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Finite element examination of the dynamic response of clamped beam grillages impacted transversely at their centres by a rigid mass
~
Pergamon
Int. J. hnpact Enttn.q Vol. 15, No. 5, pp. 687-697, 1994 Elsevier ScienceLtd Printed in Great Britain 0734-743X(93)E0009-7 0734-743x/94 $7.00+ 0.00
FINITE E L E M E N T E X A M I N A T I O N O F THE D Y N A M I C R E S P O N S E O F C L A M P E D BEAM G R I L L A G E S I M P A C T E D T R A N S V E R S E L Y AT THEIR C E N T R E S BY A RIGID MASS KAZEM KORMI, EBB! SHAGHOUEI a n d DAVID A. DUDDELL Centre for Advanced Research in Engineering, Leeds Metropolitan University, Calverley Street, Leeds LSl 3HE, U.K.
(Received 20 August 1992; in revised form 5 November 1993)
Summary The finite element package ABAQUS is used to investigate the response of metallic (aluminium and steel) beam grillages to impact loading. Modelling is based on linear elastic materials with non-linear isotropic work hardening characteristics including strain rate dependency for the steel. Two different types of model are considered, one using space beam and the other three-dimensional solid elements. Comparison of the computed values with experimental results (N. JONES et al., Int. J. Impact Engng 11,379-399 (1991)) is very encouraging.
NOTATION FEM o'~. v E P b 1, b, hi, It2 i wf
wos
Sxx, Sxr, etc.
finite element method yield stress Poisson's ratio modulus of elasticity density width of main and cross beams, respectively depth of main and cross beams, respectively depth of indentation by striker into beam permanent transverse displacement permanent transverse displacement at the centre point permanent transverse displacement at the beams' junction direct and shear stress components
1. INTRODUCTION
In the field of impact loading of structures, the major proportion of the research so far has been dedicated to the development of theory which incorporates a number of simplifying assumptions. Usually, the theory is based on the evidence of the results obtained by conducting experiments. It appears that the process can be improved by using the finite element method (FEM). One major source of inertia resisting the use of FEM would appear to be the belief that the results obtained are not necessarily reliable hence the resort to what would hitherto be considered orthodox procedures. The present paper offers further confirmation of the inappropriateness of this attitude. It is a continuation of previous studies where FEM results are compared with carefully conducted experiments [1]. In this paper FEM analysis are compared with the experimental results reported by Jones et al. [2] for impact of steel and aluminium grillages by a rigid mass. 2. M O D E L L I N G
2.1. Grillage configuration The system comprises a main beam 270 mm in length, fixed at both extremeties and trisected orthogonally by two cross beams bisected equally by the main beam and of length 180 mm as shown in Fig. 1. The ends of the cross beams are also fully encastred. The 687
K. KORMI et al.
688
b2 SECTION
//L
'
i h2
MPACT
IlL
~A
k o
N
L
L
• A
N
~
L
SECTIONAA
b0
hI FIG. 1. Schematic view of the beam grillage assembly.
TABLE 1. MODEL NAMES AND DESCRIPTIONS
Dimensions Model name MSGII5 MSGI3 MSGII2s MSGII2 MAGII2b MAGII2
Specimen number [-2]
Material
Impact velocity (m s- t)
SGII5 SGI3 SGII2 SGII2 AGII2 AGII2
Steel Steel Steel Steel Aluminium Aluminium
7.34 7.00 5.42 5.42 5.42 5.42
(ram)
bl
hi
b2
h2
Element type
6 6 6 6 6 6
6 6 6 6 6 6
10 6 10 10 10 10
6 6 6 6 6 6
B32 B32 C3D8 C3D8R B32 C3D8
specimens used in the experiments were machined from aluminium or steel sheet material 6 or 10 mm thick [2]. The actual combinations are shown for the different models in Table 1. The impact velocities for the different models are also included in Table 1. 2.2. Finite element mesh In order to give a representative set of FEM results, four experiments are chosen to be modelled and foi- two of these two different analyses are performed. The models are summarized in Table 1. All analyses are carried out using ABAQUS [3] and F E M G E N / F E M V I E W [4] is used for graphical presentation of results. In one set of analyses space beam elements (ABAQUS type B32), each of 9 mm length, are used to model one half of the grillage assembly. The cross-over junction nodes are separately generated then, to preserve kinematic compatibility, these are constrained so that all relative translations and rotations are eliminated. This particular exercise is essential for monitoring the stress and strain discontinuities which occur at the junction. The impact is simulated by a 1.5 kg mass element (to model the 3 kg striking mass), given an initial
Dynamic response of clamped beam grillages
167 1 163 ~
6
5
689
~
Y
z
×
31
and 61
81
FIG. 2. Finite element models with selected node positions.
velocity, which interacts with the elements defining the grillage by means of a single unidirectional gap element. For the other analyses, eight-noded three-dimensional (3D) solid elements (ABAQUS type C3D8 or C3D8R), each of approximately 5 mm length, 3 m m depth and 3 or 5 m m width, are used to discretize the system. Because of the existence of double symmetry only one quarter of the system is modelled. Again, at the connecting junction of the main beam and the cross beam, two sets of nodes with identical coordinates but different node numbers are generated and suitably constrained. Two mass elements each of 0.375 kg are included together with two gap elements of unidirectional type. The mesh for the C3D8 element model is shown in Fig. 2 and the positions of the three nodes to which specific reference will be made are marked. For the C3D8R element model the mesh used is similar but with slightly fewer elements along the length of each beam. Figure 2 also shows the B32 element model with the node numbers labelled at the extremeties and at the intersection.
2.3. Material properties The mild steel is modelled as linear elastic, with a non-linear work hardening characteristic. The material property values are: try = 338 MPa, v = 0.3, E = 2.09 x 105 M P a and p = 7800 kg m - 3 . The plastic work hardening characteristics are given in Table 2. In one model, MSGII2s, strain rate dependency is included. It is modelled using the C o w p e r - S y m o n d s relationship I-5] with D = 1.05 x 10 7 S-1 and p = 8.3. For the aluminium alloy the material property values used are: try = 55 MPa, v = 0.33, E = 0.68 x 105 M P a and p = 2700 kg m -3. The work hardening characteristics are given in Table 3.
690
K. KORMIet
al.
TABLE 3. STRAIN HARDENING CHARACTERISTICS FOR ALUM1N1UM
O'ya
TABLE 2. STRAIN HARDENING CHARACTERISTICS FOR
ept
(MPa)
0 0.047 0.092 0.135 0.176 0.215 0.290 0.359 0.531
55.2 78.2 92.6 100 105 108 110 111 113
STEEL
ays
epl
(MPa)
0 0.16 0.21 0.26 0.30
338 362 380 400 415
3. RESULTS AND DISCUSSION 3.1. S t r e s s w a v e s
As soon as the gap is reduced to zero the mass impacts the grillage and the transfer of energy from the mass to the assembly starts. The immediate dynamic response of the grillage system is the generation of longitudinal bending and shear elastic stress waves propagating from the struck location towards the fixed boundary of the main beam. It is not possible using ABAQUS to individually identify and monitor the different types of wave with their different velocities but it is possible to infer the presence of the waves from the stress patterns. The results for model MSGII2s show the type of behaviour which is common to all of the analyses. At a time 5/as after the mass strikes the grillage, the stress is localized around the position of impact. The syr stress component, which is responsible for the shear wave, has compressive values. The szz contours associated with the bending wave are compressive above the mid-section of the main beam and tensile below it. After 11/as propagation has begun. As more energy is delivered to the system the build up of stress wave intensity continues until it exceeds the dynamic yield limit and plastic stress wave propagation ensues. These waves, which travel more slowly, cause the distortion in the system. The first stage of this process, where yielding has just begun in the impact zone, occurs 21/as after impact. As shown in Fig. 3, a stress distribution characteristic of a plastic hinge is established around the impact region after 95/as. Figure 4 shows that after 138/as the plastic wave has reached the junction of the main and cross beams. The original zone of plasticity remains but additional ones have been established in both the main and the cross beams close to the junction. When the longitudinal bending and shear waves reach the cross beam junction they split in intensity because it is essential to maintain kinematic compatibility at this point. Bending and shear waves continue along the main beam but bending and torsional stress waves are initiated in the cross beam. Contour plots for the sxx and syz stress components clearly show the patterns of positive and negative values corresponding to bending and torsion, respectively, in the cross beam. The values are considerably lower than those measured concurrently in the main beam which is expected in that the latter is subject to the greater deformation. Finally, in addition to the bending and shearing waves there are tensile waves generated in both the main and cross beams causing their extension. Figure 5 shows the longitudinal direct stress component contours in the main beam at 1.39 ms after impact. The pattern of distribution of tensile and compressive regions arises from the combination of bending and membrane forces. The bending is confined to three zones: at the position of impact, near to the junction and at the supported end. The above discussion has highlighted the significant aspects of the stress wave patterns observed in the analyses. It should be noted that because the overall dimension of the
Dynamic
FIG. 3. The
FIG. 4. The
region
region
of the grillage
of the grillage
response
of clamped
beam
691
grillages
having levels ol von Mises effective value for MSGII2s at time 95 ps.
having levels ol von Mises effective value for MSGIIZs at time I38 PCS.
stress
stress
above
the static
above
the static
yield
yield
structure is finite there will be at any one time, once the initial stage is over, transmitted, reflected and refracted elastic and elastic-plastic stress waves moving in different directions. Because of their individual or combined strength, the stress waves at the cross-over front cause a highly complex deformation mode in the system. This paper does not attempt to describe and monitor the complete sequence of events.
692
K. KORMI et al.
~.~ ~
"
"
y
B
-48.5
A -194.
z '~ x FIG, 5. S~ stress component contours (MPa) in the main beam for MSGII2s at time 1.39 ms.
~
f
FIG. 6. Dynamic deformation mode, compared with the original configuration, for MSGI3 at 81 ps (top) and 315 ps (bottom) with magnification factors of 80 and 20, respectively.
3.2. Dynamic response The geometrical changes that take place after impact are illustrated in Fig. 6. There are two distinct stages seen here. Initially the system oscillates with the transverse (main beam) and torsional (cross beam) displacement modes predominant. These early geometry changes involve parts of the grillage assembly having displacement in the reverse direction to that of the impacting mass. As the stress intensity in the system increases beyond the elastic limit and permanent deformation takes place the frequency of this vibration changes. In the second stage the stretching mode in the main beam and stretching and torsional modes in the cross beam are the dominating factors. The large change in geometry imposed on the system takes place during this period. The dynamic response, driven by the increasing deformation at the position of impact
Dynamic response of clamped beam grillages
693
Displacement
165 and 167
10.
163 8. 6.
4. 2.
.002
.004
.006
.008
Time
Displacement
-
30.
-
167 165 163
20.
10.
.002
.006
Time
.01
FIG. 7. Variation with time of displacement "all", i.e. x/{u 2 + uy2 + u .2} (mm) for nodes at the impact position in 3D solid models MSGII2 (top) and MAGII2 (bottom).
TABLE 4. COMPOSITION OF FEM, EXPERIMENTAL AND THEORETICAL VALUES OF DISPLACEMENT (MMI
~s
~s
Model
Experiment[2]
MSGII5 MSGI3 MSGII2s MSGII2 MAGII2b MAGII2
17.28 17.04 12.12 12.12 28.02 28.02
6.90 8.64 4.29 4.29 11.67 11.67
~s
~s
Theory [21 19.78 20.51 13.76 13.76 30.44 30.44
6.39 8.29 4.28 4.28 9.90 9.90
Wos
~:
Finite element 18.30 19.00 9.79 11.10 30.10 30.07
6.38 6.90 3.45 3.96 11.10 10.06
i Experiment FEM N/A 0.38 0.30 0.30 1.10 1.I0
N/A N/A 0.29 0.26 N/A 1.00
and resisted by the strength of both main and cross beams, continues but in the case of steel the system overshoots the final stable equilibrium position. The state of disequilibrium of the system is restored by what is commonly referred to as a spring-back recovery stage. The comparison between steel and aluminium is shown by the plots of displacement versus time in Fig. 7. During the final phase the impacting mass acquires a velocity in the opposite direction to that of its initial movement. With gravity acting, subsequent impact or impacts would follow this separation. However, this is not modelled in the analyses presented here. Also evident from the displacement plots for the 3D solid element models in Fig. 7 is the depth of indentation which can be measured from the difference between the displacements for the nodes through the thickness at the position of impact. This cannot be monitored in the models which use space beam elements. The depth of indentation and final displacement values are presented in Table 4 and
694
K. KORM~et al. Strain 167 .15 .1 165
.05 i
"
~
.002
.003
.004
Time
.005
-.05 163 Strain 167
.12 .08 .04
165
.007
Time
.009
-.04 163 S~n .6
167
.5 .4 .3
165
.2 .1
-.1
' .002
' .004
' .006
.008
.01
163 T i m e
FIG. 8. Variation of the e:: strain components with time for nodes at the impact position for models MSGII2, mild steel (top), MSGII2s, strain-rate sensitive mild steel (centre) and MAGII2, aluminium alloy (bottom).
compared with the experimental and theoretical calculated values [2]. There is reasonble agreement in all cases. Although the experimental study was mainly concerned with measurement of the final deformed shape some strain values for steel grillages were also reported. The finite element analyses using 3D solid elements gives a complete record of the variation of strain with time at different positions. Figure 8 shows the variation of the longitudinal direct strain component for the nodes through the main beam thickness at the position of impact for three different models. Incomplete experimental results were obtained at this position because the strain gauges failed. The most important features of these plots are: (a) The rate of energy dissipation for steel is considerably higher than for aluminium. (b) The longitudinal neutral plane in bending is initially located at the mid-surface through the depth of the assembly. During the structural response to the impact loading the deformation causes stretching at this mid-plane position and hence a shift in the location of the neutral plane becomes apparent. The shift is dependent on the material property as seen in the comparison of the graphs for aluminium and steel.
Dynamic response of clamped beam grillages
695
Strain 0.2[] 0.15 -
Upper surface Mid surface
O
Lowersurface
0.1
T
0.05
ol -0.05.
-0.1
2'5
so
7'5
1~0
1~5
lS0 Distance I mm
FIG. 9. Variation of the e.: strain components for nodes along the main beam, in the plane bisecting it, at time 4.15 ms. Distances are measured from the supported end.
(c) The maximum levels of strain acquired are different for aluminium than for steel. (d) Comparison of the models with and without strain rate dependency is as expected. The strain in the material modelled without strain rate effect is greater than in that modelled with strain rate effect. The variation of longitudinal direct strain with position along both main and cross beams is of particular interest since it establishes where the maximum bending interaction is seen. The variation along the centre line of the main beam for the nodes on the top and bottom surfaces and at the original mid-plane position for the steel grillage model MSGII2 is shown in Fig. 9. The time chosen here is when the strain has reached maximum values. There are three regions of relatively high values at each of which there is considerable bending. In agreement with the experimental findings, the strain is lower at the support than at either the position of impact or at the beam junction. The strain at the impact position is particularly high. The values found for the beam junction region are of the same order of magnitude as those found experimentally and, also in agreement, the upper surface values are tensile and the lower surface values compressive. However, it can be seen that in this locality the bending is not centred around the junction itself but at a position nearer to the point of impact. Figure 10 shows the corresponding plot for the cross beam. Once again there is agreement with experiment. The values are now compressive. As with the main beam, the level of strain is lower at the support than at the junction. 4. C O N C L U S I O N S
In the presentation and discussion of the results the emphasis has been on a comparison where possible with the experimental measurements that have been reported. The agreement is encouraging. It should be noted that the finite element analysis also provides information that is not available from the experiments. A complete history of the deformation, rather than only the final deformed shape, is obtained. The strain history is available at any location. Finally, the distribution of stress, particularly of the residual levels, is determined. However, it is now worthwhile to comment on the scope of the finite element approach compared with that of theoretical methods. Jones et al. [2] discussed the assumptions that were made in their original theoretical treatment. They argued that for a heavy mass low-speed impact it was reasonable to assume rigid, perfectly plastic material behaviour and to neglect the inertia force. In a subsequent paper the theory was further developed to include the influence of inertia [6]. The
696
K. KORM!et al.
Strain 0.025 ////~ 0.02 10.015 0.0 / ~
IS] Uppersurface ~ Midsun'ace © Lowersurface
O:
/
"
-0.005 ~~
~
-0.01 -0.015 -0.02-0.025
2'5
50
7'5
100 Distance / mm
FIG. 10. Variation of the exx strain components for nodes along the cross beam, in the plane bisecting it, at time 4.15ms. Distances are measured from the supported end.
justification for the approach is the agreement between their theoretical and experimental results. The other major assumption which is made in such analysis is the geometry of the deformation. The deformation is approximated in that bending is only allowed at localized plastic hinges. In this particular example, as with many others, the positions at which the major changes in geometry will occur are broadly predictable and theoretical analysis is feasible in advance of experimentation. In other cases the number of plastic hinges required and their position can only be decided once experiments have been carried out. However, even if the geometry of the response can be reasonably approximated by plastic hinges the analysis self-evidently can examine neither the initial elastic response nor the final spring-back recovery stage. It can be argued that the initial elastic response causes changes in geometry which are insignificant compared with the large plastic deformation. Furthermore, for aluminium the extent of elastic spring-back is insignificant because the response is overwhelmingly plastic. However, this is not true for the elastic spring-back for the steel grillages. In contrast, the finite element method can treat the material as elastic-plastic and include the effects of inertia, strain hardening and strain rate dependency. The geometry of the deformation arises naturally from the analysis which allows a full examination of the underlying processes causing it. Thus the application of the method is not restricted to the examination of events for which circumstances are favourable. It should be emphasized that in this regard it is essential to carry out an elastic-plastic finite element analysis for modelling dynamic loading of a finite structure. This is because the positions at which yield occurs at any given time during the impact will be determined by the interaction between stress waves being generated at the position of impact and those previously generated and then subsequently reflected back from the supported ends. Finally, it can be noted that the input file for such an analysis would only be some 150 lines long and that an experienced analyst should not need much more than a day to create it. The analysis could be run in a few hours on a graphics workstation of reasonable specification. Acknowled#ements--The authors wish to express their appreciation to the staff of the Centre for Advanced
Research in Engineering and to Mrs Sue Powell for typing the manuscript. The Centre wishes to thank Hewlett Packard for their sponsorship including the provision of the Apollo DN10000 used for the computation. REFERENCES K. KORMI,D. C. WEBBand N. JONES,A FEM damage assessment of a fully clamped pipeline subjected to lateral static or impact loading. Proc. Inst. Mech. Engrs Part E, unpublished report.
Dynamic response of clamped beam grillages 2. 3. 4. 5. 6.
697
N. JONES, T. LIU, J. J. ZHENG and W. Q. SHEN, Clamped beam grillages struck transversely by a mass at the centre. Int. J. Impact Engng ll, 379-399 (1991). ABAQUS user's manual, Hibbit, Karlsson & Sorensen, Inc. (1989). FEMGEN/FEMVIEW, user manual, Femview Limited (1991). P.S. SYMONDS, Viscoplastic behaviour in response of structures to dynamic loading. Behaviour of Materials Under Dynamic Ioading (edited by N. J. HUEFINGTON), pp. 106-124. ASME, New York (1965). W.Q. SHEN and N. JOr~ES, Dynamic response of a grillage under mass impact. Int. J. Impact Engng 13, 555-565 (1993).
Pergamon
Int. J. hnpact Enttn.q Vol. 15, No. 5, pp. 687-697, 1994 Elsevier ScienceLtd Printed in Great Britain 0734-743X(93)E0009-7 0734-743x/94 $7.00+ 0.00
FINITE E L E M E N T E X A M I N A T I O N O F THE D Y N A M I C R E S P O N S E O F C L A M P E D BEAM G R I L L A G E S I M P A C T E D T R A N S V E R S E L Y AT THEIR C E N T R E S BY A RIGID MASS KAZEM KORMI, EBB! SHAGHOUEI a n d DAVID A. DUDDELL Centre for Advanced Research in Engineering, Leeds Metropolitan University, Calverley Street, Leeds LSl 3HE, U.K.
(Received 20 August 1992; in revised form 5 November 1993)
Summary The finite element package ABAQUS is used to investigate the response of metallic (aluminium and steel) beam grillages to impact loading. Modelling is based on linear elastic materials with non-linear isotropic work hardening characteristics including strain rate dependency for the steel. Two different types of model are considered, one using space beam and the other three-dimensional solid elements. Comparison of the computed values with experimental results (N. JONES et al., Int. J. Impact Engng 11,379-399 (1991)) is very encouraging.
NOTATION FEM o'~. v E P b 1, b, hi, It2 i wf
wos
Sxx, Sxr, etc.
finite element method yield stress Poisson's ratio modulus of elasticity density width of main and cross beams, respectively depth of main and cross beams, respectively depth of indentation by striker into beam permanent transverse displacement permanent transverse displacement at the centre point permanent transverse displacement at the beams' junction direct and shear stress components
1. INTRODUCTION
In the field of impact loading of structures, the major proportion of the research so far has been dedicated to the development of theory which incorporates a number of simplifying assumptions. Usually, the theory is based on the evidence of the results obtained by conducting experiments. It appears that the process can be improved by using the finite element method (FEM). One major source of inertia resisting the use of FEM would appear to be the belief that the results obtained are not necessarily reliable hence the resort to what would hitherto be considered orthodox procedures. The present paper offers further confirmation of the inappropriateness of this attitude. It is a continuation of previous studies where FEM results are compared with carefully conducted experiments [1]. In this paper FEM analysis are compared with the experimental results reported by Jones et al. [2] for impact of steel and aluminium grillages by a rigid mass. 2. M O D E L L I N G
2.1. Grillage configuration The system comprises a main beam 270 mm in length, fixed at both extremeties and trisected orthogonally by two cross beams bisected equally by the main beam and of length 180 mm as shown in Fig. 1. The ends of the cross beams are also fully encastred. The 687
K. KORMI et al.
688
b2 SECTION
//L
'
i h2
MPACT
IlL
~A
k o
N
L
L
• A
N
~
L
SECTIONAA
b0
hI FIG. 1. Schematic view of the beam grillage assembly.
TABLE 1. MODEL NAMES AND DESCRIPTIONS
Dimensions Model name MSGII5 MSGI3 MSGII2s MSGII2 MAGII2b MAGII2
Specimen number [-2]
Material
Impact velocity (m s- t)
SGII5 SGI3 SGII2 SGII2 AGII2 AGII2
Steel Steel Steel Steel Aluminium Aluminium
7.34 7.00 5.42 5.42 5.42 5.42
(ram)
bl
hi
b2
h2
Element type
6 6 6 6 6 6
6 6 6 6 6 6
10 6 10 10 10 10
6 6 6 6 6 6
B32 B32 C3D8 C3D8R B32 C3D8
specimens used in the experiments were machined from aluminium or steel sheet material 6 or 10 mm thick [2]. The actual combinations are shown for the different models in Table 1. The impact velocities for the different models are also included in Table 1. 2.2. Finite element mesh In order to give a representative set of FEM results, four experiments are chosen to be modelled and foi- two of these two different analyses are performed. The models are summarized in Table 1. All analyses are carried out using ABAQUS [3] and F E M G E N / F E M V I E W [4] is used for graphical presentation of results. In one set of analyses space beam elements (ABAQUS type B32), each of 9 mm length, are used to model one half of the grillage assembly. The cross-over junction nodes are separately generated then, to preserve kinematic compatibility, these are constrained so that all relative translations and rotations are eliminated. This particular exercise is essential for monitoring the stress and strain discontinuities which occur at the junction. The impact is simulated by a 1.5 kg mass element (to model the 3 kg striking mass), given an initial
Dynamic response of clamped beam grillages
167 1 163 ~
6
5
689
~
Y
z
×
31
and 61
81
FIG. 2. Finite element models with selected node positions.
velocity, which interacts with the elements defining the grillage by means of a single unidirectional gap element. For the other analyses, eight-noded three-dimensional (3D) solid elements (ABAQUS type C3D8 or C3D8R), each of approximately 5 mm length, 3 m m depth and 3 or 5 m m width, are used to discretize the system. Because of the existence of double symmetry only one quarter of the system is modelled. Again, at the connecting junction of the main beam and the cross beam, two sets of nodes with identical coordinates but different node numbers are generated and suitably constrained. Two mass elements each of 0.375 kg are included together with two gap elements of unidirectional type. The mesh for the C3D8 element model is shown in Fig. 2 and the positions of the three nodes to which specific reference will be made are marked. For the C3D8R element model the mesh used is similar but with slightly fewer elements along the length of each beam. Figure 2 also shows the B32 element model with the node numbers labelled at the extremeties and at the intersection.
2.3. Material properties The mild steel is modelled as linear elastic, with a non-linear work hardening characteristic. The material property values are: try = 338 MPa, v = 0.3, E = 2.09 x 105 M P a and p = 7800 kg m - 3 . The plastic work hardening characteristics are given in Table 2. In one model, MSGII2s, strain rate dependency is included. It is modelled using the C o w p e r - S y m o n d s relationship I-5] with D = 1.05 x 10 7 S-1 and p = 8.3. For the aluminium alloy the material property values used are: try = 55 MPa, v = 0.33, E = 0.68 x 105 M P a and p = 2700 kg m -3. The work hardening characteristics are given in Table 3.
690
K. KORMIet
al.
TABLE 3. STRAIN HARDENING CHARACTERISTICS FOR ALUM1N1UM
O'ya
TABLE 2. STRAIN HARDENING CHARACTERISTICS FOR
ept
(MPa)
0 0.047 0.092 0.135 0.176 0.215 0.290 0.359 0.531
55.2 78.2 92.6 100 105 108 110 111 113
STEEL
ays
epl
(MPa)
0 0.16 0.21 0.26 0.30
338 362 380 400 415
3. RESULTS AND DISCUSSION 3.1. S t r e s s w a v e s
As soon as the gap is reduced to zero the mass impacts the grillage and the transfer of energy from the mass to the assembly starts. The immediate dynamic response of the grillage system is the generation of longitudinal bending and shear elastic stress waves propagating from the struck location towards the fixed boundary of the main beam. It is not possible using ABAQUS to individually identify and monitor the different types of wave with their different velocities but it is possible to infer the presence of the waves from the stress patterns. The results for model MSGII2s show the type of behaviour which is common to all of the analyses. At a time 5/as after the mass strikes the grillage, the stress is localized around the position of impact. The syr stress component, which is responsible for the shear wave, has compressive values. The szz contours associated with the bending wave are compressive above the mid-section of the main beam and tensile below it. After 11/as propagation has begun. As more energy is delivered to the system the build up of stress wave intensity continues until it exceeds the dynamic yield limit and plastic stress wave propagation ensues. These waves, which travel more slowly, cause the distortion in the system. The first stage of this process, where yielding has just begun in the impact zone, occurs 21/as after impact. As shown in Fig. 3, a stress distribution characteristic of a plastic hinge is established around the impact region after 95/as. Figure 4 shows that after 138/as the plastic wave has reached the junction of the main and cross beams. The original zone of plasticity remains but additional ones have been established in both the main and the cross beams close to the junction. When the longitudinal bending and shear waves reach the cross beam junction they split in intensity because it is essential to maintain kinematic compatibility at this point. Bending and shear waves continue along the main beam but bending and torsional stress waves are initiated in the cross beam. Contour plots for the sxx and syz stress components clearly show the patterns of positive and negative values corresponding to bending and torsion, respectively, in the cross beam. The values are considerably lower than those measured concurrently in the main beam which is expected in that the latter is subject to the greater deformation. Finally, in addition to the bending and shearing waves there are tensile waves generated in both the main and cross beams causing their extension. Figure 5 shows the longitudinal direct stress component contours in the main beam at 1.39 ms after impact. The pattern of distribution of tensile and compressive regions arises from the combination of bending and membrane forces. The bending is confined to three zones: at the position of impact, near to the junction and at the supported end. The above discussion has highlighted the significant aspects of the stress wave patterns observed in the analyses. It should be noted that because the overall dimension of the
Dynamic
FIG. 3. The
FIG. 4. The
region
region
of the grillage
of the grillage
response
of clamped
beam
691
grillages
having levels ol von Mises effective value for MSGII2s at time 95 ps.
having levels ol von Mises effective value for MSGIIZs at time I38 PCS.
stress
stress
above
the static
above
the static
yield
yield
structure is finite there will be at any one time, once the initial stage is over, transmitted, reflected and refracted elastic and elastic-plastic stress waves moving in different directions. Because of their individual or combined strength, the stress waves at the cross-over front cause a highly complex deformation mode in the system. This paper does not attempt to describe and monitor the complete sequence of events.
692
K. KORMI et al.
~.~ ~
"
"
y
B
-48.5
A -194.
z '~ x FIG, 5. S~ stress component contours (MPa) in the main beam for MSGII2s at time 1.39 ms.
~
f
FIG. 6. Dynamic deformation mode, compared with the original configuration, for MSGI3 at 81 ps (top) and 315 ps (bottom) with magnification factors of 80 and 20, respectively.
3.2. Dynamic response The geometrical changes that take place after impact are illustrated in Fig. 6. There are two distinct stages seen here. Initially the system oscillates with the transverse (main beam) and torsional (cross beam) displacement modes predominant. These early geometry changes involve parts of the grillage assembly having displacement in the reverse direction to that of the impacting mass. As the stress intensity in the system increases beyond the elastic limit and permanent deformation takes place the frequency of this vibration changes. In the second stage the stretching mode in the main beam and stretching and torsional modes in the cross beam are the dominating factors. The large change in geometry imposed on the system takes place during this period. The dynamic response, driven by the increasing deformation at the position of impact
Dynamic response of clamped beam grillages
693
Displacement
165 and 167
10.
163 8. 6.
4. 2.
.002
.004
.006
.008
Time
Displacement
-
30.
-
167 165 163
20.
10.
.002
.006
Time
.01
FIG. 7. Variation with time of displacement "all", i.e. x/{u 2 + uy2 + u .2} (mm) for nodes at the impact position in 3D solid models MSGII2 (top) and MAGII2 (bottom).
TABLE 4. COMPOSITION OF FEM, EXPERIMENTAL AND THEORETICAL VALUES OF DISPLACEMENT (MMI
~s
~s
Model
Experiment[2]
MSGII5 MSGI3 MSGII2s MSGII2 MAGII2b MAGII2
17.28 17.04 12.12 12.12 28.02 28.02
6.90 8.64 4.29 4.29 11.67 11.67
~s
~s
Theory [21 19.78 20.51 13.76 13.76 30.44 30.44
6.39 8.29 4.28 4.28 9.90 9.90
Wos
~:
Finite element 18.30 19.00 9.79 11.10 30.10 30.07
6.38 6.90 3.45 3.96 11.10 10.06
i Experiment FEM N/A 0.38 0.30 0.30 1.10 1.I0
N/A N/A 0.29 0.26 N/A 1.00
and resisted by the strength of both main and cross beams, continues but in the case of steel the system overshoots the final stable equilibrium position. The state of disequilibrium of the system is restored by what is commonly referred to as a spring-back recovery stage. The comparison between steel and aluminium is shown by the plots of displacement versus time in Fig. 7. During the final phase the impacting mass acquires a velocity in the opposite direction to that of its initial movement. With gravity acting, subsequent impact or impacts would follow this separation. However, this is not modelled in the analyses presented here. Also evident from the displacement plots for the 3D solid element models in Fig. 7 is the depth of indentation which can be measured from the difference between the displacements for the nodes through the thickness at the position of impact. This cannot be monitored in the models which use space beam elements. The depth of indentation and final displacement values are presented in Table 4 and
694
K. KORM~et al. Strain 167 .15 .1 165
.05 i
"
~
.002
.003
.004
Time
.005
-.05 163 Strain 167
.12 .08 .04
165
.007
Time
.009
-.04 163 S~n .6
167
.5 .4 .3
165
.2 .1
-.1
' .002
' .004
' .006
.008
.01
163 T i m e
FIG. 8. Variation of the e:: strain components with time for nodes at the impact position for models MSGII2, mild steel (top), MSGII2s, strain-rate sensitive mild steel (centre) and MAGII2, aluminium alloy (bottom).
compared with the experimental and theoretical calculated values [2]. There is reasonble agreement in all cases. Although the experimental study was mainly concerned with measurement of the final deformed shape some strain values for steel grillages were also reported. The finite element analyses using 3D solid elements gives a complete record of the variation of strain with time at different positions. Figure 8 shows the variation of the longitudinal direct strain component for the nodes through the main beam thickness at the position of impact for three different models. Incomplete experimental results were obtained at this position because the strain gauges failed. The most important features of these plots are: (a) The rate of energy dissipation for steel is considerably higher than for aluminium. (b) The longitudinal neutral plane in bending is initially located at the mid-surface through the depth of the assembly. During the structural response to the impact loading the deformation causes stretching at this mid-plane position and hence a shift in the location of the neutral plane becomes apparent. The shift is dependent on the material property as seen in the comparison of the graphs for aluminium and steel.
Dynamic response of clamped beam grillages
695
Strain 0.2[] 0.15 -
Upper surface Mid surface
O
Lowersurface
0.1
T
0.05
ol -0.05.
-0.1
2'5
so
7'5
1~0
1~5
lS0 Distance I mm
FIG. 9. Variation of the e.: strain components for nodes along the main beam, in the plane bisecting it, at time 4.15 ms. Distances are measured from the supported end.
(c) The maximum levels of strain acquired are different for aluminium than for steel. (d) Comparison of the models with and without strain rate dependency is as expected. The strain in the material modelled without strain rate effect is greater than in that modelled with strain rate effect. The variation of longitudinal direct strain with position along both main and cross beams is of particular interest since it establishes where the maximum bending interaction is seen. The variation along the centre line of the main beam for the nodes on the top and bottom surfaces and at the original mid-plane position for the steel grillage model MSGII2 is shown in Fig. 9. The time chosen here is when the strain has reached maximum values. There are three regions of relatively high values at each of which there is considerable bending. In agreement with the experimental findings, the strain is lower at the support than at either the position of impact or at the beam junction. The strain at the impact position is particularly high. The values found for the beam junction region are of the same order of magnitude as those found experimentally and, also in agreement, the upper surface values are tensile and the lower surface values compressive. However, it can be seen that in this locality the bending is not centred around the junction itself but at a position nearer to the point of impact. Figure 10 shows the corresponding plot for the cross beam. Once again there is agreement with experiment. The values are now compressive. As with the main beam, the level of strain is lower at the support than at the junction. 4. C O N C L U S I O N S
In the presentation and discussion of the results the emphasis has been on a comparison where possible with the experimental measurements that have been reported. The agreement is encouraging. It should be noted that the finite element analysis also provides information that is not available from the experiments. A complete history of the deformation, rather than only the final deformed shape, is obtained. The strain history is available at any location. Finally, the distribution of stress, particularly of the residual levels, is determined. However, it is now worthwhile to comment on the scope of the finite element approach compared with that of theoretical methods. Jones et al. [2] discussed the assumptions that were made in their original theoretical treatment. They argued that for a heavy mass low-speed impact it was reasonable to assume rigid, perfectly plastic material behaviour and to neglect the inertia force. In a subsequent paper the theory was further developed to include the influence of inertia [6]. The
696
K. KORM!et al.
Strain 0.025 ////~ 0.02 10.015 0.0 / ~
IS] Uppersurface ~ Midsun'ace © Lowersurface
O:
/
"
-0.005 ~~
~
-0.01 -0.015 -0.02-0.025
2'5
50
7'5
100 Distance / mm
FIG. 10. Variation of the exx strain components for nodes along the cross beam, in the plane bisecting it, at time 4.15ms. Distances are measured from the supported end.
justification for the approach is the agreement between their theoretical and experimental results. The other major assumption which is made in such analysis is the geometry of the deformation. The deformation is approximated in that bending is only allowed at localized plastic hinges. In this particular example, as with many others, the positions at which the major changes in geometry will occur are broadly predictable and theoretical analysis is feasible in advance of experimentation. In other cases the number of plastic hinges required and their position can only be decided once experiments have been carried out. However, even if the geometry of the response can be reasonably approximated by plastic hinges the analysis self-evidently can examine neither the initial elastic response nor the final spring-back recovery stage. It can be argued that the initial elastic response causes changes in geometry which are insignificant compared with the large plastic deformation. Furthermore, for aluminium the extent of elastic spring-back is insignificant because the response is overwhelmingly plastic. However, this is not true for the elastic spring-back for the steel grillages. In contrast, the finite element method can treat the material as elastic-plastic and include the effects of inertia, strain hardening and strain rate dependency. The geometry of the deformation arises naturally from the analysis which allows a full examination of the underlying processes causing it. Thus the application of the method is not restricted to the examination of events for which circumstances are favourable. It should be emphasized that in this regard it is essential to carry out an elastic-plastic finite element analysis for modelling dynamic loading of a finite structure. This is because the positions at which yield occurs at any given time during the impact will be determined by the interaction between stress waves being generated at the position of impact and those previously generated and then subsequently reflected back from the supported ends. Finally, it can be noted that the input file for such an analysis would only be some 150 lines long and that an experienced analyst should not need much more than a day to create it. The analysis could be run in a few hours on a graphics workstation of reasonable specification. Acknowled#ements--The authors wish to express their appreciation to the staff of the Centre for Advanced
Research in Engineering and to Mrs Sue Powell for typing the manuscript. The Centre wishes to thank Hewlett Packard for their sponsorship including the provision of the Apollo DN10000 used for the computation. REFERENCES K. KORMI,D. C. WEBBand N. JONES,A FEM damage assessment of a fully clamped pipeline subjected to lateral static or impact loading. Proc. Inst. Mech. Engrs Part E, unpublished report.
Dynamic response of clamped beam grillages 2. 3. 4. 5. 6.
697
N. JONES, T. LIU, J. J. ZHENG and W. Q. SHEN, Clamped beam grillages struck transversely by a mass at the centre. Int. J. Impact Engng ll, 379-399 (1991). ABAQUS user's manual, Hibbit, Karlsson & Sorensen, Inc. (1989). FEMGEN/FEMVIEW, user manual, Femview Limited (1991). P.S. SYMONDS, Viscoplastic behaviour in response of structures to dynamic loading. Behaviour of Materials Under Dynamic Ioading (edited by N. J. HUEFINGTON), pp. 106-124. ASME, New York (1965). W.Q. SHEN and N. JOr~ES, Dynamic response of a grillage under mass impact. Int. J. Impact Engng 13, 555-565 (1993).