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The effect of prior flexural prestrain on the stability of structural steel columns
The effect of prior flexural prestrain on the stability of structural steel columns Milija N. Pavlovid Department of Civil Engineering, Imperial College, London SW7 2AZ, UK L e o n a r d K. Stevens
Department of Civil Engineering, University of Melbourne, Parkville 3052, Victoria, Australia (Received August 1980J
Prior large bending deformations may cause a significant reduction in the buckling stress of a mild steel strut subjected to direct compression. Although the work arose in investigating the properties of mangled bars, that is bars which have been deliberately deformed so as to reduce initial geometric imperfections, the conclusions are equally applicable to other instances of plastic strain reversal, such as those which might arise due to earthquake or shock waves. Particular attention is drawn to the possibly potentially dangerous situation which may arise when a severely deformed element is straightened and then subjected to loading conditions which would be regarded as safe for a previously undeformed element. Buckling stresses have been computed from material data for the prestrained specimens and these values are comoared with actual test results.
Introduction The effect of initial stresses and strains on the stability of struts has been investigated by many authors. Salmon 1 reported that, as early as 1888, it was recognized that 'previous strains' influenced the strength of columns. These 'previous strains' may arise in a number of ways. Huber and Beedle 2 showed that residual stresses induced during cooling of hot rolled sections reduce their strength in compression. Welding also causes residual stresses and allowance is normally made for this effect when assessing column strength.a, 4 The effect of strain reversal in a member which has yielded in tension before collapsing in compression was studied by Paris s who showed that a large reduction in buckling strength (sometimes greater than 50%) could result. Initial geometric crookedness also reduces the load carrying capacity of a column. This effect is marked for short and medium-length struts, 6 i.e. which, if straight, would buckle at stresses near yield (if the latter is well defmed) or near the proportional limit (if the stress-strain curve for the material is smooth). 7 In an attempt to improve the buckling strength of struts cold straightening is sometimes used; this process reduces the initial out-of-flatness of the column, but, at the same time, it introduces residual stresses which may or may not be favourable.
66 Eng. Struct., 1981, Vol. 3, April
By modifying a computer program developed by Batterman and Johnston, 7 Frey 8 studied the effect of straightening by cold bending of hot rolled double-tee sections; his method requires considerable computational effort arising from the geometry of the cross-section, the residual stress pattern due to combined hot rolling and cold bending, and the asymmetric effect of a single straightening operation. Frey found that, as far as column strength is concerned, the residual stress pattern due to cooling of the hot rolled sections was less favourable than that for the combined residual stresses due to hot rolling and cold straightening. Hence he concluded that the design curves for hot rolled sections which had been subjected to subsequent cold straightening should lie above the existing curves for elements for which the straightening operation had not been performed, both sets of curves having been derived on the assumption that the f'mal, unavoidable out-of-flatness was the same in each case. Clearly, emphasis is here placed on the effect of residual stresses on column strength, rather than initial imperfections; however, cold straightening also serves the useful purpose of reducing the latter. Another instance of cold straightening is the process of 'mangling' by which members are flexurally deformed as they pass through a set of rollers. This paper aims at obtaining an order-of-magnitude estimate of the reduction in buckling strength of initially stress-free mild steel struts 0141-0296/81/020066-05/$02.00 © 1981 IPC Business Press
Stability of structural steel columns: M. N. Pavlovid and L. K. Stevens
due to this type of imposed prestrain. Theoretical predictions can be achieved relatively simply since mangling involves a number of strain reversals which cause the properties of the rectangular cross-section used to be reasonably symmetric about the centroidal axis of the column. Tangent modulus properties can then be used in the tangent modulus column buckling formula. 9 The paper also contains a note on the effect of a single cycle of bending and straightening involving very large prestrains; here too, the investigation is contained to ordinary mild steel struts having a rectangular cross-section.
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Figure 2 Bar cross-section for elastic-perfectly plastic material subjected to differing degrees of mangling. (a) strain distribution; (b) elastic stress distribution; (c) elastic core, plastic fibres; (d) fully plastic stress distribution
R e p e a t e d p r i o r f l e x u r a l strain reversal
The mangling operation The process of 'mangling', where bars or plates are flexurally deformed as they pass through rollers, has as its main objective the attainment of a 'straight' specimen. (Originally, the aim was also the improvement of ductility, but with the advent of modern structural steels this is now, outdated.) The mangling device used consisted of two sets of rollers which do not overlap in plan. The lower set was fixed in position while the other was adjustable so that it could move downwards and overlap with the fixed set when viewed in elevation (Figure 1). In order to simulate practical conditions the whole process was entrusted to the skill of an experienced operator, who was asked to reproduce, under ordinary fabricating shop conditions, mangled specimens which would encompass the range of prestrains imposed in practice. For this purpose four structural steel bars, each 6.1 m long with 1.91 cm x 1.27 cm rectangular cros-section were used. Bar 1 was not mangled and hence provided the control or reference specimen. Increasing degrees of mangling were imposed on specimens 2 - 'light' mangling, 3 - 'intermediate' mangling, and 4 - 'heavy' mangling (an estimate of the maximum extreme fibre strain is made in the section describing the effect of a single cycle of abnormal prior flexural deformation). By adjusting the relative position of the two sets of rollers, the mangled bars were bent and then straightened; this process being repeated for several cycles until the operator judged the bars to be 'straight'. After the operation was completed, each of the four bars was cut as follows: (i) A length of 30 cm was rejected from either end since the mangling on these sections could have been different from that on the rest of the bar due to entrance and exit conditions. (ii) Specimens were cut from the mid-section of the bars (as this was deemed to be the most representative of the average state along the bar) and these were used for tensile coupon tests. The specimens were 50 cm long and a 5.08 cm gauge length was used to determine the stress-strain relationship. Adjustable set
Figure I
Elevation view of milers in mangling device
(iii) Six lengths were chosen in order to cover a wide range of 1/r values (40 -~ 200). Two specimens were cut for each of these lengths.
Theoretical estimate of the buckling load During the mangling process the bars are bent about their weak axis. If one assumes plane sections to remain plane, a linear variation of strain is developed across the cross-section (Figure 2a). When the amount of bending is small the stresses and strains will not exceed their yield values so that the distribution of stress will also be linear (Figure 2b) and will disappear on removal from the rollers. With an increasing degree of mangling the outer fibres yield, and this yielding then spreads towards the neutral axis; the stress distribution for this case is shown in Figure 2c (the stress-strain curve is assumed to be elastic-perfectly plastic). In the limit, very heavy mangling might cause the full yield condition to be approached (Figure 2d), although such a severe degree of mangling would rarely be imposed in practice. From the above, it is clear that, after mangling, the bar cross-section consists of an elastic core sandwiched between fibres possessing nonlinear stress-strain characteristics for which the Bauschinger effect 1°, n plays an important role, and is subjected to a complex set of residual stresses. In addition, the final stress state of the cross-section is further complicated by the presence of strain hardening which the simplified elastic-perfectly plastic model ignores, as well as the effect of elastic springback. The maximum curvature, and hence the strain history of any given fibre, can be worked out by simple geometry once the diameter of the rollers and their spacing and relative vertical positions are prescribed; the Bauschinger effect for that fibre may then be obtained by subjecting a virgin specimen to tension and compression following the same strain history. Allowance for the presence of residual stresses must also be included in such an analysis. On the other hand, an order-of-magnitude estimate for the overall effect on the section of the Bauschinger phenomenon together with the residual stresses can be obtained more directly by considering the average tensile stress-strain properties of the mangled cross-section (the tensile and compressive characteristics can be taken to be identical since mangling involves a number of strain reversals); for this purpose, tensile tests were performed on the representative 50 cm coupon test piece (gauge length = 5.08 cm) cut from each bar. These tensile tests were carried out in a 10-ton Amsler machine with an extensometer mounted on the test piece; these results appear in Figure 3, showing how the increasing degree of mangling causes a lowering of the yield point (proportional limit) as well as a change in the ductility of the average properties of the section.
Eng. Struct., 1981, Vol. 3, April 67
Stability o f structural steel columns: M. N. Pavlovi# and L. K. Stevens 300
0~ = 255N/ram 2
200 t~
E E
z ff
/ 100,
/ /
0
Type of prestraln 1 2 3 4
un-mangled 'light' mangled 'intermediate' mangling ' heavy' mangling
I 0 0005
I I Q 001 0 0015 0002 E Figure 3 Stress-strain characteristics in tension f o r specimens with varying degree o f mangling
The theoretical column curve for each of the four states of prestrain was obtained by using the relevant average characteristic in tension. The tangent modulus theory was used for this purpose as it provides a lower bound for the buckling load ]2,13, 9, 7 which is in good agreement with most test data 6 (Paris s obtained results closer to the reduced or double modulus theory but also concluded that scatter of results made this an unreliable criterion). The tangent modulus buckling curve is given by:
7r2Et o~ -
(1/0 2
(1)
where E t is the slope of the stress-strain curve for the material (assumed to be the same in tension as in compression), i.e.:
parative study between the relative degrees of mangling; the results appear in Figure 4. For the lower buckling stresses obtained for longer columns (l/r > 120), the value of E t was not affected by the degree of mangling; as expected, the experimental points are in good agreement with the theoretical curve. The well-known imperfection sensitivity in the intermediate 1/r range caused loads to be well below their predicted values. At lower slenderness ratios (1/r < 60) good correlation was obtained between predicted and observed loads. From Figure 4 it is clear that the relative loss in strength due to mangling is approximately the same irrespective of whether the theoretical plots or the experimental points are compared. An exception is the point corresponding to the un-mangled specimen of 1/r ~ 100 which lies considerably below the two points corresponding to light and intermediate mangling. Both un-mangled specimens corresponding to this 1/r had initial imperfections higher than those of the other struts of the same length; this clearly illustrates how the loss of strength due to mangling may be more than compensated by the reduction of column crookedness which the mangling operation aims at achieving, particularly in the sensitive medium 1/r range. It can be seen from Figure 4 that short and intermediate columns exhibit a drop of about 5% and 10% due to 'light' mangling and 10% and 20% due to 'heavy' mangling respectively. Finally, it should be pointed out that similar effects can be expected in the case of mangled plates, although in this instance transverse effects are also introduced which may be favourable in increasing the apparent yield point. As only a small number of tests were carried out on plates the limited data available does not, at this stage, permit more detailed conclusions of the type presented by previous investigators.14,15
do
Et = - -
de
(2)
These theoretical plots are shown in Figure 4.
Column tests The compression tests were carried out using a 10-ton Amsler machine. Pin-ended conditions were produced by means of standard hardened knife edges. After careful alignment and adjustment of the knife edges to reduce initial lateral displacements, the struts were failed in compression and the buckling load recorded. Readings of midpoint deflexions were taken regularly during the loadings; from these a Southwell plot 9 was constructed which permitted an estimate of the initial strut crookedness to be made. If the sole effect of mangling on the load carrying capacity of struts is to be investigated, all other factors which might have some influence on the latter should be eliminated, or at least minimized; the most relevant of these are eccentricity of loading and column crookedness, since they can cause serious reductions in buckling strength in the intermediate and low 1/r range. Careful alignment of the specimens in the machine minimized the effect of load eccentricity. Column crookedness could not be eliminated but an estimate of it was obtained from the SouthweU plot. The latter enabled a selection of struts of comparable initial small crookedness to be made for every 1/r used, and these served as a basis for the cam-
68
Eng. Struet., 1981, Vol. 3, April
Single cycle o f a b n o r m a l prior flexural d e f o r m a t i o n Bars from a batch similar to that used in the study of the effect of mangling were subjected to severe flexural deformation by being bent in a press between two curved blocks of timber to a radius of 0.3 m, as shown in Figure 5. The deformed bars were first straightened between flat plates; the residual curvature was then removed by applying additional reverse curvature over short lengths, gradually approaching the desired degree of straightness (comparable to that of the mangled bars) by alternate bending in opposing directions.
300 ~v = 255 N/mm 2
g. E E 200 ~, v ~ leo
O
Figure 4 struts
~
20
4;
n
60
Type of prestraln Theory Experiment Un-mangled 1 o 'Light' mangling 2 x 'Intermediate' mangling 3 'Heavy' mangling 4 u
1
80
Eu/er
100
150
-L_ 140
160_
180
200
l/r Buckling stress vs. slenderness ratio for mangled mild steel
Stability of structural steel columns: M. N. Pavlovi# and L. K. Stevens R=O3m
Figure 5
Bar subjected t o severe flexural deformation
/ 0"y
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C
£st
£v
a
(7
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b Figure 6 Typical stress-strain characteristics for mild steel. (a) dynamic nature o f the yield phenomenon f o r a virgin specimen; (b) Bauschinger effect and continuous yielding for a previously yielded specimen
Clearly, the specimens under investigation differ from the mangled bars in the following two respects. Firstly, the maximum fibre strain for the specimen depicted in Figure 5 exceeded that corresponding to the most heavily mangled bars. Now, it is important to recognize the dynamic nature of the yielding process in mild steel, in which the actual strain at a point on a fibre (as against the average strain ear along that fibre) is either the yield strain ey (=or/E) or the much larger strain hardening strain est (Figure 6a); hence, the term 'average strain at yield' simply implies the proportion of the length of the fibre strained by an amount est as against that strained by ey. With this in mind, it can be concluded that, in the case of abnormal prior flexuraI deformation, where the extreme fibre strain had an average value ear ~ 15-20 er, strain hardening had almost certainly taken place along the whole length of that fibre, i.e. ear ~ est. On the other hand, even in the case of heavy mangling the extreme fibre strain had a maximum value ear estimated to be between 0.25 est and 0.5 est, it probably being closer to the former value. It
may also be worth noting at this point that after the plastic range has been passed through once, either in tension or compression, a continuous stress-strain relationship is achieved without the dynamic jump encountered in the first pass; hence the change in ductility mentioned earlier. 16 This, of course, is in addition to the Bauschinger effect where further straining in the same direction raises the elastic limit, while reverse straining lowers it. Figure 6b illustrates these effects. The second difference between the mangled specimens and those subjected to the abnormally large prestrain is that in the latter case the straightening process did not involve applying a full reversal to take the extreme fibre strain to the equal and opposite strain value before flattening to the straight condition (also, the process only involved a single major cycle with some smaller alternating bending in opposite directions as required to achieve the desired straightness). However, since opposite sides have followed opposing cycles, a tension test should still provide a suitable average estimate for use as the E t value in the tangent modulus formula, even though the straightening procedure involved more uncertainties as to the final symmetry of crosssectional properties than was the case with the mangled specimens. The tensile stress-strain curve for the abnormally deformed specimen appears in Figure 7. It shows a deviation from linearity at comparatively low stresses (~ 100 N/ mm 2) and a marked reduction for stresses greater than 140 N/mm 2. Axially loaded strut tests were then carried out as before for a range of slenderness ratios from 40 to 180 with the results shown in Figure 8. This demonstrates reasonable agreement with the values predicted by the tangent modulus approach although some results may have been adversely affected by unavoidable small imperfections which had not been removed in the straightening process. However, in this instance emphasis is not being given to the prediction of the buckling load by the tangent modulus formula although, again, the correlation between predicted and observed values is in fact quite acceptable as an approximation, and the divergence at the higher stress levels - lower 1/r - is consistent with this approximation. Instead, the important feature here is the very severe reduction produced in the buckling load for the intermediate and low slenderness ratios. At 1/r = 80 the load capacity is only about 65% of that obtained for a previously underformed column and 300
or = 2 6 0 N/ram 2
200
E z
b" IO0
I 0.001
I 0002
I I I 0,003 0.004 0005 £ Figure 7 Stress-strain characteristic in tension for the specimen which has been subjected to abnormal prior flexural deformation
Eng. Struct., 1981, VoL 3, April 69
Stability o f structural steel columns: M. N. Pavlovid and L. K. Stevens
300
4" E E
0v = 2 6 O N / r u m 2 •
1
Type of prestrain None Severe
Theory Experiment 1 2
writers are also grateful to the reviewers fbr their suggestions.
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References
o
200
l 2
o
3 loo 4 0
I
I
I
I
I
I
I
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I
20
40
60
80
100
120
140
160
180
200
5
I/r Figure 8 which
Buckling stress vs. slenderness ratio for mild steel struts to abnormal prior flexural d e f o r m a t i o n
6
have been subjected
7 this marked effect extends well into the low 1/r values of about 40, an effect not previously noted with the lighter flexural prestrain developed by mangling. The buckling load is certainly improved above that if the member had been tested in the initially fully deformed state but it is far below that expected for a bar which had not been treated in this way.
8 9 10 11
Conclusions It seems that the effect o f mangling on the buckling load o f struts can be classified into two broad types: 'light' and 'heavy' mangling. The former causes a loss o f strength o f about 5% (probably less than the effect of the imperfections it removes) while the latter may cause a reduction of up to 20%; the effects being confined to the intermediate and low ranges o f slenderness ratios. A reasonable estimate o f the effect o f flexural prestrains on strut stability can be obtained by using tangent moduli curves derived from standard tension tests for the average properties o f the cross-section. Abnormal prior flexural deformation, possibly associated with accidental damage followed by straightening, can produce serious reductions in buckling load capacity (of the order of 35%). The extent o f this reduction could be a potential threat to the safety o f a system, being of the order of the allowance o f the overload factor. The effect is made more serious by the difficulty in determining if an element has been damaged and then straightened; it would appear to be prudent to exclude straightening o f severely damaged compressive elements as a permissible fabrication procedure without a thorough analysis of the particular circumstance. The above conclusions on the stability of columns do not take into account possible effects o f ageing after cold straightening. This could be the subject o f further research.
12 13 14 15 16
Notation The following symbols are used in this paper: e o E Et 1/r
R
Acknowledgements
b
The authors wish to thank Professor B. B. Hundy for his interest and valuable advice on the mangling operation. The
st
Eng. Struct., 1981, Vol. 3, April
engineering strain engineering stress Young's modulus tangent modulus slenderness ratio, where 1 = effective length of strut = actual length when b o t h strut ends are pinned; r = radius of gyration of strut cross-section radius o f curvature
Subscripts: av
70
Salmon, E. H. 'Columns', Oxford Tech. Pub., 1921 Huber, A. W. and Beedle, L. S. 'Residual stress and compressive strength of steel', Weld. J. 1954, 33 (no. 12), 589 Johnston, B. G. (Ed), 'Column Research Council guide to design criteria for metal compression members', Wiley, New York, 1966 Tall, L. 'Recent developments in the study of column behaviour', J. Inst. Engrs. Australia 1964, 36 (no. 12), 319 Paris, P. C. 'The Bauschmger effect on columns', J. Appl. Mech. 1956, 23,479 Bleich, F. 'Buckling strength of metal structures', McGrawHill, New York, 1952 Batterman, R. H. and Johnston, B. G. 'Behaviour and maximum strength of metal columns', J. Struct. Div. ASCE 1967, 93 (ST2), 205 Frey, F. 'Effet du dressage/~ froid des profil6s lamin6s en double t6 sur leur force portante', I A B S E 1969, 29 (II), 101 Hoff, N. J. 'The analysis of structures', Wiley, New York, 1956 Bauschinger, W. 'Gerichtc Verfestigung', Civilingenieur 1881, 27,299 Bauschinger, W. '0ber die Ver~inderung der Elasticit~itsgrenze und des Festigkeit des Eisens und Stahls durch Strecken und Quetschen, durch Erw~irmung und Abkiihlen und durch oftmal wiederholte Beanspruchung', Mitt. mech. tech. Lab. Miinch. 1866, 13, 1 Shanley, F. R. 'The column paradox', J. A ero. ScL 1946, 13 (no. 12), 678 Shanley, F. R. 'Inelastic column theory', J. Aero. Sci. 1947; 14 (no. 5), 261 (see also Discussion by Th. Von K~rman, 267) Pascoe, K. J. 'Strength of cold-formed cylindrical steel plates', J. Strain Anal. 1971,6 (no. 3), 167 Pascoe, K. J. 'Directional effects of prestrain in steel', J. Strain Anal 1971,6 (no. 3), 181 Jevons, I. D. 'The metallurgy of deep drawing and pressing', Chapman & Hall, London, 1949
Y
average buckling strain hardening yield
Department of Civil Engineering, University of Melbourne, Parkville 3052, Victoria, Australia (Received August 1980J
Prior large bending deformations may cause a significant reduction in the buckling stress of a mild steel strut subjected to direct compression. Although the work arose in investigating the properties of mangled bars, that is bars which have been deliberately deformed so as to reduce initial geometric imperfections, the conclusions are equally applicable to other instances of plastic strain reversal, such as those which might arise due to earthquake or shock waves. Particular attention is drawn to the possibly potentially dangerous situation which may arise when a severely deformed element is straightened and then subjected to loading conditions which would be regarded as safe for a previously undeformed element. Buckling stresses have been computed from material data for the prestrained specimens and these values are comoared with actual test results.
Introduction The effect of initial stresses and strains on the stability of struts has been investigated by many authors. Salmon 1 reported that, as early as 1888, it was recognized that 'previous strains' influenced the strength of columns. These 'previous strains' may arise in a number of ways. Huber and Beedle 2 showed that residual stresses induced during cooling of hot rolled sections reduce their strength in compression. Welding also causes residual stresses and allowance is normally made for this effect when assessing column strength.a, 4 The effect of strain reversal in a member which has yielded in tension before collapsing in compression was studied by Paris s who showed that a large reduction in buckling strength (sometimes greater than 50%) could result. Initial geometric crookedness also reduces the load carrying capacity of a column. This effect is marked for short and medium-length struts, 6 i.e. which, if straight, would buckle at stresses near yield (if the latter is well defmed) or near the proportional limit (if the stress-strain curve for the material is smooth). 7 In an attempt to improve the buckling strength of struts cold straightening is sometimes used; this process reduces the initial out-of-flatness of the column, but, at the same time, it introduces residual stresses which may or may not be favourable.
66 Eng. Struct., 1981, Vol. 3, April
By modifying a computer program developed by Batterman and Johnston, 7 Frey 8 studied the effect of straightening by cold bending of hot rolled double-tee sections; his method requires considerable computational effort arising from the geometry of the cross-section, the residual stress pattern due to combined hot rolling and cold bending, and the asymmetric effect of a single straightening operation. Frey found that, as far as column strength is concerned, the residual stress pattern due to cooling of the hot rolled sections was less favourable than that for the combined residual stresses due to hot rolling and cold straightening. Hence he concluded that the design curves for hot rolled sections which had been subjected to subsequent cold straightening should lie above the existing curves for elements for which the straightening operation had not been performed, both sets of curves having been derived on the assumption that the f'mal, unavoidable out-of-flatness was the same in each case. Clearly, emphasis is here placed on the effect of residual stresses on column strength, rather than initial imperfections; however, cold straightening also serves the useful purpose of reducing the latter. Another instance of cold straightening is the process of 'mangling' by which members are flexurally deformed as they pass through a set of rollers. This paper aims at obtaining an order-of-magnitude estimate of the reduction in buckling strength of initially stress-free mild steel struts 0141-0296/81/020066-05/$02.00 © 1981 IPC Business Press
Stability of structural steel columns: M. N. Pavlovid and L. K. Stevens
due to this type of imposed prestrain. Theoretical predictions can be achieved relatively simply since mangling involves a number of strain reversals which cause the properties of the rectangular cross-section used to be reasonably symmetric about the centroidal axis of the column. Tangent modulus properties can then be used in the tangent modulus column buckling formula. 9 The paper also contains a note on the effect of a single cycle of bending and straightening involving very large prestrains; here too, the investigation is contained to ordinary mild steel struts having a rectangular cross-section.
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E
E
a
o=EE
b
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0v
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c
o't,
d
Figure 2 Bar cross-section for elastic-perfectly plastic material subjected to differing degrees of mangling. (a) strain distribution; (b) elastic stress distribution; (c) elastic core, plastic fibres; (d) fully plastic stress distribution
R e p e a t e d p r i o r f l e x u r a l strain reversal
The mangling operation The process of 'mangling', where bars or plates are flexurally deformed as they pass through rollers, has as its main objective the attainment of a 'straight' specimen. (Originally, the aim was also the improvement of ductility, but with the advent of modern structural steels this is now, outdated.) The mangling device used consisted of two sets of rollers which do not overlap in plan. The lower set was fixed in position while the other was adjustable so that it could move downwards and overlap with the fixed set when viewed in elevation (Figure 1). In order to simulate practical conditions the whole process was entrusted to the skill of an experienced operator, who was asked to reproduce, under ordinary fabricating shop conditions, mangled specimens which would encompass the range of prestrains imposed in practice. For this purpose four structural steel bars, each 6.1 m long with 1.91 cm x 1.27 cm rectangular cros-section were used. Bar 1 was not mangled and hence provided the control or reference specimen. Increasing degrees of mangling were imposed on specimens 2 - 'light' mangling, 3 - 'intermediate' mangling, and 4 - 'heavy' mangling (an estimate of the maximum extreme fibre strain is made in the section describing the effect of a single cycle of abnormal prior flexural deformation). By adjusting the relative position of the two sets of rollers, the mangled bars were bent and then straightened; this process being repeated for several cycles until the operator judged the bars to be 'straight'. After the operation was completed, each of the four bars was cut as follows: (i) A length of 30 cm was rejected from either end since the mangling on these sections could have been different from that on the rest of the bar due to entrance and exit conditions. (ii) Specimens were cut from the mid-section of the bars (as this was deemed to be the most representative of the average state along the bar) and these were used for tensile coupon tests. The specimens were 50 cm long and a 5.08 cm gauge length was used to determine the stress-strain relationship. Adjustable set
Figure I
Elevation view of milers in mangling device
(iii) Six lengths were chosen in order to cover a wide range of 1/r values (40 -~ 200). Two specimens were cut for each of these lengths.
Theoretical estimate of the buckling load During the mangling process the bars are bent about their weak axis. If one assumes plane sections to remain plane, a linear variation of strain is developed across the cross-section (Figure 2a). When the amount of bending is small the stresses and strains will not exceed their yield values so that the distribution of stress will also be linear (Figure 2b) and will disappear on removal from the rollers. With an increasing degree of mangling the outer fibres yield, and this yielding then spreads towards the neutral axis; the stress distribution for this case is shown in Figure 2c (the stress-strain curve is assumed to be elastic-perfectly plastic). In the limit, very heavy mangling might cause the full yield condition to be approached (Figure 2d), although such a severe degree of mangling would rarely be imposed in practice. From the above, it is clear that, after mangling, the bar cross-section consists of an elastic core sandwiched between fibres possessing nonlinear stress-strain characteristics for which the Bauschinger effect 1°, n plays an important role, and is subjected to a complex set of residual stresses. In addition, the final stress state of the cross-section is further complicated by the presence of strain hardening which the simplified elastic-perfectly plastic model ignores, as well as the effect of elastic springback. The maximum curvature, and hence the strain history of any given fibre, can be worked out by simple geometry once the diameter of the rollers and their spacing and relative vertical positions are prescribed; the Bauschinger effect for that fibre may then be obtained by subjecting a virgin specimen to tension and compression following the same strain history. Allowance for the presence of residual stresses must also be included in such an analysis. On the other hand, an order-of-magnitude estimate for the overall effect on the section of the Bauschinger phenomenon together with the residual stresses can be obtained more directly by considering the average tensile stress-strain properties of the mangled cross-section (the tensile and compressive characteristics can be taken to be identical since mangling involves a number of strain reversals); for this purpose, tensile tests were performed on the representative 50 cm coupon test piece (gauge length = 5.08 cm) cut from each bar. These tensile tests were carried out in a 10-ton Amsler machine with an extensometer mounted on the test piece; these results appear in Figure 3, showing how the increasing degree of mangling causes a lowering of the yield point (proportional limit) as well as a change in the ductility of the average properties of the section.
Eng. Struct., 1981, Vol. 3, April 67
Stability o f structural steel columns: M. N. Pavlovi# and L. K. Stevens 300
0~ = 255N/ram 2
200 t~
E E
z ff
/ 100,
/ /
0
Type of prestraln 1 2 3 4
un-mangled 'light' mangled 'intermediate' mangling ' heavy' mangling
I 0 0005
I I Q 001 0 0015 0002 E Figure 3 Stress-strain characteristics in tension f o r specimens with varying degree o f mangling
The theoretical column curve for each of the four states of prestrain was obtained by using the relevant average characteristic in tension. The tangent modulus theory was used for this purpose as it provides a lower bound for the buckling load ]2,13, 9, 7 which is in good agreement with most test data 6 (Paris s obtained results closer to the reduced or double modulus theory but also concluded that scatter of results made this an unreliable criterion). The tangent modulus buckling curve is given by:
7r2Et o~ -
(1/0 2
(1)
where E t is the slope of the stress-strain curve for the material (assumed to be the same in tension as in compression), i.e.:
parative study between the relative degrees of mangling; the results appear in Figure 4. For the lower buckling stresses obtained for longer columns (l/r > 120), the value of E t was not affected by the degree of mangling; as expected, the experimental points are in good agreement with the theoretical curve. The well-known imperfection sensitivity in the intermediate 1/r range caused loads to be well below their predicted values. At lower slenderness ratios (1/r < 60) good correlation was obtained between predicted and observed loads. From Figure 4 it is clear that the relative loss in strength due to mangling is approximately the same irrespective of whether the theoretical plots or the experimental points are compared. An exception is the point corresponding to the un-mangled specimen of 1/r ~ 100 which lies considerably below the two points corresponding to light and intermediate mangling. Both un-mangled specimens corresponding to this 1/r had initial imperfections higher than those of the other struts of the same length; this clearly illustrates how the loss of strength due to mangling may be more than compensated by the reduction of column crookedness which the mangling operation aims at achieving, particularly in the sensitive medium 1/r range. It can be seen from Figure 4 that short and intermediate columns exhibit a drop of about 5% and 10% due to 'light' mangling and 10% and 20% due to 'heavy' mangling respectively. Finally, it should be pointed out that similar effects can be expected in the case of mangled plates, although in this instance transverse effects are also introduced which may be favourable in increasing the apparent yield point. As only a small number of tests were carried out on plates the limited data available does not, at this stage, permit more detailed conclusions of the type presented by previous investigators.14,15
do
Et = - -
de
(2)
These theoretical plots are shown in Figure 4.
Column tests The compression tests were carried out using a 10-ton Amsler machine. Pin-ended conditions were produced by means of standard hardened knife edges. After careful alignment and adjustment of the knife edges to reduce initial lateral displacements, the struts were failed in compression and the buckling load recorded. Readings of midpoint deflexions were taken regularly during the loadings; from these a Southwell plot 9 was constructed which permitted an estimate of the initial strut crookedness to be made. If the sole effect of mangling on the load carrying capacity of struts is to be investigated, all other factors which might have some influence on the latter should be eliminated, or at least minimized; the most relevant of these are eccentricity of loading and column crookedness, since they can cause serious reductions in buckling strength in the intermediate and low 1/r range. Careful alignment of the specimens in the machine minimized the effect of load eccentricity. Column crookedness could not be eliminated but an estimate of it was obtained from the SouthweU plot. The latter enabled a selection of struts of comparable initial small crookedness to be made for every 1/r used, and these served as a basis for the cam-
68
Eng. Struet., 1981, Vol. 3, April
Single cycle o f a b n o r m a l prior flexural d e f o r m a t i o n Bars from a batch similar to that used in the study of the effect of mangling were subjected to severe flexural deformation by being bent in a press between two curved blocks of timber to a radius of 0.3 m, as shown in Figure 5. The deformed bars were first straightened between flat plates; the residual curvature was then removed by applying additional reverse curvature over short lengths, gradually approaching the desired degree of straightness (comparable to that of the mangled bars) by alternate bending in opposing directions.
300 ~v = 255 N/mm 2
g. E E 200 ~, v ~ leo
O
Figure 4 struts
~
20
4;
n
60
Type of prestraln Theory Experiment Un-mangled 1 o 'Light' mangling 2 x 'Intermediate' mangling 3 'Heavy' mangling 4 u
1
80
Eu/er
100
150
-L_ 140
160_
180
200
l/r Buckling stress vs. slenderness ratio for mangled mild steel
Stability of structural steel columns: M. N. Pavlovi# and L. K. Stevens R=O3m
Figure 5
Bar subjected t o severe flexural deformation
/ 0"y
°
I'"
I
C
£st
£v
a
(7
£
b Figure 6 Typical stress-strain characteristics for mild steel. (a) dynamic nature o f the yield phenomenon f o r a virgin specimen; (b) Bauschinger effect and continuous yielding for a previously yielded specimen
Clearly, the specimens under investigation differ from the mangled bars in the following two respects. Firstly, the maximum fibre strain for the specimen depicted in Figure 5 exceeded that corresponding to the most heavily mangled bars. Now, it is important to recognize the dynamic nature of the yielding process in mild steel, in which the actual strain at a point on a fibre (as against the average strain ear along that fibre) is either the yield strain ey (=or/E) or the much larger strain hardening strain est (Figure 6a); hence, the term 'average strain at yield' simply implies the proportion of the length of the fibre strained by an amount est as against that strained by ey. With this in mind, it can be concluded that, in the case of abnormal prior flexuraI deformation, where the extreme fibre strain had an average value ear ~ 15-20 er, strain hardening had almost certainly taken place along the whole length of that fibre, i.e. ear ~ est. On the other hand, even in the case of heavy mangling the extreme fibre strain had a maximum value ear estimated to be between 0.25 est and 0.5 est, it probably being closer to the former value. It
may also be worth noting at this point that after the plastic range has been passed through once, either in tension or compression, a continuous stress-strain relationship is achieved without the dynamic jump encountered in the first pass; hence the change in ductility mentioned earlier. 16 This, of course, is in addition to the Bauschinger effect where further straining in the same direction raises the elastic limit, while reverse straining lowers it. Figure 6b illustrates these effects. The second difference between the mangled specimens and those subjected to the abnormally large prestrain is that in the latter case the straightening process did not involve applying a full reversal to take the extreme fibre strain to the equal and opposite strain value before flattening to the straight condition (also, the process only involved a single major cycle with some smaller alternating bending in opposite directions as required to achieve the desired straightness). However, since opposite sides have followed opposing cycles, a tension test should still provide a suitable average estimate for use as the E t value in the tangent modulus formula, even though the straightening procedure involved more uncertainties as to the final symmetry of crosssectional properties than was the case with the mangled specimens. The tensile stress-strain curve for the abnormally deformed specimen appears in Figure 7. It shows a deviation from linearity at comparatively low stresses (~ 100 N/ mm 2) and a marked reduction for stresses greater than 140 N/mm 2. Axially loaded strut tests were then carried out as before for a range of slenderness ratios from 40 to 180 with the results shown in Figure 8. This demonstrates reasonable agreement with the values predicted by the tangent modulus approach although some results may have been adversely affected by unavoidable small imperfections which had not been removed in the straightening process. However, in this instance emphasis is not being given to the prediction of the buckling load by the tangent modulus formula although, again, the correlation between predicted and observed values is in fact quite acceptable as an approximation, and the divergence at the higher stress levels - lower 1/r - is consistent with this approximation. Instead, the important feature here is the very severe reduction produced in the buckling load for the intermediate and low slenderness ratios. At 1/r = 80 the load capacity is only about 65% of that obtained for a previously underformed column and 300
or = 2 6 0 N/ram 2
200
E z
b" IO0
I 0.001
I 0002
I I I 0,003 0.004 0005 £ Figure 7 Stress-strain characteristic in tension for the specimen which has been subjected to abnormal prior flexural deformation
Eng. Struct., 1981, VoL 3, April 69
Stability o f structural steel columns: M. N. Pavlovid and L. K. Stevens
300
4" E E
0v = 2 6 O N / r u m 2 •
1
Type of prestrain None Severe
Theory Experiment 1 2
writers are also grateful to the reviewers fbr their suggestions.
o ×
References
o
200
l 2
o
3 loo 4 0
I
I
I
I
I
I
I
I
I
20
40
60
80
100
120
140
160
180
200
5
I/r Figure 8 which
Buckling stress vs. slenderness ratio for mild steel struts to abnormal prior flexural d e f o r m a t i o n
6
have been subjected
7 this marked effect extends well into the low 1/r values of about 40, an effect not previously noted with the lighter flexural prestrain developed by mangling. The buckling load is certainly improved above that if the member had been tested in the initially fully deformed state but it is far below that expected for a bar which had not been treated in this way.
8 9 10 11
Conclusions It seems that the effect o f mangling on the buckling load o f struts can be classified into two broad types: 'light' and 'heavy' mangling. The former causes a loss o f strength o f about 5% (probably less than the effect of the imperfections it removes) while the latter may cause a reduction of up to 20%; the effects being confined to the intermediate and low ranges o f slenderness ratios. A reasonable estimate o f the effect o f flexural prestrains on strut stability can be obtained by using tangent moduli curves derived from standard tension tests for the average properties o f the cross-section. Abnormal prior flexural deformation, possibly associated with accidental damage followed by straightening, can produce serious reductions in buckling load capacity (of the order of 35%). The extent o f this reduction could be a potential threat to the safety o f a system, being of the order of the allowance o f the overload factor. The effect is made more serious by the difficulty in determining if an element has been damaged and then straightened; it would appear to be prudent to exclude straightening o f severely damaged compressive elements as a permissible fabrication procedure without a thorough analysis of the particular circumstance. The above conclusions on the stability of columns do not take into account possible effects o f ageing after cold straightening. This could be the subject o f further research.
12 13 14 15 16
Notation The following symbols are used in this paper: e o E Et 1/r
R
Acknowledgements
b
The authors wish to thank Professor B. B. Hundy for his interest and valuable advice on the mangling operation. The
st
Eng. Struct., 1981, Vol. 3, April
engineering strain engineering stress Young's modulus tangent modulus slenderness ratio, where 1 = effective length of strut = actual length when b o t h strut ends are pinned; r = radius of gyration of strut cross-section radius o f curvature
Subscripts: av
70
Salmon, E. H. 'Columns', Oxford Tech. Pub., 1921 Huber, A. W. and Beedle, L. S. 'Residual stress and compressive strength of steel', Weld. J. 1954, 33 (no. 12), 589 Johnston, B. G. (Ed), 'Column Research Council guide to design criteria for metal compression members', Wiley, New York, 1966 Tall, L. 'Recent developments in the study of column behaviour', J. Inst. Engrs. Australia 1964, 36 (no. 12), 319 Paris, P. C. 'The Bauschmger effect on columns', J. Appl. Mech. 1956, 23,479 Bleich, F. 'Buckling strength of metal structures', McGrawHill, New York, 1952 Batterman, R. H. and Johnston, B. G. 'Behaviour and maximum strength of metal columns', J. Struct. Div. ASCE 1967, 93 (ST2), 205 Frey, F. 'Effet du dressage/~ froid des profil6s lamin6s en double t6 sur leur force portante', I A B S E 1969, 29 (II), 101 Hoff, N. J. 'The analysis of structures', Wiley, New York, 1956 Bauschinger, W. 'Gerichtc Verfestigung', Civilingenieur 1881, 27,299 Bauschinger, W. '0ber die Ver~inderung der Elasticit~itsgrenze und des Festigkeit des Eisens und Stahls durch Strecken und Quetschen, durch Erw~irmung und Abkiihlen und durch oftmal wiederholte Beanspruchung', Mitt. mech. tech. Lab. Miinch. 1866, 13, 1 Shanley, F. R. 'The column paradox', J. A ero. ScL 1946, 13 (no. 12), 678 Shanley, F. R. 'Inelastic column theory', J. Aero. Sci. 1947; 14 (no. 5), 261 (see also Discussion by Th. Von K~rman, 267) Pascoe, K. J. 'Strength of cold-formed cylindrical steel plates', J. Strain Anal. 1971,6 (no. 3), 167 Pascoe, K. J. 'Directional effects of prestrain in steel', J. Strain Anal 1971,6 (no. 3), 181 Jevons, I. D. 'The metallurgy of deep drawing and pressing', Chapman & Hall, London, 1949
Y
average buckling strain hardening yield