 Open Access
 Total Downloads : 978
 Authors : Sanjoy Das Neogi, Amit Karmakar, Dipankar Chakravorty
 Paper ID : IJERTV1IS9398
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Finite Element Analysis of CornerSupported Composite Skewed Hypar Shell Roof Impacted by a Solid Striker
Sanjoy Das Neogi* 1, Amit Karmakar2, Dipankar Chakravorty3

Civil Engineering Department, Meghnad Saha Institute of Technology, Kolkata700107,India

Mechanical Engineering Department, Jadavpur University, Kolkata700032,India

Civil Engineering Department, Jadavpur University, Kolkata700032, India
Abstract
Hyperbolic paraboloid shell is a preferred roofing unit in many practical situations demanding large column free spaces. Laminated composite has become a natural choice of different industrial sectors for its huge specific strength, specific weight, and good weathering resistance, is now being used by civil engineers too. The low transverse shear strength under the action of impact load prompted the researchers to study the response of the composite shells under such loads. Shells on point supports have wide applications in car parks and theatres etc. In the present study, a finite element code is applied to investigate the impact induced response of pointsupported laminated composite hypar shells for different impact velocities. Contact behavior is described by modified Hertzian contact law and time dependent equations are solved using Newmarks method in present analysis.

Introduction
Hyperbolic paraboloid shell bounded by straight edges (commonly called hypar shell) is aesthetically elegant, easy to fabricate, being a doubly ruled surface and is preferred as roofing units in many practical situations demanding large column free space. In civil engineering applications of hypar shells as a roofing unit, their direct discrete support on columns is often needed. Car parks and theatres, where beams are normally absent in view of headroom restriction, often require largespan shell roofs supported on point supports. Moreover beams are sometimes undesirable in theatres from the viewpoint of the acoustics also. Introduction of laminated composite as an improved material, from the second half of the last century, in different branches of technology including civil engineering have prompted the researchers to study the different aspects of laminated composite hypar shells. In spite of the different advantages laminated composites are found to be vulnerable to sudden impact due to its low transverse shear capacity. It is obvious that an accurate modeling of contact behavior of target as well as the striker is the most important part of impact analysis. The classical contact law between elastic solids derived by Hertz [1] was found to be insufficient for composite materials and was modified
by Tan and Sun [2]. Time histories of contact force and displacement were reported by Sun and Chen [3] for simply supported initially stressed plate under impact using steel ball as an impactor. Impact analysis of shell structure was first reported by Toh et al.[4] for an orthotropic laminated cylindrical shell under low velocity impact generated by a solid striker. Shim et al.[5] studied an elastic response of glass/epoxy laminated composite ogival shells subjected to low velocity impact at any arbitrary location by a solid striker. They reported an analytic biharmonc polynomial solution. A finite element model, with and without geometric nonlinearity, was presented by Kistler and Waas [6] for a laminated composite cylindrical shell subjected to transverse central impact. A parametric study of impact response and the resulting damage of laminated composite shell impacted by a metallic impactor were studied using finiteelement method and Fourier series by Krishnamurthy et al.[7] for cylindrical curved panel. Karmakar et al.[8,9] undertook a transient dynamic finite element analysis to study the response of centrally impacted delaminated composite pretwisted cylindrical shells and rotating cylindrical shells due to lowvelocity impact. Effects of transverse shear deformation and rotary inertia were included in their study. Based on the elastoplastic mechanics, the damage analysis and dynamic response of elastoplastic laminated composite shallow spherical shell of orthotropic material under lowvelocity impact were studied by Yiming et al.[10] Laminated composites which are now often used as a roofing component in different shell forms by civil engineers, often subjected to impact load by wind born debris, snowfall and in many other situations. These impacts cause hidden injury in the composites and may lead to total collapse due to gradual loss of stiffness. A close look through the literature shows the impact analysis of civil engineering shell forms are not addressed properly. A parallel review in the area of composite shells reveal that the industrially important hypar shells (Fig1) need more indepth study although some important aspects of these shell forms were reported recently by Sahoo and Chakravorty [11,12]. The only paper reported on impact response of composite hypar shells was due to Neogi et.al.[13] where they discussed
some aspects of such response of twolayered shells for simply supported boundary condition. But the impact analysis for point supported hypar remained untouched. keeping in mind the industrial requirements, the impact response of pointsupported composite hypar shells has been chosen for investigation in the present research.

Mathematical formulation
The basic mass and stiffness matrices of the skewed hypar shell (Fig1) adopted in the present paper follows the equations and relations as reported by Sahoo and Chakravorty[11,12].
contact law accounted the permanent indentation after unloading cycle i.e. collisions upon the rebound of the target structure after the first period of contact were considered. The modified version of the above mentioned contact law was proposed by Tan and Sun
[2] was utilised by Sun and Chen [3]. The contact force model following Sun and Chen [3] is incorporated in the present finite element formulation. If k is the contact stiffness and m is the maximum local indentation, the contact force Fc during loading is given byFc=k1.5 0< m (4)
The indentation parameter depends on the difference of the displacements of the impactor and the target structure at any instant of time, and so also the contact force. The values of are changing with time because of time varying displacements of both the rigid impactor and the target structure. At an instant the maximum indentation takes place and as a result maximum contact force is also obtained. At this instant displacement of the impactor also attains the maximum value (Goldmith, W.,)[15]. There after the displacement of the impactor gradually decreases, but the target point displacement keeps on changing and
Surface equation: z
4c (x
ab
a / 2) ( y
b / 2)
finally increases to a maximum value and some point of time these two displacements become equal
Figure1 Surface of a skewed hypar shell and degrees of freedom
The dynamic equilibrium equation of the target shell for low velocity impact is given by the following equation:
(Goldmith, W.)[15]. This leads to zero value of indentation and eventually the contact force becomes zero. At this instant the impactor loses the contact with the target. The process after attaining the maximum contact force till the reduction of contact force to zero value is essentially referred as unloading [3]. If the
equation: M
k F (1)
mass of the impactor is not very small, a second impact may occur upon the rebound of the target structure
where M] and [K] are global mass and elastic stiffness matrices, respectively.{} is the global displacement vector. For the impact problem,{F} is given as
{F}={0 0 0 .FC0 0 0}T (2)
leading to a same phenomenon of contact deformation and attainment of maximum contact force. This process is known as reloading. If Fm is the maximum contact force at the beginning of unloading and m is the maximum indentation during loading, the contact force Fc for unloading and reloading are expressed as [3].
2.5
Here FC is the contact force given by the indentation law and the equation of motion of the rigid impactor is
Unloading phase: Fc
Fm 0
(5)
given as:
..
m F 0
Reloading phase: Fc Fm
m 0
1.5
0
(6)
i i C
(3) m 0
where mi and
..
i are the mass and
where 0 denotes the permanent indentation in a loadingunloading cycle.
acceleration of the impactor respectively. Evaluation of the contact force depends on a contact law which relates the contact force with indentation. A power law was proposed by Yang and Sun [14] based on static indentation tests using steel ball as an indentor. This
0=c (mp) if m>cr (7)
0=0 if m<cr (8) where c is a material dependent constant and cr is the critical indentation beyond which permanent
indentation occurs, and the values are 0.094 and
.01667cm respectively for graphiteepoxy composite (Sun and Chen 1985). Equations (1) and (3) are solved using Newmark constant acceleration time integration
algorithm in the present analysis. Equation (1) may be expressed in iteration form at each time step.
Table1 Non dimensional natural frequencies for three layer graphite epoxy twisted plates / /
of twist (deg) 0
15
30
45
60
75
90
15 Qatu and Lessia[16] 1.0035
0.9296
0.7465
0.5286
0.3545
0.2723
0.2555
Present formulation 0.9990
0.9257
0.7445
0.5279
0.3542
0.2720
0.2551
30 Qatu and Lessia[16] 0.9566
0.8914
0.7205
0.5149
0.3443
0.2606
0.2436
Present formulation 0.9490
0.8842
0.7181
0.5142
0.3447
0.2613
0.2444
Angle
E11= 138 GPa, E22=8.96 GPa, G12 = 7.1GPa, 12 = 0.3: a/b = 1, a/h = 100
t 2 .x 20cm. x 0.269cm.; ply orientation = [ 00/450/00/
K i 1 F i
M b (9) 0 0
t t 4 t t i
Where
45 /0 ]2s Velocity of impactor = 3m/s; Mass density of impactor = 7.96×105 Nsec/cm4
t 2 (10)
K K
4
b t t t
M
t 2 (11)
t 4 t
The same solution scheme is also utilized for solving the equation of motion of the impactor, i.e. Equation (3). In Equation (9), i is the number of iterations within a time step. It is to be noted that a modified
F
t t
.
t t
contact force i obtained from the previous iteration is used to solve the current response i 1
The iteration procedure is continued until the
equilibrium criterion is met.

Numerical example
Problems are solved with two different objectives. The present formulation is applied to solve natural frequencies of graphiteepoxy twisted plates, for problems appearing in published literature which are structurally similar to skewed hypar shells. This problem is expected to validate both the stiffness and mass matrix formulation of present finite element code. Another problem, solved earlier by Sun and Chen[3] regarding the impact response of composite plate , is taken up as the second benchmark to validate the impact formulation. The details of the benchmark problems are furnished along with Table 2 and Fig2
E11=120 GPa, E22= 7.9 GPa, G12= G23=G13=5.5 GPa,
12 =0.30, =1.58x 105 Nsec2/cm4 plate size=20cm
Figure2.Contact force history of simply supported plate
Apart from the problems mentioned above, impact response of skewed hypar shells being impacted at the central point are also studied for point supported boundary condition for different impact velocities. The details of the problems which are the authors own are given below.

Boundary condition : Point supported (PS)

Lamination: +450/450 (AP)

Velocity of impact (m/s): 1, 3, 5, 10

Details of shell geometry :a = 1.0m, b =1.0m, t=0.02m, c=0.2m

Material details :E11=120Gpa,E22= 7.9 Pa, G12= G23= G13= 5.5GPa
12 = 0.30, = 1.58×105Nsec2/cm4
Figure3 Nodes at which deflections measured


Results and discussions
The results of Table1 shows that the fundamental frequency values of the twisted plates obtained by the present formulation for a shell with cross curvature agree very closely to those reported by Qatu and Lessia [14]. This agreement validates the correct incorporation of stiffness and mass matrix formulation in the present code. Fig.2 shows the time variation of the contact force induced in a composite plate under low velocity impact previously reported by Sun and Chen [3]. The values obtained by the present formulation are also presented graphically in the same figure in a different style. Here again excellent agreement of results is observed which establishes the correctness of impact formulation.
To study the impact response of point supported (PS) angle ply (AP) shell figure 5 to 8 and Table2 are studied. All the results of contact force and displacement that are presented in either graphical or tabular form are arrived at after a study of time step convergence. The finite element mesh adopted is also based on force and displacement convergence criteria.
When low velocity normal impact response of simply supported angle ply shell is studied being struck by the spherical impactor centrally , it is observed that the contact force shows a sort of parabolic variation with a single peak. After a given time interval which is 100s or less the contact force converges to a null value. It is interesting to note that higher the impactor velocity higher is the contact force as expected, but the force dies down to a null value earlier. This behavior may be attributed to the fact that the higher the velocity more rapid is the elastic rebound of the impactor followed by detachment which causes contact force to decay out faster. It is also very interesting to observe that the time instant corresponding to peak contact force and that for peak displacement do not match. This is because the resultant displacement at any time instant is a cumulative effect of the instantaneous
contact force value and the inertia effect of the previous instant. The figure showing the transient displacement reflects the fact that vibration continues even after the force dies down with successively occurring peaks, though the peak values are less in magnitude than the highest peak which occurs a bit after the instant of maximum contact force but before the full decay of it.
To estimate the equivalent static load (ESL) corresponding to a particular impactor velocity , a concentrated load at the centre (point of impact) is applied and adjusted the yield a central displacement equal to the maximum dynamic displacement. The magnitude of the central displacement when the peak contact force is applied at the point of impact as a static concentrated load is also calculated. The central displacement obtained under such a load when divides the maximum dynamic displacements yields dynamic magnification factor (DMF). The variations of maximum contac force, maximum dynamic displacement and equivalent static load (ESL) with impactor velocity are almost linear and of three above mentioned values are increasing functions of impactor velocity (Fig9). However the dynamic magnification factor (DMF) and the impactor velocity shows a logarithmic relation and the DMF is a decreasing function of the velocity of the impactor (Fig9).
Apart from the central point displacements of some other points (Fig3) are also calculated but the point of impact i.e. the central node is found to be the most critical one showing the highest magnitude of displacement.
350
Contact force (N)
300
250
200
150
100
50
0
50
0 100 200 300 400 500
2.00E04
1.50E04
1.00E04
Displacement (m)
Time(Âµs)
1400
Contact force (N)
1200
1000
800
600
400
200
0
200
0 100 200 300 400 500
Time (Âµs)
600
400
200
Point1 Point8 Point9
Point10
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0
0.0001 0
0.0002
Time(Âµs)
600
400
200
0
5.00E05
0.00E+00
Point10
5.00E05
Point1
Point8
Point9
Displacement (m)
Time (Âµs)
Time (Âµs)
600
400
200
Point1
Point2
Point3
Point4
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0
0.0001 0
0.0002
Time (Âµs)
5.00E05
Point1
Point2
Point3
Point4
400 600
200
0
2.00E04
1.50E04
1.00E04
5.00E05
0.00E+00
Displacement (m)
Displacement (m)
2.00E04
Displacemrnt (m)
1.50E04
1.00E04
5.00E05
Point1 Point5
Point1
Point5
Point6
Point7
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0
0.0001 0
0.0002
0.0003
Displacement (m)
Point6
600
400
200
0.00E+00
Point7
5.00E05
1.00E04
0 200 400 600
Time (Âµs)
Time (Âµs)
Figure6 Impact response of clamped angle ply
Figure5 Impact response of clamped angle ply (PS/AP) composite hypar shells for impact velocity 1m/s
(PS/AP) composite hypar shells for impact velocity 3m/s
2500
Contact force (N)
2000
1500
1000
500
0
500
0 100 200 300 400 500
Time (Âµs)
600
400
200
0
0.0005 0
0.001
Point1
Point8
Point9
Point10
Time(Âµs)
600
400
200
Point1 Point8 Point9
Point10
0.001
0.0008
0.0006
0.0004
0.0002
0
0.0002 0
0.0004
Displacement (m)
Displacement (m)
Time (Âµs)
6000
5000
Contact force (N)
4000
3000
2000
1000
0
1000
0 100 200 300 400 500
0.002
0.0015
0.001
0.0005
Time (Âµs)
Time (Âµs)
600
400
200
Point2
Point3
Point4
0.0015
0.001
0.0005
0
0.0005 0
0.001
Point1
0.002
Time (Âµs)
600
400
200
Point1 Point2 Point3
Point4
0.001
0.0008
0.0006
0.0004
0.0002
0
0.0002 0
0.0004
Displacement (m)
Displacement (m)
0.001
Point1
0.002
Point1
0.0008
Displacement (m)
Point5
0.0006 Point6
0.0004 Point7
0.0002
0.0015 Point5
Displacement (m)
0.001 Point6
0.0005 Point7
0
0
0.0002
0.0004
0 200 400 600
Time (Âµs)
0.0005
0.001
0 200 400 600
Time (Âµs)
Figure7 Impact response of clamped angle ply (PS/AP) composite hypar shells for impact velocity 5m/s
Dynamic magnification factor
2.5
2
1.5
1
0.5
0
0 2 4 6 8 10 12
Velocity of impact (m/s)
Figure8 Impact response of clamped angle ply (PS/AP) composite hypar shells for impact velocity
10m/s
Equivalent static load (N)
Figure9 Variation of dynamic magnification factor and equivalent static load with velocity for point supported cross ply (PS/CP) composite hypar
12000
10000
8000
6000
4000
2000
0
0
5
10
15
Velocity of impact (m/s)
Table 2 Maximum contact force, maximum dynamic displacement, equivalent static load, dynamic magnification factor for different velocities
Boundary condition and ply orientation
Velocity(m/s)
Maximum Maximum Equivalent Dynamic
load(N)
(N)
factor
Point 1
309.4212
0.000169
1.089 x103
1.949
Supported 3
1166.147
0.000511
3.292 x103
1.436
+450/450 5
2162.702
0.000861
5.547 x103
1.346
10
5013.577
0.001772
11.416×103
1.055
impact
displacement(m)
static load
magnification

Conclusion
The following conclusions may be derived from the present study.

The close agreement of the results obtained by the present method with those available in the published literature establishes the correctness of the approach used in the present investigation.

Under the influence of normal low velocity impact the contact force shows a parabolic combined loading and unloading curve with a single peak for the practical class of shells considered here. Higher magnitude of impact velocity results in higher value of the peak contact force. However, due to a sharp elastic rebound the total duration of contact force is less for higher velocity of impactor.

The time instants at which the maximum contact force and the maximum dynamic displacement occur show a phase difference and interestingly in some cases the maximum displacement and hence stresses may occur even after the contact force dies down totally
.Thus it is concluded that the study should be continued only after when the major peaks of the dynamic displacement die down and not after the full decay of the contact force only.

The maximum contact force, the peak dynamic displacement and the equivalent static load are all increasing functions of impactor velocity, the relations being almost linear. However, the dynamic magnification factor shows a logarithmically decreasing tendency with increase of the velocity of impact
6. References

Hertz, H. On the contact of elastic solids, Journal fur die reine und angewandte Mathematik, (1881),92,156171

Tan,T.M., Sun, C.T., Use of statical indentation laws in the impact analysis of laminated composite plate. Journal of Applied Mechanics. (1983),52,6 12

Sun, C.T., Chen, J.K., On the impact of initially stressed laminates, Journal of Composite Material. (1985), 19, 490503

Toh S.L., Gong S.W., Shim V.P.W. Transient stress generated by low velocity impact on orthotropic laminated cylindrical shell, Composite Structures, (1995), 31(3),213228

Shim, V.P.W., Toh S.L., Gong S.W., The elastic impact response of glass/epoxy laminated ogival shells, International journal of Impact Engineering, (1996), 18(6) 633655

Kistler, L.S., and Waas, A.M., Impact response of cylindrically curved including a large deformation scaling study, International Journal of Impact Engineering, (1998),21(12)6175

Krishnamurthy K.S, Mahajan P, Mittal R.K, A parametric study of the impact response and damage of laminated cylindrical composite shells, Composites Science and Technology,(2001), 61(12) 16551669

Karmakar Amit, Kishimoto Kikuo Transient Dynamic Response of Delaminated Composite Cylindrical Shells Subjected to Low Velocity Impact, Key Engineering Material, (2005),297300 12851290

Karmakar Amit, Kishimoto Kikuo Transient Dynamic Response of Delaminated Composite Rotating Shallow Shells Subjected to Low Velocity Impact, Shock and Vibration(2006), 13, 619628.

Yiming Fu, Yiqi Mao, Yanping Tian Damage analysis and dynamic response of elastoplastic laminated composite shallow spherical shell under low velocity impact, International Journal of Solids and Structures, (2009), 47 (1) 126137

Sahoo, S .. Chakravorty, D., Finite element bending behaviour of composite hyperbolic paraboloidal shells with various edge conditions, Journal of Strain Analysis for Engineering Design, (2004), 39 (5) 499513

Sahoo, S.. Chakravorty, D. Finite element vibration characteristics of composite hypar shallow shells with various edge supports, Journal of Vibration and Control, 2005,11(10) 12911309.

Das Neogi, S., Karmakar, A., Chakravorty, D. Impact response of simply supported skewed hypar shell roofs by finite element, Journal of Reinforced Plastics Composites. (2011), 30(21),17951805

Yang, S.H., Sun, C.T., Indentation law for composite laminates, Composite Materials: Testing and Design, ASTME STP(1985),787,425446

Goldmith, W. IMPACT The theory and physical behavior of colliding solids, (2001), Dover Publication, Inc., New York,

Qatu, M. S., Leissa, A. W., Natural frequencies for cantilevered doublycurved laminated composite shallow shells, Computers and Structures, (1991),17(3) 227256