Note on long columns

Aug.,IS87.]

Note an Long Columns.

i2 9

N O T E ON L O N G "COLUMNS. By WM. CAIN, Prof. Math. and Eng., S. C. Mil. Academy.

( ConcZudedfrom page 6r.) ANOTHER FORMULA, INVOI.VING THE BREAKING STRESS MATERIAL.

O F THE

To point out the difficulties of the subject, it may be well to deduce a formula that shall include the breaking stress of the material, and that will lead, on certain suppositions, to the wellknown Rankine form, whose insufficiency can thus more clearly be realized. As in practice there is nearly always some eccentricity in the position of tim load, whose influence is very marked, particularly in the case of short columns, where theoretically there can be no beading except for this eccentricity, we shall include it from the start in tim formula. Thuq. in the figure, let the 'load, jo act at a distance, k~ to the left of the centre line of the column, A B, which, after strain, bends to the curve, A r D.B ; then calling, as before, f--= C D the deflecti0n at the centre from the axis, the moment of P with respect to ]) will now be P ( f + k). W e shall further designate b y Q the area of the cross section at D~ by z the distance from the centre of gravity of the section to the most compressed fibre, and by .E~ /'and r the modulus, m o m e n t of inertia and radius of gyration of the section about a horizontal axis through D, perpendicular to the plane of the paper. Now, if we conceive applied at .D, the centre of gravity of the cross section, two opposed forces, both equal to and parallel to _P, then the downward force at / ) causes a uniformly distributed stress over the section Q of intensity, P and the upward force at D with the load ]~, k inches to the left of A, form a couple, whose moment is P ( f + k), which causes a WHOLE NO. VOL. CXXIV.

(TtlIRD SERIES, Vol. xciv.)

9

x 3o

C~in :

[ I, F. I,,

uniformly varying stress in the section, whose greatest i n t e n s i t y , at the most compressed edge, is p2= ~

1

The greatest compression per square unit at the most c o m p r e s s e d edge of the section at D is, therefore,

P

P ( f -I--k)z

in which expression, if 2}) represents the breaking load, P o represents the ultimate compressive unit stress, supposing, a p p r o x i mately, the law " u t tensio sic vis," holds up to the failing p o i n t . Putting I = Q P, we can write the above formula,

p0 =

(1 +

Now we shall introduce our f i r s t a2proMmation by s u p p o s i n g the curve A' D B to be an arc of',z circle of radius p, so t h a t ~ve have, calling l the length of the column,

y(2 p--y)=

z . ' . ,o =

l2

The last relation is sufficiently true for flat arcs, and is e x a c t t y true for the radius of curvature at D, if we regard the a r c A r D / ~ as parabolic. Next let us consider two cross sections after strain a t _D, d , apart along the axis, then if a - - change in length o f a fibre parallel to the axis, one unit long and at the distance z f r o m the axis of the column, due to the bending m o m e n t alone, w e g a v e the well-known relation, p:ds::z:ads

". P =

z

Equating t h k value of p with the previous one, and n o t i n g

that

~ -~.~ we h a v e .LM

.'.

z

12

a

2f f--

2 z'~

2l~Ep~ z .

(11}

Aug., x887.]

No/e on Long

Co!umns,

I3r:

which, substituted in the value for p above, gives _P

. . . . . . . . . . .

po

...........

o2)

This formula, within the limits of elasticity, is perfectly correct, provided the elastic curve is taken approximately as a circle or parabola; but as we know neither k nor p~, we may introduce errors just at this point by regarding them both as constant for all length ratios; for although k may be nearly constant for the same shapes and end connections and care in fitting, y e t ?,- = P ( f + k ) z varies with f and probably is not constant for all

length ratios.

Placing (1 -}- ~ . ] ==a, and P2 --:b, (12),can be \ r~, ' 2E

written, P . - - - - ......... P~' Q a+b

-- -

(18)

tn the well known Gordon-Rankine formula the load is not supposed eccentrically placed, so that k - - o and a = 1 ; also P2 is regarded as constant (which is doubtless erroneous) and b is treated as a constant to be determined by experiment. The writer first pointed out in Uan N o s t r a n d ' s Magazine, for November, [877 (in criticising the Hatzel formula), that b was not necessarily a cgnstant, though it is remarkable how near the results agree in a general way with experiments at the breaking limit for values of-/ varying from 30 to 600, upon the supposition that b T

is a constant. It is much more rational though not to assume the eccentricity of the load zero, but determine, from the average of the great number of experiments that have been made, an .average value for a, and likewise ascertain whether b can be treated practically as a constant or otherwise (see Weisbach's Mechanics, vol. i, P- 545, for a similar determination). Now, a difficulty presents itself at once in pursuing this plan, namely, the determination of p0 for wrought iron, for its value has been given all the way from 36,000 to 6o,ooo pounds. I believe the value 36,000, given in Gordon's formula, to be

Cain :

~32

[j. F. 1,,

erroneous, since Fairbairn and others have shown b y e x p e r i m e n t s on transverse breaking of wrought-iron beanas, properly d e s i g n e d at the compression flange, so as to give way there, if at all, b y direct crushing, and not by bending, that the tensile unit stress (exerted at the lower flange) was practically equal to the c o m p r e s sive unit stress exerted in the top flan~e. If the tensile unit strength of the iron was, say, ~5,ooo, then we should p l a c e Pu -~65,000 in the formula. In fact, I find that for P0 = 56,000, a ~ 1"4 and b = ~z~-~y0 for wrought-iron columns, with fixed ends, t h a t the fornmla strikes an excellent average of the results given o n Mr. Wilson's diagram (see Trans. Am..%c. Uiv. Legs., vol, x v , ) for length ratios varying from o up to 240. F o r higher l e n g t h ratios, b should gradually increase from 2-g 6g~" i to T;gVgF 1 (see M r . J o h n son's paper, mentioned further.on). The formula is very f l e x i b l % for not knowing P0, a or b, we can determine them to s u i t the results between any limits, t h o u g h of course the Jorm~da 3ecomes largely emtOirical by so doinZ. Thus the values i~o ~ 50,000, a ~--- t ' 2 and b = gffgg01 for wroug/zt-zron columns withfi.ved ends w i l l a g r e e fairly with experiments fl'om / == 0 to ! -~ r

7'

500, t h o u g h

the

agreement for the smaller length ratios is not so good a s o n the first assumption. Similarly the values P0 =: 50~000, a ~--1"1 and b ~-- xa-0~vl will approximate to the average of the e x p e r i m e n t s on wroug/zt.iron columns, hing'ed ends,* though not nearly so w e l 1 for / ~ 24o as for higher length ratios where a good average is s t r u c k . r F o r cast-iron columns with fixed ends, the values a s s u m e d w e r e P0 ~--- 100,000, a -=- 1"2, and b i which gave e x c e l l e n t r e s ~ h s for all length ratios. Some of the principal results are recorded in the f o l l o w i n g table : Wrough/-fron Columns, ffixed ]:2nd.~.

-~r/-.

.

.

.

.

.

.

.

.

@

.

0 .

.

.

.

.

.

.

.

.

.

.

.

.

50 .

.

41667

.

.

.

.

.

.

.

.

Ioo .

377oc

.

.

.

.

.

.

.

.

150 .

.

.

.

.

.

.

.

.

.

e94oo ] ~50o

2oo ~oo l....... L ..... L _4o0 1 x56c,o

~5c:o

8717o t 5~1.3o i F~'5~5o

The eccentricity of the load in the case of " h i n g e d e n d s " i s o n l y realized when friction between the pin and column is exerted, and i t s *.v/~-e~,'.:e ~}m7 is easily found, as in the case of trunni0ns, by k:nown la~,vs.of ~naecl-Lanic~..

Note on Long Columns.

Rug., 1887.]

I33

W,'ought-[ran G)lumns, iZinged Ends.

Gist-Iron Columns, Fired Ends.

~,

1 83333 l 58~oo ] 313oo [ 17(~,o] Io9oo i 5=oo [ 3ooo [ 2ooo I

I

t

t

I

I

I

I r 2

For ~ --= 1'2, as found for fixed ends, we have k ~---0"2 -~-, from which we can determine the eccentricity of the load for an assumed cross section. Thus, on substituting the well-known values of r, the radius of gyration, and z, the distance fi'om the axis to the most compressed edge of the column, we find that for .rcctang'nlar sections, k = ,al0- depth ; for solid ~ylindrical columns, k ----:: l diameter ; for thin, ko//ozo cylinders, k ~---~x0- diameter ; for thin, hollow, square columns, k : lg1 d e p t h ; and tot common columns, formed of two channels connected b y latticing, which give way in the direction of the latticing, k - - T} depth of latticing. Although the eccentricity is thus found to be small, it can be objected that the formula gives different eccentricities t o different shapes whose ratio is seemingly arbitrary, for which there can be no good reason. A separate formula, then, for each shape would alone answer the objection. T h e same objection holds, however, with respect to any one formula t h a t m a y be devised for all shapes, and is not very serious when a r o u g h average of the whole is alone desired. For the hinged ends, where ct - - 1"1, k has one-half the values given above. In cases where the direction of P is not parallel to the axis, or even crosses it, k would still represent the eccentricity at the dangerous section. 1

oo o ,Ua ,

wrou.h

(;)

zero, that the averaffe intensity of stress on the cross section

P

approaches the value 41,667 pounds, which is only a fractional part of the assumed greatest intensity, 5o,oo0 pounds¢as it should be. Similarly for cast iron. -/ The form proposed would, therefore, seem to be far preferable to the ordinary Gordon form in this respect, though b o t h labor under the objection of being semi-empirical in character.

124

C~z)z :

[ j. ~-. I..

!

It may theti be inquired whether a simple, purely empirica~ formula cann0~ be devised from the experiments that will give better results than the Gordon and allied forms ? In the discussion on Mr. Wilson's paper, before mentioned, there were certain ~mpirical formul,~ proposed, as embodying the average results 6~ experiments on wrought iron, by E. T h a c h e ,~, :" Z M.Am.SocC.E., which, within the limits, )" -----20 to 200, a r e f o u n d to give as go6i] or better results than more complicated formala:'. This whoi~ subject, however, has been so fully presented by Thomas H. J01!nson, M.Am.Soc.C.E., in a paper " On tlxe S t r e n g t l : of Colum ns, " :read at the Annual Convention of the Arnericar, Society of Cii}il Engineers, June 26, I,g8-~, that one has b u t to glance at the admirable diagrams given in that paper, to see thal :,t the kind of formul~ proposed by Mr. Johnson represent a b o u t as nearly as is pogsible for any set of form.ul~eto represent, the a v e r a g e of the great n~imber of experiments that have been m a d e o n the breaking weights of columns. In the numerous diagrams given, the abscissas represettt the ratio of the l~r~gth to the least radius of gyration

iF

p ordinates, th~;corresponding average breaking stress ~"

and

the

on

the

cross section'. Mr. John~0n found for all materials that "(x), that p a r t o f the line corresponding to the higher length ratios, is a curve, t h e equat,on of whmh:~s Euler s formula ; (-), that part of the line corresponding to the lower length ratios, is a straight line t a n g e n t t o tt:~e aforesaid c u ~ , and intersecting the vertical axis at a point which is constant fo*' each material in all the varieties of end b e a r i n g s : (3), the ordio~tte at the point of tangency is one-third of t h a t p a r t of the vertical ~:~is intercepted by the tansent.~ " The sec&~d law above is in agreement witl~ the p r o p o s e d :7 formula of ~Mr. Thacher for wrought iron, as far as e x p r e s s i n g the average breaking stress as a linear function of the length rati, os is concerned. :Mr. Johnson has drawn curves for m a x i m u m ant minimum, a~well as average values, and expressed likewise b y th.' 7

formula gi~i~h below, for various materials, the average s t r e s s <1

Aug., I887.1

Note on Long Colu~nns.

I35

on the cross section at the breaking limit for various ratios o f / to r. For the sake of brevity, let us call, P breakhz~/oad in pounds. P ~ (~ - - cross st,cttoTz bz inc]les. I r z'-----(17)"

length of cofu~Jzn in inches. radi~ts o/ oyration q/ cross sectio~z in inches. abscissa of point of ,angency below whicl~ the equation of the taugent is zlsed and f o r greater values of x tlze curve of Eulcr's equation.

We give in the table below the special forms of the equations for various matcrials and end bearings, observing that the experiments on oak and oolitic limestone are comparatively meagre ; but it is understood in formul~ of this kind that the constants are to be amended from time to time as experience dictates. For wrought iron, the formulae embrace the average results of a very large number of experiments for every style of cross section and extending up to 5oo radii of gyration. On looking over these formub~, which largely e m b o d y the results of experiments, the question naturally occurs, W h y does Euler's formula give the crippling load for very long columns, but not for the shorter ones ? The answer is, in a general way, that in the first case the limit of elasticity has not been exceeded when bending just occurs ; in the last case it has been. Thus, for columns hinged or round at both ends, we have for f----- o from (7), since the quantity in brackets is equal to zero, and making i ~ ], 2a

,I

e . --~

ff

nearly,

P P in which e - - ~ - Q ~= shortening of column per unit of length. For wrought iro~¢, take E -= 27,000,000, and at the elastic limit the P unit stress,~--_--2i,0)0, which, in the above formula, gives 2a 7''

--t00

nearly.

Now, if we suppose all the terms of the

Cam:

136

MA'rRKIALS.

l!;nd Bearings.

[j.

xt . . . . .

Equation of Tangent,

L

F.

Equation

of

C u rye,

Flat,

z¢

:¢)auchl Iron. ttinged, p ,.: 4 2 o o o - - 2~ 3 .~t' iJ

R o u nd,

v38"°

/') ,

9 0 ,c.~.. " - e- 6 6 , 4--.#2"

' 7 9 :c !

D5.~c

p _

666p)o,o,,~

P ~

4-44"r~5-0-'c~cL

[

i Flal,

z = 525oo-

Hinged,

:c =

5 2 5 o 0 - - 220 :c ! .i

:59'3

1~ =

52500

i23" 3

Mild .ged. (Carbon 1'2;

oua,)

27,ooo,oco

Round,

a 8 4 :c

[

[! . . . . . . . .

p ~

~66L49o,~

~r-'

t

Flat,

a; , = 8 0 o 0 0 - - 3 3 7 :c

x58-o

l~ =

666-o9o,c.c>'; 2r2

( C a r b o n == 0"36. )

Hinged,

1~ =

8c*oc~ - - 4~4 :c.

~9"o

p=

L4¢,_Zs_ Q,°m

Js' ~= 27,000,0oo

Round,

i) ~

8 0 o 0 o - - 534 x

99"9

~.~

=66, 49°.°°°.

I[ard Steel.

.......................

G,st

z'-'

.

. . . . . . . . . . . . . . Flat,

i) == 8 o o o o - - 4 3 8

Hinged,

p =

.r:2

Iron. 8oooo--

537

99"3

9=

26 ~,2oo,c-o~

7To

~ =

z57,9~o,c
1": ~, ~6,ooo,ooo Round.

)~

8ocoo -- 693 Z

. . . . . . . . . . . .

Flat.

p~54oo

--

28x

lqat,

~)~9

--

3;;

]

. . . . t28'x

2 = ~ : 6 L * , _,,5,%

Oak.

Oolitic

Limestont.

ooo

I89"I

l~0=

r

1 ~' ~ 4,35o,oco

i!

~o7,314,5o

o

Aug., ~887.]

Note on Lon~ £:otumns.

'I37

formula to remain constant but the length of column 2 a, we see that an increase of length in formula (7) will give some deflection for the same _P, so that bending will j'z/.st begin for a less uniform compressive stress than the 27,ooo assumed ; therefore, when the column begins to fail by bending for 2 a ~ 100, the limit of elasT

ticity has not been passed, at the instant when bending begins, so that Euler's formula should exactly apply, provided all the theoretical hypotheses (of a straight column of homogeneous material, with a load acting along the axis and i = 1) are exactly realized. 2a The reverse holds for shorter columns; thus, for - ~ _ - 7 7 , we find P that -Q2 == 4,5,~)00 pounds,or the elastic limit has been long exceeded before the column, theoretically, begins to bend at all.

If 45,ooo 2a. pounds was the crushing unit stress, we see for this ratio, - - 77, and for all lower length ratios that the column should, theoretically, give way by crushing entirely, of uniform intensify over the whole cross section, as n o bending is possible if the original hypotheses are realized. In a similar manner, we find the least values of ~ 2a for which r the elastic limit has not bee'n passed when bending is just about to begin, and Euler's formula becomes app~icable~for mild steel, 87 ; ]zarct steel, 67 ; cast iron, 70, using the m0duli given in the table above, and regarding the elastic limits in order: 35,ooo, 6o,ooo and 32,000 pounds. For columns withfixed ends, these ratios are all doubled, as we easily deduce from the formul~ corresponding to fixed ends, given above. These results show, too, in a general way, that the inapplicability of Euler's formulm to usual column lengths, as shown in the table above, is not entirely, or even principally, due to " the innate perversity of inanimate things," but rather that the hypotheses of perfect elasticity, etc., upon which the theory is founded, no longer, even approximately, holds. The empirical formuke proposed answer all the needs of practice, though it is possible that formula (13) may answer b e t t e r for

I38

CaiJz ."

[J. F. I..

any one shape ; but where a general average only is desired, the linear equations above cover nearly all practical cases, and t h e y are, of course, preferable to Fuler's formula:, which give values entirely too large for small length ratios; also, to t h e G o r d o n Rankine form, not only oil account of their simplicity, but because for very small length ratios they are more accurate, and for m e d i u m length ratios equally as accurate, as tile partly empirical formulx. they are intended to replace ; on which accounts the above empirical formulm are cordially recommended for practice.

APPENDIX.

In /L~'ecu¢#;cDoc~ment 12, 47th Congress, ISt Session, is given the Watertown experiments on posts of white and yellow pine, varying in size from about 5'2 inches square to a cross s e c t i o n of 8. 5 x I6. 5 inches and up to 62 diameters in length. ( A l s o , see Lanza's Applied J~fechanics, pp. 511- 5 I9.) On plotting the ascertained unit stresses corresponding t o given ratios of length to least side of the rectangular cross section -~[ , it is found that the average unit stresses -XP are a p p r o x i mately given by the following formulm up to l = 60 d n e a r l y : for white ~ilze, -P~ = 3000 - - ~100 - (/).and 00 for yellow pine, -P~ = 5 5 0 0 - - 2--3

(') -d- ,

in pounds per square inch of cross section. The mbdmum u n i t stresses are about 400 pounds less f o r corresponding values of l with but few exceptions. T h e r e were d sixty-six experiments on white pine single sticks (which a l o n e were considered), and seventy-two tests of the yellow pine posts. It is hardly probable that in practice the end fittings a r e so perfect as in the Watertown tests, so that, although t h e above formulm probably give us the average of the most e o m p t e t e tests that have yet been made on pine posts, yet a certain d i s c r e t i o n naust be used in applying them.

Aug.. [887.]

Note on Loug Columns:

r3~Y

According to Mr. Johnson's rule, it will be found that for while

;~ine, E u l e r ' s f o r m u l a w o u l d h a v e t o b e u s e d f o r ( - l - ) g r e a t e r t h a n 60, for which length ratios, P _4 3,600,000

and for yellow pine the limit is dl = 55, beyond which, I ~ - - 5,544,825 \ d/ But it must be carefully borne in mind that no experiments can. be appealed to to substantiate these last values, which are much higher than sometimes given. It should be noted that for " r o u n d " ends, Mr. Johnson used, Euler's exact formula in the table above, but the experiments on. iron showed that for flat ends, the coefficient 4 zr2 = 39"48 of Euler should be changed to 24"67, which is probably on account of the unknown eccentricity of the load.

Tt~ F U T U R E

WATER

SUPPLY oF P H I L A D E L P H I A .

In Chief Ogden's report of the operations of the W a t e r Department of. the City of Philadelphia, for the year 1886, which was submitted to Councils, on Thursday, July i4th, is given the final. report of R u d o l p h Hering, engineer in charge of the '" Surveys for the Future VTater Supply." These investigations were begun in May, [883, and continued during the years I884, I885 and until~ July, I886. There was expended for the surveys $81,547.96. In the course of his recapitulation of " conclusions that h a v e been arrived at from the examinations," Mr. Hering says: "In m a k i n g these investigations, it has been taken for granted from the o u t s e t that the water from any point in the Schuylkill' River, and from any point in the Delaware River, below Trenton,. will not be of a suffidient good quality to furnish a future supply for the city, although the fact has been admitted that, at present,.