colum n strut

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{SOM- I} [Unit – VI] Columns And Struts

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INTRODUCTION:Column or strut is defined as a member of a structure, which is subjected to axial compressiveload. If the member of the structure is vertical and both of its ends are fixed rigidly whilesubjected to axial compressive load, the member is known as column, for example avertical pillar between the roof and floor. If the member of the structure is not vertical and one oboth of its ends are hinged or pin joined, the bar is known as strut i.e. connecting rods, pistonrods etc.

FAILURE OF A COLUMN:The failure of a column takes place due to the anyone of the following stresses set up in the

columns:a !irect compressive stresses.b "uckling stresses.c Combined of direct compressive and buckling stresses.

Failure of a Short Column: # short column of uniform cross-sectional area #, subjected to an axial compressive load $, asshown in %ig. The compressive stress induced is given by& p=PA

If the compressive load on the short column is gradually increased, a stage will reach when thecolumn will be on the point of failure by crushing. The stress induced in the columncorresponding to this load is known as crushing stress and the load is called crushing load.'et, $c ( Crushing load,

)c( Crushing stress, and

# ( #rea of cross-section

#II short columns fail due to crushing.!"=P" A

Failure of a Lon# Column: # long column of uniform cross-sectional area # and of length l, subjected to an axiacompressive load $, is shown in %ig. # column is known as long column, if the length othecolumn in comparison to its lateral dimensions, is very large *uch columns do not fail bycrushing alone, but also by bending +also known buckling as shown in figure. The buckling loadat which the column just buckles, is known as buckling or crippling load. The buckling loadis less than the crushing Ioad for a long column. #ctually the value of buckling load folong columns is low whereas for short columns the value of buckling load is relatively high.



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'et I ( 'ength of a long column$ ( 'oad +compressive at which the column has just buckled

# ( Cross-sectional area of the columne ( aximum bending of the column at the centre)o ( *tress due to direct load ( $-#)b ( *tress due to bending at the centre of the column ( +$ x e -

/here, ( *ection modulus about the axis of bending.

The extreme stresses on the mid-section are given by:Ma$imum %tre%% = !o & !' andMinimum %tre%% = !o ( !'

The column will fail when maximum stress +i.e., )o 0 )b is more than the crushing stress )c. "uin case of long columns, the direct compressive stresses are negligible as compared to bucklingstresses. 1ence very long columns are subjected to buckling stresses only.

A%%umption% ma)e in the Euler*% Column theor+:The following assumptions are made in the 2uier3s column theory:

4. The column is initially perfectly straight and the load is applied axially.5. The cross-section of the column is uniform throughout its length.

6. The column material is perfectly elastic, homogeneous and isotropic and obeys1ooke3s law.7. The length of the column is very large as compared to its lateral dimensions.8. The direct stress is very small as compared to the bending stress.9. The column will fail by buckling alone.. The self -weight of column is negligible.

En) "on)ition% for Lon# Column%:In case of long columns, the stress due to direct load

is very small in comparison with the stress due to buckling. 1ence the failure of long columns





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takes place entirely due to buckling +or bending. The following four types of end conditionsof the columns are important:4. "oth the ends of the column are hinged +or pinned.5. ;ne end is frxed and the other end is free.

6. "oth the ends of the column are frxed.7. ;ne end is frxed and the other is pinned.

For a hin#e) en), the )efle"tion i% -ero. For a fi$e) en) the )efle"tion an) %lope are

-ero. For a free en) the )efle"tion i% not -ero.

Si#n Con/ention%:The following sign conventions for the bending of the columns will be used :4. # moment which will bend the column with its convexity towards its initial central line as

shown in %ig. +a is taken as positive. In %ig +a, #" represents the initial centre line of acolumn. /hether the column bends taking the shape #"3 or #"<, the moment producing thistype of curvature is positive.

5. # moment which will tend to bend the column with its concavity towards its initial centreline as shown in %ig. +b is taken as negative.

E$pre%%ion for "ripplin# loa) 0hen 'oth the en)% of the Column are hin#e):The load at which the column just buckles +or bends is called crippling load. Consider acolumn #" of length l and uniform cross-sectional area, hinged at both of its ends # and ". 'et$ be the crippling load at which the column has just buckled. !ue to the crippling load, thecolumn wiII deflect into a curved form #C" as shown in %ig.





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Consider any section at a distance r from the end #.'et y ( !eflection +lateral displacement at the section.

The moment due to the crippling load at the section ( -$ x y +- ve sign is taken due tosign convention

/here C4 and C5 are the constants of integration and the values are obtained asfollows: #t #, x ( = and y ( =

#s if C4 ( /, then if C5 is also e>uals to ?ero, then from 2>n

no. +i, we will find that y ( /. This

means that the bending of the column will be ?ero or the column will not bend at all, which is notrue.

Taking the least practical value.

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E$pre%%ion for "ripplin# loa) 0hen one en) of the "olumn i% fi$e) an) theother i% free:Consider a column #", of length l and uniform cross-sectional area, fixed at the end # andfree at the end ". The free end will sway sideways when load is applied at free end andcurvature in the length I will be similar to that of upper half of the column whose both ends arehinged.'et $ is the crippling load at which the column has just buckled. !ue to the crippling load $, thecolumn will deflect as shown in %ig., in which #" is the original position of the column and #"3is

the deflected position due to crippling load$.

Consider any section at a distance x from the fixed end #.'et y ( !eflection +or lateral displacement at the section

a ( !eflection at the free end "3Then moment at the section due to the crippling load ( $ +a @y +0ve. sign is taken due to sign convention



The solution of the !ifferential 2>uation is:

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/here C4 and C5 are the constants of integration and the values are obtained from the

boundary conditions, which are as follows:

i. #t fixed end, the deflection as well as slope will be ?ero.

"ut at the free end of the column, x ( l and y ( a,

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E$pre%%ion for the "ripplin# loa) 0hen 'oth en)% of the "olumn are fi$e):Consider a column #" of length I and uniform cross-sectional area fixed at both ends # and "as shown in %ig. 'et $ is the crippling load at which the column has buckled. !ue to the

crippling load $, the column will deflect as shown. !ue to fixed ends, there will be fixed endmoments say / at the ends # and ". The fixed end moments will be acting in such direction

so that slope at the fixed ends becomes ?ero.Consider a section at a distance x from the end #. 'et the deflection of the column at thesection is y. #s both the ends of the column are fixed and the column carries a crippling loadthere will be some fixed end moments at # and ".

'et / ( %ixed end moments at # and ".


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*ubstituting the value x ( = and y ( = in e>uation no +i, we get.

!ifferentiating e>uation +i, with respect to x, we get.


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E$pre%%ion for the "ripplin# loa) 0hen one en) of the "olumn i% fi$e) an)the other en) i% hin#e) 1or pinne)2:Consider a column #" of length I and uniform cross-sectional area, fixed at the end #and

hinged at the end " as shown in %ig. 'et $ is the crippling load at which the column hasbuckled. !ue to the crippling load $, the column will deflect as shown in %ig. There will be fixedend moment +/ at the fixed end #. This will try to bring back the slope of deflected column?ero at #. 1ence it will be acting anticlockwise at #. The fixed end moment / at # is to bebalanced. This will be balanced by a hori?ontal reaction +1 at the top end " as shown in %ig.

Consider a section at a distance x from the end #'et y ( !eflection of the column at the section,

/ (1 (

%ixed end moment at #, and1ori?ontal reaction at ".




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The solution"' of the above diff er ential equation is

H + p(/-x) ... (i)

where C1 and C2 are constants of integration and their values are o btained from bouridm-yconditions. oundary conditions ar e !

(i} "t the fi#ed end A. x ( /. y $ and also %& ( /

(ii) "t the hinged end B, x '' l and y ( /.

ubstituting the value x ( / andy ( in equation (i ), we get

o ( C1

# 40 C 2

# ; * H A @ O> ' c1 t -P

c+ (@ P H

. t ... ( ii )

,iff er entiating the equation (i) w.r .t. x, we get

CZ ( -1 , 4 sin ( x . J %z 0 2 cos ( x. J ; ) . [Gi- - 1/,

( @ 1 sin ( x: .

J ;1 )

fu* C ~ cos ( x.

/:i / . 0-1 -

~

d y"t A, x ( / and dx ( /.

/ ( @ c I x $ 0 2 . 1 . J El - 1:(.2 sin$!c$+co+)'1)

( 2 V t E r: - p H or -2 ( H p

!E p

".

u bstituting the values of -1 ( @ B6

l and 2 ( F J ~in equation U

,, we

get

y ' - p . l cos & x \ ) E " ) 0 # C # sin x 3 El 0 p ( I - x i

"t the end ,!#! ' l andy ( /.

4ence the above equation become+1+ as

$ ' - p l cos l l C EI 0 p ...! # sm $! EI P

_ H 5 (5 fl6/ 0 H ~ sin (1 f76/ 0 /

or 1:, ~ ~ sin (5 / :I )= ~ l cos c l D &Ior sin ( z D ;I ) ' & . l EA B # D ;I · cos ( l D :& /

' l. ,I ;I . cos ( l. D ;I )or tan ( l J :I j <3 l · D ;I ·

3 . .,

(l - l)

The solution to the a bove equation is, l . / ;I ' 8.9 radiansz2 .

:I 8.92 ( 2/.29

l H .l

:I ) .

H ( (P ) H ~ · (5 tP ) 0 H

H ' r$F3I H I E" ' /P J

( - P cos ..!Ei ) # v # v Ei

P



(quaring both sides, we get

(

:'2.29 <<<<?5

ut a;;r o#iinately 2.29 ' 21t<=

5:rr 5 E" P ( 52

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Effe"ti/e len#th 1or e3ui/alent len#th2 of a "olumn:The effective length of a given column with given end conditions is the length of ane>uivalent column of the same material and cross-section with hinged ends and having thevalue of the crippling load e>ual to that of the given column. 2ffective length is also called

e>uivalent length. 'et 'e ( 2ffective length of a columnI ( #ctual length of the column and$ ( Crippling load for the column

Then the crippling load for any type of end condition is given by

The crippling load +$ in terms of actual length and effective length and also the relationbetween effective length and actual length are given in Table below.

There are two values of moment of inertia i.e., Ixx and Iyy.The value of I +moment of inertia in the above expressions should be taken as the least value othe two moments of inertia as the column will tend to bend in the direction of least momentof inertia.

Cripplin# %tre%% in Term% of Effe"ti/e Len#th an) Ra)iu% of 4+ration:


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Slen)erne%% Ratio:The ratio of the actual length of the column to the least radius of gyration of the column isknown as slenderness ratio.

Limitation% of the Euler*% Formula:

if the slenderness ratio i.e. +l-k is small the crippling stress +or the stress at failure will be high"ut for the column material the crippling-stress cannot be greater than the crushingstress. 1ence when the slenderness ratio is less than a certain limit 2uler3s formula gives avalue of crippling stress greater than the crushing stress. In the limiting case we can find thevalue of l-k, for which the crippling stress is e>ual to crushing stress.

%or example, for a mid steel column with both ends hinged.Crushing stress ( 66/ G-mm

5

Houng3s modulus, 2 ( 5.4 x 4/8

G-mm5

2>uating the crippling stress to the crushing stress corresponding to the minimum value oslenderness ratio, we get

5en"e if the %len)erne%% ratio i% le%% than 67 for mil) %teel "olumn 0ith 'oth en)% hin#e), the Euler*%formula 0ill not hol) #oo).





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Pro'lem8. # hollow mild steel tube 9 m long 7 cm internal diameter and 9 mm thick is used asa strut with both ends hinged. %ind the crippling load and safe load taking factor osafety as 6. Take 2 ( 5 x 4/

8G - mm

5.

Pro'lem9. # simply supported beam of length 7m is subjected to a uniformly distributed load o6/ G-m over the whole span and deflects 48 mm at the centre. !eterminethe crippling load the beam is used as a column with the following conditions:+i ;ne end fixed and another end hinged+ii "oth the ends pin jointed.





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Ranine*% Formula:/e have learnt that 2uler3s formula gives correct results only for very long columns. "ut whahappens when the column is a short or the column is not very long . ;n the basis of results oexperiments performed by Jankine, he established an empirical formula which is applicableto all columns whether they are short or long. The empirical formula given by Jankine is knownas JankineKs formula, which is given as

%or a given column material the crushing stress )c is a constant. 1ence the crushing load $+which is e>ual to )c x # will also be constant for a given cross-sectional area of the column. Ine>uation +i, $c is constant and hence value of $ depends upon the value of $2. "ut for a givencolumn material and given cross-sectional area, the value of $2 depends upon the effectivelength of the column.

+i If the column is a short, which means the value of 'e is small, then the value of $will be large. 1ence the value of 4-$2 will be small enough and is negligible ascompared to the value of 4-$C . Geglecting the value of 4-$2 in e>uation +i, we get,

1ence the crippling load by Jankine3s formula for a short column is approximatelye>ual to crushing load. #lso we have seen that short columns fail due to crushing.

+ii If the column is long, which means the value of 'e is large. Then the value of $2 wibe small and the value of 4-$2 will be large enough compared with 4-$C.1ence thevalue of 4-$C may be neglected in e>uation +i.

+iii 1ence the crippling load by Jankine3s formula for long columns is approximatelye>ual to crippling load given by 2uler3s formula.

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Pro'lem: # hollow cylindrical cast iron column is 7 m long with both ends fixed. !etermine theminimum diameter of the column if it has to carry a safe load of 58/ G with afactor of safety of 8. Take the internal diameter as /.L times the external diameterTake)C ( 88/ G -mm

5and. a ( 4-49// in Jankine3s formula.




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colum n strut

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