one reading can be taken out from the fixed

ends experiments because there is no rotating angle at the fixed ends. Then

experiments is repeated using a fixedpin ends support. Variations of load that is

applied here is the same as the experiment before but the only differences is that

in this experiment there is two readings that need to be taken down. First one is

the bendings form the middle gauge and secondly is the angle of rotation at the

gauge that is located at the pin support. After readings of both experiments has

been taken down, need not to forget to measure the beam properties that is latter

use to measure Inertia of the beam.


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At the third experiments, we have the same set ups as the first one, but the only

different thing is that there is a moment that is located in the middle of the beam.

With moment arms up to 80 mm. Variations of loads is the same where 2N, 4N,

6N, 8N and 10N of loads is place to produce moment in the middle of the span. At

the fixedpin end support set up there is two readings that need to be taken out,

first one is the deflection in shown in the middle gauge and the second reading is

taken out from the other gauge that is place on the pin support. Things to note is

that we need to always reset the gauge reading to 0 before load is place and

moment act on the middle of the span. There is a slight different for the forth

experiment, reading from the last set up (fixed ends support) is only taken at one

gauge which is the middle gauge.

A brief explanation for the first experiment is that reading are taken down in roder

to find the modulus of elasticity (E) obtain in the pratical. This modulus of

elasticity value are later compared with the theoretical value of mudulus of

elasticity which is 200000 Mpa, unfortunately in this experiment there is no

perfect result and therefore there is always a relatives error for this experiment.

At the second experiment, we obtained the value of k (constant) which is also

later compared to the theoretical value of k, the same situation happened here,

where we could not expect a perfect value of k where later we found out that our

reading is slightly different from the theoretical value, therefore there is also a

relatives error for this test.

Therefore in total we have 4 experiment that is carried out from the laboratory.


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B. Calculation Analysis

At the first experiment with fixed pin end condition, value of modulus of

elasticity is compared between the result obtain from the experiment (laboratory

experiment) with its ideal value. There is two value of E that is obtained from the

first experiment as two reading is taken out from the gauge. First one is the E due

to rotation angle and the second is E due to deflection.

At the first experiment, value of E due to deflection obtained is 352826 MPa and

Value of E due to rotation angle obtained from the pin support is 277299.7 MPa.

If we compared these results with the standard value of E which is 200000 MPa

therefore there is a relative error of both of the result. Relatives error obtained

from the E value due to deflection and the relatives error of E due to rotation

angle at the pin support is 43.22734% and 24.54791001 %

Secondly, is observation on experiment with fixed ends set up. The modulus of

elasticity (E) obtained from deflection formula and with the help of the readings

taken during the laboratory experiment is 402914.8 MPa. Therefore if wecompare this result with the standard value of E (Modulus of elasticity) that is

200000 Mpa, there is relative error for this experiment. Relatives error that is

obtained from the calculation when we compare our reading with the standard

reading is about 101.4574%

At the third experiment which is the fixed pin end support, calculation result

that is obtained is the value of constant (K). There is two K value of obtained in

this experiment: KMiddle and KRight. After calculation is carried out, Kmiddle obtained

is 14.25841 and KRightobtained is 5.582061. Of course there is also relatives error

of K for both KMiddle and KRight. Both relatives error are: 98.70365033% and

78.118433% respectively.

Finally at the last experiment which is in Fixed ends condition, value of K

obtained at the middle of the span is 67.40480691. and the relatives error obtained

in this experiment is 60.57706869%.


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C. Error Analysis

Relatives error is obtained in all of the experiment we took in the laboratory. The

highest relative error percentage obtained is from the last experiment of the fixed

ends set up. From the first experiment for both E due to deflection and E due to

rotation are: 43.22734% and 24.54791001% respectively. From the third

experiment relatives error of k (constant) from the middle and right gauge are:

98.70365033% and 78.118433% respectively. And from the last experiment the

relative error obtained are: 60.57706869%.

Relatvies error could be obtained from several mistakes taken in lab. Firstly in can

be human error. This error cause by us, the practitioners as we have lacked of

experienced to carry out this experiment. Faultness that can be done from human

error are the in accurate reading of the gauge itself.

Secondly error can be obtained from instrumental error. The instrument such as

gauge used in laboratory are too old that some of them do not function well. Some

dial gauge are very sensitive, not to mention during the unloading process of theload, dial gauge didnt go back to zero. This show that the dial gauge reading is

not consistent. During our experiment, we tried applying the same load but all of

them produce different readings. The same thing happen when we accidentally

shake off the set ups or apply a little shock to be set ups, dial gauge can straight

away changes its value.

Third, may be the parallex error, Parallax error occur when practitioners did not

take reading from the dial gauge properly. Meaning to say during the process of

reading the instrument, the eye of the practitioners is not parallel with dial gauge

and therefore error can occur.


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IX. Conclusion

1. In this experiment we could straightaway tell that there is deflection on the

beam when load is applied on to it.

2.

Also there is rotation in the pin when load is applied to structure that is

equipped with pin support.

3. Deflection due to fixed ends set up is smaller than the deflections due in

the fixed and pin ends support.

4. Relatives error is obtained because there is human, instrumental and

parallax error done during the practical.

5. At the first experiment E due and E due are 277299.7 MPa and 352826

MPa.

6. At the second experiment E due is 402914.8 Mpa.

7. At the third experiment Ka is 14.25891 and Kd is 5.582061

8. At the last experiment Ka is 14.25891.

X. References

1.

Tim Penyusun Pedoman Praktikum Analisa Struktur, Pedoman praktikumAnalisa Struktur, Jurusan Sipil FT UI Depok.

2. Hibbeler, R.C.Mechanics Of Materials, Prentice-Hall, Inc. 2003

3. Hibbeler, R.C. Structural Analysis,6thEdition, Prentice-Hall, Inc. 2006


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XI. Attachment

Pitcure XI.1 Side Gauge

Pitcure XI.2 Experimental Gauge

Pitcure XI.3 Middle Gauge

b theory of a statically indeterminate bending beam

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