Mechanics of material lab report

MECHANICS OF MATERIAL LAB REPORT

Experiment #01 Objective: To determine the defection at a point o application o orce on cantilever beam.

Apparatus: 

Straight beam deflection apparatus.

escription o apparatus: !trai"ht beam defection apparatus: It consists of a light, stable frame made of aluminium. The various supports are fastened to the lower girder with clamping levers. The The dial gauges are fastened to the upper upper girder with holders.

t a r a p p a n o i t c e l f e d m a e b t h g i a r t S . s u Following are the complete description of bar bending device:

1


The load weights weights are attached attached to the bar bar via movable movable riders riders . The riders riders can can be locked in position. The load can be adjusted in incre-ments of 2 ! using weight blocks. The articulated supports are fitted with dynamometers . The height of the support can be adjusted using a threaded spindle . The support can be locked in position b" the screw . This compensates deformation of the bar b" its own weight or deflection of the support caused b" spring e#cursion of the d"namometer. In staticall" undetermined s"stems$ it is possible to demonstrate the influence of support deflection on load distribution.

s r e d i r e l b a v o m h t i % r a $

r e t e m o m a n # 

The scales on on the d"namometers d"namometers rotate to enable taring. The bar bar is fi#ed in the support with clamp b" means of a clamping clamping plate . The height of the dial gauges can be adjusted on on their holders.

2


e " u a " l a i 

r e t e m o m a n # d n o e l a c !

p m a l c h t i % t r o p p u !

Theory of an experiment:

Classification of beam based on supports:

1. Simply supported beam: % beam: % beam supported on the ends which are free to rotate & have no moment resistance. resistance.

m a e b d e t r o p p u s # l p m i !

2. Fixed Fixed beam beam : % beam supported at both end and restrained from rotation.

3


m a e b d e x i &

. !"er hang: % simple beam e#tending be"ond its support on one end.

" n a h r e v O

#. $ouble o"erhang beam: % simple beam e#"ending be"ond its both support.

! a e b  n a h r e v o e l b u o D

%. Continuous beam: % beam e#tending over more than two supports.

4


m a e b s u o u n i t n o '

&. Cantile"er beam: % projection beam fi#ed onl" at one end.

m a e b r e v e l i t n a '

'. Trussed : % beam is strengthened b" adding cable or rod to form a truss.

d e s s u r T

(ased on geometry: ). Straight beam: 'eam with straight profile.

"


m a e b t h " i a r t !

(. Cur"ed beam: 'eam with curved profile.

m a e b d e v r u '

(. Tapered beam: beam with tapered cross section.

m a e b d e r e p a T

(ased on the shape of cross section: I. II. III.

*+beam: beam with )I* cross section T+beam: beam with )T* cross section C+beam: beam with )c* cross section

#


m a e b ) T

m a e b ) (

m a e b ) '

(ased on euilibrium conditions:

i.

Statically determinate beam: . It is anal"sed just b" the use of basic e+uilibrium e+uations. '" this anal"sis$ the unknown reactions are found for the further determination of stresses.

m a e b e t a n i m r e t e 

ii.

Statically indeterminate beam:

when the static e+uilibrium e+uations are insufficient for determining the internal forces and reactions on that structure.

$


m a e b e t a n i m r e t e d n (

(ased on materials: a, b, c, d, e, f,

Timber beam Steel beam (oncrete beam %luminium beam (upper beam 'rass beam

$eflection : It is the degree to which a structural element is displaced under a load . It ma" refer : istance • %ngular •

$eflection measuring euipment :



-lectronic gauges : lectronic devices of various t"pes have been emplo"ed more fre+uentl" in recent "ears. These devices can be e#pensive .the" have man" advantages in that data can be obtained without having to get near the bridge & projection can be showed on computer.

%


e " u a " c i n o r t c e l E 

$ial gauges: ial gauges are fairl" common$ however e#tensive e#perience indicates that these devices are e#tremel" slow .

e " u a " l a i 

Finely graduated measuring stic:

e " u a " * c i t !

Tape measure:

e " u a " e p a T

Factors effecting beam strength: &


'ending strength ma" be limited b" :   

/aterial strength 0ateral-torsional buckling 0ocal buckling

(ending : It is the stress induced in the beam because of the application of loads perpendicular to its principal a#is. (ucling 1hen member is subjected to a#ial compression know as buckling. when it is occur majorit" of the chances is that the structure will collapse.

$ifference between bending /bucling: The major the difference is that in bending beam undergoes internal stresses but then recover back after the load is removed while buckiling is a permanent failure of the structure it is normall" occur in colums due to its self-weight. (eam strength depends on its: /a#imum moment  Shear capacit" %nd these both depend upon :  elevant allowable stresses 

(eam fail in three ways :  Strength failure of beam : plastic hinge form when bending 

stress reaches the material "ield strength . 0ateral 3torsional failure of beam : it is due to compression



developed in part of the beam cross section due to bending. 0ocal buckling of beams: it produce at an" pont of the beam it ma" be web buckling or flange buckling.

The aim of this e#periment is to check the mathe-maticall" determined deflection of the cantilever bardetermined manner. The e+uation for the deflection of the bar at the point of application of force is: 4

f = F L

4 E I y

1'


eflection is proportional to the load F and inver-sel" proportional to the modulus of elasticit"  and planar moment of intertia 56/I, I". The length of the bar is cubed. 1here$ I" 7 b h4 892

+rocedure o an experiment: The e#periment is set up as shown in the diagram.

The following e+uipment is re+uired: 9. Steel bar  # 2; # 9;;; mm 2. ider for weight 4. Suspender for weights <. 4 weights ! . ial gauge with holder . Support pillar with clamp =. Fasten the support pillar to the frame >. (lamp the bar in the support pillar ?. 6lace the rider on the bar and lock in the re+uired position 9;. Fasten the dial gauge to the frame with the holder in such a wa" that the tracer pin is touching the flattened part of the rider bolt 99. Set the dial gauge to @ero with the bar unloa-ded. To do so$ adjust the holder and rotate the scale for precise adjustment 92. Suspend the load weight$ read the deflection on the dial gauge and record

Observation , calculation:

11


The following table compares the results of the e#periment with the results of the mathematical calculation. Table:

0enght 0 mm

$eflection mm

Calculated deflection mm

Sample calculation: Formula: F 7 %pplied load 5!, 7 07 length of the bar 5mm, 7 7 /odulus of elasticit" 5!8mm2, 7 b 7 width 5mm, 7 h7height 5mm, 7

/oment of inertia 7I"7 bh4892 7 4

f = F L

4 E I y

12


'onclusion :

(((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( ((((((((((((((((((((((((((((((((((((((((((((((((((

Experiment #0Objective: 13


To calculate the supportin" orces on a bar havin" t%o supports

Apparatus: 


Theor o an experiment:

(ar on two supports: It act as simpl" supported beam having two supports at their ends .

s t r o p p u s o % t n o r a $

This e#periment determines the supporting forces for a bar depending on the point of application of the load #. The supporting forces % and ' can be determined via balances of moments.

(alance of moments around support ': Σ M B = ; = F (L − x ) − A L

Supporting force %

14


x

A = F ( 9− L )

'alance of moments around support %

Σ M A = ; = B L − F x

Supporting force '

x

B = F L .

+rocedure o an experiment: The e#periment is set up as shown in the diagram. The following e+uipment is re+uired: 

Steel bar  # 2; # 9;;; mm



ider for weight



Suspender for weights



4 weights !$ 9 weight 2. !



2 articulated supports with d"namometer

1"


s t r o p p u s o % t n o r a $

9.

Fasten the articulated supports at a distance of 9;;; mm 2. 6ush the rider for the weight suspender onto the bar and place the bar on the supports. 4. 0oosen the locking screw on the support . %djust the height of the support using the rotar" knob until the bar is hori@ontal. <. e-secure the support using the locking screw . Set the scale on the d"namometer to @ero b" twisting . Suspend the weight and load the bar ead the supporting forces on the d"namomters and record reading.

Observation , calculation: The measured supporting forces are ver" consistent with the calculated values :

Table :

1#


istance x rom support A mm/

&orce on support A /

&orce on support $ /

!ample calculation : &ormula : A= support at A point F= load (N) = L= length(mm) = X= distane x from point A (mm) =

A = F(! " L X ) =

B = FL X =

'onclusion :

(((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( 1$


(((((((((((((((((((((((((((((((((((((((((((((((((( ((((((((((((((((((((((((((((((((((((((((((((((((((

Experiment #0 Objective: To dra% the elastic line o a cantilever beam. Apparatus: 

Straight beam deflection apparatus

Theor o an experiment: -lastic line for cantile"er: This e#periment measures the elastic line of a cantilever bar and compares it with the result of the mathematical calculation.

1%


The euation for the elastic line of a cantilever bar loaded with a single force is as follows for the loaded section II with ; ≤ x 2 ≤ a

In the unloaded section I between the point of application of the force and the free end$ the de-flection is a linear function of the length and the inclination α in the point of application of force. This is not bending$ but slanting


Steel bar  # 2; # 9;;; mm

1&




ider for weight












9 dial gauges with holder

t a r a p p a n o i t c e l f e d m a e b t h g i a r t S . s u The load remains constant and is applied in the centre at a7 ;; mm

9. The deflection of the bar is measured with the dial gauge at intervals of 9;; mm. 2. ial gauges measure the deflection due to the d"namometer . 4. elieve the bar <. %ppl" the dial gauge at the re+uired position and set to @ero . 0oad the bar . ead the deflection value and record =. elieve the bar and move the dial gauge to the ne#t position

2'


>. The d"namometers e#perience spring e#cursion under load. In order to prevent measurement errors as a result of this additional deflection f$ the supports should be returned to their original position. ?.

read the deflection value from the dial gauge and record

elieve the bar$ move the dial gauge to the ne#t position$ and repeat the measurement.


Table : !2 #

3 mm/

efection in %x/ mm/

'alculated defection %x/ mm/

21


!ample calculation:

4raph :

22


'onclusion :

(((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( ((((((((((((((((((((((((((((((((((((((((((((((((((

Experiment #05 Objective: To dra% the elastic line or center loadin" o a bar on t%o support Apparatus: 

Straight beam deflection apparatus

Theor o an experiment: -lastic line for centre loading: This e#periment measures the elastic line of a bar on two supports and compares it with the mathe-maticall" calculated result.The euation for the elastic line of a bar loaded in the centre with a single force is as follows: 4

FL

f

.

=

<>

E I y

23



Steel bar  # 2; # 9;;; mm



ider for weight













The load remains constant and is applied in the centre at #7 ;; mm

24


t a r a p p a n o i t c e l f e d m a e b t h g i a r t S . s u

9;. The deflection of the bar is measured with the dial gauge at intervals of 9;; mm. 99. Two dial gauges on the support measure the deflection due to the d"namometer . 92.elieve the bar 94. 0oosen the locking screw on the support. 9<. %djust the height of the support using the rotar" knob until the dial gauges read @ero. 9. Fasten the supports using the locking screw

9. 6lace the dial gauge in the re+uired position and set to @ero 9=.0oad the bar 9>.The d"namometers e#perience spring e#cursion under load. In order to prevent measurement errors as a result of this additional deflection f$ the supports should be returned to their original position. 9?. 0oosen the locking screw on the support. 2;. aise the support using the rotar" knob until the dial gauges read @ero. 29. Fasten the support using the locking screw

22.

read the deflection value from the dial gauge and record

elieve the bar$ move the dial gauge to the ne#t position$ and repeat the measurement.

2"



Table : !2 #

3 mm/

efection in %x/ mm/

'alculated defection %x/ mm/

!ample calculation:

2#


4raph :

'onclusion :

(((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( ((((((((((((((((((((((((((((((((((((((((((((((((((

2$


Experiment #06 Objective: To determine the 7ax%ell)$etti8s infuence coe9cient on a bar havin" t%o supports Apparatus: 


Theor o an experiment: 3axwell+(etti4s influence coefficients and law: Influence coefficients link the deflection at a certain point in the bar to the loading of forces Fj as follows: This e#periment is onl" intended to e#amine the effect of a force on points # 9 and #2 on the deflecti-on at points #9 and #2

# 9 = a99 F 9

# 9 = a92 F 2

# 2 = a29 F 9

# 2 = a22 F 2

2%


%ccording to the Max#ell"Betti transposition law$ deflection at point #9 as a result of the force on point #2 is just as large as the deflection at point # 2 caused b" an identical force on point #9. This correlation is described b" the following formula:

# 9 = a92 F 2 = # 2 = a29 F 9

a =a 29

92

In general$ according to Max#ell"Betti $ the follo-wing applies

a =a i$

$i

The two influence coefficients a99 and a22 indi-cate the deflection at the point of force.


Steel bar  # 2; # 9;;; mm



ider for weight













2&


t a r a p p a n o i t c e l f e d m a e b t h g i a r t S . s u

9. The distance between the supports is 9;;; mm.The load of 2; ! remains constant and is applied at #97 4;; mm and #2 7 ;; mm 2. The deflection of the bar at points #9 and #2 is measured with the dial gauge.

4. Two dial gauges on the supports measure the deflection caused b" the d"namometers and serve to compensate it. The procedure is the same as described in the previous e#periment. <. 0oad the bar at point #9 and measure the deflection at #9 and #2. . 0oad the bar at #2 and measure the deflection at #9 and #2.

Observation , calculation : Table:

3'


!r #

efectio n at poin mm/

&orce at point /

efectio n % mm/

(nfuence coe9cien t mm/

!ample calculation:

'onclusion :

(((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((( 31

Mechanics of material lab report

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