RE-EXAMINATION OF YLINEN AND OTHER COLUMN EQUATIONS

RE-EXAMINATION OF YLINEN AND OTHER COLUMN EQUATIONS By John J. Zahn 1 ABSTRACT: In 1991, Ylinen s column equation was adopted for the design of wood columns in the United States. Ylinen originally derived this equation as a plastic buckling criterion, using a nonlinear compressive stress-strain law as the starting point. The stress-strain law contained a parameter c < 1, which controlled the degree of nonlinearity. Linearity was achieved at c = 1. This same parameter appeared in the final column equation. At c = 1. the column equation reduced to elementary theory, i.e., to a perfectly plastic-perfectly elastic failure criterion. Column design equations are viewed in this paper as interaction equations between two modes of failure: crushing and buckling. Ylinen s column equation is found to result from the addition of a cross-product term to the linear interaction equation. This enables us to view c as measuring the degree of departure from the assumptions of elementary theory, including such phenomena as inhomogeneity of material. imperfections of shape, and accidental eccentricity of loading. in addition to nonlinearity of the stress-strain law. Column design equations from wood design codes in Europe and Canada are compared and discussed from this point of view. INTRODUCTION The adoption of Ylinen s equation (Ylinen 1956) for design of wood columns was long advocated by Malhotra (1972) and more recently by the writer (Zahn 1986). In 1991, it was adopted by the National Forest Products Association (NFPA) for the National Design Specification for Wood Construction (NDS) (National Design 1991), replacing the FPL fourth-power parabola that had been the basis of column design in the United States for more than 40 years. It has also been selected for use in the Load and Resistance Factor Design Specification for Engineered Wood Construction (LRFD), a joint standard of a NFPA and the ASCE. At the time of this writing, the LRFD standard was undergoing the ASCE s review and acceptance process. Column design equations in the intermediate-length range are ordinarily viewed as empirical equations that express the compressive capacity as a function of slenderness. In this paper, we adopt a fresh point of view: that the form of the column design equation essentially expresses the interaction of two failure modes, namely crushing and buckling. The failure of an intermediate-length column is regarded as a mixed-mode failure, whose pure failure modes are perfect plasticity (crushing) and perfect elasticity (elastic buckling). Of course. this is not quite true: the intermediate column failure is one of inelastic buckling, in which the stress-strain law is neither perfectly plastic nor perfectly elastic. This is how Ylinen (1956) approached the problem, as discussed later in this paper. However. real wood columns are even more complex: they are inhomogeneous (knots and grain irregularities) and the lumber-grading rules permit a certain amount of crook. The inhomogeneities are located unsym 1 Res. Gen. Engr., U.S. Dept. of Agr.. Forest Service, Forest Products Lab.. Madison, WI 53705-2398. Note. Discussion open until March 1. 1993. 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 October 21, 1991. This paper is part of the Journal of Structural Engineering, Vol. 118, No. 10, October, 1992. ASCE, ISSN 0733-9445/92/0010-2716/$1.00 + $.15 per page. Paper No. 2803. 2716

metrically on various cross sections. Therefore, we do not specify the details of the failure mode interaction in this paper, but simply model it empirically. The resulting design equation has certain advantages: 1. It permits easier reliability analysis [see Zahn (1990) for the effect of correlated properties]. 2. It covers the whole range of slenderness in a single equation, whereas other design criteria often have three separate equations for short, intermediate, and long columns. 3. It enables us to express the degree of interaction in a single, adjustable parameter, which models total interaction from all causes. This also makes the equation easy to modify to reflect future changes in the material resource base or to accommodate new wood-based materials. EMPIRICAL COLUMN EQUATIONS AS INTERACTION EQUATIONS The essential form of an interaction equation is a relationship between failure ratios that exhibits proper limiting behavior, namely, that if either failure ratio is zero. the other failure ratio is required to be unity. For example in which P = mixed-mode failing load; P c = failing load for pure crushing: and P e = failing load for pure elastic buckling. For a wood column with a solid section, P e is given by the Euler formula (1) in which E = modulus of elasticity: I = moment of inertia of cross section; and L = effective length. In general. the interaction equation need not be linear. Selection of the form of the equation should be guided by available data. Recent data for lumber columns (Buchanan 1984; Johns and Buchanan 1982; Malhotra 1972; Neubauer 1972: Forest Products Laboratory unpublished) are plotted in the P/P c, P/P e plane in Fig. 1. Markers in this figure have the following meaning: is spruce-pine-fir (Johns and Buchanan 1982; Buchanan 1984); is douglas fir (Forest Products Lab unpublished; is eastern spruce (Malhotra 1972): and is Douglas fir (Neubauer 1972). Marker size is congruent with sample size: The largest sample was 100 and smallest was 1. Linear interaction between crushing and buckling (1) shown for comparison. Note symmetry about line of equal failure ratios. These data support the idea that the equation should be symmetric in the two failure ratios. Because the simplest symmetric equation is the linear interaction equation (1), that equation should be examined first. Eq. (1) has often been employed as a conservative design expedient in other cases of failure interaction. When Compared with data, however, it is seen to be extremely conservative (Fig. 1). There is an historic design equation that comes close to being (1). We consider it next. 2717 (2)

FIG. 1. Recent Lumber Column Data from Various Sources Rankine-Gordon Equation The Rankine-Gordon equation dates back to the 1860s. Timoshenko (1936) says it was originated by Tredgold, adapted by Gordon, and given its final form by Rankine. See Todhunter and Pearson (1886). Ylinen calls it the Navier-Schwarz equation and references Salmon (1921). The equation, in its final form, can be derived from the maximum stress theory of failure with an assumed eccentricity equal to in which c = distance from the neutral axis to the remotest fiber in compression and = an assumed constant. This assumption yields the equation (3) in which = radius of gyration of cross section. Rewriting (3) in the form of an interaction of failure ratios. it becomes in which A = cross-sectional area. Note that unless (4) does not have the proper limiting behavior. If is less, the Rankine-Gordon equation will exceed the Euler equation for sufficiently long members. However. if & is not less. the equation has been found to be too conservative [at = 1, it reduces to (1)]. We therefore attempt to modify (1) to make it less conservative. Cross-Product Equation The column data in Fig. 1 indicate that the sum of the failure ratios should exeed unity when neither ratio is zero. A natural way to achieve this is to add another term to the right side of (1). The simplest symmetry-preserving term is the product of the two ratios 2718 (4)

However. this has the surprising result of eliminating interaction. To see this. write (5) in either of the forms (5) or (6a) (6b) The solution of (6a) is P/P e = 1, P/P c arbitrary, and the solution of (66) is P/P c = 1, P/P e arbitrary, that is. noninteraction. In Fig. 1. noninteraction is represented by the lines that form the top and right side of the figure. The addition of the cross-product term to (1) has moved the equation from the diagonal line to a pair of lines beyond the data. This suggests that interaction could be reintroduced by multiplying the cross product (P/P c )(P/ P e ) by a factor c < 1. We show next that the resulting equation turns out to be the Ylinen equation, which he obtained by another method. Ylinen Equation Modifying the cross-product term in (5) with a parameter c yields In (7), the case c = 1 yields noninteraction. When (7) is solved for P, the result is (7) This is Ylinen s equation, derived here as a nonlinear interaction of crushing and buckling. Slenderness Ratio At this point. one could substitute for P e. from (2), but it is more conimplicit through P e. In fact, plotting (8) as PIP, versus yields venient to leave (8) in its present form, with the dependence on slenderness a column design curve similar to the usual empirical design curves that plot P/P c versus slenderness ratio. Using as a universal slenderness ratio makes it easy to plot data for members of different sizes and properties on the same figure. If the members should all obey the same design equation. then they can all be plotted on the same figure and compared with the universal design equation (8) (Fig. 3). In addition. (8) is easily adapted for use as a beam design equation by merely replacing axial loads with moments. P c. becomes M b. the ultimate bending moment. and P e becomes M e, which is calculated from an appropriate formula for elastic lateral-torsional buckling. In that way, serves as a universal beam slenderness. Note that these universal slenderness 2719 (8)

FIG. 2. Ylinen Column Equation with c = 0.80 Compared with Same Column Data as in Fig. 1 ratios make it possible to compare design equations as well as data, after the equations have been rewritten in terms of rather than Ylinen, of course. did not consider his equation to be an interaction equation. He ignored inhomogeneities and crookedness and focused only on the nonlinearity of the stress-strain law, using the tangent modulus buckling theory, which we discuss next. Tangent Modulus Buckling Theory To model inelastic buckling, Engesser proposed the tangent modulus buckling theory and later the double modulus buckling theory (Engesser 1889, 1895). Shanley (1946, 1947) showed that these two theories are actually lower and upper bounds, respectively, on the true inelastic buckling load. Careful applications of Shanley s complex theory by Larsson (1956) showed that the true buckling load was nearly always a very close lower bound to the tangent modulus buckling load. This led Ylinen to propose that the tangent modulus buckling load be used as the inelastic buckling load. and that inelastic buckling be used as the basis of column design. He cleverly devised the following equation to represent the variation of the tangent modulus with compressive stress: in which = compressive stress: E t = tangent modulus; = compressive strain, and = stress at the plastic yield point. Using (9) is equivalent to assuming the following stress-strain law. obtained by integration of (9) (9) (10) Ylinen s derivation of (8) can now be completed by substituting E t (9) for E in Euler s equation and taking P c. = from 2720

Parker (1964) investigated the application of the tangent modulus buckling theory to wood columns of structural grade. Parker included (9) (wrongly attributing it to Timoshenko) as one of the analytical expressions he fitted to compressive stress-strain data. obtaining the values c = 0.8 for southern pine and c = 0.9 for douglas fir. Parker concluded that the tangent modulus buckling theory works reasonably well for southern pine but predicts loads that exceed experimental values for douglas fir. In truth. his samples contained only six specimens, too few to produce a statistically significant confirmation or denial of the tangent modulus buckling theory. Subsequent studies. using larger samples, have shown that c = 0.8 applies well to structural grades of several species (Fig. 2). Equal-Power Equation Note that it is possible to construct other symmetric nonlinear interaction equations with a single parameter. An obvious one is (11) in which a > 1 is the adjustable parameter. Here. the case of noninteraction interaction. Furthermore. it is not possible to associate a nonlinear stressstrain law with (11) by using the Engesser tangent modulus theory, as Ylinen did in his original derivation of (8). EMPIRICAL MEASUREMENT OF PARAMETER C Meaning of c Because c < 1 implies interaction, we now have its true meaning: it is a parameter that governs the degree of interaction between crushing and buckling. At c = 1 interaction vanishes. Noninteraction means that P = min{p c, P e }; that is, noninteraction corresponds to using ideal failure criteria. These ideals are an upper bound on real mixed-mode failure. The ideal upper bound can only be reached if all the following conditions, assumed by elementary theory, are met: 1. The material is perfectly homogeneous and straight-grained (orthotropic). 2. The member is perfectly straight. and the load is perfectly concentric. 3. The stress-strain behavior of the material is linear elastic to the point of crushing and perfectly plastic thereafter. Because no real columns satisfy these ideal conditions. all real columns should have c < 1. Greater departures from these ideal conditions should result in smaller values of c. For solid wood members, conditions 1 and 2 are more difficult to satisfy than 3. Whereas Ylinen s derivation identified c as a parameter in a nonlinear stress-strain law. that is. as associated only with condition 3. this interaction-based derivation shows that c can be associated with departures from all three of the assumed conditions of elementary theory. Parameter c should not be inferred from the stress-strain data. as Parker attempted to do. but should be seen as a generalized interaction parameter governing the failure of wood columns and inferred from column test data. 2721

Best Slenderness Value for Measurement of c Noninteraction means that P = min{p c, P e }. The greatest sensitivity to c occurs when Ylinen s equation differs most from noninteraction. As seen in Fig. 3, this occurs when P c = P e. This is intuitively appealing, for it says that the two failure modes interact most when their failing loads are equal. It is a necessary result of the symmetry that we imposed after noting the symmetry of the data of Fig. 1. Value of c for Lumber Solving (8) for c yields (12) No sample on Fig. 1 has P c and P e exactly equal, but the three that come closest yield the three values c = 0.736, 0.820, 0.857; that is, c is approximately 0.8. Fig. 2 shows that this gives a reasonably good fit to all the data. Selecting c in Absence of Data In the absence of data. one can resort to code calibration or to arguments that appeal to the meaning of c. Calibrating c to the column design criterion of the 1986 NDS, namely the so-called FPL fourth-power parabola. yields c = 0.96. In writing the 1991 edition of the NDS (National Design 1991), the committee assigned c values to various types of wood columns as follows. Although columns designed under the previous editions of the NDS were apparently adequate, the recent lumber column data cited here indicate that lumber columns are not as reliable as formerly believed. As for other types of wood columns, namely glulam and poles and piles. there was a desire to keep the level of reliability uniform within the new edition. Therefore. the 1991 NDS specifies the following c values: FIG. 3. Comparison of Ylinen Equation and Noninteraction. Dotted Line Shows Point of Maximum interaction, Where Ylinen Equation is Most Sensitive to Value of c 2722

1. For solid lumber columns, c = 0.80, based on recent data. 2. For beams, c = 0.95, based on code calibration. 3. For glulam columns, c = 0.9, based on a compromise between code calibration and the value for lumber. Here. it is argued that the laminating effect makes glulam straighter and more homogeneous than lumber. 4. For poles and piles used as columns. c = 0.85, based on a compromise between the values for glulam and lumber. Here, it is argued that the inhomogeneities (principally knots) are more symmetrically arrayed in poles than in sawn lumber. Because interaction has been a fruitful way to view Ylinen s column equation, we conclude by examining the wood column design equations used in Europe and Canada from this point of view. EUROPEAN AND CANADIAN DESIGN CRITERIA Europe The Perry-Robertson equation is the basis of design in the most recent European draft standard (Eurocode No. 5 1988). In terms of axial loads, the equation is (13) in which m = = compressive strength; F b = bending strength; and = an adjustable parameter. Eq. (13) was derived using the maximum stress theory of failure for combined bending and compression and assuming an eccentricity e = in which c = distance from the neutral axis to the remotest fiber in bending. In (13), P c = to compensate for the fact that the European crushing strength is obtained from a test with a specified eccentricity. The draft standard specifies that = 0.006 for lumber and = 0.004 for glulam. The resemblance to Ylinen s equation is striking. Replacing with P c. (13) can be rationalized and simplified to (14) in which = e/3. This is the interaction equation implied by the Perry- Robertson equation. We see that when combined bending and compression are viewed as interacting linearly, the effective eccentricity is only one-third as great. Like the Ylinen equation, interaction was not the starting point of the derivation, nor is the result usually displayed in this form. Note, however, that if the eccentricity e = 0, we recover the condition of noninteraction. In other words. all interaction has effectively been attributed to eccentricity. The essential similarity between the Ylinen and Perry-Robertson equations is most clearly displayed when comparing them as interaction equations (7) and (14). Both models generalize the linear interaction equation with a cross-product term. The Perry-Robertson equation does so in a way that produces noninteraction, as we saw in (6), but then adds a bending term to model an assumed eccentricity. We conclude that the assumed eccentricity 2723

is actually a geranlzed parameter that effectively models causes of interaction between crushing and buckling. Regarding the bending term, the assumption of a linear interaction between crushing and bending failure is not supported by data. For solid-sawn lumber, a more accurate model for that interaction is parabolic (Zahn 1986) (15) Canada Neubauer (1973) proposed a column equation, which he called the cubic Rankine-Gordon equation. of the form (16) in which d = depth of cross section; and = an adjustable parameter. Eq. (16) has been adopted for wood column design in the most recent Canadian design standard (Engineering Design 1989) with = F c/(35e); F c = P c/a. Eq. (16) suffers the same defect as the Rankine-Gordon equation; namely, it dies not exhibit the proper limiting behavior. Whereas the Rankineby Gordon equation can be made to exhibit the proper limiting behavior choosing (at the expense of becoming too conservative), the Neubauer equation, because of its cubic form, cannot. Like the Rankine-Gordon equation, (16) may exceed the Euler curve (upper-bound limit) for a part of its range. It can be shown mathematically that (16) is tangent to the Euler curve at L/d = (16) exceeds the Euler limit for a part of its range. For the = F c /(35E) given by the Canadian standard, implies FIG. 4. Theory Comparison of Neubauer and Rankine-Gordon Equations with Elementary 2724

FIG. 5. Comparison of Ylinen Equation with c = 0.80, Perry-Robertson Equation with Properties of Spruce-Pine-Fir, Neubauer Equation with Properties of Spruce- Pine-Fir, and Same Column Data as in Fig. 1 FIG. 6. Comparison of Ylinen Equation, Recalibrated Perry-Robertson Equation, and Same Column Data as in Fig. 1 that E/F c < 326. For graded lumber. E/F c typicaly ranges from 300 to 375 (Green and Evans 1987; Mechanical Properties 1988). In Fig. 4, the Neubauer equation is compared with the Rankine-Gordon equation. On this figure. the Neubauer equation was calculated using E/F c = 300 so as to maximally exceed the Euler limit, and the Rankine-Gordon equation was calculated with chosen so as to make the two equations agree at P c = P e. The results are plotted along with the upper bound imposed by elementary theory. Clearly, Neubauer s equation is a great improvement over the Rankine-Gordon equation despite its flawed theoretical basis. 2725

Comparison Fig. 5 compares the Perry-Robertson. Neubauer, and Ylinen equations by plotting them against universal slenderness, along with the same column data as in Fig. 1. Note that Ylinen s equation produces the best fit to the data. Neubauer s equation does nearly as well, achieving a better fit than the Perry-Robertson equation. However, Neubauer s equation lacks a sound theoretical basis and is neither as adaptable nor as accurate as the Ylinen equation. Therefore. the Ylinen equation should be preferred. The Perry-Robertson equation can be made to fit the data better by ignoring the specification that = 0.006 and instead using the value 0.003. Fig. 6 compares this version of the Perry-Robertson equation with the Ylinen equation. Although still too conservative for very short members, this recalibrated Perry-Robertson equation now yields a better fit to the data. However. the Ylinen equation still produces the best fit and has a more straightforward interpretation. CONCLUSIONS Empirical column equations. which present column capacity as a function of slenderness. can be viewed as empirical interaction equations between crushing and buckling failure modes. Interaction between crushing and buckling is caused by any departure from the assumptions of elementary elastic-plastic theory, that is, by nonlinear stress-strain behavior, inhomogeneity, crookedness, and accidental eccentricity. Of the column equations employed in current design codes. the Ylinen equation appears to be the best candidate for the design of solid wood members. It follows the trend of column data best and offers the most straightforward interpretation and application. In the Ylinen design model, maximum interaction occurs when the interaction ratios are equal. as it should intuitively. and the degree of interaction can be easily adjusted to a variety of wood products. The degree of interaction is modeled by Ylinen s parameter c. This parameter should be less than l. by an amount that reflects the degree of departure from the conditions assumed in elementary theory. The experimental determination of the parameter c is best done at a slenderness value for which the two single-mode failure loads are equal. That is. member length and cross-section geometry should be chosen such that the elastic buckling load equals the failure load of a similar member under full lateral support. This is the point of maximum interaction. If the length is varied, the range should center around this point. Although it is traditional to design laterally unsupported beams with the same slenderness effect as for columns. the two interactions are not necessarily identical. In beams. only the part above the neutral axis is under compression. and torsion introduces another failure variable. Whether the same interaction equation applies to both beams and columns is yet to be experimentally determined. APPENDIX I. REFERENCES Buchanan, A. H. (1984). Strength model and design methods for bending and axial load interaction in timber members. PhD thesis, University of British Columbia, Vancouver. British Columbia, Canada. Eurocode No. 5: Common unified rules for timber structures. (1988). Commission 2726

of the European Communities, Office for Official Publications of the European Communities. Luxembourg. Engesser. F. (1889). Uber die knicksestigkeit gerader stäbe. Zeitschrift des Architekten- und Ingenieur-Vereins zu Hannover, 455. Engesser, F. (1895). Schweizerische Bauzeitung. Vol. 26, 24. Engineering Design in Wood. (1989). CAN/CSA-086.1-M89. Canadian Standards Association. Toronto, Ontario. Canada. Green, D. W., and Evans, J. W. (1987). Mechanical properties of visually graded lumber: a summary. Vol. 1, National Technical Information Service, Springfield, Va., U.S. Department of Agriculture, Forest Service. Forest Products Laboratory, Madison, Wis. Johns, K. C., and Buchanan, A. H. (1982). Strength of timber members in combined bending and axial loading. Proc. of the Int. Union of Forestry Research Organizations, Wood Engineering Group S5.02, May, 343-368. Larsson, H. (1956). (Title of paper unavailable). J. Aeronaut. Sci., 23, 867-873. Malhotra, S. K. (1972). A rational approach to the design of solid timber columns. Applications of solid mechanics, Study No. 7, University of Waterloo, Waterloo, Ontario, Canada. Mechanical properties of visually graded lumber: a summary. (1988). Canadian Wood Council, Ottawa, Ontario. Canada. National design specification for wood construction. (1991). National Forest Products Association. Washington, D.C. Neubauer, L. W. (1972). Full-size stud tests confirm superior strength of squareend columns." Trans. ASAE, 15(2), 346-349. Neubauer. L. W. (1973). A realistic and continuous wood column design formula. Forest Products. 23(3), 38-44. Parker, J. E. (1964). A study of the strength of short and intermediate wood columns by experimental and analytical methods. Research Note FPL-028, U.S. Department of Agriculture, Forest Service. Forest Products Laboratory, Madison, Wis. Salmon, E. H. (1921). Columns. Oxford Technical Publications, London, U.K. Shanley, F. R. (1946). The column paradox. J. Aeronaut. Sci., 13, 678. Shanley, F. R. (1947). Inelastic column theory. J. Aeronaut. Sci., 14, 261. Todhunter, I., and Pearson, K. (1886). History of the theory of elasticity. Vol. 1, Univ. of Cambridge Press. 105. Timoshenko, S. (1936). Theory of elastic stability. McGraw-Hill, New York, N.Y. 183. Ylinen, A. (1956). A method of determining the buckling stress and the required cross sectional area for centrally loaded straight columns in elastic and inelastic range. Publications of the International Association for Bridge and Structural Engineering, Zurich, Switzerland, 16, 529-550. Zahn, J. J. (1986). Design of wood members under combined load. J. Struct. Engrg., ASCE, 112(9), 2109-2126. Zahn, J. J. (1990). Empirical failure criteria with correlated resistance variables. J. Struct. Engrg., ASCE, 116(11), 3122-3137. APPENDIX II. NOTATION The following symbols are used in this paper: A = area of cross section; a = parameter in equal-power equation: c = Ylinen parameter for columns; d = depth of cross section; E = modulus of elasticity; e = eccentricity; F c = compressive strength: I = moment of inertia of cross section: 2727

length: bending moment; ultimate bending moment; lateral-torsional buckling moment: ultimate column load: crushing load (ultimate load under full lateral support); elastic buckling load; Neubauer parameter; Perry-Robertson parameter universal slenderness, radius of gyration of cross section; and Rankine-Gordon parameter. Printed on Recycled Paper 2728