stability of built‐up columns
TRANSCRIPT
STABILITY OF BUILT-UP COLUMNS
By Atle Gjelsvik,1 Member, ASCE
ABSTRACT: The buckling of columns with finite shear stiffness such as built-up and laced columns is investigated. The methods of analysis associated with Engesser and Haringx are compared. The difference between the methods can be traced to the direction of the axial force and the shear force used in the analysis. It is concluded that when the usual shear stiffness of the column is used, the Engesser method is the correct one for columns modeled as continuous Timoshenko shear beams. The appropriate method to use for helical springs is not investigated. The effect of the shear stiffness on the column buckling load for the usual standard boundary conditions is also presented analytically, and in the form of graphs.
INTRODUCTION
The buckling analysis of columns with finite shear stiffness has a long history in engineering science. In structural engineering the problem occurs in the analysis of various built-up and laced columns. In mechanical engineering, it shows up in the analysis of helical and elastomeric springs and bearings. Laced columns are usually treated as continuous disregarding the discontinuous form of the lacing. Springs are also usually treated this way. Within this analytical framework there is no physical or mathematical difference between columns and springs. It is therefore interesting and perhaps surprising that two schools of thought as to the proper formulation of this buckling problem have developed and remained side by side up to the present time. Structural engineers follow typically, but not without exception, what will be called the Engesser school. Mechanical engineers follow what will be called the Haringx school. Engesser's classical work goes all the way back to 1889 (Engesser 1889, 1891). He was followed by many investigators all essentially following or rediscovering his original method [see, e.g., Bleich (1952) and Timoshenko and Gere (1961)]. The spring stability problem was first treated by Biezeno and Koch in 1924 (Bienzo and Koch 1925) using Engesser's approach. At this time, a discrepancy became apparent between the theoretical and experimental results for highly compressed very short helical springs in that it was found experimentally that if the spring is sufficiently stubby it will never buckle. This is contrary to Engesser's theory, according to which the buckling load will approach the shear stiffness as the column gets very short. To remedy this difficulty Haringx (1948) modified the Engesser theory and obtained results which agreed well with the experimental values. He was followed by a number of investigators right up to the present. Haringx's method was also used, apparently independently, by Lin et al. (1970) in their analysis of laced columns and is also used in the Guide to Stability Design Criteria for Metal Structures (1976).
It will be shown why in this author's opinion the Engesser school is cor-
'Prof., Dept. of Civ. Engrg., Columbia Univ. in the City of New York, New York, NY 10027-6699.
Note. Discussion open until November 1, 1991. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on March 20, 1989. This paper is part of the Journal of Engineering Mechanics, Vol. 117, No. 6, June, 1991. ©ASCE, ISSN 0733-9399/91/0006-1331/S1.00 + $.15 per page. Paper No. 25890.
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rect within the assumptions of Timoshenko shear beam theory and that Har-ingx's approach in its present form is not a suitable approximation for a continuous column. For a helical spring, however, the situation is different and Haringx's approach may indeed be correct.
BUCKLING LOADS
For a simply supported column with bending stiffness EI, shear stiffness K and length L the buckling load Pc obtained by the Engesser school is
PE Pc = 77 (1)
K
where
TT2£/ P^ir (2)
is the Euler load for the column, i.e., the buckling load for the column neglecting the effect of shear deformation. The shear stiffness K is defined as in the shear beam theory. For a solid cross section, for example, K is AG In where A is the cross sectional area, G the shear modulus, and n a numerical factor depending on the shape of the cross section [see Timoshenko and Gere (1961)]. The buckling load, (1), is the solution of the differential equation
P EIW + w = 0 (3)
'-3 where w = the displacement of the column axis, as shown in Fig. 1. A prime is used to indicate differentiation with respect to the axial coordinate x.
The buckling load obtained by the Haringx school is
^ = f [ ( 1 + ? ) 1 / 2 - 1 j (4)
which is the solution of the differential equation
EIw" + p( 1 + - )w = 0 (5)
For comparison the differential equation for the corresponding Euler column is
EIW + Pw = 0 (6)
Both schools use the same boundary conditions
w = 0 at x = 0, x = L (7)
For columns very stiff in shear, i.e., for very large Ks, the three equations
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FIG. 1. Deflected Column
converge, and the buckling loads converge to
Pc = PB (8)
For columns very flexible in shear, the Engesser load becomes
Pc = K (9)
and the Haringx's load
Pc = VKk
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The two buckling loads, (1) and (4), depend on the ratio
(11) PE TI2EI
K ~ L2K
which can be conveniently written as
where
c=yJEi (13)
c = the dimension of length and is a characteristic length for a shear beam; a = a convenient nondimensional parameter. In terms of a the Engesser load can be written as
Pc 1 (14a)
~E ' ' " ' " PE 1 + a2
or •3 Pc a2
— = : (14fe) K 1 + a2
and the Haringx load as
Pc (1 + 4a2)1/2 - 1 (15a)
PE 2a2
or
^ = i [(1 + 4a 2 )" 2 - 1] (15*) K 2
These results are plotted in Figs. 2 and 3. It is clear that for small values of a , i.e., for columns very stiff in shear, the difference between the two solutions is quite small. For large values of a , i.e., for columns very flexible in shear, however, the difference is clearly significant. To get a feeling for a typical numerical value of a consider a simply supported, uniformly loaded shear beam of such proportions that the bending and shear contribution to the central deflections are equal. In this case it turns out that
a = 1.0 (16)
and
^ ^ E . _ L 2 5 (17) P 1 CEngesser
This kind of value could easily occur in a laced column, which indicates that the difference between the two approaches is of some practical importance.
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1.0
0.8 UJ
a a° a 0.6
2 0.4
OQ 0 .2
i
---^iHARINGX
•~~-^ENGESSER
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2.0
SHEAR BENDING PARAMETER a = 2 ^
FIG. 2. Effect of Shear Stiffness on Buckling Load for Simply Supported Columns; a Small
To understand the two schools it is necessary to look at the derivation of (3) and (5). The classical analysis is based on what is now usually called the Timoshenko (1921) shear beam theory even though the basic ideas of this theory without being explicitly stated go back at least to Engesser (1889, 1891). The primary assumption in this theory is that plane sections do remain plane when the beam is deformed, but these sections do not remain normal to the deformed centerline. This kinematic scheme is shown in Fig. 4. The rotation <|) of the cross section about the positive y-axis is
<|> = -y - w' (18)
where 7 = the nominal shear strain and (—w') = the rotation of the column axis about the positive v-axis. The bending moment M and shear force Q are related to the displacement by
M = EI$' (19a)
G = * 7 (19*)
To show the derivation of (3) and (5) it is sufficient to analyze a simply supported column. Statically indeterminate columns will be treated next. The derivation is taken from Timoshenko (1961) and Haringx (1948).
For a simply supported column (see Fig. 1) the bending moment is
M = Pw (20)
The crucial statement made by the Engesser school is that the shear force
Q = Pw' (21)
The physical reasoning behind this equation will be discussed next. Eqs. (196) and (21) can be combined to give
P 7 = - w' (22a)
K
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2.0
1.8
1.6
,o
D < O _J
O
z o QQ
.4
1.2
1.0
0.8
0.6
0.4
0.2
•\ENG
\HARINGX
JESSEF ? \
0 0.2 0.4 0.6 0.8 1.0
INVERSE SHEAR BENDING PARAMETER 3 = ^
FIG. 3. Effect of Shear Stiffness on Buckling Load for Simply Supported Columns; a Large
or
y = — w K
(22b)
The derivative of (18)
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Axis
ry w , z
FIG. 4. Displacements in Timoshenko Shear Beam
4>' = y - w"
can be combined with (19a), (20), and (222?) to
Pw P — = - - 1 w" EI \K or
(23)
(24a)
EIw" + P
w = 0 (24fc)
which is the result obtained by Engesser. Haringx's approach is identical up to and including (20). Then, instead
of (21) the shear force is given by
Q= -/>4> (25)
The reason behind this choice will also be discussed next. Eqs. (19b) and (25) can be combined to give
(26a) y =
or
V =
p — 4,
K
p — 4,'
K (26b)
This result can be substituted into (23) to give
1 + £ ) * ' " ~W" (27)
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„X
'•^3 DISPLACED AND ROTATED CROSS SECTION
y w > z
FIG. 5. Internal Forces According to Engesser
w, z
FIG. 6. Parallel Internal Forces
which can in turn be combined with (19a) and (20) to give Haringx's result
(28) EIw" + P[ 1 + - J w = 0
The only difference in the two approaches lies in the form of the shear force Q used in (21) and (25). The correctness of either approach therefore depends on a scrutiny of the internal forces in the column, in particular the shear force. Consider Engesser's approach first. The subscripts E and H will be used for Engesser and Haringx to distinguish the two methods. The shear force QB is a convected force in the sense that it follows the material points of the displaced and rotated cross section. The axial force NE is also a convected force and it follows the material points of the displaced centerline of the column, as shown in Fig. 5. NE and QE therefore do not remain at right angles as the column deforms. Since QE and NE are convected forces they obey the constitutive relationship (19b)
QE = KEy (29)
These internal forces are, however, not the only internal forces which can
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be defined. For clarity a second set of forces N, Q acting in the column is defined to follow the displacement of the column, but to keep their original directions, i.e., parallel to the x and z directions as shown in Fig. 6. For small displacement the two sets of forces are related by
N = NE-QE$ (30a)
Q = QE - NEw' (30b)
Since QE is small compared to N, the second term in (30a) is small compared to the first and can be neglected. Then
N = NE = P (31)
which can be combined with (30b)
Q = QE-PW' (32)
For a simply supported column, Q is zero, and from (32)
QE = PW (33)
which confirms that this is Engesser's approach. Haringx, on the other hand, defines the internal forces such that NH re
mains normal to the displaced and rotated cross section whereas QH remains parallel to this cross section, as shown in Fig. 7. NH and QH are therefore normal to each other but NH is not lying along the axis of the deformed column. For the Haringx column therefore
N = NH-QH$ = P (34a)
Q = QH + AM> = QH + P* (34fc)
when Q is zero
2« = -P4> (35)
which is identical to (25). This is therefore Haringx's approach. When comparing (32) and (34b) it is clear that for QE and QH to be statically equivalent
QH = QE ~ P(w' + 4>) (36)
or in view of (18)
QH = QE-Py (37)
From (19b) in turn
QH = (KE-P)y (38)
Therefore, for a consistent result, the shear stiffness to be used in the Haringx approach is
KH = KE-P (39)
The shear modulus KH is not simply a property of the column, but depends on the axial force in the column. If KH is used instead of K in (4) and (5) the Haringx equation and buckling load become identical to those of En-gesser. Thus, in principle, there is no difference in the two approaches provided the correct shear moduli are used. This has already been pointed out
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/>y
'^r»
w, z
FIG. 7. Internal Forces According to Harlngx
,,x
M + M'cfx.
y w,z
FIG. 8. Deformed Column Element
by Bazant (1971), where he shows that the Engesser and Haringx approaches respectively correspond to known formulations in the theory of elasticity. Unfortunately, the same shear modulus K is used by both schools. Haringx's approach is not confirmed for a continuous column since it appears to be based on an erroneous assumption either as to the direction of the axial force in the deformed column or to the correct form of the shear modulus. It may, however, be correct for a helical spring where the kinematic assumption, (18), is not really relevant. Engesser's result has also been confirmed using energy methods (Bienzo and Koch 1925) and variational methods. It has also been found to be a good approximation when compared to a three-dimensional elasticity solution (Nanni 1971). Haringx's equation, (28), has the further difficulty that if P is negative with a value larger than K, the coefficient of w is positive. Physically this means that the column can also buckle in tension, which is contrary to all experience.
Since the literature using Engesser's approach appears to deal only with
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statically determinate columns, it is of some interest to derive the general equation governing buckling of a shear beam-column. Fig. 8 shows the forces acting on a deformed column element. The curvature of the axis has no effect on the equilibrium equations and is not shown for clarity.
The distributed transverse load q acting on a column is assumed to remain parallel to its original direction when the column deforms. Equilibrium in the z-direction gives
Q' + q = 0 (40)
which can be combined with (32) to
Q = Pw" - q (41)
Moment equilibrium of the column element in Fig. 8 gives
Q + Pw' - M' = 0 (42)
which can also be combined with (32)
Q = M' (43)
Eq. 43 is identical to the moment equilibrium equation for an undeformed column element, i.e., a beam element. By further combining (18), (19), and (43)
w' = -<() + c24>" (44)
where c = the characteristic length given by (13). This equation relates the transverse displacement to the rotation of the cross section for a shear beam or a column. Eq. (41) can be combined with (43) and (18a)
EW" - Pw" = -q (45)
which finally can be combined with (39)
P Elty" + <!>' = - ? (46)
Together with (44) this is the governing equation for a shear beam-column. Together these equations form a fourth-order differential equation needing four boundary conditions.
For a column q is zero, and the solution to the buckling problem is the solution to the set of equations
4>'" + X24>' = 0 (47a)
w' = -<|> + c V (47b)
where
i p
X2 = (48)
As an auxiliary equation it is also convenient to have an explicit expression
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for Q. This is obtained by combining (19a), (42), (47fo), and (48)
Q = (^ V + <\>)p (49)
The appropriate boundary conditions can best be obtained by looking at the virtual work done if the cross section of the column is given a virtual displacement w and <J>. This work is
W = M<$> + Qw (50)
One of the factors in each of the two products must be given at a boundary. Four possibilities exist.
1. Simple support:
w = 0 and M = 0 (51)
or in terms of the displacements using (19a)
w = 0 and 4>' = 0 (52)
2. Built-in:
w = 0 and <|> = 0 (53)
3. Free to sway and rotate:
M = 0 and Q = 0 (54)
This can also be written in terms of the displacements using (19a) and (49)
<)>' = 0 and <\>" + \24> = 0 (55)
4. Free to sway without rotation
<|> = 0 and Q = 0 (56)
This can also be written in terms of the displacements using (49)
<t> = 0 (57a)
<|>" = 0 {51b)
The governing equations and the boundary conditions are coupled in <f> and
w and it is impossible to reduce the general problem to one involving w
only. The solution to a particular buckling problem, i.e., (47), with the appro
priate boundary conditions can always be written in the form
As before PE is the Euler load for a simply supported column. The nondi-mensional quantity XL is the eigenvalue of (47) with the appropriate bound-
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XL = 2TT
XL = TT/2
XL = IT
tan XL = XL
1+(f-)(XL)
FIG. 9. Eigenvalues for Typical Boundary Conditions
ary conditions. The values of XL for the usual five column boundary conditions are given in Fig. 9. The first four are characterized by having Q zero. The eigenvalue is then the same as for the corresponding Euler column. In the built-in, simply supported column Q is different from zero and the value of \L on c/L.
The importance of the reduction in shear strength is best seen by looking at the ratio PC/PB where PB is the Euler load for the column with the relevant boundary conditions. For a built-in column, for example
4ir2£/ (59)
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< O.f
0.6
0.4
0.2
c
1 1 1 —-\
i
A
\^
i i i V \ *
f B
7
~s~
r / i i \ \ c
>
7~~^-~»
h\B s A
£
D~~
1
/c S77
i 1
ID-
I ^
0.2 0.4 0.6 0.8 1.0 1.2 1.4
SHEAR BENDING PARAMETER a = ^-
2.0
FIG. 10. Effect of Shear Stiffness on Buckling Load; a Small
1.0
0.8
s _ J
O z _ J
o D CO
0.6
0.4
0.2
D
A
^
0.2 0.4 0.6 0.8 1.0 1.2 INVERSE SHEAR BENDING PARAMETER £= jp£
FIG. 11. Effect of Shear Stiffness on Buckling Load; a Large
The result is plotted in Fig. 10. In Fig. 11, the values for P/K are also plotted. It is clear that a built-in column is most effected by a decrease in the shear stiffness.
It is perhaps of interest to note that for the four cases where Q is zero, the buckling load can be written in terms of the Rankine interaction equation
J 1_ 1 Pc~ PB K
(60)
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whereas for the case where Q is different from zero, this is only approximately true.
The use of the Timoshenko shear beam theory for laced columns has one drawback. In this theory it is inherently impossible to suppress the shear deformations at the ends of the column. Laced columns often have relatively rigid stay plates at their ends. These plates have the double benefit of increasing the rigidity and strength at the ends of the column and suppressing the shear deformation in the laced part of the column at the connection to the stay plates. A boundary condition of y equal to zero should be enforced here. In the Timoshenko shear beam theory, this is not a possible boundary condition. An improvement of the Engesser theory is explored by Gjelsvik (1990).
CONCLUSION
The Engesser type solution to the buckling of a shear column is confirmed. Buckling loads for statically indeterminate columns are also presented.
APPENDIX. REFERENCES
Bazant, Z. P. (1971). "A correlation study of formulations of incremental deformations and stability of continuous bodies." J. Appl. Mech. Trans. ASME, Dec, 919.
Biezeno, C. B., and Koch, J. J. (1925). "Knickung von Schraubenfedern." Zeit-schrift der Angewandte Mathematik und Mechanik, 5, 279.
Bleich, F. (1952). Buckling strength of metal structures. McGraw-Hill Book Co., N.Y., 23, 167.
Engesser, F. (1889). "Die Knickfestigkeit gerader Stabe." Zeitschrift des Architekten und Ingenieur Vereins zu Hannover, 35, 455.
Engesser, F. (1891). "Die Knickfestigkeit gerader Stabe." Zeutralblatt der Bauver-waltung, 11(49), 483.
Gjelsvik, A. (1990). "Buckling of built-up columns with or without stay plates." J. Engrg. Mech., ASCE, 116(5), 1142.
Guide to stability design criteria for metal structures. (1976). John Wiley and Sons, New York, N.Y., 359.
Haringx, J. A. (1948). "On highly compressible helical springs and rubber rods, and their application for vibration—free mountings." Part I, Philips Res. Report 3, Philips Gloeilampenfabrieken, Eindhoven, the Netherlands, 104.
Lin, F. J., Glauser, E. C , and Johnston, B. G. (1970). "Behaviour of laced and battened structural members." J. Struct. Div., ASCE, 7, 1377.
Nanni, J. (1971). "Das Eulersche Knickproblem unter Beriicksichting der Quer-krafte." Zietschrift fiir Angewandte Mathematik und Physik, Birkhauser, Basel, Switzerland, 22, 156.
Timoshenko, S. P. (1921). "On the correction for shear of the differential equation of transverse vibrations of prismatic bars." Philos. Mag., 21, 747.
Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability. McGraw-Hill Book Co., New York, N.Y., 132, 142.
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