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Chapter, , Theory of Columns, , , , , , 7.1 Intrropuction, A long slender bar under compression is called a column. Usually the word column is, used for vertical bars, while the word strut is used for short members under compression., when a column ora strut is subjected to some compressive force, then the compressive, stress included; i.e.;, P, oo, A, where => P - Compressive force, A = Cross-Sectoral area of the column., Ifthe force or load is gradually applied, the column reaches a stage, where it is subjected, to the ultimate crushing stress, beyond that the column fails. The load corresponding to, the crushing stress is called crushing load., Sometimes a compression member does not fail entirely by crushing . but fails by bending, (buckling). The load at which the column starts bulkling is called as buckling load,, critical load or crippling load., Bulkling load is less for long column and high for short columns., 7.2__EuLer’s CoLuMN THEORY, , , , The theory depicts the buckling load of long columns based on the bending stress., , It may be justified with the statement that the direct stress induced in a Jong column is, , negligible as compared to the bending stress., , It may be noted that the Euler’s formula cannot be used in the case of short columns., , because the direct stress is considerable and hence cannot be neglected, , Scanned with CamScanner
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co, , rs en ana 4, , ae, , e, , , , ’, tee, , i, , oo @, , pally the column is perfectly straight and the load applied iy truly axial, ere cross-section of the column ts uniform through out is lengsh, poe column matenal is perfectly elastic, homogeneous and wattage: ond thes obe, yooke’s law The tength of the column is very lange a8 Compaired to it's (res sectional dimension, The shortening of column, duc to direct compression (very small) ix neglected, The failure of column occurs due to buckling alone, , |, i, , P, , , , 7 ZY, (a}-positive (by negative, , A moment, which tends to bend the column with, convexity towards it’s initial central line as shown in fig, , , , ‘a’ as taken as positive., 4 moment, which tends to bend the column with its concavity towards its initial central, ine as shown. in figure ‘b’ is taken as negative., , Types of Exp ConpItion OF COLUMNS, , There are four types of end conditions, Both end hinged, , Both end fixed, , One end is fixed and other hinged, (ne end is fixed and other free, , Both End Hinged, , - ceri € 1, Comdering x colamn AB olen "1 ine Pn ee acre hws, *cntical load at 'B’ Asa result of loading. let the column ¢ . ‘, , ‘the figure, Let X be the any section at 2 distance x from A, , ‘Scanned with CamScanner
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250 Mechanics of Soyiy, 2 Mids, , P > Critical load on the colin, y © Deflection of the column at X,, ° Moment due to critical load’?, M Py, 2?, aC ; Lo, :, El- a= Py (Minus si rh due to concavily ), t, , Jx, , Wy, EI—-+P.y = 0, dx* ,, , dy Pp, , ~—+— y= 0, dat”, , , , The general solution of the above shores equation is, , | D . Pp - ,, da Acos{s ry. osinfx | ( where — A, B are constant of integration,, , We know that, whenx =0, y= 0, Therefore, A= 0, Similarly when x= ¢, y=0, , P, Therefore, 0 = B sn(¢ *), , Pp, . So either ‘B’ is equal to zero or sin [4 ; is equal to zero,, , o If we consider ‘B’ to be equal to zero, then it indicates that the column has not bent, atall., , Pp, But, if sin (E)- oO. then;, , , , , , , , Scanned with CamScanner
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yhes, , 93., , wot Columns, , , , Both Ends Fixed, , Considering a column AB of length * ¢* fixed, both of its ends ‘A’ and *B’ and carrying a, critical load at B. As a result of loading, let, , the column deflects as shown in fig ., , Let X be the any section at distance x from A, P= Critical load on the column, y = Deflection of the column at ‘X’, , Let M, = Fixed end moment at ‘A’ and ‘B’, , , , Kee i, , «, Moment due to critical load ‘P’;, , M=- P.y, dy, Ee =My-Py («: ‘~ due to concavity), dy P M,, —t+oys, dx* El El, , The general solution of the above differential equation is, , lm me, , where A and B are the constants of integration, , we know when x=0,y =0, , , , M, therefore, A= a, , dy P.( [P fe ff, Seok | — |+B,/—cos| x,|—, te afEsn(x[}+ El El, , We also known that when x = 0; then, #9 0=B E, ae Therefore, 0 = El, , . P, “. 80 either *B’ = 0 or la =0, , Scanned with CamScanner, , tea, , is
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=, , Mechanic: Of Soha, , M s , ”, Substituting value at A Ser and B = 0 in equation (i); we get :, , M, [P).M, M, fe, ei x (a [pee =— | I-cos} x,/—, ye D cs | P P El, , We also know that when x = ¢, then y = 0, , Therefore, o= Mo 1-cos} @ je, P El, P [P, : LjJ— |= — =0=2n =4n=6n=......., eas E) or l 7 0=2n=4n=6n, , ‘ 8 4n’El, taking the least significant value, ¢ = =2n or P= zp, , , , 7.3.3, Oneand Fixed and Other Free, , Considering a column AB of length ‘¢’ fixed at ‘A’, and free at B and carrying a critical load at B. As a, result of loading the column deflects as shown in the, fig, such that the free end B moves a horizontal distance, of ‘a’ to B,., , Let X be any section at a distance ‘x’ from A, , P =critical load on the column, , , , y = Deflection of the column at X., , -. Moment due to the critical load ‘P’, , -. M=+P(a-y) = P.a-Py(‘ +? due to convexity), d’y dy P pa, E]—-=Pa-P. ag a, ox? y eee, , The general solution of the above differential equation is, , y =Acos [x] msi xf Joa eal (i), , (.. A, B integration constant), We know when x=0, y=0, Therefore, A = ~ a, Now differentiating the above equation., , Scanned with CamScanner
