0\"DS (N>t) atton n a i x t i O'O i r O'OB C O S O'Ot\" O ' K (N>i) aacn luixu F i g u r e 55 - P r e d i c t e d i n t e r a c t i o n diagram f o r end moments compared w i t h t e s t r e s u l t s f o r 38x89mm s i z e 167 r e s u l t s as d e s c r i b e d i n Chapter 5. The f i t of the p r e d i c t e d curves to the data i s reasonable. In general the model tends to underestimate the f a i l u r e moment, but the di s c r e p a n c y i s not l a r g e . No d e f i n i t e e x p l a n a t i o n i s a v a i l a b l e f o r t h i s underestimate, but some c o n t r i b u t i n g f a c t o r s w i l l be d i s c u s s e d . One p o s s i b i l i t y i s that the computer model assumes that s t r e n g t h and s t i f f n e s s are uniform along each board, and that f a i l u r e w i l l always occur at mid-span. In p r a c t i c e , the f a i l u r e s were o f t e n at a de f e c t away from mid-span, r e q u i r i n g a l a r g e r load and moment than would have been necessary had the de f e c t been at mid-span. A r e l a t e d e x p l a n a t i o n i s that the input v a l u e s of a x i a l t e n s i o n and compression s t r e n g t h f o r the model have been m o d i f i e d to the len g t h of the column, assuming that the f u l l l e n g t h of the column i s sub j e c t e d to uniform s t r e s s . T h i s i s a c o n s e r v a t i v e assumption f o r columns which d e f l e c t c o n s i d e r a b l y before f a i l u r e , and may c o n t r i b u t e to the underestimate of the model. The accuracy of the f i t i s con s i d e r e d to be q u i t e a c c e p t a b l e , and i f the p r e d i c t e d curves are to be used as a design t o o l they g e n e r a l l y represent a s l i g h t l y c o n s e r v a t i v e p r e d i c t i o n of s t r e n g t h . 168 7.2.4 Long Column Interaction Curves for Mid-Span Moments Figure 56 shows a comparison of the model prediction and the data, using interaction curves of ax i a l load and mid-span moment. A l l of the curves and data points show much larger moments than in figure 55 because the moments have been amplified due to deformations within the members. The f i t of the data to the model i s not as good in this case, as the model sometimes underestimates the moment at f a i l u r e by a considerable amount. The factors discussed in the previous section s t i l l apply, but another point must be included. Each data point i s calculated from the measured mid-span deflection at maximum ax i a l load. In most cases the f a i l u r e i s an i n s t a b i l i t y f a i l u r e associated with rapid increase in l a t e r a l deflections at very small changes in load. The load was not applied very slowly, so the measured deflections could be considerably larger than that which would just cause f a i l u r e under steady state conditions. A similar problem has been reported in tests on concrete members (Nathan 1983b). In view of these problems, more emphasis should be given to end moments than mid-span moments when checking the v a l i d i t y of the computer model. 169 F i g u r e 56 - P r e d i c t e d i n t e r a c t i o n d i a g r a m f o r m i d - s p a n moments c o m p a r e d w i t h t e s t r e s u l t s f o r 38x89mm s i z e 170 7.2.5 Axial Load - Slenderness Curves Figure 57 shows the axi a l load vs. slenderness curves for the 38x89mm boards at the 5th percentile, mean and 95th percentile l e v e l s . The top (dotted) l i n e i s the predicted curve for concentric loading (not tested), and the next five l i n e s are for the test e c c e n t r i c i t i e s , defined in the key to the data points. The data points are the same points from the test results presented previously. The f i t of the model to the data i s very good. In some cases the model underestimates the a x i a l load recorded in the tests. 7.3 38x14 0mm BOARDS 7.3.1 Short Column Interaction Curves Figure 58 shows the ultimate interaction diagrams from the model, calibrated to mean values of test results for the shortest length (0.914m) of 38x140mm,boards. A l l the parameters used for thi s prediction are as for the 34x89mm material with, one exception. The stress-d i s t r i b u t i o n parameter in tension, which relates the tension strength to the bending strength, i s d i f f e r e n t . The value of k3=7.0 used for the smaller size has been changed to k3=9.0 to provide a better f i t to the test data\u00b0for the larger cross section. The model predictions for the 5th percentiles, mean, and 95th percentiles have been made using the same parameters. General observations about the goodness of f i t in this figure are similar to those described for the smaller size material. Note that there are only two points in the tension region, one each at the 5th percentile and mean l e v e l s . As (a) 5 th % i l e (b) Mean (c) 95th % i l e F i g u r e 57 - P r e d i c t e d a x i a l l o a d - s l e n d e r n e s s c u r v e s compared w i t h t e s t r e s u l t s f o r 38x89mm s i z e 172 Length 0.91 Am. \\ Mean \/ \/ \/ \/ \/ 9 5 t h %ile I 1 1 1 1 1 1 1 1 1 1 1 1 T -0.0 ID 2.D 3.0 4.0 S.0 6.0 7.0 noriENT cKN.ro 8.0 F i g u r e 58 - U l t i m a t e i n t e r a c t i o n diagram c a l i b r a t e d t o t e s t r e s u l t s f o r 38x140mm s i z e d e s c r i b e d i n Chapter 4, the t e s t method f o r d e r i v i n g these p o i n t s was not v e r y s a t i s f a c t o r y , so they s h o u l d be viewed w i t h c a u t i o n , a l t h o u g h they do tend t o support the p r e d i c t e d c u r v e s . 173 7.3.2 Long Column Interaction Curves Figure 59 shows the predicted long column interaction curves for end moments compared with the test r e s u l t s . Figure 60 shows the predicted long column interaction curves for mid-span moments compared with the test r e s u l t s . In both cases the comparison of the model and test r e s u l t s i s of similar form to that previously shown for the 38x89mm size, but the f i t i s not as good and the model i s seen to be very conservative in many cases. No s p e c i f i c explanation i s available other than the points raised in discussion of the 38x89mm re s u l t s . 7.3.3 Axial Load-Slenderness Curves Figure 61 shows the predicted curves of a x i a l load vs. slenderness compared with the test results for the 38x140mm boards. The l i n e s are as described for the smaller size, and the same comments apply. 7.4 REPRESENTATIVE STRENGTH PROPERTIES When investigating possible design methods in the next chapter i t w i l l be convenient to refer to material with certain representative strength properties, rather than s p e c i f i c grades, species, and sizes of timber. The load capacity of timber members is influenced by one or more of the following, depending on the nature of the loads. tension strength compression- strength bending strength modulus of e l a s t i c i t y Figure 59 - Predicted interaction diagram for end moments compared with test results for 38x140mm size 175 (NX) ouoi \"ibixti (NX) ami it i ixti Figure 60 - Predicted interaction diagram for mid-span moments compared with test results for 38x140mm size cn (a) 5th % i l e (b) Mean (c) 95th % i l e F i g u r e 61 - P r e d i c t e d a x i a l l o a d - s l e n d e r n e s s c u r v e s compared w i t h t e s t r e s u l t s f o r 38x140mm s i z e 1 7 7 The r a t i o between tension and compression strengths has a major influence on the shape of the ultimate interaction diagram for cross section strength. The r a t i o of tension to bending strength also a f f e c t s the shape of the ultimate interaction diagram because th i s r a t i o i s an estimator for the st r e s s - d i s t r i b u t i o n parameter. For long columns the modulus of e l a s t i c i t y has a major influence on the load capacity. A small survey was carr i e d out on the results of the tests reported in Chapter 5 and on the results of several other in-grade tests carried out at the University of B r i t i s h Columbia. For the r a t i o of in-grade tension strength to i n -grade compression strength, most results were in the range of 0.55 to 0.85. For the r a t i o of tension strength to bending strength, corrected for length e f f e c t s , most results were in the range of 0.6 to 0.8. The modulus of e l a s t i c i t y was between 300 and 350 times the a x i a l compression strength in most cases. These figures are representative of 5th percentile results for SPF and Hem-fir material in sizes 39x89mm to 38x235mm, grades Number 2 to Select Structural, and moisture content approximately 15%. On the basis of this small survey, three groups of representative properties have been chosen, one average group and two groups representing the l i m i t s of observed results, these having properties that produce the largest and the smallest \"nose\" in the compression region of the a x i a l load moment interaction diagram. Figure 62 shows ultimate interaction diagrams for material of these three representative groups, non-178 AVERAGE STRONG W E A K MOMENT Figure 62 - Ultimate interaction diagrams for representative strength properties dimensionalized to a compression strength of 1.0. Material that i s r e l a t i v e l y weak in tension and in bending produces a large \"nose\" in the compression region. This is representative of low grade material, and w i l l be referred to as weak material in the next chapter. Material that is r e l a t i v e l y strong in tension and in bending produces a smaller \"nose\" in the compression region. This i s representative of high grade material, and w i l l be referred to as strong material. The curves representing weak and strong material ( should be considered upper and lower bounds for the material described. Material represented by the central l i n e on figure 179 62 w i l l be referred to as average material. The ratios and r e l a t i v e strengths are summarized in Table I I I . In a l l cases Weak Average Strong Ratios Ft\/Fc .55 .70 .85 Ft\/Fb .80 .70 .60 Relat ive Fc 1.0 1 .0 1 .0 Strengths Ft .55 .70 .85 Fb .69 1 .0 1 .42 Table III - Material property ratios for representative groups the r a t i o of modulus of e l a s t i c i t y to compression strength has been taken as 300. Figure 63 shows interaction diagrams of end moments, for columns of three slenderness r a t i o s , and material of the three representative strength groups. The curves have been non-dimensionalized so that the shapes of the curves can be compared. Figure 64 shows a x i a l load - slenderness curves for the three representative strength groups, for columns with several end e c c e n t r i c i t i e s . Again these plots have been non-dimensionalized so that r e l a t i v e shapes can be compared. 180 Figure 64 - Axial representative strength load-slenderness curves for properties (non-dimensionalized) 181 7.5 APPLICABILITY OF STRENGTH MODEL The s t r e n g t h model d e s c r i b e d i n t h i s t h e s i s i s a d e t e r m i n i s t i c model w h i c h does no s t a t i s t i c a l c a l c u l a t i o n s . To use t h e model t o c a l c u l a t e s t r e n g t h a t any p e r c e n t i l e i n t h e d i s t r i b u t i o n , i t i s n e c e s s a r y t o use i n p u t d a t a f o r t h a t p e r c e n t i l e , and t o c a r r y o u t a d e t e r m i n i s t i c c a l c u l a t i o n . The t e s t r e s u l t s d e s c r i b e d i n C h a p t e r 5 showed v e r y l a r g e v a r i a b i l i t y i n s t r e n g t h . In t h i s c h a p t e r t h e s t r e n g t h model has been c a l i b r a t e d u s i n g mean t e s t r e s u l t s , and has g i v e n a good p r e d i c t i o n of s t r e n g t h t h r o u g h o u t t h e d i s t r i b u t i o n . T h i s d e m o n s t r a t e s t h e v e r s a t i l i t y o f t h e model, and shows t h a t the p r e d i c t e d b e h a v i o u r of t i m b e r members under combined b e n d i n g and a x i a l l o a d depends o n l y on the'-values of t h e i n p u t d a t a , and n o t t h e l o c a t i o n i n t h e d i s t r i b u t i o n of s t r e n g t h . T h e s e f i n d i n g s a r e s i g n i f i c a n t b e c a u s e t h e y i n d i c a t e t h a t t h e s t r e n g t h model can p o t e n t i a l l y be used f o r t i m b e r from any s o u r c e , a t any l e v e l i n a d i s t r i b u t i o n of s t r e n g t h , p r o v i d e d t h a t i n p u t d a t a i s a v a i l a b l e and t h a t s u i t a b l e c a l i b r a t i o n i s c a r r i e d o u t . 7.6 SUMMARY T h i s c h a p t e r has b r o u g h t t o g e t h e r t h e t e s t r e s u l t s d e s c r i b e d i n C h a p t e r 5 and t h e s t r e n g t h model d e s c r i b e d i n C h a p t e r 6. The model has been c a l i b r a t e d u s i n g o n l y mean t e s t r e s u l t s f o r t h e s h o r t e s t b o a r d s t e s t e d . The s t r e n g t h model c a l i b r a t e d i n t h i s way has been u s e d t o p r o v i d e a r e a s o n a b l y a c c u r a t e p r e d i c t i o n of member s t r e n g t h , f o r a wide range of 182 V member l e n g t h s , throughout the d i s t r i b u t i o n of s t r e n g t h v a l u e s . T h i s i s c o n s i d e r e d t o be q u i t e a s e v e r e t e s t f o r the model. D i s c r e p a n c i e s between the p r e d i c t e d c u r v e s and the t e s t r e s u l t s a re g e n e r a l l y not l a r g e , and where they occur they t e n d t o be c o n s e r v a t i v e . On the b a s i s of th e s e r e s u l t s the model w i l l be used i n the next c h a p t e r t o i n v e s t i g a t e a number of p o s s i b l e d e s i g n methods f o r t i m b e r members s u b j e c t e d t o combined bending and a x i a l l o a d i n g . 183 VIII. DESIGN METHODS FOR COLUMNS AND BEAM COLUMNS 8.1 INTRODUCTION Structural designers require a simple yet e f f e c t i v e method for s i z i n g a member to safely r e s i s t prescribed a x i a l loads and bending moments. One of the objectives of this study is to propose design methods which can be used to check whether a member of selected dimensions has s u f f i c i e n t load carrying capacity. This chapter consists of a very brief review of design philosophy and current code requirements for combined bending and a x i a l loading in timber, steel and concrete, followed by several proposals for new design methods for timber members. 8.1.1 Allowable Stress Design For many years structural design was based on the behaviour of structures under working load conditions and e l a s t i c behaviour. Design codes specified allowable stresses that were not to be exceeded when the structure was loaded with the maximum anticipated loads, without consideration of behaviour at ultimate loads. The allowable stresses were derived from strength tests on materials using large factors of safety. 8.1.2 R e l i a b i l i t y - B a s e d Design A s i g n i f i c a n t advance in structural design was made when design codes began to place more emphasis on the behaviour of structures loaded to conditions near f a i l u r e . \"Ultimate strength\" design of reinforced concrete and \" p l a s t i c \" design of steel structures involved checking that a structure would 1 8 4 not actually collapse when loaded with maximum anticipated loads increased by a substantial \"load factor\". \"Limit states\" design codes consider structural f a i l u r e as an ultimate l i m i t state to be investigated along with a number of s e r v i c e a b i l i t y l i m i t states. In recent years there has been a lot of attention given to quantifying the amount of structural safety provided by various design methods. Design codes for steel and reinforced concrete have been improved in an attempt to provide a certain minimal pr o b a b i l i t y of f a i l u r e over the l i f e of a structure. Design codes based on t h i s approach may be c a l l e d \"probability-based\" or \" r e l i a b i l i t y -based\" l i m i t states design codes. Accurate cal c u l a t i o n of prob a b i l i t y of f a i l u r e requires detailed knowledge of the d i s t r i b u t i o n s of both load and resistance in the overlapping region. This information is not readily available, so approximate methods have beeen developed to calculate a \" r e l i a b i l i t y index\" from just the mean and standard deviation of load and resistance. There have been many developments in t h i s area since early work by Cornell(1969) and others. Recent developments towards consistent design codes for di f f e r e n t materials in North America are summarized by Galambos et al.(1982). Despite the best attempts to quantify structural safety, observations of structural f a i l u r e s have demonstrated that even when best available estimates of load and resistance d i s t r i b u t i o n s are used, some allowance must, be made for other factors more d i f f i c u l t to quantify such as poor workmanship, 185 design mistakes and unforseen circumstances (Blockley 1980). 8.1.3 Reliability-Based Design of Timber Rel i a b i l i t y - b a s e d design methods for timber structures are less advanced than for other materials. The ca l c u l a t i o n of loads i s e s s e n t i a l l y the same for structures of any material, with a large amount of uncertainty in most cases. The c a l c u l a t i o n of resistance depends on detailed knowledge of the strength properties and behaviour of the structural material. The strength of timber under various loading conditions i s not well defined and i s much more variable than for other materials. Sexsmith and Fox (1978) were among the f i r s t to demonstrate how r e l i a b i l i t y - b a s e d l i m i t states design methods could be applied to glued laminated beams. Foschi(1979) discussed some potential problems with application of the r e l i a b i l i t y index concept to timber structures. Goodman et a l . ( l 9 8 l ) have compared the l e v e l of safety implied by code sp e c i f i c a t i o n s in several international codes. Another investigation of r e l i a b i l i t y - b a s e d design of timber members i s described by Ellingwood(1981). Goodman et al.(l983) summarize a large investigation into r e l i a b i l i t y - b a s e d design of wood transmission l i n e structures. Malhotra(1983) has investigated the r e l i a b i l i t y index implied by several alternative design methods for timber compression members. 186 8.1.4 Scope The results of t h i s study w i l l be used in this chapter to propose design methods for checking that a selected member has the capacity to r e s i s t specified loads. In this context i t is not important whether the loads are specified by a design code in a working stress format or in a l i m i t states format. The actual load and resistance factors in each case must be determined by others. It i s beyond the scope of this study to develop r e l i a b i l i t y - b a s e d design methods for timber members. A p r i n c i p a l contribution of t h i s study has been to produce information on the strength d i s t r i b u t i o n of timber members under, various combinations of a x i a l and f l e x u r a l loadi-ng, which w i l l be useful input to the eventual development of r e l i a b i l i t y - b a s e d design methods. 8.2 EXISTING DESIGN METHDOS 8.2.1 Canadian Timber Code The current Canadian Code (CSA 1980) i s based on allowable design stresses providing a certain factor of safety against f a i l u r e under specified working loads (Wilson 1978). The Timber Design Manual (1980) provides a working guide to the code with many design aids. The background to these requirements have been described e a r l i e r in t h i s thesis. 187 a. Concentrically Loaded Columns For concentrically loaded columns design i s based on three d i f f e r e n t slenderness classes, as described in Chapter 2 and shown in figure 8. Slenderness r a t i o i s defined as the r a t i o of e f f e c t i v e length, L, to the appropriate cross sectional dimension, d. The allowable stress, fa, in a column i s given by L\/d < 10 f a = fca ( 8- 1> 10 < L\/d < K f a = f c J l - l \/ 3 ( ^ f ] (8.2) L\/d > K f - , 2 3 3 E (8.3) 3 (L\/d) 2 The t r a n s i t i o n between intermediate length and long columns i s at a slenderness r a t i o K, given by K = 0.591 | \u2014 (8.4) ca where E i s the modulus of e l a s t i c i t y and fca i s the allowable stress p a r a l l e l to the grain in a short column. These formulae include safety factors. The notation has been changed from that in the code for consistency. Although quite simple in concept, the design formulae are unwieldy and awkward to use because there i s a d i f f e r e n t formula for each 1 8 8 of three ranges of slenderness. A number of continuous column formulae which could be used for a l l slenderness ratios w i l l be described later in t h i s chapter. The experimental phase of thi s study did not include concentrically loaded columns, so direc t v e r i f i c a t i o n of the code column formula i s not possible. However, the strength model has been calibrated to results of tests for a large number of e c c e n t r i c i t i e s , and can with s l i g h t extrapolation, be used to predict behaviour at zero e c c e n t r i c i t y . Figure 65 compares the code formulae with behaviour predicted by the model for concentrically loaded columns. The plot has been non-dimensionalized to maximum load using a modulus of e l a s t i c i t y 300 times the material strength. There is a moderate discrepancy between the two curves for intermediate column lengths, the code formula overestimating column strength in t h i s range. b. Combined Axial Load and Bending For members subjected to combined a x i a l compression and bending the code sp e c i f i e s a linear interaction between the a x i a l load capacity of a concentrically loaded column and the moment capacity in pure bending. The formula i s UA + *l\u00b1 < i (8.5) fa fb where P i s the a x i a l load, A i s the area of cross section, fa is the allowable a x i a l stress under concentric loading for the pa r t i c u l a r slenderness r a t i o , M i s the bending moment 189 nODEL PREDICTION = ~l i i i I 1 1 1 1 1 1 1 1 0-D 8.0 16.0 24.0 32.0 40.0 48.0 SLENDERNESS (L\/d) Figure 65 - Code column formula compared with model prediction (non-dimensionalized) including that due to a x i a l loads, S is the section modulus, and fb is the allowable bending stress. The code provides no guidance for ca l c u l a t i n g the bending moment due to a x i a l load and column deflections, but the Timber Design Manual (1980) suggests equations for simple cases and a t r i a l and error method for others. For short columns equation 8 .5 i s consistent with assumptions of linear e l a s t i c behaviour, a l i m i t i n g value of compression stress, with no consideration of a f a i l u r e in the tension zone. E a r l i e r chapters have shown that these assumptions are not consistent with observed behaviour. As an example of the inadequacies of the code provisions for short 190 columns, refer to figure 23. The form of equation 8.5 i s represented by the straight dotted l i n e which i s seen to be very conservative compared with the inner s o l i d l i n e sketched through the test r e s u l t s . For longer columns refer to the four parts of figure 56. The inner dotted curves are the model predictions of 5th percentile strength. Equation 8.5 would be represented by straight l i n e s with the same axis intercepts, which would again be very conservative. A similar linear interaction formula i s spec i f i e d for combined bending and a x i a l tension. c. Summary To summarize, present Canadian code requirements are unsatisfactory for the following reasons: 1. For concentrically loaded columns of intermediate lengths, the code formula overestimates strength. The awkward formulae for intermediate and long columns could be replaced by a single continuous formula for a l l slenderness r a t i o s . 2. For combined bending and a x i a l compression the code interaction equation i s a poor representation of actual behaviour, grossly underestimating strength in many cases. 3. Moment magnification due to slenderness e f f e c t s is not adequately provided for. 191 8.2.2 NFPA Timber Code A timber design code widely used in the United States i s that produced by the National Forest Products Association (NFPA 1982). This code i s also based on allowable stresses. a. Concentrically Loaded Columns Design of concentrically loaded columns i s similar to the Canadian code requirements of equations 8.1 to 8.4. The t r a n s i t i o n between short and intermediate columns is at a slenderness r a t i o of 11 rather than 10, and the safety factors are s l i g h t l y d i f f e r e n t . b. . Combined Axial Load and Bending For combined a x i a l compression and bending, a general formula is used which includes both eccentric a x i a l loads and transverse loads. ' The derivation by Newlin(l940) and Wood(l950), described b r i e f l y in Chapter 2, used assumptions more applicable to clear wood than to sawn timber. The formula i s P\/A M M\/S + P\/A(6 + 1.5 J) ( e \/ d ) { 8 > 6 ) T~ + f - P\/A * 1 W H E R E J = M^S* . 0< J < 1 (8-7) and M\/S i s the bending stress resulting from l a t e r a l loads, fb is the allowable stress in bending, e is the e c c e n t r i c i t y of the a x i a l load. A l l other terms are as defined in the previous section. 192 The code formula w i l l be compared with behaviour predicted by the strength model for e c c e n t r i c a l l y loaded columns, assuming that the allowable stresses in the code are representative of actual behaviour, and ignoring safety factors. For the case of no transverse loads (M\/S=0) the moment allowed by the code formula can be obtained from equation 8.6 as f. - JP\/A a where M i s now the end moment P times e. This has been plotted for comparison with the model prediction in figure 66. For low slenderness r a t i o (L\/d=lO) the code formula i s seen to be conservative at low a x i a l loads and unsafe at high a x i a l loads. For columns with L\/d=20 the code formula is again very conservative for low a x i a l loads, mainly because i t predicts only 80% of actual bending strength when a x i a l load is zero (This can be seen by puttting P=0 and J=1 in equation 8.8). The small unsafe region near the v e r t i c a l axis is due to the discrepancy between the code formula and the model prediction for t h i s slenderness r a t i o , as shown in figure 65. For slender columns (L\/d=40) the code i s again conservative for low a x i a l loads. The comparison i l l u s t r a t e d in figure 66 has been made for material with \"average\" strength properties as defined in Chapter 7. For \"strong\", or \"weak\" material the model predicts very d i f f e r e n t interaction curves as shown in figure 1 9 3 riODEL PREDICTION NFPA FORriULA U N H f l G N I F I E D FlOHENT Figure 66 - NFPA formula compared with model prediction (non-dimensionalized) 63, but equation 8.8 produces almost i d e n t i c a l curves. This comparison has not been plotted, but a comparison of figures 66 and 63 shows that the NFPA formula w i l l result in large unsafe areas for strong material and large conservative areas for weak material. c. Summary 1 Compared with the Canadian code, the NFPA code i s an improvement in that moment magnification i s incorporated into a r e l a t i v e l y simple formula that handles both eccentric a x i a l loads and transverse loads. However, by comparing the NFPA formula with the prediction of the strength model, i t has been shown that the formula does not accurately predict the 194 s t r u c t u r a l behaviour of timber compression members at ultimate loads. 8.2.3 Code Requirements for Steel The current Canadian code for steel design (CSA 1978) i s in a l i m i t states format. Structural members are designed such that their load capacity, reduced by a performance factor, i s not exceeded by spec i f i e d factored loads. The Handbook of Steel Construction (1980) provides a commentary and working guide to the code with many design aids. The Canadian s t e e l code i s based on similar p r i n c i p l e s to many other codes for steel design. For concentrically loaded columns, design follows the same general procedure as for timber, with a parabolic t r a n s i t i o n from material f a i l u r e in short columns to Euler buckling in long columns. For combined bending and a x i a l loading the code spe c i f i e s that strength and s t a b i l i t y be checked separately. When designing a member to r e s i s t some combination of a x i a l load P and bending moment M, the s t a b i l i t y requirement i s s a t i s f i e d in i t s simplest form i f P , 1 M . , \/ n \u2014 + T=W7f- M~ < 1 (8.9) u e u where Pu i s the a x i a l strength of the column under concentric loading (including possible e f f e c t s of buckling), Mu i s the moment capacity and Pe i s the Euler buckling load from equation 2.6. This formulation assumes that the interaction 195 diagram for ax i a l load and mid-span moment in slender beam columns i s the straight l i n e Pu-Mu on Figure\" 67, which is an empirical relationship v e r i f i e d by experimental studies MOMENT Figure 67 - Axial load-moment interaction diagram for steel members (Galambos 1968). To s a t i s f y t h i s requirement, a design combination of P and M must l i e to the l e f t of the curved dotted l i n e Pu-Mu, the horizontal distance between the dotted l i n e and the s o l i d l i n e representing moment magnification caused by l a t e r a l deflections in the member. The moment magnification factor i s given by F = 1 - P\/P, (8.10) which i s a close approximation to the exact expression for 196 linear e l a s t i c behaviour (Timoshenko and Gere 1961). For short columns in which the a x i a l strength i s not reduced by s t a b i l i t y e f f e c t s the interaction diagram becomes the straight l i n e Pa-Mu on Figure 67, where Pa i s the axi a l compression strength of the column material. If the beam-column i s loaded with d i f f e r e n t moments at each end the design equation becomes r + I=P7P- M- < 1 { 8 - 1 1 ) u e u where Cm i s a factor to account for the d i s t r i b u t i o n of moments along the member which can be approximated by C m = 0.6 + 0.4 M2\/M1 > 0.4 (8.12) where M, and M2 are the larger and smaller end moments respectively. Galambos(1968) and Johnston(1976) show that th i s i s a reasonable approximation for a wide range of loading cases. The product of Cm and M is an equivalent uniform moment that should lead to the same long column strength as the actual moment diagram. For some unsymmetrical loading conditions the load capacity of a beam-column w i l l be governed by the moment capacity of a p l a s t i c hinge at one end. In thi s case the load capacity w i l l be governed by strength rather than by s t a b i l i t y considerations, and the design must be checked against an interaction diagram for cross section strength. The interaction formula for strong-axis bending i s 1 97 M < 1.18 (1 - !-) , M < M (8.13) u u which i s shown by the l i n e Pa-B-Mu in figure 67. The b i l i n e a r shape appears because the web of an I-section can carry some ax i a l load without reducing the p l a s t i c moment capacity of the flanges. The shaded l i n e on figure 67 shows the resulting design envelope for a possible situation with unsymmetrical end moments. The possible e f f e c t s of l a t e r a l - t o r s i o n a l buckling in bending can be accommodated by using a bending strength value Mu that represents the strength at which buckling occurs. B i a x i a l behaviour i s included by extending the interaction formula for s t a b i l i t y to C M C M , ' x L_ + rc* x + m y _ Z _ < i (8.14) P 1 - P \/ P M 1 - P \/ P M u e x u x e y u y where the subscripts x and y refer to actions about the x and y axes., A similar formula is specified for material strength under b i a x i a l loading. These formulae assume linear interaction between x-axis and y-axis behaviour. This assumption i s very conservative (Johnston 1976), but an appendix to the code provides more detailed formulae for more accurate design in certain cases. It can be seen that the steel requirements have some s i m i l a r i t i e s to those for timber, but are s i g n i f i c a n t l y better because 1. Strength and s t a b i l i t y effects are considered separately. 198 2. The specified interaction between a x i a l load and moment i s more representative of actual behaviour at ultimate loads. 3. Moment magnification due to second-order effects i s included. 4. The effect of unequal end moments i s included. 5. B i a x i a l e f f e c ts are accounted for. 6. The possible e f f e c t s of l a t e r a l t o rsional buckling can be included. 8.2.4 Canadian Concrete Code The current Canadian code (CSA 1977) for reinforced concrete design is written in a l i m i t states format, often referred to as \"ultimate strength design\". The Canadian code is similar to many other reinforced concrete codes. As with s t e e l , s t r u c t u r a l members are designed such that their ultimate load capacity, reduced by a capacity reduction factor, i s not exceeded by sp e c i f i e d factored loads. A detailed background to the column provisions is given by MacGregor et al. ( l 9 7 0 ) , and is used as the basic reference for this section. The design method for reinforced concrete compression members i s s i g n i f i c a n t l y d i f f e r e n t from the methods described above for steel and timber. The code provisions are much less extensive than for s t e e l , partly because long slender members occur much less frequently in concrete than in s t e e l , and because members with hinged end conditions are rare. No s p e c i f i c provision is made for concentrically loaded compression members. A l l members must be designed to r e s i s t a minimum nominal bending moment due to an e c c e n t r i c i t y of 10% 199 of the member's dimension about either axis. . Although an accurate second-order s t r u c t u r a l analysis i s recommended, slenderness effects are generally accounted for by magnifying the bending moment from a l l sources by a magnification factor, F, given by C F = m 1-P\/4>P (8.15) where P i s the factored a x i a l load, Pe is the Euler buckling load, 0 is a capacity reduction factor, and Cm i s the c o e f f i c i e n t for di f f e r e n t end moments exactly as used in the steel code. The member i s then sized and reinforced such that the factored a x i a l load combined with the magnified factored bending moment does not cause a material f a i l u r e at any cross section. The strength of the cross section is determined from f i r s t p r i n c i p l e s by standard methods or from published design charts (ACI 1970). The shape of the ultimate interaction diagram for cross section strength is very similar to that obtained for timber members in t h i s study. This method of design assumes that a l l f a i l u r e s are material f a i l u r e s , not i n s t a b i l i t y f a i l u r e s . The design method considers i n s t a b i l i t y only i n d i r e c t l y , in that equation 8.15 can only be used sensibly for a x i a l loads less than the Euler buckling load. Equation 8.15 i s an approximation based on linear e l a s t i c theory, but real behaviour of reinforced concrete i s non-linear. To overcome th i s problem the code 2 0 0 includes an empirical expression for the s t i f f n e s s EI which leads to approximately correct results for members which have material f a i l u r e s . These approximations are apparently not serious d e f i c i e n c i e s in the design method because very few reinforced concrete columns are very slender. For members f a i l i n g in an i n s t a b i l i t y mode (slender prestressed members for example) the apparent s t i f f n e s s EI would have to be further reduced empirically, or a more rati o n a l method method used (Nathan 1983a). The expression for EI also includes a factor to allow for the p o s s i b i l i t y of creep under long duration loading. The p o s s i b i l i t y of a similar provision for timber members is discussed in Chapter 9. The p r i n c i p a l advantages of the reinforced concrete method are that 1. A single design procedure can be used, for a l l types of compression members. 2. No concentric loading column formula is required. 3. The short column interaction curve used for a l l design can be derived from f i r s t p r i n c i p l e s , or simply obtained from published graphs or tables. The primary disadvantage i s that the design method for slender columns is not an accurate representation of real behaviour. The approximations introduced are sati s f a c t o r y for reinforced concrete but may not be for more slender columns made of timber. ( 201 8.2.5 Limit States Timber Codes A proposal for a l i m i t states design format for timber structures was made by SexsmitM1979). Two new codes contain provisions similar to some of those suggested by Sexsmith. The Canadian code proposal (CSA 1983) i s almost i d e n t i c a l to the exis t i n g working stress code previously described, with new load factors and resistance factors which change the format of the design equations without any s i g n i f i c a n t conceptual changes. A second code in l i m i t s states format i s the Ontario Highway Bridge Design Code (OHBDC 1982) referred to as OHBDC in t h i s chapter. The design provisions of OHBDC for compression members represent a major change from existing requirements. There are several improvements but the requirements are s t i l l lacking in some respects. The main change i s the deletion of a design method for concentric loading, recognizing that i t i s impossible to load a timber member with zero end e c c e n t r i c i t y . For a x i a l loading a minimum end e c c e n t r i c i t y of 0.05 times the cross section dimension i s speci f i e d , together with an i n i t i a l bow (or crook) at mid-length of 1\/500 times the e f f e c t i v e length. The design equation is a e u where a l l the terms are the same as defined previously for steel and concrete. This equation assumes that a l l f a i l u r e s 2 0 2 are material f a i l u r e s ; i n s t a b i l i t y f a i l u r e s are only considered i n d i r e c t l y in that the equation can only be used for a x i a l loads less than the Euler buckling load. To investigate the possible consequences of t h i s s l i g h t l y incorrect approach, a comparison with the more correct formulation of the steel code w i l l be made. Consider figure 68 which i s an interaction diagram of a x i a l load vs. moment. MOMENT Figure 68 - Axial load-moment interaction diagram from OHBDC Pa i s the concentric a x i a l load capacity of a short column, Pu is the concentric a x i a l load capacity of the long column under consideration, and Mu is the bending capacity. Figure 68 has been constructed, to scale, to i l l u s t r a t e the behaviour of a slender column whose concentric load capacity Pu i s half of 203 the short column load capacity Pa. The curve Pu-C-Mu has been obtained from the OHBDC formula (equation 8.16). The curve Pu-B-Mu has been constructed from the more correct steel formula (equation 8.11). The steel code suggests that t h i s member, subjected to a x i a l load A, can just carry an end moment represented by point B, which corresponds to a magnified mid-span moment represented by point D. OHBDC suggests that this member can carry a larger end moment represented by point C which corresponds to a magnified mid-span moment causing material f a i l u r e at point E. The OHBDC approach i s seen to be conceptually incorrect and s l i g h t l y unsafe. The OHBDC formula i s compared with the prediction of the strength model in figure 69. To make a v a l i d comparison i t has been necessary to work backwards from the linear interaction diagram for magnified moments, to compare the unmagnified moments that the designer begins with, shown as dotted l i n e s . The dotted lines are a l l curved toward the ori g i n because the moment value of each point on the linear interaction diagram has been reduced by the code magnification factor to give a design moment. For short columns (L\/d=l0) the OHBDC formula i s seen to be very conservative, except at high a x i a l loads, even more so than the NFPA formula. For longer columns the formula becomes more accurate. Figure 69 has been plotted for \"average\" strength material, but the OHBDC curves w i l l not change for \"strong\" or \"weak\" material, so a comparison with figure 63 shows that the formula w i l l be 204 o cr o DODEL PREDICTION - OHBDC FORrlllLH \\ \\\\ \\ \\ CO _ C3 \\ CO _ \\ \\ , \\ \\ \\ . V \\ x \\ A A U 1 0 \\ _ \\ ^ \\ ^ J V d = 20 \\ X. \\ \\ Ol _ \\ \\ \\ L\/d = 40 \\ \u2022= = ' * \u2014 \/ a \" - ^ ^ \\ \/ <=>' I I I I 1 I I 1 I I I I I 0.0 0.2 0.4 0.6 0.8 I.D 1.2 UNHflGNIFIED HOHENT Figure 69 - OHBDC formula compared with model prediction (non-dimensionalized) very conservative for weak material. The OHBDC formula does not accurately represent behaviour at ultimate loads. The linear interaction diagram is conservative and the moment magnification factor w i l l be shown to be unconservative, but these two effects cancel out somewhat to produce the curves shown in figure 69. For b i a x i a l loading the OHBDC speci f i e s a linear interaction formula almost i d e n t i c a l to that in the steel code. The s p e c i f i c a t i o n of minimum end e c c e n t r i c i t y and i n i t i a l bow suggests that b i a x i a l loading should be considered in every case but the code i s not clear on this matter. In summary, the OHBDC represents a major development in 205 tha t i t s p e c i f i e s minimum moments and i n c l u d e s a des ign equa t ion fo r a m p l i f i c a t i o n of a l l moments, but some of i t s assumpt ions are not c o n s i s t e n t wi th the r e s u l t s of t h i s s tudy . 8.3 COLUMN CURVES FOR CONCENTRIC LOADING Some of the des ign methods to be proposed fo r beam-columns w i l l r e q u i r e a method fo r c a l c u l a t i n g the s t r eng th of the column under c o n c e n t r i c l o a d i n g . A b r i e f i n t r o d u c t i o n in Chapter 2 d e s c r i b e d some of the d i f f i c u l t i e s in c a l c u l a t i n g the s t r e n g t h of columns of m a t e r i a l s w i th non- l i nea r p r o p e r t i e s . Any accu ra te approach r e q u i r e s d e t a i l e d knowledge of the s t r e s s - s t r a i n r e l a t i o n s h i p s , and such i n fo rma t i on i s not a v a i l a b l e fo r t imber members, e s p e c i a l l y when d e a l i n g wi th t y p i c a l p r o p e r t i e s of p o p u l a t i o n s of members r a the r than w i th i n d i v i d u a l spec imens. For s l ende r columns the Eu l e r equa t ion can be used , but fo r s h o r t e r l eng ths some form of e m p i r i c a l formula must be used . There are s e v e r a l e m p i r i c a l formulae a v a i l a b l e that can be used fo r any s l ende rness r a t i o . The best formula i s l i k e l y to be the one wi th the s i m p l e s t computa t iona l form produc ing a good f i t to expe r imen ta l r e s u l t s . F i g u r e 70 shows a compar ison of s e v e r a l c u r v e s . The s o l i d l i n e i s the column behav iour p r e d i c t e d by the computer model . The do t t ed l i n e s are i d e n t i f i e d in the key, and w i l l be d i s c u s s e d b r i e f l y h e r e . The l i n e marked CODE i s the c u r r e n t code formula (CSA 1980, NFPA 1982), which has been d i s c u s s e d e a r l i e r wi th r e f e r ence to equa t ions 8.1 to 8 .3 . 206 16.0 24.0 32.0 SLENDERNESS (L\/d) 48.0 Figure 70 \u2014 Comparison of column curves for concentric loading (non-dimensionalized) The curve marked MALHOTRA i s almost a perfect f i t to the model prediction. This curve has been proposed by Malhotra and Mazur (1970) by using tangent modulus theory based on the str e s s - s t r a i n relationship of equation 2.1. The resulting column formula is TT2E + f c ( L \/ r ) 2 2c(L\/r) 2 [TT 2E + f c ( L \/ r ) 2 ] 2 - 4TT2E f c(L\/r) 2 4c 2(L\/r) 4 (8.17) where fu i s the maximum a x i a l stress that the column can support, fc i s the failure, stress for a short column of the same material, and r i s the radius of gyration. The term c, defining the shape of the s t r e s s - s t r a i n relationship in 207 equation 2.1, i s taken as 0.90 both here and by Malhotra. The formula has a sound theoretical background but i s cumbersome to use. The l i n e marked CUBIC RANKINE i s a cubic modification to the t r a d i t i o n a l Rankine formula, proposed by Neubauer(1973). The formula i s f f = \u00a3 U i + f c ( W d ) 3 ( 8 . 1 8 ) E 4 0 where d i s the cross sectional dimension in the d i r e c t i o n that gives the largest slenderness r a t i o L\/d, and a l l other terms are as used above. The number 40 has been chosen to provide the best f i t to the model prediction, compared with a value of 50 used by Neubauer to give a good f i t to tests on small clear Douglas-fir columns. Neubauer describes the theore t i c a l development and the subsequent simplification, for use with timber, but the formula must be considered empirical because one parameter has been obtained from a c u r v e - f i t t i n g exercise. The l i n e marked PERRY-ROBERTSON i s the curve obtained from the Perry-Robertson formula which i s the basis of B r i t i s h and European codes. The formula i s _ f e + ( ^ i ) , . _ r*e + c n + i ) . \u2022 ( 8 . 1 9 ) u 2 \/ v 2 ' c e , ca where n = \u2014 Terms not previously defined are fe, the stress in the column 208 due to application of the Euler buckling load; c, the assumed i n i t i a l deviation from straightness and a, the distance\" of the extreme f i b r e on the concave side from the neutral axis. Robertson(1925) and Sunley(l955) report that p r a c t i c a l l y a l l experimental values l i e between the two curves obtained by taking n=0.00l L\/r and n=0.003 L\/r. The figure of 0.001 has been used to give the best f i t in t h i s case. Allen and Bulson (1980) describe how the Perry-Robertson formula must be regarded as an empirical formula despite the logic by which i t was derived, because of the inadequacy of several assumptions. If one of these curves is to be selected for design purposes, there i s l i t t l e between them as they a l l give a very good f i t to the strength predicted by the computer model, and several have previously been v e r i f i e d experimentally for timber columns. The one with the simplest computational form therefore becomes favoured, and this is the cubic Rankine formula. Figure 71 shows the curve resulting from the proposed formula, compared with the model prediction. The curve i s not an exact f i t , but compared with ' the present formula i t i s no less accurate, and is a l o t simpler to use. 8.4 PROPOSED DESIGN METHODS FOR ECCENTRICALLY LOADED COLUMNS This section w i l l describe several alternative design methods for e c c e n t r i c a l l y loaded timber columns, based on the experimental and a n a l y t i c a l results of e a r l i e r chapters. 209 ex o _J _J o ' cr i\u2014i x cr MODEL P R E D I C T I O N PROPOSED FORriULR i i r i i i i i O.D B.O 16.0 24.0 32.0 SLENDERNESS (L\/d) 40.0 48.0 F i g u r e 71 - Design p r o p o s a l f o r c o n c e n t r i c l o a d i n g compared with model p r e d i c t i o n 8.4.1 Type of Loading and A n a l y s i s T h i s chapter i s concerned with columns su b j e c t e d to e c c e n t r i c a x i a l loads, with no t r a n s v e r s e loads a p p l i e d to the member. Transverse loads w i l l be d i s c u s s e d i n Chapter 9. At t h i s stage, assume that the a p p l i e d a x i a l l o a d has equal e c c e n t r i c i t i e s at each end. The case of unequal end e c c e n t r i c i t i e s w i l l be i n t r o d u c e d a t ' a l a t e r stage i n t h i s c h a p t e r . The design methods presented i n t h i s chapter are f o r s i n g l e members, f o r which the a x i a l loads and end moments are known. These methods are intended f o r design without the use of a second order s t r u c t u r a l a n a l y s i s . Chapter 9 w i l l d i s c u s s 210 how the results of thi s thesis can be used when a second order analysis is avai l a b l e . 8.4.2 Input Strength Properties This section assumes that certain input information i s available for the t r i a l member which is to be designed. The information is 1. The a x i a l tension capacity Tu, of the member, which i s the product of the cross section area A, and the a x i a l tension f a i l u r e stress f t . 2. The a x i a l compression capacity Pa, of the member, which i s the product of the cross section area A, and the ax i a l compression f a i l u r e stress fc, for a short column. 3. The bending moment capacity Mu, of the member, which is the product of the section modulus S, and the modulus of rupture f r . 4. The modulus of e l a s t i c i t y , E. The tension, compression and bending strength values referred to above are a l l assumed to depend on the length and depth of the member, as described in Chapter 3 and summarized in Chapter 9. Modulus of e l a s t i c i t y i s assumed to be independent of member siz e . 8.4.3 Design Approaches This section b r i e f l y categorizes the six design methods that w i l l be described in the remainder of this chapter. Suppose that a column i s to be designed to r e s i s t a given a x i a l load P with equal end e c c e n t r i c i t i e s e. If a t r i a l member i s selected, a design method i s required to check 211 whether the member has s u f f i c i e n t s t r e n g t h to r e s i s t the a p p l i e d l o a d . The d e f l e c t e d shape of a t y p i c a l member i s shown i n f i g u r e 39. The a x i a l l o a d throughout the member i s ' P. The bending moment M at the ends of the member (and the \"unmagnified\" moment at mid-span) i s P times e. The \"magnified\" moment at midspan i s FM=P(e+A), where F i s a m a g n i f i c a t i o n f a c t o r due to the member d e f l e c t i o n A. With r e f e r e n c e to f i g u r e 72, proposed design methods 1, 2 and 3 w i l l provide an approximate method of c a l c u l a t i n g the m a g n i f i c a t i o n f a c t o r F, hence the magnified moment FM at mid-span, and w i l l compare that magnified moment with an i n t e r a c t i o n diagram (or f a i l u r e envelope).. In both methods 1 and 2, comparisons w i l l be made with the c o r r e c t i n t e r a c t i o n diagram shown by the s o l i d l i n e . - The d i f f e r e n c e between Methods 1 and 2 w i l l be d i f f e r e n t approximations to the same i n t e r a c t i o n diagram. C o n c e p t u a l l y t h i s i s - t h e same approach as used i n the s t e e l code. In method 3 the magnified moment w i l l be compared with the u l t i m a t e i n t e r a c t i o n diagram r e p r e s e n t i n g m a t e r i a l f a i l u r e , shown by the dotted curve i n f i g u r e 72. Conc e p t u a l l y t h i s i s a l e s s a c c u r a t e approach, which i s i s used f o r r e i n f o r c e d concrete design, and i s s i m i l a r to the OHBDC pro p o s a l f o r timber members. Methods 4 and 5 w i l l compare end moments, or unmagnified moments, with the corresponding i n t e r a c t i o n diagram (or f a i l u r e envelope) f o r end moments. These two methods do not 212 o o Z p u a I p M F M M u M o m e n t Figure 72 - Typical interaction diagram for a x i a l load and magnified moment involve c a l c u l a t i o n of a magnification factor, because the effe c t s of moment magnification have been considered in construction of the interaction diagram. Method 4 w i l l propose obtaining the interaction diagram from the strength model using published graphs, while method 5 w i l l propose an approximate formula. Method 6 w i l l take a d i f f e r e n t approach by proposing the use of a x i a l load vs. slenderness formulae for e c c e n t r i c a l l y loaded columns. 8.4.4 Moment Magnification Factor Methods 1, 2 and 3 w i l l require a method for cal c u l a t i n g a moment magnification factor. Most design codes use the e l a s t i c moment magnification factor given by equation 8.10. For the timber columns investigated in- this study, the actual amplification of mid-span moment at f a i l u r e i s considerably Material failure Mid - span moments at failure v. 213 more than predicted by this equation, because of non-linear figures 73(a) (b) and (c), which are ty p i c a l non-dimensionalized interaction diagrams of a x i a l load vs. moment for members with slenderness ra t i o s of 10, 20 and 40, respectively. In each case the outer s o l i d curve i s the ultimate interaction diagram for a cross section. The next curve (chain dotted) represents combinations of a x i a l load and mid-span moment causing f a i l u r e for that p a r t i c u l a r length. The inner curve ( s o l i d line) i s the combinations of a x i a l load and end moment causing f a i l u r e . The horizontal distance between these two curves represents the magnification of moment due to deformations in the member. The intermediate curve (small dots) has been obtained from the curve of end moments by amplifying the moments using equation 8.10. It i s apparent that the actual amplification at f a i l u r e i s considerably more than predicted by this equation, so that i f the other parameters in t h i s method could be quantified i t would be unsafe to use equation 8.10 for cal c u l a t i n g the moment magnifier, and a more accurate expression would be necessary. Suppose that the magnification factor i s to be increased by an empirical factor A at maximum load, with no increase at zero a x i a l load, and linear interpolation between these two extremes. The equation becomes material behaviour in compression. For example, consider F 1-P\/P 1 [l + (A-l) |-] (8.20) e e 214 Figure 73 - Interact ion diagram showing t r a d i t i o n a l moment magnifier 215 T r i a l calculations show that a suitable value for A i s 2.0, producing \"a magnification factor of 1 + P\/P 1 - P\/P e e (8.21) The previous interaction diagrams replotted using this equation are shown in figure 74. Although rather conservative for slender columns, t h i s i s considered to be a reasonable empirical approach for design purposes which w i l l be used for the rest of thi s study. 8.4.5 METHOD 1: Bi l i n e a r Interaction Diagram Figure 75(a) shows a t y p i c a l interaction diagram for ax i a l load vs. magnified moments, superimposed with straight l i n e s passing from concentric a x i a l column strength to a point on the horizontal axis at a r a t i o of B times the moment capacity. These li n e s can be used as a design approximation for the interaction diagram, terminating at the v e r t i c a l l i n e through the moment capacity, which cuts off the \"nose\". This approximation i s conservative as shown by the shading, but is quite simple. The resulting design equation, providing a s t a b i l i t y check for any length of column, i s C F M \u2014 < B(l - !-) , C F M < M (8.22) M v P ' m u u u where F i s the magnification factor from equation 8.21, Cm is the factor for unequal end e c c e n t r i c i t i e s from equation 8.12 216 F i g u r e 74 I n t e r a c t i o n diagram showing proposed moment mag n i f i e r 217 O HRGNIFIED HOHENT (b) W e a k 25 ^ 30 *\"*\" 1 ______ 40 *~ \" - \u2014 \u2014 i *~ \u2022 . y ~~ \u2014\u2014\u2014_~\u2014 \u2014 50 \u2014 1 1 1 1 r t i i i i i -1 1 1 1 1 1 1 I r\u2014 o.o ' 0.2 a.\" o.c 0.1) 1.0 HRGNIFIED HOHENT (c) S t r o n g HRGNIFIED HOHENT Figure 75 for - B i l i n e a r approximation to interaction diagram magnified moments (non-dimensionalized) 218 and B i s the h o r i z o n t a l a x i s i n t e r c e p t , which depends on the shape of the u l t i m a t e i n t e r a c t i o n diagram. I t can be shown that a s u i t a b l e s e m i - e m p i r i c a l value f o r B can be given by 1.35 = F7f~ (8.23) t C where f t \/ f c i s the r a t i o of a x i a l t e n s i o n s t r e n g t h to a x i a l compression s t r e n g t h f o r the m a t e r i a l . The d e r i v a t i o n of t h i s e x p r e s s i o n i s given i n appendix B. F i g u r e s 75(b) and (c) show t h i s approximation f o r weak and strong m a t e r i a l , and i t i s seen to be reasonable i n most cases. For s i t u a t i o n s where the two end moments are d i f f e r e n t and the l o a d c a p a c i t y i s governed by the s t r e n g t h of the c r o s s s e c t i o n at one end of the member, a s t r e n g t h check must be made, as s p e c i f i e d i n the s t e e l code, using the approximation f o r the outer curve i n f i g u r e 75 TT <