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Energy absorption of dynamically and statically tested mild steel beams under conditions of gross deformation
Int. J. Mech. 8ci. Pergamon Press Ltd. 1967. Vol. 9, pp. 633-649. Printed in Great Britain
E N E R G Y A B S O R P T I O N OF D Y N A M I C A L L Y A N D STATICALLY T E S T E D MILD STEEL BEAMS U N D E R CONDITIONS OF GROSS D E F O R M A T I O N ]~. RAVCLI:NGS* Department of Civil Engineering, The University of Sydney, Australia (Received 5 April 1967)
Summary--An account is given of dynamic and static tests on mild steel members deformed under conditions of gross geometry change. An evaluation is made of the rigid-plastic theory, taking account of change of geometry, and also the elastic-plastic theory, assuming deformation to occur in the static mode, in predicting the behaviour of the members. i. I N T R O D U C T I O N
MILD steel has become accepted as the principal material for fabricating a great variety of engineering products. A number of reasons m a y be quoted for this, including its relative ease and cheapness of manufacture, its high elastic modulus, high strength to weight ratio and large ductile range. Considerable advantage has been taken of the ductility of mild steel for m a n y years in the fabrication of pressed components, and, more recently, in civil engineering, where plastic methods of analysis and design of steel structures have been evolved to take account of the capacity of flexural members to develop plastic hinges and consequently to enable redistribution of stress to take place throughout frames. However, such structures never attain this condition except through misadventure, as design is based on a load factor and, in most cases, under working loads the structure is everywhere elastic. Associated with the high ductility of mild steel is its capacity to absorb a very considerable amount of energy, which is dissipated as heat during the plastic and strain-hardening phases of loading. It is very surprising that, despite a knowledge of this fact for very m a n y years, little rational use has been made of the energy-dissipating properties of steel. One significant exception to this, in the Second World War, was the development of the Morrison table shelter, in which a mild steel angle frame was so proportioned that, under the effects of dynamic loading from collapsing structure above, the deformation was not sufficiently great as to injure the occupants taking refuge inside. 1 One other example where some attention has been paid to the use of steel deforming plastically is in wharf installations using steel fenders. * Nevertheless, as mentioned above, mild steel is an inexpensive material, and could be used for disposable members which could absorb the energy of motor vehicles or • Now at : Department of Civil and Structural Engineering, University of Sheffield. 42 633
634
B. RAmbLINGS
falling elevators, for example, which got out of control. Furthermore, by suitable design it is possible to arrange for such a member to have any forceretardation characteristic desired. As energy-dissipating members would normally be subjected to loading applied at a rate considerably faster than normal static rates it is necessary to consider their behaviour in the light of knowledge of the post-elastic dynamic behaviour as both material properties and structural behaviour are influenced significantly by the rate of loading.S-~ DynamSc plasticity is quite a recent study, in fact, and a t t e n t i o n so far has been confined to an e x a m i n a t i o n of beams a n d o t h e r simple m e m b e r s in which g e o m e t r y changes are small. Consequently the potentialities of such elements to dissipate e n e r g y u n d e r conditions of gross d e f o r m a t i o n have not been examined. A few e x p e r i m e n t a l studies h a v e been m a d e of the d y n a m i c postelastic response of steel structures ;s-10 however, v i r t u a l l y none e x c e p t those c o n d u c t e d b y t h e a u t h o r were i n s t r u m e n t e d to provide t r a n s i e n t records of applied force and the corresponding displacement. Thus the d y n a m i c rigidplastic t h e o r y , which ignores elastic response, b u t provides the only really t r a c t a b l e means of analysis, has not been fully e v a l u a t e d experimentally. Two criteria h a v e been suggested for determining t h e v a l i d i t y of this t h e o r y , n-iS the one which is easier to a p p l y requires t h a t t h e e n e r g y absorbed b y plastic d e f o r m a t i o n be of a higher order t h a n the m a x i m u m elastic strain energy which can be stored. I n this paper, an account is given of a series of tests which were carried out on mild steel members, b o t h u n d e r static a n d impulsive loading, in order to assess their l o a d - d e f o r m a t i o n and e n e r g y absorption characteristics. The results are e x a m i n e d in t e r m s of ( a ) t h e d y n a m i c rigid-plastic t h e o r y , taking into a c c o u n t the v e r y large g e o m e t r y changes which arise during the testing process, and ( b ) t h e elastic-plastic t h e o r y , assuming the elastic response to occur in the static mode. An evaluation is m a d e of the work done in deforming the specimens, t o g e t h e r with the e n e r g y recoverable u p o n unloading, a n d from this, an assessment is m a d e of the v a l i d i t y of t h e rigid-plastic t h e o r y to these tests. I n this study, a t t e n t i o n is confined to V-shaped m e m b e r s in which the arms are closed b y the action of equal and opposite forces acting a t the cantilevered ends. 2. T H E O R E T I C A L
B E H A V I O U R OF S P E C I M E N S
2.1. Rigid--~lastic analysis
Consider the V-shaped specimen having the initial configuration .4 0 Co B0 as shown in Fig. 1. Under the action of the applied loads, P, acting at the tip the specimen will deform once the moment at the elbow reaches the full plastic value, given by n~ = Po do
(1)
At some later stage of deformation, assuming the mass of the members to be small, and ignoring strain h~rdening, the hinge will still be located at the point of maximum moment C. The configuration A C B will be that shown by broken lines, Fig. 1. If the new tip separation A B is denoted by 2b, and the new moment arm by d, the force P required to cause continued hinge action will have been reduced to p = n~--.-z~ d
(2)
E n e r g y absorption of d y n a m i c a l l y a n d statically tested mild steel b e a m s
635
W r i t i n g the tip closure A = 2~ = 2(bo - b ) a n d m a k i n g use of the fact t h a t axial strains a r e negligible, L 2 = A B 2 = A o B ~ = b 2 + d ~ = b~+d~
Thus d = ~/(d~ + 2b 0 ~ - r ~)
(3)
As d 0, b 0 are d e t e r m i n e d b y the initial shape of the specimen it is a simple m a t t e r to e v a l u a t e the n e w lever a r m in terms of the tip closure a n d to plot the load-deflection
7 zf
Io
131~ line
F i e . 1. Specimen configuration.
I
/I
,
--'~}_/lire 'or
tip closure (a)
tip closure (b)
FIo. 2. Load-deflection relationships. (a) Rigid-plastic analysis; (b) elastic-plastic analysis. characteristics of a m e m b e r as shown in Fig. 2(a). As the m e m b e r deforms as two rigid links the hinge r o t a t i o n is given b y
/ ~ b0 2~tanXo-tan- 1 ~)b~
(~)
The work done, U, in deforming the specimen, which is simply the area u n d e r the load-deflection curve, is given b y 2 f:Pd7
= 2
F~'
= 2m~[sin -1
d7 y-bo
bo
636
B. RAWLINGS
2.2. Elastic~lastic analysis I f account is taken of the finite elastic modulus of the material prior to the development of a plastic hinge at the section of m a x i m u m moment, the initial response of the specimen will be an elastic one, the load-deflection relationship being
La 2PLd~ A = 2 P c o s 2 ¢ 0 . 3 E I = 3EI
(5)
where ¢0 is the semi-angle at the elbow, and E I is the flexural rigidity of the section. I f the development of partially plastic zones of material is ignored, the line given by equation (5) then m a y be drawn from the origin to intersect the rigid-plastic line, as shown in Fig. 2(b). By similar reasoning, the elastic unloading line at any stage, having the relationship 2PLd 2 3EI can be drawn in, as shown in Fig. 2(b). The work U, expended in deforming such a member to any given degree of closure, is given by the area under the curve atvaa, and the recoverable elastic energy R, equal to the area under the triangle va T, and given by
R-
Lm~ 3EI
(6)
Consequently, the relationship between the work and tip closure in this case is similar to that for the rigid-plastic case, except for the area of the zone af]e, and the dissipated energy ( U - R) is obtained by deducting the additional term given by equation (6). 3. E X P E R I M E N T A L
WORK
In order to examine the energy absorption of mild steel specimens under conditions of gross deformation, both static and impact tests were conducted. The material used for the tests was ~ in. square black mild steel bar, this size being selected because it proved convenient for the testing equipment, and it could be bent by hand tools into the desired configurations. The material had the following chemical composition: C 0.13%
S 0.048%
P 0.038%
Si 0.06%
Mn 0.43%
Ni 0.02%
Cr 0.03%
Mo 0.01 ~)
The material was subjected to a considerable degree of cold working in the process of manufacture, and it was decided to do some tests on it in this condition, firstly to examine the properties when effects of upper yield stress would be eliminated because of previous cold working, and secondly because it was felt that the characteristics would more closely simulate those of a material with less significant strain-hardening characteristics than normalized mild steel taken into the early strain-hardening range. Furthermore, provided the material has sufficient ductility, the energy absorption per unit volume is greater with material having higher yield stress values, and, in practice, the most efficient energyabsorbing structure would appear to be the one having the greatest yield stress, provided the ductility were not sacrificed. 3.1. Fabrication of specimens Specimens were of V shape, with straight sides, and had included angles of 60 °, 90 ° and 120 ° . The dimensions were such that, in their original configuration at the commencement of each test, the lever arm of the forces closing the tips, as measured to the hinge at the elbow, was in all cases 4 in. Each specimen was carefully fabricated from straight lengths by bending it in a jig around a mandrel, 0-198 in. dis. to the appropriate angle. Care was taken to ensure that the material was not deformed beyond the angle required. The specimens were then marked out and drilled in order to take the pins for connexion to the supports.
Energy absorption of dynamically and statically tested mild steel beams
637
3.2. Static tests on specimens Static tests were performed in an Amsler hydraulically operated testing machine on the 200 lb range of loading. The specinlens were pinned at the ends to fittings which were aligned, and measurements were taken of the central lateral deflection, 8, and the closing of the ends, A, as shown in Fig. 3. Frequent unloading and reloading curves were plotted
com.~esting
machine
gauge measuring e
FIG. 3. Static test details. during the course of the tests, which were continued either to the stage where the tips had closed together or to fracture of the extreme tensile fibres if this occurred previously. A hydraulic machine was selected as in m a n y cases the load-deflection characteristic of the specimen was a falling one, and the use of a rigid type of loading device enabled load to be applied at a constant strain-rate. Static control tests were also conducted on straight lengths of material, which was loaded transversely on an 8 in. simply supported span by a concentrated central load. These furnished information on the full plastic moment of resistance and strain-hardening characteristic of each material length from which the specimens were manufactured, and enabled all results to be reduced to a common value of full plastic moment. 3.3. D y n a m i c tests on specimens Groups of three specimens, identical to those described above, were tested under impulsive loading in a machine designed for applying dynamic loads to structures and defe~-tiontr'~rc~cer
breech ~o~
valve
~o~,e
~: II
timercrosswires
g®
3==~1
pressc¢~gauge
f~rce ~ e v
Ib
movingcrosshead
@
~1
~Jmen
~.
m~r~t,e
ar r~ceiver CR.Q
trigger~l ~
deflection~
~r~ns ~_
FIG. 5. Diagrammatic representation of apparatus. structural elements. A photograph of the equipment is shown in Fig. 4, and a diagrammatic representation is given in Fig. 5. Essentially the equipment consists of a pneumatically operated gun, together with a projectile and loading mechanism which moves and deforms the specimen. The gun itself consists of a 10 ft length of precision-drawn steel
638
B. RAWLINGS
tube, 1.117 in. internal dia. and 0.192 in. wall thickness, s u p p o r t e d at close intervals on a double-channel frame. Pressure is applied to the gun b y air from a large air receiver, o p e r a t i n g at up to 225 lb/in ~ pressure, t h r o u g h a plug v a l v e w i t h large ports, which has been found to give consistent p e r f o r m a n c e in operation. I n the present series of tests, a large projectile has been used, consisting of a 3 ft length of steel rod, weighing 9.33 lb, t u r n e d to 1.094 in. dia. a n d h a v i n g r u b b e r O-rings as seals. A p h o t o g r a p h of it, resting alongside the gun, is shown in Fig. 4. The projectile, which was lubricated b y light m a c h i n e oil, could be set at a n y desired position prior to a test, b y unscrewing t h e breech cap a n d inserting a rod into the barrel. I t will be a p p r e c i a t e d t h a t , b y v a r y i n g the pressure, the initial position of t h e projectile and its mass, a v e r y wide range of muzzle velocities could be obtained. The a p p a r a t u s was originally designed to h a v e considerable v e r s a t i l i t y in the loading m e c h a n i s m arrangements, b u t for this series of experiments, these t o o k the form of a fixed frame w i t h four guides, on which a t u b u l a r t u p connected to a crosshead slides when s t r u c k b y the projectile. The specimen is held b y pins to two spring steel cantilever m e m b e r s , one at each end, to which are a t t a c h e d electric resistance strain gauges. These cantilevers are connected, one to t h e crosshead, a n d one to t h e s t a t i o n a r y b a c k plate, so t h a t the line of action of the force on the specimen coincides w i t h the axis of the gun. These p a r t i c u l a r loading a r r a n g e m e n t s were selected because t h e y p e r m i t t e d gross deformations of the specimens to develop w i t h o u t u n d u l y restrictive limitations on travel. 3.4. Instrumentation As m e n t i o n e d above, each specimen was s u p p o r t e d at its ends by spring steel cantilever transducers, which h a d a n a t u r a l f r e q u e n c y of v i b r a t i o n of the order of 4000 c/s. E a c h had a t t a c h e d to the faces, two electric resistance strain gauges a r r a n g e d as an active, d u m m y pair. The gauges of t h e crosshead t r a n s d u c e r were connected to a d y n a m i c strain bridge, h a v i n g a carrier f r e q u e n c y of 5000 e/s, t h e o u t p u t of this being fed into one channel of a double-beam c a t h o d e - r a y oscilloscope. The t r a n s i e n t deflection was d e t e c t e d b y a linear resistance displacement transducer, consisting of a long coil of wire located w i t h a horizontal axis, and s i t u a t e d vertically o v e r the centre line of the target. A spring-loaded sliding contact, a t t a c h e d to t h e crosshead, established a variable e a r t h p o i n t on t h e coil. The t r a n s d u c e r was o p e r a t e d from direct current, the o u t p u t being fed into t h e second channel of the oscilloscope. The horizontal sweep of the oscilloscope was a c t u a t e d b y t h e fracture of a fine wire s t r e t c h e d across the muzzle of t h e gun. P h o t o g r a p h i c records of t h e oscilloscope traces were m a d e b y a P o l a r o i d c a m e r a m o u n t e d in front of t h e screen. A typical record is shown in Fig. 6. I n order to d e t e r m i n e t h e incident velocity, and consequently the m o m e n t u m and kinetic energy of the projectile, a t i m i n g device was installed in its path, at a small distance in front of the tup. This comprised two pairs of fine cross-wires, 4 in. apart, each pair consisting of two skew wires lying in planes n o r m a l to the projectile and ~ in. apart. Short-circuiting b e t w e e n t h e first pair of wires f r o m t h e passing of the projectile initiated a microsecond c o u n t e r - t i m e r which was stopped b y the second pair. 3.5. Test procedure A f t e r setting up the specimen in the testing apparatus, care was t a k e n to ensure t h a t the crosshead a n d t u p were free of friction, and baselines for t h e force a n d deflection traces were established. The projectile was set to its correct position, receiver charged to t h e r e q u i r e d pressure, a n d t h e i m p a c t test performed. This pressure was k e p t a t q u i t e a low value, e.g. 15 lb/in ~, in order t h a t after the test, t h e static residual force on t h e projectile was small w h e n c o m p a r e d w i t h the specimen resistance. A f t e r t h e test, t h e projectile was t h e n r e t r a c t e d from t h e t u p and a double exposure t a k e n to establish the final positions of the load and deflection traces in order to d e t e r m i n e the p e r m a n e n t set of the specimen. 3.6. Calibrations B o t h the force t r a n s d u c e r and displacement p o t e n t i o m e t e r were calibrated i m m e d i a t e l y after each test was completed. The force t r a n s d u c e r was r e m o v e d and r e m o u n t e d in an
(b)
FIG.
4. (a) General view of apparatus; (b) close-up member, transducers and crosshead.
view
showing
f. y. 638
FIG. 6. Typical C.R.O.
trace.
FIG. 7. Statically tested members.
FIG. lO(a). Section of tested member showing cross-sectional distortion at elbow.
FIG. lO(b). Dimensions of cross-section at hinge of a 90° specimen after testing.
FIG. 15. Dynamically
ksterl
members.
Energy absorption of dynamically and statically tested mild steel beams
639
elevated position, and subjected to incremental dead weight loading from a weight carrier suspended from the connexion pin. This gave, on the oscilloscope, a series of parallel lines which were photographed a n d subsequently used to establish the scale of the transient load trace. The deflection transducer was also calibrated b y moving the crosshead through incremental distances covering the range of deformation, and photographing the traces. The time scale was established b y superimposing on one photograph a 50 c/s waveform. 4. R E S U L T S 4.1. Static tests Photographs of the specimens after testing, with identical specimens before testing, are shown in Fig. 7. I t will be seen that even under conditions of gross deformation, plasticity has been confined to an area very close to the elbow, so that the assumption of point hinge action is valid. The experimental relationships between applied load and lateral deflection, and applied load and end closure are shown in Fig. 8, together with
loo I
loo 1
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loo I
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,, .,
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.,o
o 1 2 34 56
deflection - irt 12@
d e f l e c t ~ n - irt
deflect}on - in.
9@
6O °
/
FIe. 8(a). Experimental relationships between applied load and deflection--static tests.
--~4
¢
i
~2
~3 32
¢2
2 4 6 tip closure - in. 120 °
8
10
()
2 4 6 tip closure - in. 90 ~
O" 2 4 tip closur~ - in. 6O°
Fro. 8(b). Experimental and theoretical deflection--tip closure relationships. unloading and reloading curves. One point of interest is the variation in slopes between the unloading lines a t various stages of testing, due to the large geometry changes occurring during the course of the test. I t is also of interest to compare the experimental relationship between the lateral deflection and end closure and the theoretical relationship based upon equation (3), as shown b y the full and dotted lines, respectively, in Fig. 8(b). The close agreement in all cases again confirms that the specimen deformed almost as two rigid links, which indicates that strain hardening was not significant enough to increase the moment of resistance to a greater extent than the applied moment gradient along each arm of the specimen.
640
B. RAWLI:NGS
Making use of the knowledge t h a t t h e m e m b e r s did in fact act as two rigid links, relationships b e t w e e n the full plastic m o m e n t of resistance and t h e change in angle A C B (Fig. 1) f r o m t h e original v a l u e A 0 Co B0 h a v e been p l o t t e d in Fig. 9. I t will be clear t h a t the steeply rising region n e a r t h e origin is in each case due to t h e elastic response, after which plasticity developed at the elbow a n d c o n t i n u e d hinge r o t a t i o n occurred at v i r t u a l l y c o n s t a n t m o m e n t of resistance until t h e end of the test. As the a s s u m p t i o n is m a d e in t h e simple plastic t h e o r y of bending t h a t plane sections r e m a i n plane and t h a t fibres in the b e a m cross-section b e h a v e independently, it is of interest to observe j u s t how high the c u r v a t u r e s at the elbow become, a n d to see the gross distortion of the cross-sectional dimensions. The material, as m e n t i o n e d previously, was ¼ in. square in cross-section, but the cross-section at a typical hinge, after completion of a test, had t h e dimensions shown in Fig. 10.
~400
~3oo[
~3oo
~oo
b°I
i~2oo
{lo0
"-~1oo
20
40 60 80 100 in angle- degr,ees 120~
d
""100
~
20
40 60 in angledeles
20 cl"angein angle -
FIa. 9. l~elationship b e t w e e n full plastic m o m e n t and angle c h a n g e - - s t a t i c tests. Straight b e a m s were also tested, as m e n t i o n e d in Section 3.2, ill order to establish t h e full plastic m o m e n t of resistance of t h e p a r e n t material, and these were loaded until a considerable degree of hinge r o t a t i o n had occurred. The load-deflection curve for t h e l e n g t h f r o m which t h e other static specimens were cut is shown in Fig. l l ( a ) . As t h e large central deflection was a c c o m p a n i e d b y pull-in of the roller supports, which was measured, it was necessary to correct for this w h e n establishing the m o m e n t of resistance; c o n s e q u e n t l y t h e relationship b e t w e e n m o m e n t of resistance and central deflection is shown p l o t t e d as t h e c u r v e in Fig. l l ( b ) . Making the same a s s u m p t i o n as in t h e tests on t h e V-shaped specimens t h a t all deformation was confined to r o t a t i o n at t h e section of m a x i m u m m o m e n t , this c u r v e could also be r e d r a w n as a m o m e n t - h i n g e r o t a t i o n c u r v e for an initially straight m e m b e r , as shown in Fig. l l ( e ) . I t is n o w possible to represent all four curves, in Figs. 9 and l l ( e ) , on t h e one set of axes, b y c o m m e n c i n g at the a p p r o p r i a t e angle to represent t h e initial elbow angle of t h e specimen, as shown in Fig. 12. These indicate quite clearly t h a t it was a close a p p r o x i m a tion to assume t h a t t h e full plastic m o m e n t of resistance could be t a k e n as a c o n s t a n t 365 lb-in, t h r o u g h o u t t h e whole range of loading, until t h e specimen was b e n t v i r t u a l l y double. Consequently the a s s u m p t i o n to be made, in considering t h e v a l i d i t y of t h e rigid-plastic theory, is t h a t the m a t e r i a l has the properties shown in Fig. 13. F u r t h e r m o r e , if a t t e n t i o n is directed b a c k to t h e load-closure relationships for each of the three b e n t specimens, as shown in Fig. 8, it is possible simply to s u b s t i t u t e in the a b o v e assumed v a l u e of m~ in e q u a t i o n s (2) and (3) and to plot t h e theoretical rigid-plastic curves shown in d o t t e d lines in Fig. 14, t h e unloading a n d reloading lines in all specimens being, of course, v e r t i c a l in this case. These lines are p l o t t e d to the limiting condition in which t h e two arms of t h e specimen become parallel; indeed this value would h a v e to be reduced by the thickness of the material.
Energy absorption of dynamically and statically tested mild steel beams
200 160 12C
!
0
80
(a}
40 O'
i
0:4 ' 0:8 ' 1.2 ' 1:6 ' 2:0' central deflection- in.
c' .(3 I
eu 400
c
$ 3oc
L
o .~ 200 c-
(b}
©
E
~ 100
0
'0'4'0:8' 1:2 ' 1.6' 2tO central deflection- in.
FIG. 11. Static tests on straight lengths. (a)Load-deflection relationship; (b) central moment-deflection relationship.
° _
--~400
~300S ~ 200
{el
i lOO
o
Ib 2b 3b 40
hinge Potation- degrees
50
Fro. l l(c). Moment-hinge rotation relationship.
641
642
B. RAWLINGS J540C
/
30C
20C
~
100
0
E 0 18
,
,
160
J
,
,
,
,
,
,
140 120 100 80 60 40 included angle of specimen - degrees
,
,
,
,
20
0
FIG. 12. M o m e n t - i n c l u d e d angle r e l a t i o n s h i p s - - s t a t i c tests.
~365 JD
C
18o hinge Potation - degrees
Fro. 13. A s s u m e d m o m e n t - h i n g e r o t a t i o n relationship. 100
__I0(
4100 I
I
80 60
"'"
60
40
4O
40
°20
20
2c
o
' :~ ' 4 ' 6 ' 8 deflection - in. 12ff
'16
' 1~ '14(~
2 4 deflection - in 90"
6
8
2 4 deflection - in 60 ~
FIG. 14. Theoretical load-deflection curves based on rigid-plastic
theory. 4.2. Dynamic tests The d a t a relating to the various d y n a m i c tests are g i v e n in Table 1. P h o t o g r a p h s of d y n a m i c a l l y tested specimens after testing, shown in Fig. 15, exhibit the s a m e b e h a v i o u r w i t h v i r t u a l l y all d e f o r m a t i o n occurring at t h e elbow, as t h e static specimens u n d e r test. The loa~l-time a n d d e f l e c t i o n - t i m e traces recorded in a typical test are shown in Fig. 6. I t will be seen t h a t t h e load increased v e r y rapidly to a m a x i m u m value, accompanied b y a certain degree of vibration which rapidly subsided, after which t h e load g r a d u a l l y diminished. The first stage in the process of analysing t h e records was to redraw the curves carefully on g r a p h paper, w i t h uniform scales for force, t i m e a n d deflection. This was done b y superimposing t h e force and deflection calibration lines on t h e record, t o g e t h e r w i t h t h e 50 c/s t i m e marks, establishing a n u m b e r of points precisely on the g r a p h and i n t e r p o l a t i n g as carefully as possible. F r o m these rectified f o r c e - t i m e
120 120 120 90 90 90 60 60 60
7 10 33 13 21 22 18 19 20
13.92 13.88 13-86 8.30 8-10 8.12 4.90 4.90 4.91
(3)
(in.)
(degrees)
(2)
Initial spacing
Angle a t elbow
(1)
Specimen No.
7.0 7.0 7.0 7.0 7.0 7.0 5.0 4.0 4-0
(4)
(ft)
Projectile travel
15.0 15.0 15.0 15.0 15-0 15.0 15.0 15.0
15.0
(5)
(lb/in ~)
16.70 15.15 17-70 16-66 17.89 18-92 16.66 13-55 13.55
(6)
(ft/sec)
Velocity v0
8.66 7-86 9.28 8.94 9.58 10.18 9.29 7.66 7-66
(7)
(ft/sec)
v2
Velocity after impact,
R E S U L T S OF DYI~'AMIC TESTS
Projectile pressure
TABLE ].
9.78 9.10 10.40 9.82 12.20 11-18 11-37 6-93 6-93
(8)
(ft/sec)
v2
Velocity after i m p a c t as measured from C.R.O. t r a c e ,
11.25 12.04 10-90 5.60 4.86 4-72 2-20 3.06 3.05
(9)
(in.)
AB
Final spacing o f ends,
2.67 1-84 2.96 2.70 3.25 3-40 2.70 1.84 1.86
(10)
(in.)
Permanent set as d e t e r m i n e d experimentally
o
644
B. R A W L I ~ S
and d e f l e c t i o n - t i m e curves it was t h e n possible to draw a load-deflection curve for each specimen. I t was r a t h e r difficult to establish precisely t h e slope of the elastic rising line, because of t h e v e r y small r i s e - t i m e of t h e force pulse, and this accounts for t h e v a r i a t i o n b e t w e e n records in the curves shown in Fig. 16. A v a r i a t i o n occurred also in the magnit u d e s of t h e force ordinates during t h e yielding phase, due p a r t l y to differences between specimens and p a r t l y to errors in reducing t h e record. The largest difference occurred b e t w e e n specimen 33 a n d t h e o t h e r two 120 ° specimens, the v a r i a t i o n from the m e a n of the m a x i m u m force in this case being 18 per cent. I n the case of the 90 ° and 60 ° specimens, however, this v a r i a t i o n a b o u t the m e a n a m o u n t e d to 8 per cent.
lOO
100
10C
cimen 0
8C
18
:)ecimen 21 80
80
"--'--.
-~I 60i
_~ 4c
2¢
2O
2C
2 ~,~ 4 d 2 4 def~:tion - irt deflection -JR defL-:~ztion- in 9O o 120° 6o ° Fro. 16. Force-deflection r e l a t i o n s h i p - - d y n a m i c tests.
I f m e a n lines are d r a w n for each group, and c o m p a r e d w i t h the l o a d - t i p closure curves o b t a i n e d f r o m t h e corresponding static tests, it will be seen t h a t t h e loads in t h e d y n a m i c tests are e n h a n c e d b y a p p r o x i m a t e l y 12 per cent o v e r those of the static tests, due to t h e s t r a i n - r a t e sensitivity effects of the material. This a g r e e m e n t b e t w e e n d y n a m i c and static results is r a t h e r closer t h a n t h a t which m i g h t h a v e been expected, for t h e particular t y p e of material, b u t is due to t h e severe degree of cold working in t h e m a n u f a c t u r e of the m a t e r i a l and fabrication of t h e specimen; this has reduced the s t r a i n - r a t e sensitivity considerably. I n fact, tests on identical specimens of normalized material, which will be r e p o r t e d elsewhere, h a v e e x h i b i t e d significantly greater s t r a i n - r a t e sensitivity t h a n these results.
4.3. Work done on specimens during deformation The w o r k r e q u i r e d to deform each of the statically loaded specimens to any given degree of tip closure, represented b y the typical area ae~'aa in Fig. 17, has been p l o t t e d in Fig. 18, t o g e t h e r w i t h t h e recoverable energy available u p o n unloading, represented b y t h e triangular area, ~a~. I f the a s s u m p t i o n is m a d e t h a t the d y n a m i c a l l y loaded specimens deform in t h e static mode, b u t t h a t t h e y h a v e a d y n a m i c full plastic m o m e n t of resistance which is 12 per cent higher t h a n t h e static value, the w o r k done, Us, in deforming a d y n a m i c a l l y loaded specimen would be, to a close a p p r o x i m a t i o n , the value o b t a i n e d f r o m Fig. 18, multiplied b y 1.12. H o w e v e r , it is possible to e v a l u a t e the kinetic energy of t h e projectile, t u p and crosshead, i m m e d i a t e l y a f t e r impact, in each test, and by m a k i n g a correction for the additional work done b y the n e t projectile force (pA -Fir ) during t h e deformation, to d e t e r m i n e the work done in deforming the specimen to its m a x i m u m e x t e n t b y observations during each test. These e x p e r i m e n t a l values are shown t a b u l a t e d in Table 2, column 7, and are denoted by UE. B y e q u a t i n g Ug and Us it is possible to d e t e r m i n e t h e m a x i m u m deformation of each specimen, a n d this is shown t a b u l a t e d in c o l u m n 8. The elastic recovery, being 1.12 times t h e corresponding value from t h e static test, m a y be s u b t r a c t e d from this m a x i m u m d e f o r m a t i o n to give t h e derived v a l u e of p e r m a n e n t set, listed in column 9. I t will be seen b y c o m p a r i n g these
Energy absorption of dynamically and statically tested mild steel beams
645
with results in columns 4 and 3, that in general the agreement between the observed and predicted results for both m a x i m u m closure and permanent set agree within 12 per cent for the m a x i m u m closure and 25 per cent for the permanent set.
r~
6°I
-~40
-- 20
0
deflection
in.
-
17Io. 17. L o a d - d e f l e c t i o n
600
curve--static
test..
/ .
work requiredto
J /
500 400
/ /
400
~400
r, I
1 O0
/
i=
. . . . . . . . . . 2
4
6
deflection - irt 120'
8
10C
10
12
i00
O'
2
4
. . . . .
(J
defied.ion - in. 90'
2
4
deflection - in. 60~
FIO. 18. Work-deflection curves--static tests. 600 500
60C work required to /
deform,specim~//, 50C
/
500
/ i
z/
400
40C
/
--•300
400~
e"
///
I
100 /~///
10C
S' /
O'
2
4
6
deflection - In. 120'
8
~ 300 I
,
pOioo,i Jl II ,
,
2
,
4
.
deflection - In.
90"
.
6
.
l/I
/
.
~
2
~
deflection - in.
ecr
FIe. 19. Work-deflection curves--rigid-plastic theory. If, on the other hand, the assumption is made that the material has rigid-plastic characteristics, the dynamic full plastic moment being 12 per cent above the static value, or 409 lb-in., the curves relating work to tip closure take the form shown in Fig. 19. Denoting this work term b y U2p it is possible to equate ORe to UE and to derive a
(2)
120 120 120 90 90 90 60 60 60
7 10 33 13 21 22 18 19 20
(degrees)
Angle
(1)
81)ocimen No.
2-67 1.84 2.96 2.70 3.25 3.40 2-70 1.84 1.86
(3)
(in.)
Permanent set as determined experimentally
4.5 3.8 4.5 4-0 4.5 5.0 3.7 3.1 2.9
(4)
(in.)
Maximum closure during impact, from C.R.O. trace
E x p e r i m e n t a l results
T A B L E 2.
251 207 288 268 308 348 288 196 196
(5)
(lb-in.)
K . E . of moving parts after impact,
26 18 29 23 30 37 30 20 19
(6)
(lb-in.)
Work done by projectile after impact
277 225 317 291 338 385 318 216 215
(7)
(lb-in.)
UE
Total energy applied to specimen
4.3 3-5 4.9 3-8 4.4 5.0 3.9 2-7 2.6
(8)
(in.)
Maximum closure
3.1 2.3 3.4 2.8 3.3 4.2 3.2 1.9 1.8
(9)
(in.)
Permanent set
D e r i v e d results based on factored values of Static Tests
C O M P A R I S O N OF T H E O R E T I C A L A N D E X P E R I M E N T A L R E S U L T S
3.4 2-7 4.0 3.3 3.9 4.5 3-4 2.2 2-2
(10)
(in.)
Closure
Theoretical results from rigidplastic theory
4.1 3.3 4-7 3-7 4.3 4.9 3.7 2.5 2.5
(11)
(in.)
Maximum closure
2.7 2.0 3.2 2-9 3.8 4.3 3.1 1.9 1.9
(12)
(in.)
Permanent set
Theoretical results from elastic-plastic theory
G~
Energy absorption of dynamically and statically tested mild steel beams
647
theoretical value of tip closure, as shown in Table 2, column 10. As rigid-plastic behaviour is assumed, there would be no elastic recovery, on this assumption. I t will be seen t h a t in all cases the theoretical values are significantly lower than the experimentally recorded m a x i m u m closures, lying in general approximately midway b e t w e e n the m a x i m u m closure and the final permanent set. Finally, if the assumption is made t h a t the material is elastic-plastic, with full plastic m o m e n t of 409 lb-in., as shown in Fig. 20, it is possible to plot modified curves to allow for the initial elastic response, as shown by the dotted lines in Fig. 19. Denoting these
I
409t !
rotation - degrees FIG. 20. Assumed elastic-plastic m o m e n t - r o t a t i o n relationship. energy terms by UEp, and equating UEp to UE, the theoretically derived values of m a x i m u m tip closure are those shown in column 11 of Table 2, and assuming elastic recovery to occur in the static mode, the permanent set values are tabulated in column 12. Examining again the experimental values in columns 4 and 3 the agreement between experimental and theoretical values has improved to the same order as those predicted from factored values of the static tests, the worst error being a 25 per cent over-estimation of permanent set in the case of specimen 22. 4. C O N C L U S I O N S Simple flexural elements of mild steel m a y be fabricated to absorb significant amounts of energy, whilst undergoing gross plastic deformation. I n order to evaluate the static load-deformation characteristics it is necessary to take account of the actual m o m e n t rotation characteristics of the plastic hinge. However, if strain hardening is not so severe as to cause hinge displacement and the elastic component of deformation is small when compared with the hinge rotation component the normal rigid-plastic assumptions of behaviour lead to a close prediction of the load-deflection relationship, provided account is taken of the constantly changing structural geometry. Corresponding dynamically tested specimens have similar load~leflection characteristics to statically tested specimens, provided some account is taken of the increase in full plastic moment of resistance due to the strain-rate sensitivity of the material. The deformed shapes of both statically and dynamically tested specimens indicate a similar degree of spread of the plastic zones, and similar appearance of local deformation of material at the elbow. Allowing for the errors associated with the reduction of results from force-time and deflection-time oscilloscope traces, the elastic loading and unloading lines correspond with the assumption of specimen deformation in the static mode, and the same falling load~leflection characteristic over the plastic deformation region as in the static tests. I n evaluating the rigid-plastic theory, it is convenient to compare the total energy applied to the specimen with the absolute m a x i m u m elastic strain energy which could be stored, if the specimen were in a state of uniform flexure. In the specimens tested the m a x i m u m possible elastic strain energy which could be stored ranged from 1/4 to 1/1.6 of the total energy applied; the actual ratios, allowing for linearly varying moment conditions, v a r y from 1/12 to 1/4.8. The rigid-plastic theory led to predictions of closure which were intermediate between the actually recorded m a x i m u m closure and permanent set values ; however, the errors were of the order of 20-30 per cent in most cases.
648
B. RAWL~NGS
When allowance was made for the finite elastic modulus of the material in the theory, the agreement between theoretical predictions and experimentally observed values of m a x i m u m closure and permanent set improved significantly, the average error being of the order of 10 per cent. As more rigorous evaluation of the dynamic rigid-plastic theory would require tests over a wider range of conditions than those described above, the present series of tests must be regarded as being but a first stage ; however, further work is being planned on specimens in which the elastic energy will be significantly less important in relation to the plastic work of deformation, and this should give data for the determination of limits of validity.
Acknowledgements--This work was carried out in the Materials and Structures Laboratory of the School of Civil Engineering, University of Sydney, the Professor of Civil Engineering and Head of the School being Professor J. ~ . Roderick. The research was financed from funds made available by the Australian Research Grants Committee. The author wishes to record his sincere thanks to Professor Roderick, and acknowledge the assistance given by Messrs. W. J. Cottam and J. McEnnally both in the experimental work and reducing the results. REFERENCES 1. J. F. BAKER, The Civil Engineer in War, Vol. 3, p. 30, Institution of Civil Engineers, London (1948). 2. G. E. DENT, Struct. Engr, 42, 39 (1964); A. G. SENIOR, Struct. Engr, 42, 345 (1964). 3. H. G. HOPKINS, Appl. Mech. Rev. 14, 417 (1961). 4. B. RAWLINGS, Trans. Instn Engrs Aust. CES, p. 89 (1963). 5. S. R. BODNER and P. S. SYMONDS, Plasticity, p. 488. Pergamon Press, Oxford (1960). 6. B. RAWLI~GS, Proc. R. Soc. A, 275, 528 (1963). 7. R. J. ASPDE~ and J. D. CAMPBELL, Proc. R. Soc. A, 290, 266 (1966). 8. E. W. PARKES, Proc. R. Soc. A, 228, 462 (1955). 9. B. RAWLINGS, Proc. Instn civ. Engrs, 29, 389 (1964). 10. B. RAWLI~GS, J. mech. Engng Sci. 6, 327 (1964). l l . H. H. BLEICH and M. G. SALVADORI, Trans. Am. Soc. civ. Engrs, 120, 499 (1955). 12. R. C. ALVERSO~, J. appl. Mech. 23, 411 (1956). 13. J. A. SEILER, B. A. COTTER and P. S. SYMONDS,J. appl. Mechs, 23, 515 (1956). APPENDIX
By making use of the principle of conservation of m o m e n t u m it is a simple matter to check the velocity of the tup immediately after impact against the value independently obtained from measurements of the oscilloscope trace. Furthermore, the distance through which the specimen deforms can be readily checked against the measured value. initial position , of specimen^ i, D
nt~l
~ t i o n of projectile
~
L
a
Fro. 21. Analysis of motion of tup. I t is convenient to represent the testing arrangement diagrammatically as shown in Fig. 21. Adopting the following notation: a acceleration of projectile prior to impact A area of projectile d deceleration of projectile after impact D spacing between "start" and "stop" pulse wires
E n e r g y absorption of d y n a m i c a l l y and statically tested m i l d steel b e a m s
649
-FI~ Me N/~ p Pl P2 P S1
frictional force opposing m o t i o n (assumed constant) mass of t u p a n d crosshead mass of projectile pressure on projectile initial pressure on projectile final pressure on projectile specimen force distance f r o m front of initial position of projectile to point m i d w a y b e t w e e n t i m i n g wires $2 distance from front of initial position of projectile to t u p T i n t e r v a l of t i m e t a k e n b y projectile to t r a v e r s e t i m i n g wires Assuming t h e frictional force FI~ opposing the m o t i o n to be c o n s t a n t t h r o u g h o u t t h e test, the e q u a t i o n of m o t i o n of t h e projectile prior to i m p a c t is a
p A --F~T My
(i)
I f Pl a n d p , are nearly equal, for low piston velocities it m a y be assumed t h a t the pressure p is constant, h a v i n g the v a l u e {Pl +P2)/2, so t h a t the p r o b l e m simplifies to the case of c o n s t a n t acceleration conditions. The velocity v 0 at the stage shown in Fig. 21, w h e n the projectile has t r a v e l l e d distance S 1, is g i v e n b y v~ = 2aS 1 (ii) whilst the velocity v 1 i m m e d i a t e l y prior to i m p a c t is given b y v~ = 2aS2
(iii)
I t can be shown t h a t , for the ease when D / S 1 is small, the velocity v 0 is given b y D I n t h e tests outlined in this paper, D/S~ was of the order of ~-~, so t h a t t h e error in assuming D / T = v 0 was negligible. Thus, from equations (ii) and (iv) it was possible to e v a l u a t e a, and consequently (pA---Fsr ), and t h e velocity v 1 could be c o m p u t e d , allowing for friction. A t the m o m e n t of i m p a c t it is assumed t h a t the projectile a n d t u p coalesce, so t h a t , f r o m conservation of m o m e n t u m , the c o m m o n velocity v, i m m e d i a t e l y after i m p a c t is given b y M~ v~ M~ + M~ vl (v) -
43
E N E R G Y A B S O R P T I O N OF D Y N A M I C A L L Y A N D STATICALLY T E S T E D MILD STEEL BEAMS U N D E R CONDITIONS OF GROSS D E F O R M A T I O N ]~. RAVCLI:NGS* Department of Civil Engineering, The University of Sydney, Australia (Received 5 April 1967)
Summary--An account is given of dynamic and static tests on mild steel members deformed under conditions of gross geometry change. An evaluation is made of the rigid-plastic theory, taking account of change of geometry, and also the elastic-plastic theory, assuming deformation to occur in the static mode, in predicting the behaviour of the members. i. I N T R O D U C T I O N
MILD steel has become accepted as the principal material for fabricating a great variety of engineering products. A number of reasons m a y be quoted for this, including its relative ease and cheapness of manufacture, its high elastic modulus, high strength to weight ratio and large ductile range. Considerable advantage has been taken of the ductility of mild steel for m a n y years in the fabrication of pressed components, and, more recently, in civil engineering, where plastic methods of analysis and design of steel structures have been evolved to take account of the capacity of flexural members to develop plastic hinges and consequently to enable redistribution of stress to take place throughout frames. However, such structures never attain this condition except through misadventure, as design is based on a load factor and, in most cases, under working loads the structure is everywhere elastic. Associated with the high ductility of mild steel is its capacity to absorb a very considerable amount of energy, which is dissipated as heat during the plastic and strain-hardening phases of loading. It is very surprising that, despite a knowledge of this fact for very m a n y years, little rational use has been made of the energy-dissipating properties of steel. One significant exception to this, in the Second World War, was the development of the Morrison table shelter, in which a mild steel angle frame was so proportioned that, under the effects of dynamic loading from collapsing structure above, the deformation was not sufficiently great as to injure the occupants taking refuge inside. 1 One other example where some attention has been paid to the use of steel deforming plastically is in wharf installations using steel fenders. * Nevertheless, as mentioned above, mild steel is an inexpensive material, and could be used for disposable members which could absorb the energy of motor vehicles or • Now at : Department of Civil and Structural Engineering, University of Sheffield. 42 633
634
B. RAmbLINGS
falling elevators, for example, which got out of control. Furthermore, by suitable design it is possible to arrange for such a member to have any forceretardation characteristic desired. As energy-dissipating members would normally be subjected to loading applied at a rate considerably faster than normal static rates it is necessary to consider their behaviour in the light of knowledge of the post-elastic dynamic behaviour as both material properties and structural behaviour are influenced significantly by the rate of loading.S-~ DynamSc plasticity is quite a recent study, in fact, and a t t e n t i o n so far has been confined to an e x a m i n a t i o n of beams a n d o t h e r simple m e m b e r s in which g e o m e t r y changes are small. Consequently the potentialities of such elements to dissipate e n e r g y u n d e r conditions of gross d e f o r m a t i o n have not been examined. A few e x p e r i m e n t a l studies h a v e been m a d e of the d y n a m i c postelastic response of steel structures ;s-10 however, v i r t u a l l y none e x c e p t those c o n d u c t e d b y t h e a u t h o r were i n s t r u m e n t e d to provide t r a n s i e n t records of applied force and the corresponding displacement. Thus the d y n a m i c rigidplastic t h e o r y , which ignores elastic response, b u t provides the only really t r a c t a b l e means of analysis, has not been fully e v a l u a t e d experimentally. Two criteria h a v e been suggested for determining t h e v a l i d i t y of this t h e o r y , n-iS the one which is easier to a p p l y requires t h a t t h e e n e r g y absorbed b y plastic d e f o r m a t i o n be of a higher order t h a n the m a x i m u m elastic strain energy which can be stored. I n this paper, an account is given of a series of tests which were carried out on mild steel members, b o t h u n d e r static a n d impulsive loading, in order to assess their l o a d - d e f o r m a t i o n and e n e r g y absorption characteristics. The results are e x a m i n e d in t e r m s of ( a ) t h e d y n a m i c rigid-plastic t h e o r y , taking into a c c o u n t the v e r y large g e o m e t r y changes which arise during the testing process, and ( b ) t h e elastic-plastic t h e o r y , assuming the elastic response to occur in the static mode. An evaluation is m a d e of the work done in deforming the specimens, t o g e t h e r with the e n e r g y recoverable u p o n unloading, a n d from this, an assessment is m a d e of the v a l i d i t y of t h e rigid-plastic t h e o r y to these tests. I n this study, a t t e n t i o n is confined to V-shaped m e m b e r s in which the arms are closed b y the action of equal and opposite forces acting a t the cantilevered ends. 2. T H E O R E T I C A L
B E H A V I O U R OF S P E C I M E N S
2.1. Rigid--~lastic analysis
Consider the V-shaped specimen having the initial configuration .4 0 Co B0 as shown in Fig. 1. Under the action of the applied loads, P, acting at the tip the specimen will deform once the moment at the elbow reaches the full plastic value, given by n~ = Po do
(1)
At some later stage of deformation, assuming the mass of the members to be small, and ignoring strain h~rdening, the hinge will still be located at the point of maximum moment C. The configuration A C B will be that shown by broken lines, Fig. 1. If the new tip separation A B is denoted by 2b, and the new moment arm by d, the force P required to cause continued hinge action will have been reduced to p = n~--.-z~ d
(2)
E n e r g y absorption of d y n a m i c a l l y a n d statically tested mild steel b e a m s
635
W r i t i n g the tip closure A = 2~ = 2(bo - b ) a n d m a k i n g use of the fact t h a t axial strains a r e negligible, L 2 = A B 2 = A o B ~ = b 2 + d ~ = b~+d~
Thus d = ~/(d~ + 2b 0 ~ - r ~)
(3)
As d 0, b 0 are d e t e r m i n e d b y the initial shape of the specimen it is a simple m a t t e r to e v a l u a t e the n e w lever a r m in terms of the tip closure a n d to plot the load-deflection
7 zf
Io
131~ line
F i e . 1. Specimen configuration.
I
/I
,
--'~}_/lire 'or
tip closure (a)
tip closure (b)
FIo. 2. Load-deflection relationships. (a) Rigid-plastic analysis; (b) elastic-plastic analysis. characteristics of a m e m b e r as shown in Fig. 2(a). As the m e m b e r deforms as two rigid links the hinge r o t a t i o n is given b y
/ ~ b0 2~tanXo-tan- 1 ~)b~
(~)
The work done, U, in deforming the specimen, which is simply the area u n d e r the load-deflection curve, is given b y 2 f:Pd7
= 2
F~'
= 2m~[sin -1
d7 y-bo
bo
636
B. RAWLINGS
2.2. Elastic~lastic analysis I f account is taken of the finite elastic modulus of the material prior to the development of a plastic hinge at the section of m a x i m u m moment, the initial response of the specimen will be an elastic one, the load-deflection relationship being
La 2PLd~ A = 2 P c o s 2 ¢ 0 . 3 E I = 3EI
(5)
where ¢0 is the semi-angle at the elbow, and E I is the flexural rigidity of the section. I f the development of partially plastic zones of material is ignored, the line given by equation (5) then m a y be drawn from the origin to intersect the rigid-plastic line, as shown in Fig. 2(b). By similar reasoning, the elastic unloading line at any stage, having the relationship 2PLd 2 3EI can be drawn in, as shown in Fig. 2(b). The work U, expended in deforming such a member to any given degree of closure, is given by the area under the curve atvaa, and the recoverable elastic energy R, equal to the area under the triangle va T, and given by
R-
Lm~ 3EI
(6)
Consequently, the relationship between the work and tip closure in this case is similar to that for the rigid-plastic case, except for the area of the zone af]e, and the dissipated energy ( U - R) is obtained by deducting the additional term given by equation (6). 3. E X P E R I M E N T A L
WORK
In order to examine the energy absorption of mild steel specimens under conditions of gross deformation, both static and impact tests were conducted. The material used for the tests was ~ in. square black mild steel bar, this size being selected because it proved convenient for the testing equipment, and it could be bent by hand tools into the desired configurations. The material had the following chemical composition: C 0.13%
S 0.048%
P 0.038%
Si 0.06%
Mn 0.43%
Ni 0.02%
Cr 0.03%
Mo 0.01 ~)
The material was subjected to a considerable degree of cold working in the process of manufacture, and it was decided to do some tests on it in this condition, firstly to examine the properties when effects of upper yield stress would be eliminated because of previous cold working, and secondly because it was felt that the characteristics would more closely simulate those of a material with less significant strain-hardening characteristics than normalized mild steel taken into the early strain-hardening range. Furthermore, provided the material has sufficient ductility, the energy absorption per unit volume is greater with material having higher yield stress values, and, in practice, the most efficient energyabsorbing structure would appear to be the one having the greatest yield stress, provided the ductility were not sacrificed. 3.1. Fabrication of specimens Specimens were of V shape, with straight sides, and had included angles of 60 °, 90 ° and 120 ° . The dimensions were such that, in their original configuration at the commencement of each test, the lever arm of the forces closing the tips, as measured to the hinge at the elbow, was in all cases 4 in. Each specimen was carefully fabricated from straight lengths by bending it in a jig around a mandrel, 0-198 in. dis. to the appropriate angle. Care was taken to ensure that the material was not deformed beyond the angle required. The specimens were then marked out and drilled in order to take the pins for connexion to the supports.
Energy absorption of dynamically and statically tested mild steel beams
637
3.2. Static tests on specimens Static tests were performed in an Amsler hydraulically operated testing machine on the 200 lb range of loading. The specinlens were pinned at the ends to fittings which were aligned, and measurements were taken of the central lateral deflection, 8, and the closing of the ends, A, as shown in Fig. 3. Frequent unloading and reloading curves were plotted
com.~esting
machine
gauge measuring e
FIG. 3. Static test details. during the course of the tests, which were continued either to the stage where the tips had closed together or to fracture of the extreme tensile fibres if this occurred previously. A hydraulic machine was selected as in m a n y cases the load-deflection characteristic of the specimen was a falling one, and the use of a rigid type of loading device enabled load to be applied at a constant strain-rate. Static control tests were also conducted on straight lengths of material, which was loaded transversely on an 8 in. simply supported span by a concentrated central load. These furnished information on the full plastic moment of resistance and strain-hardening characteristic of each material length from which the specimens were manufactured, and enabled all results to be reduced to a common value of full plastic moment. 3.3. D y n a m i c tests on specimens Groups of three specimens, identical to those described above, were tested under impulsive loading in a machine designed for applying dynamic loads to structures and defe~-tiontr'~rc~cer
breech ~o~
valve
~o~,e
~: II
timercrosswires
g®
3==~1
pressc¢~gauge
f~rce ~ e v
Ib
movingcrosshead
@
~1
~Jmen
~.
m~r~t,e
ar r~ceiver CR.Q
trigger~l ~
deflection~
~r~ns ~_
FIG. 5. Diagrammatic representation of apparatus. structural elements. A photograph of the equipment is shown in Fig. 4, and a diagrammatic representation is given in Fig. 5. Essentially the equipment consists of a pneumatically operated gun, together with a projectile and loading mechanism which moves and deforms the specimen. The gun itself consists of a 10 ft length of precision-drawn steel
638
B. RAWLINGS
tube, 1.117 in. internal dia. and 0.192 in. wall thickness, s u p p o r t e d at close intervals on a double-channel frame. Pressure is applied to the gun b y air from a large air receiver, o p e r a t i n g at up to 225 lb/in ~ pressure, t h r o u g h a plug v a l v e w i t h large ports, which has been found to give consistent p e r f o r m a n c e in operation. I n the present series of tests, a large projectile has been used, consisting of a 3 ft length of steel rod, weighing 9.33 lb, t u r n e d to 1.094 in. dia. a n d h a v i n g r u b b e r O-rings as seals. A p h o t o g r a p h of it, resting alongside the gun, is shown in Fig. 4. The projectile, which was lubricated b y light m a c h i n e oil, could be set at a n y desired position prior to a test, b y unscrewing t h e breech cap a n d inserting a rod into the barrel. I t will be a p p r e c i a t e d t h a t , b y v a r y i n g the pressure, the initial position of t h e projectile and its mass, a v e r y wide range of muzzle velocities could be obtained. The a p p a r a t u s was originally designed to h a v e considerable v e r s a t i l i t y in the loading m e c h a n i s m arrangements, b u t for this series of experiments, these t o o k the form of a fixed frame w i t h four guides, on which a t u b u l a r t u p connected to a crosshead slides when s t r u c k b y the projectile. The specimen is held b y pins to two spring steel cantilever m e m b e r s , one at each end, to which are a t t a c h e d electric resistance strain gauges. These cantilevers are connected, one to t h e crosshead, a n d one to t h e s t a t i o n a r y b a c k plate, so t h a t the line of action of the force on the specimen coincides w i t h the axis of the gun. These p a r t i c u l a r loading a r r a n g e m e n t s were selected because t h e y p e r m i t t e d gross deformations of the specimens to develop w i t h o u t u n d u l y restrictive limitations on travel. 3.4. Instrumentation As m e n t i o n e d above, each specimen was s u p p o r t e d at its ends by spring steel cantilever transducers, which h a d a n a t u r a l f r e q u e n c y of v i b r a t i o n of the order of 4000 c/s. E a c h had a t t a c h e d to the faces, two electric resistance strain gauges a r r a n g e d as an active, d u m m y pair. The gauges of t h e crosshead t r a n s d u c e r were connected to a d y n a m i c strain bridge, h a v i n g a carrier f r e q u e n c y of 5000 e/s, t h e o u t p u t of this being fed into one channel of a double-beam c a t h o d e - r a y oscilloscope. The t r a n s i e n t deflection was d e t e c t e d b y a linear resistance displacement transducer, consisting of a long coil of wire located w i t h a horizontal axis, and s i t u a t e d vertically o v e r the centre line of the target. A spring-loaded sliding contact, a t t a c h e d to t h e crosshead, established a variable e a r t h p o i n t on t h e coil. The t r a n s d u c e r was o p e r a t e d from direct current, the o u t p u t being fed into t h e second channel of the oscilloscope. The horizontal sweep of the oscilloscope was a c t u a t e d b y t h e fracture of a fine wire s t r e t c h e d across the muzzle of t h e gun. P h o t o g r a p h i c records of t h e oscilloscope traces were m a d e b y a P o l a r o i d c a m e r a m o u n t e d in front of t h e screen. A typical record is shown in Fig. 6. I n order to d e t e r m i n e t h e incident velocity, and consequently the m o m e n t u m and kinetic energy of the projectile, a t i m i n g device was installed in its path, at a small distance in front of the tup. This comprised two pairs of fine cross-wires, 4 in. apart, each pair consisting of two skew wires lying in planes n o r m a l to the projectile and ~ in. apart. Short-circuiting b e t w e e n t h e first pair of wires f r o m t h e passing of the projectile initiated a microsecond c o u n t e r - t i m e r which was stopped b y the second pair. 3.5. Test procedure A f t e r setting up the specimen in the testing apparatus, care was t a k e n to ensure t h a t the crosshead a n d t u p were free of friction, and baselines for t h e force a n d deflection traces were established. The projectile was set to its correct position, receiver charged to t h e r e q u i r e d pressure, a n d t h e i m p a c t test performed. This pressure was k e p t a t q u i t e a low value, e.g. 15 lb/in ~, in order t h a t after the test, t h e static residual force on t h e projectile was small w h e n c o m p a r e d w i t h the specimen resistance. A f t e r t h e test, t h e projectile was t h e n r e t r a c t e d from t h e t u p and a double exposure t a k e n to establish the final positions of the load and deflection traces in order to d e t e r m i n e the p e r m a n e n t set of the specimen. 3.6. Calibrations B o t h the force t r a n s d u c e r and displacement p o t e n t i o m e t e r were calibrated i m m e d i a t e l y after each test was completed. The force t r a n s d u c e r was r e m o v e d and r e m o u n t e d in an
(b)
FIG.
4. (a) General view of apparatus; (b) close-up member, transducers and crosshead.
view
showing
f. y. 638
FIG. 6. Typical C.R.O.
trace.
FIG. 7. Statically tested members.
FIG. lO(a). Section of tested member showing cross-sectional distortion at elbow.
FIG. lO(b). Dimensions of cross-section at hinge of a 90° specimen after testing.
FIG. 15. Dynamically
ksterl
members.
Energy absorption of dynamically and statically tested mild steel beams
639
elevated position, and subjected to incremental dead weight loading from a weight carrier suspended from the connexion pin. This gave, on the oscilloscope, a series of parallel lines which were photographed a n d subsequently used to establish the scale of the transient load trace. The deflection transducer was also calibrated b y moving the crosshead through incremental distances covering the range of deformation, and photographing the traces. The time scale was established b y superimposing on one photograph a 50 c/s waveform. 4. R E S U L T S 4.1. Static tests Photographs of the specimens after testing, with identical specimens before testing, are shown in Fig. 7. I t will be seen that even under conditions of gross deformation, plasticity has been confined to an area very close to the elbow, so that the assumption of point hinge action is valid. The experimental relationships between applied load and lateral deflection, and applied load and end closure are shown in Fig. 8, together with
loo I
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deflection - irt 12@
d e f l e c t ~ n - irt
deflect}on - in.
9@
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FIe. 8(a). Experimental relationships between applied load and deflection--static tests.
--~4
¢
i
~2
~3 32
¢2
2 4 6 tip closure - in. 120 °
8
10
()
2 4 6 tip closure - in. 90 ~
O" 2 4 tip closur~ - in. 6O°
Fro. 8(b). Experimental and theoretical deflection--tip closure relationships. unloading and reloading curves. One point of interest is the variation in slopes between the unloading lines a t various stages of testing, due to the large geometry changes occurring during the course of the test. I t is also of interest to compare the experimental relationship between the lateral deflection and end closure and the theoretical relationship based upon equation (3), as shown b y the full and dotted lines, respectively, in Fig. 8(b). The close agreement in all cases again confirms that the specimen deformed almost as two rigid links, which indicates that strain hardening was not significant enough to increase the moment of resistance to a greater extent than the applied moment gradient along each arm of the specimen.
640
B. RAWLI:NGS
Making use of the knowledge t h a t t h e m e m b e r s did in fact act as two rigid links, relationships b e t w e e n the full plastic m o m e n t of resistance and t h e change in angle A C B (Fig. 1) f r o m t h e original v a l u e A 0 Co B0 h a v e been p l o t t e d in Fig. 9. I t will be clear t h a t the steeply rising region n e a r t h e origin is in each case due to t h e elastic response, after which plasticity developed at the elbow a n d c o n t i n u e d hinge r o t a t i o n occurred at v i r t u a l l y c o n s t a n t m o m e n t of resistance until t h e end of the test. As the a s s u m p t i o n is m a d e in t h e simple plastic t h e o r y of bending t h a t plane sections r e m a i n plane and t h a t fibres in the b e a m cross-section b e h a v e independently, it is of interest to observe j u s t how high the c u r v a t u r e s at the elbow become, a n d to see the gross distortion of the cross-sectional dimensions. The material, as m e n t i o n e d previously, was ¼ in. square in cross-section, but the cross-section at a typical hinge, after completion of a test, had t h e dimensions shown in Fig. 10.
~400
~3oo[
~3oo
~oo
b°I
i~2oo
{lo0
"-~1oo
20
40 60 80 100 in angle- degr,ees 120~
d
""100
~
20
40 60 in angledeles
20 cl"angein angle -
FIa. 9. l~elationship b e t w e e n full plastic m o m e n t and angle c h a n g e - - s t a t i c tests. Straight b e a m s were also tested, as m e n t i o n e d in Section 3.2, ill order to establish t h e full plastic m o m e n t of resistance of t h e p a r e n t material, and these were loaded until a considerable degree of hinge r o t a t i o n had occurred. The load-deflection curve for t h e l e n g t h f r o m which t h e other static specimens were cut is shown in Fig. l l ( a ) . As t h e large central deflection was a c c o m p a n i e d b y pull-in of the roller supports, which was measured, it was necessary to correct for this w h e n establishing the m o m e n t of resistance; c o n s e q u e n t l y t h e relationship b e t w e e n m o m e n t of resistance and central deflection is shown p l o t t e d as t h e c u r v e in Fig. l l ( b ) . Making the same a s s u m p t i o n as in t h e tests on t h e V-shaped specimens t h a t all deformation was confined to r o t a t i o n at t h e section of m a x i m u m m o m e n t , this c u r v e could also be r e d r a w n as a m o m e n t - h i n g e r o t a t i o n c u r v e for an initially straight m e m b e r , as shown in Fig. l l ( e ) . I t is n o w possible to represent all four curves, in Figs. 9 and l l ( e ) , on t h e one set of axes, b y c o m m e n c i n g at the a p p r o p r i a t e angle to represent t h e initial elbow angle of t h e specimen, as shown in Fig. 12. These indicate quite clearly t h a t it was a close a p p r o x i m a tion to assume t h a t t h e full plastic m o m e n t of resistance could be t a k e n as a c o n s t a n t 365 lb-in, t h r o u g h o u t t h e whole range of loading, until t h e specimen was b e n t v i r t u a l l y double. Consequently the a s s u m p t i o n to be made, in considering t h e v a l i d i t y of t h e rigid-plastic theory, is t h a t the m a t e r i a l has the properties shown in Fig. 13. F u r t h e r m o r e , if a t t e n t i o n is directed b a c k to t h e load-closure relationships for each of the three b e n t specimens, as shown in Fig. 8, it is possible simply to s u b s t i t u t e in the a b o v e assumed v a l u e of m~ in e q u a t i o n s (2) and (3) and to plot t h e theoretical rigid-plastic curves shown in d o t t e d lines in Fig. 14, t h e unloading a n d reloading lines in all specimens being, of course, v e r t i c a l in this case. These lines are p l o t t e d to the limiting condition in which t h e two arms of t h e specimen become parallel; indeed this value would h a v e to be reduced by the thickness of the material.
Energy absorption of dynamically and statically tested mild steel beams
200 160 12C
!
0
80
(a}
40 O'
i
0:4 ' 0:8 ' 1.2 ' 1:6 ' 2:0' central deflection- in.
c' .(3 I
eu 400
c
$ 3oc
L
o .~ 200 c-
(b}
©
E
~ 100
0
'0'4'0:8' 1:2 ' 1.6' 2tO central deflection- in.
FIG. 11. Static tests on straight lengths. (a)Load-deflection relationship; (b) central moment-deflection relationship.
° _
--~400
~300S ~ 200
{el
i lOO
o
Ib 2b 3b 40
hinge Potation- degrees
50
Fro. l l(c). Moment-hinge rotation relationship.
641
642
B. RAWLINGS J540C
/
30C
20C
~
100
0
E 0 18
,
,
160
J
,
,
,
,
,
,
140 120 100 80 60 40 included angle of specimen - degrees
,
,
,
,
20
0
FIG. 12. M o m e n t - i n c l u d e d angle r e l a t i o n s h i p s - - s t a t i c tests.
~365 JD
C
18o hinge Potation - degrees
Fro. 13. A s s u m e d m o m e n t - h i n g e r o t a t i o n relationship. 100
__I0(
4100 I
I
80 60
"'"
60
40
4O
40
°20
20
2c
o
' :~ ' 4 ' 6 ' 8 deflection - in. 12ff
'16
' 1~ '14(~
2 4 deflection - in 90"
6
8
2 4 deflection - in 60 ~
FIG. 14. Theoretical load-deflection curves based on rigid-plastic
theory. 4.2. Dynamic tests The d a t a relating to the various d y n a m i c tests are g i v e n in Table 1. P h o t o g r a p h s of d y n a m i c a l l y tested specimens after testing, shown in Fig. 15, exhibit the s a m e b e h a v i o u r w i t h v i r t u a l l y all d e f o r m a t i o n occurring at t h e elbow, as t h e static specimens u n d e r test. The loa~l-time a n d d e f l e c t i o n - t i m e traces recorded in a typical test are shown in Fig. 6. I t will be seen t h a t t h e load increased v e r y rapidly to a m a x i m u m value, accompanied b y a certain degree of vibration which rapidly subsided, after which t h e load g r a d u a l l y diminished. The first stage in the process of analysing t h e records was to redraw the curves carefully on g r a p h paper, w i t h uniform scales for force, t i m e a n d deflection. This was done b y superimposing t h e force and deflection calibration lines on t h e record, t o g e t h e r w i t h t h e 50 c/s t i m e marks, establishing a n u m b e r of points precisely on the g r a p h and i n t e r p o l a t i n g as carefully as possible. F r o m these rectified f o r c e - t i m e
120 120 120 90 90 90 60 60 60
7 10 33 13 21 22 18 19 20
13.92 13.88 13-86 8.30 8-10 8.12 4.90 4.90 4.91
(3)
(in.)
(degrees)
(2)
Initial spacing
Angle a t elbow
(1)
Specimen No.
7.0 7.0 7.0 7.0 7.0 7.0 5.0 4.0 4-0
(4)
(ft)
Projectile travel
15.0 15.0 15.0 15.0 15-0 15.0 15.0 15.0
15.0
(5)
(lb/in ~)
16.70 15.15 17-70 16-66 17.89 18-92 16.66 13-55 13.55
(6)
(ft/sec)
Velocity v0
8.66 7-86 9.28 8.94 9.58 10.18 9.29 7.66 7-66
(7)
(ft/sec)
v2
Velocity after impact,
R E S U L T S OF DYI~'AMIC TESTS
Projectile pressure
TABLE ].
9.78 9.10 10.40 9.82 12.20 11-18 11-37 6-93 6-93
(8)
(ft/sec)
v2
Velocity after i m p a c t as measured from C.R.O. t r a c e ,
11.25 12.04 10-90 5.60 4.86 4-72 2-20 3.06 3.05
(9)
(in.)
AB
Final spacing o f ends,
2.67 1-84 2.96 2.70 3.25 3-40 2.70 1.84 1.86
(10)
(in.)
Permanent set as d e t e r m i n e d experimentally
o
644
B. R A W L I ~ S
and d e f l e c t i o n - t i m e curves it was t h e n possible to draw a load-deflection curve for each specimen. I t was r a t h e r difficult to establish precisely t h e slope of the elastic rising line, because of t h e v e r y small r i s e - t i m e of t h e force pulse, and this accounts for t h e v a r i a t i o n b e t w e e n records in the curves shown in Fig. 16. A v a r i a t i o n occurred also in the magnit u d e s of t h e force ordinates during t h e yielding phase, due p a r t l y to differences between specimens and p a r t l y to errors in reducing t h e record. The largest difference occurred b e t w e e n specimen 33 a n d t h e o t h e r two 120 ° specimens, the v a r i a t i o n from the m e a n of the m a x i m u m force in this case being 18 per cent. I n the case of the 90 ° and 60 ° specimens, however, this v a r i a t i o n a b o u t the m e a n a m o u n t e d to 8 per cent.
lOO
100
10C
cimen 0
8C
18
:)ecimen 21 80
80
"--'--.
-~I 60i
_~ 4c
2¢
2O
2C
2 ~,~ 4 d 2 4 def~:tion - irt deflection -JR defL-:~ztion- in 9O o 120° 6o ° Fro. 16. Force-deflection r e l a t i o n s h i p - - d y n a m i c tests.
I f m e a n lines are d r a w n for each group, and c o m p a r e d w i t h the l o a d - t i p closure curves o b t a i n e d f r o m t h e corresponding static tests, it will be seen t h a t t h e loads in t h e d y n a m i c tests are e n h a n c e d b y a p p r o x i m a t e l y 12 per cent o v e r those of the static tests, due to t h e s t r a i n - r a t e sensitivity effects of the material. This a g r e e m e n t b e t w e e n d y n a m i c and static results is r a t h e r closer t h a n t h a t which m i g h t h a v e been expected, for t h e particular t y p e of material, b u t is due to t h e severe degree of cold working in t h e m a n u f a c t u r e of the m a t e r i a l and fabrication of t h e specimen; this has reduced the s t r a i n - r a t e sensitivity considerably. I n fact, tests on identical specimens of normalized material, which will be r e p o r t e d elsewhere, h a v e e x h i b i t e d significantly greater s t r a i n - r a t e sensitivity t h a n these results.
4.3. Work done on specimens during deformation The w o r k r e q u i r e d to deform each of the statically loaded specimens to any given degree of tip closure, represented b y the typical area ae~'aa in Fig. 17, has been p l o t t e d in Fig. 18, t o g e t h e r w i t h t h e recoverable energy available u p o n unloading, represented b y t h e triangular area, ~a~. I f the a s s u m p t i o n is m a d e t h a t the d y n a m i c a l l y loaded specimens deform in t h e static mode, b u t t h a t t h e y h a v e a d y n a m i c full plastic m o m e n t of resistance which is 12 per cent higher t h a n t h e static value, the w o r k done, Us, in deforming a d y n a m i c a l l y loaded specimen would be, to a close a p p r o x i m a t i o n , the value o b t a i n e d f r o m Fig. 18, multiplied b y 1.12. H o w e v e r , it is possible to e v a l u a t e the kinetic energy of t h e projectile, t u p and crosshead, i m m e d i a t e l y a f t e r impact, in each test, and by m a k i n g a correction for the additional work done b y the n e t projectile force (pA -Fir ) during t h e deformation, to d e t e r m i n e the work done in deforming the specimen to its m a x i m u m e x t e n t b y observations during each test. These e x p e r i m e n t a l values are shown t a b u l a t e d in Table 2, column 7, and are denoted by UE. B y e q u a t i n g Ug and Us it is possible to d e t e r m i n e t h e m a x i m u m deformation of each specimen, a n d this is shown t a b u l a t e d in c o l u m n 8. The elastic recovery, being 1.12 times t h e corresponding value from t h e static test, m a y be s u b t r a c t e d from this m a x i m u m d e f o r m a t i o n to give t h e derived v a l u e of p e r m a n e n t set, listed in column 9. I t will be seen b y c o m p a r i n g these
Energy absorption of dynamically and statically tested mild steel beams
645
with results in columns 4 and 3, that in general the agreement between the observed and predicted results for both m a x i m u m closure and permanent set agree within 12 per cent for the m a x i m u m closure and 25 per cent for the permanent set.
r~
6°I
-~40
-- 20
0
deflection
in.
-
17Io. 17. L o a d - d e f l e c t i o n
600
curve--static
test..
/ .
work requiredto
J /
500 400
/ /
400
~400
r, I
1 O0
/
i=
. . . . . . . . . . 2
4
6
deflection - irt 120'
8
10C
10
12
i00
O'
2
4
. . . . .
(J
defied.ion - in. 90'
2
4
deflection - in. 60~
FIO. 18. Work-deflection curves--static tests. 600 500
60C work required to /
deform,specim~//, 50C
/
500
/ i
z/
400
40C
/
--•300
400~
e"
///
I
100 /~///
10C
S' /
O'
2
4
6
deflection - In. 120'
8
~ 300 I
,
pOioo,i Jl II ,
,
2
,
4
.
deflection - In.
90"
.
6
.
l/I
/
.
~
2
~
deflection - in.
ecr
FIe. 19. Work-deflection curves--rigid-plastic theory. If, on the other hand, the assumption is made that the material has rigid-plastic characteristics, the dynamic full plastic moment being 12 per cent above the static value, or 409 lb-in., the curves relating work to tip closure take the form shown in Fig. 19. Denoting this work term b y U2p it is possible to equate ORe to UE and to derive a
(2)
120 120 120 90 90 90 60 60 60
7 10 33 13 21 22 18 19 20
(degrees)
Angle
(1)
81)ocimen No.
2-67 1.84 2.96 2.70 3.25 3.40 2-70 1.84 1.86
(3)
(in.)
Permanent set as determined experimentally
4.5 3.8 4.5 4-0 4.5 5.0 3.7 3.1 2.9
(4)
(in.)
Maximum closure during impact, from C.R.O. trace
E x p e r i m e n t a l results
T A B L E 2.
251 207 288 268 308 348 288 196 196
(5)
(lb-in.)
K . E . of moving parts after impact,
26 18 29 23 30 37 30 20 19
(6)
(lb-in.)
Work done by projectile after impact
277 225 317 291 338 385 318 216 215
(7)
(lb-in.)
UE
Total energy applied to specimen
4.3 3-5 4.9 3-8 4.4 5.0 3.9 2-7 2.6
(8)
(in.)
Maximum closure
3.1 2.3 3.4 2.8 3.3 4.2 3.2 1.9 1.8
(9)
(in.)
Permanent set
D e r i v e d results based on factored values of Static Tests
C O M P A R I S O N OF T H E O R E T I C A L A N D E X P E R I M E N T A L R E S U L T S
3.4 2-7 4.0 3.3 3.9 4.5 3-4 2.2 2-2
(10)
(in.)
Closure
Theoretical results from rigidplastic theory
4.1 3.3 4-7 3-7 4.3 4.9 3.7 2.5 2.5
(11)
(in.)
Maximum closure
2.7 2.0 3.2 2-9 3.8 4.3 3.1 1.9 1.9
(12)
(in.)
Permanent set
Theoretical results from elastic-plastic theory
G~
Energy absorption of dynamically and statically tested mild steel beams
647
theoretical value of tip closure, as shown in Table 2, column 10. As rigid-plastic behaviour is assumed, there would be no elastic recovery, on this assumption. I t will be seen t h a t in all cases the theoretical values are significantly lower than the experimentally recorded m a x i m u m closures, lying in general approximately midway b e t w e e n the m a x i m u m closure and the final permanent set. Finally, if the assumption is made t h a t the material is elastic-plastic, with full plastic m o m e n t of 409 lb-in., as shown in Fig. 20, it is possible to plot modified curves to allow for the initial elastic response, as shown by the dotted lines in Fig. 19. Denoting these
I
409t !
rotation - degrees FIG. 20. Assumed elastic-plastic m o m e n t - r o t a t i o n relationship. energy terms by UEp, and equating UEp to UE, the theoretically derived values of m a x i m u m tip closure are those shown in column 11 of Table 2, and assuming elastic recovery to occur in the static mode, the permanent set values are tabulated in column 12. Examining again the experimental values in columns 4 and 3 the agreement between experimental and theoretical values has improved to the same order as those predicted from factored values of the static tests, the worst error being a 25 per cent over-estimation of permanent set in the case of specimen 22. 4. C O N C L U S I O N S Simple flexural elements of mild steel m a y be fabricated to absorb significant amounts of energy, whilst undergoing gross plastic deformation. I n order to evaluate the static load-deformation characteristics it is necessary to take account of the actual m o m e n t rotation characteristics of the plastic hinge. However, if strain hardening is not so severe as to cause hinge displacement and the elastic component of deformation is small when compared with the hinge rotation component the normal rigid-plastic assumptions of behaviour lead to a close prediction of the load-deflection relationship, provided account is taken of the constantly changing structural geometry. Corresponding dynamically tested specimens have similar load~leflection characteristics to statically tested specimens, provided some account is taken of the increase in full plastic moment of resistance due to the strain-rate sensitivity of the material. The deformed shapes of both statically and dynamically tested specimens indicate a similar degree of spread of the plastic zones, and similar appearance of local deformation of material at the elbow. Allowing for the errors associated with the reduction of results from force-time and deflection-time oscilloscope traces, the elastic loading and unloading lines correspond with the assumption of specimen deformation in the static mode, and the same falling load~leflection characteristic over the plastic deformation region as in the static tests. I n evaluating the rigid-plastic theory, it is convenient to compare the total energy applied to the specimen with the absolute m a x i m u m elastic strain energy which could be stored, if the specimen were in a state of uniform flexure. In the specimens tested the m a x i m u m possible elastic strain energy which could be stored ranged from 1/4 to 1/1.6 of the total energy applied; the actual ratios, allowing for linearly varying moment conditions, v a r y from 1/12 to 1/4.8. The rigid-plastic theory led to predictions of closure which were intermediate between the actually recorded m a x i m u m closure and permanent set values ; however, the errors were of the order of 20-30 per cent in most cases.
648
B. RAWL~NGS
When allowance was made for the finite elastic modulus of the material in the theory, the agreement between theoretical predictions and experimentally observed values of m a x i m u m closure and permanent set improved significantly, the average error being of the order of 10 per cent. As more rigorous evaluation of the dynamic rigid-plastic theory would require tests over a wider range of conditions than those described above, the present series of tests must be regarded as being but a first stage ; however, further work is being planned on specimens in which the elastic energy will be significantly less important in relation to the plastic work of deformation, and this should give data for the determination of limits of validity.
Acknowledgements--This work was carried out in the Materials and Structures Laboratory of the School of Civil Engineering, University of Sydney, the Professor of Civil Engineering and Head of the School being Professor J. ~ . Roderick. The research was financed from funds made available by the Australian Research Grants Committee. The author wishes to record his sincere thanks to Professor Roderick, and acknowledge the assistance given by Messrs. W. J. Cottam and J. McEnnally both in the experimental work and reducing the results. REFERENCES 1. J. F. BAKER, The Civil Engineer in War, Vol. 3, p. 30, Institution of Civil Engineers, London (1948). 2. G. E. DENT, Struct. Engr, 42, 39 (1964); A. G. SENIOR, Struct. Engr, 42, 345 (1964). 3. H. G. HOPKINS, Appl. Mech. Rev. 14, 417 (1961). 4. B. RAWLINGS, Trans. Instn Engrs Aust. CES, p. 89 (1963). 5. S. R. BODNER and P. S. SYMONDS, Plasticity, p. 488. Pergamon Press, Oxford (1960). 6. B. RAWLI~GS, Proc. R. Soc. A, 275, 528 (1963). 7. R. J. ASPDE~ and J. D. CAMPBELL, Proc. R. Soc. A, 290, 266 (1966). 8. E. W. PARKES, Proc. R. Soc. A, 228, 462 (1955). 9. B. RAWLINGS, Proc. Instn civ. Engrs, 29, 389 (1964). 10. B. RAWLI~GS, J. mech. Engng Sci. 6, 327 (1964). l l . H. H. BLEICH and M. G. SALVADORI, Trans. Am. Soc. civ. Engrs, 120, 499 (1955). 12. R. C. ALVERSO~, J. appl. Mech. 23, 411 (1956). 13. J. A. SEILER, B. A. COTTER and P. S. SYMONDS,J. appl. Mechs, 23, 515 (1956). APPENDIX
By making use of the principle of conservation of m o m e n t u m it is a simple matter to check the velocity of the tup immediately after impact against the value independently obtained from measurements of the oscilloscope trace. Furthermore, the distance through which the specimen deforms can be readily checked against the measured value. initial position , of specimen^ i, D
nt~l
~ t i o n of projectile
~
L
a
Fro. 21. Analysis of motion of tup. I t is convenient to represent the testing arrangement diagrammatically as shown in Fig. 21. Adopting the following notation: a acceleration of projectile prior to impact A area of projectile d deceleration of projectile after impact D spacing between "start" and "stop" pulse wires
E n e r g y absorption of d y n a m i c a l l y and statically tested m i l d steel b e a m s
649
-FI~ Me N/~ p Pl P2 P S1
frictional force opposing m o t i o n (assumed constant) mass of t u p a n d crosshead mass of projectile pressure on projectile initial pressure on projectile final pressure on projectile specimen force distance f r o m front of initial position of projectile to point m i d w a y b e t w e e n t i m i n g wires $2 distance from front of initial position of projectile to t u p T i n t e r v a l of t i m e t a k e n b y projectile to t r a v e r s e t i m i n g wires Assuming t h e frictional force FI~ opposing the m o t i o n to be c o n s t a n t t h r o u g h o u t t h e test, the e q u a t i o n of m o t i o n of t h e projectile prior to i m p a c t is a
p A --F~T My
(i)
I f Pl a n d p , are nearly equal, for low piston velocities it m a y be assumed t h a t the pressure p is constant, h a v i n g the v a l u e {Pl +P2)/2, so t h a t the p r o b l e m simplifies to the case of c o n s t a n t acceleration conditions. The velocity v 0 at the stage shown in Fig. 21, w h e n the projectile has t r a v e l l e d distance S 1, is g i v e n b y v~ = 2aS 1 (ii) whilst the velocity v 1 i m m e d i a t e l y prior to i m p a c t is given b y v~ = 2aS2
(iii)
I t can be shown t h a t , for the ease when D / S 1 is small, the velocity v 0 is given b y D I n t h e tests outlined in this paper, D/S~ was of the order of ~-~, so t h a t t h e error in assuming D / T = v 0 was negligible. Thus, from equations (ii) and (iv) it was possible to e v a l u a t e a, and consequently (pA---Fsr ), and t h e velocity v 1 could be c o m p u t e d , allowing for friction. A t the m o m e n t of i m p a c t it is assumed t h a t the projectile a n d t u p coalesce, so t h a t , f r o m conservation of m o m e n t u m , the c o m m o n velocity v, i m m e d i a t e l y after i m p a c t is given b y M~ v~ M~ + M~ vl (v) -
43