- Email: [email protected]
Influence of blast pressure modeling on the dynamic response of conical and hemispherical shells
Thin-Walled Structures 49 (2011) 604–610
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Influence of blast pressure modeling on the dynamic response of conical and hemispherical shells K. Kowal-Michalska n, T. Kubiak, J. Swiniarski Department of Strength of Materials, Technical University of Lodz, ul. Stefanowskiego 1/15 PL 90-924 Lodz, Poland
a r t i c l e in f o
a b s t r a c t
Keywords: Blast pressure Shells Strain energy Effective stresses
The paper aims to investigate the effect of blast pressure modeling, accounting for the ground reflections, on conical and hemispherical shells behavior. Steel shells, of constant height, rigidly connected with a steel plate are considered. The uniformly distributed external overpressure and true material stress–strain curve are assumed in calculations. The problem is solved by FEM using ANSYS software. The time relations of strain energy and effective stresses for different simulations of blast pressure are determined. & 2010 Elsevier Ltd. All rights reserved.
1. Introduction Conical and hemispherical shells due to their advantages (easiness of manufacturing, lightweight and what is important in some applications—esthetic shape) are widely used in many constructions: from roofing of large dimensions through pressure vessels to structures designated as explosion protections. Modeling of structures absorbing blast energy, resulted from industrial accident or terrorist attack, is the subject of interest of many engineers. The problem of building protection against explosions was discussed in [6]. The investigations are also conducted in field of aircraft [2] and automotive industry for vehicles moving in dangerous zones. The idea of applying thin shells as absorbers of the explosive energy brought the research, which has been presented by authors in paper [4]. It is well known that thin-walled plate and shell structures are good energy absorbers especially when they lose their stability in the elastic range and then the plastic deformations appear leading to failure. When the structures are subjected to pulse load (as it is in case of blast pressure) the different phenomena may occur depending on the so called ‘‘pulse intensity’’ [7]—impact for pulses of high amplitudes and durations in range of microseconds or quasi-static behavior if amplitude is low and duration is two or more times of fundamental natural vibrations period. For pulses of intermediate intensity (amplitudes in range of static buckling load and durations close to the period of fundamental natural vibrations) the phenomenon of dynamic buckling occurs. The problem of shells dynamic buckling is well described in literature (e.g. [7,8]). The assessment of blast loading effects is required for design of structures to withstand the explosion. The blast loading time n
Corresponding author. E-mail address: [email protected] (K. Kowal-Michalska).
0263-8231/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2010.09.008
relation, obtained empirically, is exponential [2,5,6] and consists of two phases: positive (overpressure) of duration in range of milliseconds and negative one (subpressure) of duration in range of seconds. In the numerical analysis of explosion protection structures this relation is often represented by a right angle triangle and furthermore the important effects as multiple blast wave reflections, the Mach effect, and negative phase of the blast wave are usually neglected. Generally, it is assumed that explosions on surface or in air are free blast propagations without any contact with the ground or adjacent structures. Neglecting reflections of the blast wave could lead to overestimation or underestimation of structure effective stresses and its strain energy. It should be mentioned that the characteristics of reflected blast pressure can never be achieved empirically but only by numerical simulations (see [2,5,6]). In many cases the interaction of explosion wave and surrounding can result in magnification of the pressure impulse. The blast loading time relation may be of different character—it can be similar to the incident wave but of higher peak value or assuming perfect ground reflection and reflected wave overlaying the incident one, the blast loading time relation can be obtained by superimposing additional linearly varying loading. This paper aims to analyze the response of conical and hemispherical shells under different courses of blast pressure and to find out the direct effect of blast pressure modeling on shells strain energy and effective stress values. The problem has been solved by the finite element method using ANSYS software.
2. Assumed blast pressure relations The exponential blast loading time relation, obtained empirically, consists of two phases: positive (overpressure) of duration in range of milliseconds and negative one (subpressure) of
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
duration in range of seconds. In the numerical analysis of protection structures this relation is often represented by a triangle of the same duration Tp and peak value p0. From numerical reasons it has been assumed that this peak value is reached at t¼kTp where k ¼0.1 (Fig.1). In order to consider the effect of blast wave reflection three types of pressure characteristics are considered (Fig. 2). It has been assumed that in all cases pulse duration is the same and kept constant and equal to 0.5 ms. For all types the peak value of blast pressure equals p0 and the impulses equal each other. The pulse denoted by ‘‘I’’ is a triangle commonly used in calculations of structures subjected to blast pressure. The pulses denoted by ‘‘II’’ and ‘‘III’’ represent the reflected blast waves in which the value of overpressure and impulse decreases linearly until certain limit when the pressure begins to increase due to the effects of reflections. For pulse ‘‘II’’ the maximum value of overpressure p0 is reached at 0.05 ms and it decreases to p ¼0.2p0 at t ¼0.2525 ms and next the pressure increases and equals p¼0.8p0 at t ¼0.2975 ms and finally it linearly decreases to zero at time t ¼Tp ¼0.5 ms. For pulse ‘‘III’’ the first peak value equals 0.8p0 and it decreases to p ¼0.2p0 at t ¼0.2525 ms and next the pressure increases to the maximum value p0 at t ¼0.2975 ms and next linearly tends to zero at t¼Tp ¼0.5 ms.
605
presents the finite element models (way of discretization) for conical and hemispherical shells. The applied element enables to model linear and nonlinear material properties. For the assumed uniform pressure distribution the load was applied to all elements with the same value in a given time step. In the numerical analysis the dynamic responses of shell structures loaded by pulse pressure have been searched for. At the first stage the modal analysis has been performed with the aim to determine the period of natural frequency Tps. It has been found that for both considered structures Tps E0.1 ms. In the dynamic analysis the equilibrium equation has the following form: ::
:
fPg ¼ ½MUfug þ ½CUfug þ½KUfug
ð1Þ
where [M] is a structural mass matrix, [C] is a structural damping matrix and [K] is a stiffness matrix. In the analysed cases the damping can be neglected and then Eq. (1) can be written as follows: ::
fPg ¼ ½MUfug þ ½KUfug
ð2Þ
Substituting time derivative of displacement {¨ u} by increment of displacement {u} in consecutive discrete instant of time t the new equilibrium equations included inertia forces that are obtained in each time step. For the equation obtained the solution
3. Geometrical model of shells
pulse I pulse II pulse III
p0 0.8·p0 p(t)
The present paper deals with shells in shape of hemisphere and truncated cone closed by flat circular plate (Fig. 3). Both shells are of the same thickness (h¼0.5 mm) and height (40 mm) and are rigidly connected with a square plate of dimensions: in case of hemisphere 80 mm 80 mm and thickness h¼0.5 mm and in case of cone 60 mm 60 mm and thickness equal to 2 mm. The boundary conditions are shown in Fig. 3. The plates have been used to model the boundary conditions for shells as close as possible to real ones. The displacements in three perpendicular directions and rotations about three perpendicular axes have been set to zero for all nodes lying along bottom plate edges. The structures are loaded by blast pressure (shown in Fig. 2) uniformly distributed on the shell surface.
0.2·p0 4. Finite element models To solve the problem of dynamic response of shell structures the ANSYS [9] software based on the finite element method has been employed. For discretization the four-node shell element SHELL 43 [9] of six degrees of freedom has been applied. Fig. 4
0
0.1
0.2
0.3
0.4 0.5 0.6 time [ms]
0.7
0.8
Fig. 2. Assumed simulations of reflected blast pressure.
Fig. 1. Blast pressure representation: (a) empirical and (b) triangular approximation.
0.9
1
606
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
Fig. 3. Dimensions and boundary conditions of analysed conical and hemispherical shells.
Fig. 4. The finite element model of analysed shells.
algorithms used in static analysis can be employed. In ANSYS software the Newmark method is used to perform the integration over time and for equations solution in a consecutive time step the Newton–Raphson algorithm is applied. In the calculations the material stress–strain curve shown in Fig. 5 has been implemented with following strength properties: Young’s modulus E¼200 GPa, Poisson’s ratio n ¼0.3, density r ¼7850 kg/m3 and initial yield stress s0 ¼200 MPa.
5. Results of numerical calculations The results of numerical calculations are presented in figures showing the plots of maximal effective stresses and strain energy in time for assumed maximal values of blast pressure distributed accordingly to the relations described in Section 3. In Figs. 6–8 the time dependent plots of strain energy and effective stresses are presented for a conical shell loaded by three assumed distributions of blast pressure. The pressure maximal value p0 equals 4.5, 5.0 and 5.5 MPa, respectively.
Fig. 5. Material stress–strain relation obtained in laboratory test.
It is found that in case of pulse ‘‘I’’, when p0 increases from 4.5 to 5.5 MPa, the maximal value of strain energy increases almost ten times and the structure works in the elastic range for
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
607
Fig. 6. Strain energy and effective stresses time relations for conical shell when p0 ¼ 4.5 MPa.
Fig. 7. Strain energy and effective stresses time relations for conical shell when p0 ¼ 5.0 MPa.
Fig. 8. Strain energy and effective stresses time relations for conical shell when p0 ¼ 5.5 MPa.
p0 ¼ 4.5 MPa and when pressure peak value equals 5.5 MPa large regions become plastic. The similar behavior is observed for the pulse ‘‘II’’—in this case the energy increases (from p0 ¼4.5 to 5.5 MPa) about five times. For the third type of blast pressure distribution the plastic deformation appear in the second stage of pulse duration (when p0 ¼5.5 MPa) and the maximum value of strain energy increases only three times. In Fig. 9 the distributions of deformations (Fig. 9a, c) and effective stresses (Figs. 9b, d) are shown for conical shell subjected to triangular shape of blast pressure (pulse I) for two values of pulse peak (p0 ¼5 MPa – Fig. 8a, b and p0 ¼6 MPa – Fig. 8c, d). These pictures confirm the remarks made earlier and it can be seen that the shell collapses at its bottom base. This behavior conforms to the experimental results concerning static buckling of truncated conical shells, presented in paper [3]. Additionally, in arbitrarily chosen point of conical shell, loaded by pulse ‘‘I’’, the deflection has been registered for different values of peak overpressure values p0. The results are composed in Fig. 10. It should be underlined that the relations presented
in Fig. 10 do not correspond to the shell maximal deflection for the assumed value of p0. Three curves in Fig. 10 showing the deflection in chosen point registered at t ¼0.08 ms (‘‘crosses’’), t ¼0.3 ms (‘triangles’’) and the third curve (solid line) presents the maximal deflection in this point that appeared in whole range of calculation time (equal to 1 ms). The depicted relations allow estimating the value of peak overpressure p0 for which the deflections start to grow rapidly and it can be assumed that for p0 larger than 5.8 MPa the structure subjected to pulse ‘‘I’’ will fail. For the same values of p0 (4.5, 5.0, 5.5 MPa) the calculations of strain energy and effective stresses have been conducted for the hemispherical shell loaded by three different blast pressure characteristics (Figs. 11–13). It can be seen that the character of curves is similar to that found for conical shell (Figs. 6–8). When the maximal value p0 equals 4.5 MPa the shell is almost entirely elastic and the plastic regions increase with p0 growth for each type of blast pressure. Attention should be paid to the fact that for triangular pulse ‘‘I’’ of peak value equal to 5.5 MPa the shell
608
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
collapses—the effective stresses reach the ultimate strength (compare material stress–strain curve in Fig. 5 with effective stresses plots shown in Fig. 13). In Fig. 14 the distributions of deformations and effective stresses for two values of p0 are shown (the pulse loading is of type ‘‘I’’). It is found that the hemispherical shell destroys similarly to the conical one (compare Figs. 14 and 9). Comparing the collapse mode with those presented in paper [1] it can be seen that in case of the analysed hemispherical shell the buckling phenomenon does not occur. The overpressure peak value (p0 ¼5.5 MPa) is much lower than static bifurcation collapse pressure.
be more conservative but at the same time it is much safer when the analysed structure is to serve a purpose of explosion protection.
6. Conclusions On the basis of the presented calculation results it can be stated that the effect of reflection accounted for in the blast loading characteristic strongly influences the dynamic response of considered shell structures. Although the assumed pulses are of the same impulse value, peak overpressure and duration time the triangular distribution (pulse I) is the most dangerous and results in greater values of strain energy and effective stresses. Therefore, the traditional approach of blast pressure time relation seems to
Fig. 10. Relation between deflection and peak overpressure value for pulse ‘‘I’’ in arbitrary point of conical shell, registered in different time of pulse duration.
Fig. 9. Distributions of deformations and effective stresses for p0 ¼ 5 MPa (a, b) and for p0 ¼ 6 MPa (c, d).
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
strain energy [Nm]
0.7 0.6
250
pulse I pulse II pulse III
effective stress [MPa]
0.8
0.5 0.4 0.3 0.2 0.1 0
609
pulse I pulse II pulse III
200 150 100 50 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [ms]
0
0.2
0.4 0.6 time [ms]
0.8
1
Fig. 11. Strain energy and effective stresses time relations for hemispherical shell when p0 ¼ 4.5 MPa.
1.4 strain energy [Nm]
1.2 1
pulse I pulse II pulse III
0.8 0.6 0.4 0.2 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [ms]
time [ms]
Fig. 12. Strain energy and effective stresses time relations for hemispherical shell when p0 ¼ 5.0 MPa.
strain energy [Nm]
5
pulse I pulse II pulse III
4 3 2 1 0
300 effective stress [MPa]
6
pulse I pulse II pulse III
250 200 150 100 50 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [ms]
0
0.2
0.4 0.6 time [ms]
0.8
1
Fig. 13. Strain energy and effective stresses time relations for hemispherical shell when p0 ¼ 5.5 MPa.
It should be also noted that from two proposed shell structures the conical one can sustain pulses of greater peak overpressure than the hemisphere. At last some general remarks:
In calculations the pulses duration is kept constant and equal to 0.5 ms while the period of fundamental natural vibrations equals approximately 0.1 ms. Therefore, according to the definition given in Section 1 the phenomenon should be
treated as quasi-static. In authors’ opinion this can be true if the pulse shape is rectangular but for other shapes, especially triangular (almost right triangle) the shell behavior is dynamic not static. The assumption of damping neglecting should be verified in further investigations. In this paper it was assumed that for each time step the blast pressure is uniformly distributed on shell surface. Perhaps it would be worth-while effort to determine pressure values depending on time and location on the shell surface.
610
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Fig. 14. Distributions of deformations and effective stresses for p0 ¼4.5 MPa (a, b) p0 ¼5.5 MPa (c, d).
References [1] B"achut J, Galletly GD. Buckling strength of imperfect hemispheres. Journal of Thin-walled Structures 1995;23:1–20. [2] Dae Kwan K, Jae-Huang H. Establishment of gun blast wave model and structural analysis for blast load. Journal of Aircraft 2006;43(4):1159–68. [3] Golzan BS, Showkati H. Buckling of thin-walled conical shells under uniform external pressure. Thin-walled Structures 2008;46:516–29. [4] Kubiak T, Kolakowski Z, Kowal-Michalska K, Mania RJ, Swiniarski J. Dynamic response of conical and spherical shell structures subjected to blast pressure. In: Batista E, Vellasco P, de Lima L, editors. Proceedings of SSDS’Rio 2010; 2010.
[5] Le Blanc G, Adoum M, Lapoujade V. External blast load on structures—empirical approach. In: Proceedings of the fifth European LS-DYNA conference, Ulm, Germany; 2006. [6] Remennikov AM, Rose TA. Modelling blast loads on buildings in complex city geometries. Computers and Structures 2005;83:2197–205. [7] Simitses GJ. Dynamic stability of suddenly loaded structures. New York: Springer Verlag; 1990. [8] Simitses GJ. Buckling of moderately thick laminated cylindrical shell: a review. Composites Part B 1996;27 B:581–7. [9] User’s guide ANSYS 11sp1, Ansys, Inc., Houston, USA.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Influence of blast pressure modeling on the dynamic response of conical and hemispherical shells K. Kowal-Michalska n, T. Kubiak, J. Swiniarski Department of Strength of Materials, Technical University of Lodz, ul. Stefanowskiego 1/15 PL 90-924 Lodz, Poland
a r t i c l e in f o
a b s t r a c t
Keywords: Blast pressure Shells Strain energy Effective stresses
The paper aims to investigate the effect of blast pressure modeling, accounting for the ground reflections, on conical and hemispherical shells behavior. Steel shells, of constant height, rigidly connected with a steel plate are considered. The uniformly distributed external overpressure and true material stress–strain curve are assumed in calculations. The problem is solved by FEM using ANSYS software. The time relations of strain energy and effective stresses for different simulations of blast pressure are determined. & 2010 Elsevier Ltd. All rights reserved.
1. Introduction Conical and hemispherical shells due to their advantages (easiness of manufacturing, lightweight and what is important in some applications—esthetic shape) are widely used in many constructions: from roofing of large dimensions through pressure vessels to structures designated as explosion protections. Modeling of structures absorbing blast energy, resulted from industrial accident or terrorist attack, is the subject of interest of many engineers. The problem of building protection against explosions was discussed in [6]. The investigations are also conducted in field of aircraft [2] and automotive industry for vehicles moving in dangerous zones. The idea of applying thin shells as absorbers of the explosive energy brought the research, which has been presented by authors in paper [4]. It is well known that thin-walled plate and shell structures are good energy absorbers especially when they lose their stability in the elastic range and then the plastic deformations appear leading to failure. When the structures are subjected to pulse load (as it is in case of blast pressure) the different phenomena may occur depending on the so called ‘‘pulse intensity’’ [7]—impact for pulses of high amplitudes and durations in range of microseconds or quasi-static behavior if amplitude is low and duration is two or more times of fundamental natural vibrations period. For pulses of intermediate intensity (amplitudes in range of static buckling load and durations close to the period of fundamental natural vibrations) the phenomenon of dynamic buckling occurs. The problem of shells dynamic buckling is well described in literature (e.g. [7,8]). The assessment of blast loading effects is required for design of structures to withstand the explosion. The blast loading time n
Corresponding author. E-mail address: [email protected] (K. Kowal-Michalska).
0263-8231/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2010.09.008
relation, obtained empirically, is exponential [2,5,6] and consists of two phases: positive (overpressure) of duration in range of milliseconds and negative one (subpressure) of duration in range of seconds. In the numerical analysis of explosion protection structures this relation is often represented by a right angle triangle and furthermore the important effects as multiple blast wave reflections, the Mach effect, and negative phase of the blast wave are usually neglected. Generally, it is assumed that explosions on surface or in air are free blast propagations without any contact with the ground or adjacent structures. Neglecting reflections of the blast wave could lead to overestimation or underestimation of structure effective stresses and its strain energy. It should be mentioned that the characteristics of reflected blast pressure can never be achieved empirically but only by numerical simulations (see [2,5,6]). In many cases the interaction of explosion wave and surrounding can result in magnification of the pressure impulse. The blast loading time relation may be of different character—it can be similar to the incident wave but of higher peak value or assuming perfect ground reflection and reflected wave overlaying the incident one, the blast loading time relation can be obtained by superimposing additional linearly varying loading. This paper aims to analyze the response of conical and hemispherical shells under different courses of blast pressure and to find out the direct effect of blast pressure modeling on shells strain energy and effective stress values. The problem has been solved by the finite element method using ANSYS software.
2. Assumed blast pressure relations The exponential blast loading time relation, obtained empirically, consists of two phases: positive (overpressure) of duration in range of milliseconds and negative one (subpressure) of
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
duration in range of seconds. In the numerical analysis of protection structures this relation is often represented by a triangle of the same duration Tp and peak value p0. From numerical reasons it has been assumed that this peak value is reached at t¼kTp where k ¼0.1 (Fig.1). In order to consider the effect of blast wave reflection three types of pressure characteristics are considered (Fig. 2). It has been assumed that in all cases pulse duration is the same and kept constant and equal to 0.5 ms. For all types the peak value of blast pressure equals p0 and the impulses equal each other. The pulse denoted by ‘‘I’’ is a triangle commonly used in calculations of structures subjected to blast pressure. The pulses denoted by ‘‘II’’ and ‘‘III’’ represent the reflected blast waves in which the value of overpressure and impulse decreases linearly until certain limit when the pressure begins to increase due to the effects of reflections. For pulse ‘‘II’’ the maximum value of overpressure p0 is reached at 0.05 ms and it decreases to p ¼0.2p0 at t ¼0.2525 ms and next the pressure increases and equals p¼0.8p0 at t ¼0.2975 ms and finally it linearly decreases to zero at time t ¼Tp ¼0.5 ms. For pulse ‘‘III’’ the first peak value equals 0.8p0 and it decreases to p ¼0.2p0 at t ¼0.2525 ms and next the pressure increases to the maximum value p0 at t ¼0.2975 ms and next linearly tends to zero at t¼Tp ¼0.5 ms.
605
presents the finite element models (way of discretization) for conical and hemispherical shells. The applied element enables to model linear and nonlinear material properties. For the assumed uniform pressure distribution the load was applied to all elements with the same value in a given time step. In the numerical analysis the dynamic responses of shell structures loaded by pulse pressure have been searched for. At the first stage the modal analysis has been performed with the aim to determine the period of natural frequency Tps. It has been found that for both considered structures Tps E0.1 ms. In the dynamic analysis the equilibrium equation has the following form: ::
:
fPg ¼ ½MUfug þ ½CUfug þ½KUfug
ð1Þ
where [M] is a structural mass matrix, [C] is a structural damping matrix and [K] is a stiffness matrix. In the analysed cases the damping can be neglected and then Eq. (1) can be written as follows: ::
fPg ¼ ½MUfug þ ½KUfug
ð2Þ
Substituting time derivative of displacement {¨ u} by increment of displacement {u} in consecutive discrete instant of time t the new equilibrium equations included inertia forces that are obtained in each time step. For the equation obtained the solution
3. Geometrical model of shells
pulse I pulse II pulse III
p0 0.8·p0 p(t)
The present paper deals with shells in shape of hemisphere and truncated cone closed by flat circular plate (Fig. 3). Both shells are of the same thickness (h¼0.5 mm) and height (40 mm) and are rigidly connected with a square plate of dimensions: in case of hemisphere 80 mm 80 mm and thickness h¼0.5 mm and in case of cone 60 mm 60 mm and thickness equal to 2 mm. The boundary conditions are shown in Fig. 3. The plates have been used to model the boundary conditions for shells as close as possible to real ones. The displacements in three perpendicular directions and rotations about three perpendicular axes have been set to zero for all nodes lying along bottom plate edges. The structures are loaded by blast pressure (shown in Fig. 2) uniformly distributed on the shell surface.
0.2·p0 4. Finite element models To solve the problem of dynamic response of shell structures the ANSYS [9] software based on the finite element method has been employed. For discretization the four-node shell element SHELL 43 [9] of six degrees of freedom has been applied. Fig. 4
0
0.1
0.2
0.3
0.4 0.5 0.6 time [ms]
0.7
0.8
Fig. 2. Assumed simulations of reflected blast pressure.
Fig. 1. Blast pressure representation: (a) empirical and (b) triangular approximation.
0.9
1
606
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
Fig. 3. Dimensions and boundary conditions of analysed conical and hemispherical shells.
Fig. 4. The finite element model of analysed shells.
algorithms used in static analysis can be employed. In ANSYS software the Newmark method is used to perform the integration over time and for equations solution in a consecutive time step the Newton–Raphson algorithm is applied. In the calculations the material stress–strain curve shown in Fig. 5 has been implemented with following strength properties: Young’s modulus E¼200 GPa, Poisson’s ratio n ¼0.3, density r ¼7850 kg/m3 and initial yield stress s0 ¼200 MPa.
5. Results of numerical calculations The results of numerical calculations are presented in figures showing the plots of maximal effective stresses and strain energy in time for assumed maximal values of blast pressure distributed accordingly to the relations described in Section 3. In Figs. 6–8 the time dependent plots of strain energy and effective stresses are presented for a conical shell loaded by three assumed distributions of blast pressure. The pressure maximal value p0 equals 4.5, 5.0 and 5.5 MPa, respectively.
Fig. 5. Material stress–strain relation obtained in laboratory test.
It is found that in case of pulse ‘‘I’’, when p0 increases from 4.5 to 5.5 MPa, the maximal value of strain energy increases almost ten times and the structure works in the elastic range for
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
607
Fig. 6. Strain energy and effective stresses time relations for conical shell when p0 ¼ 4.5 MPa.
Fig. 7. Strain energy and effective stresses time relations for conical shell when p0 ¼ 5.0 MPa.
Fig. 8. Strain energy and effective stresses time relations for conical shell when p0 ¼ 5.5 MPa.
p0 ¼ 4.5 MPa and when pressure peak value equals 5.5 MPa large regions become plastic. The similar behavior is observed for the pulse ‘‘II’’—in this case the energy increases (from p0 ¼4.5 to 5.5 MPa) about five times. For the third type of blast pressure distribution the plastic deformation appear in the second stage of pulse duration (when p0 ¼5.5 MPa) and the maximum value of strain energy increases only three times. In Fig. 9 the distributions of deformations (Fig. 9a, c) and effective stresses (Figs. 9b, d) are shown for conical shell subjected to triangular shape of blast pressure (pulse I) for two values of pulse peak (p0 ¼5 MPa – Fig. 8a, b and p0 ¼6 MPa – Fig. 8c, d). These pictures confirm the remarks made earlier and it can be seen that the shell collapses at its bottom base. This behavior conforms to the experimental results concerning static buckling of truncated conical shells, presented in paper [3]. Additionally, in arbitrarily chosen point of conical shell, loaded by pulse ‘‘I’’, the deflection has been registered for different values of peak overpressure values p0. The results are composed in Fig. 10. It should be underlined that the relations presented
in Fig. 10 do not correspond to the shell maximal deflection for the assumed value of p0. Three curves in Fig. 10 showing the deflection in chosen point registered at t ¼0.08 ms (‘‘crosses’’), t ¼0.3 ms (‘triangles’’) and the third curve (solid line) presents the maximal deflection in this point that appeared in whole range of calculation time (equal to 1 ms). The depicted relations allow estimating the value of peak overpressure p0 for which the deflections start to grow rapidly and it can be assumed that for p0 larger than 5.8 MPa the structure subjected to pulse ‘‘I’’ will fail. For the same values of p0 (4.5, 5.0, 5.5 MPa) the calculations of strain energy and effective stresses have been conducted for the hemispherical shell loaded by three different blast pressure characteristics (Figs. 11–13). It can be seen that the character of curves is similar to that found for conical shell (Figs. 6–8). When the maximal value p0 equals 4.5 MPa the shell is almost entirely elastic and the plastic regions increase with p0 growth for each type of blast pressure. Attention should be paid to the fact that for triangular pulse ‘‘I’’ of peak value equal to 5.5 MPa the shell
608
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
collapses—the effective stresses reach the ultimate strength (compare material stress–strain curve in Fig. 5 with effective stresses plots shown in Fig. 13). In Fig. 14 the distributions of deformations and effective stresses for two values of p0 are shown (the pulse loading is of type ‘‘I’’). It is found that the hemispherical shell destroys similarly to the conical one (compare Figs. 14 and 9). Comparing the collapse mode with those presented in paper [1] it can be seen that in case of the analysed hemispherical shell the buckling phenomenon does not occur. The overpressure peak value (p0 ¼5.5 MPa) is much lower than static bifurcation collapse pressure.
be more conservative but at the same time it is much safer when the analysed structure is to serve a purpose of explosion protection.
6. Conclusions On the basis of the presented calculation results it can be stated that the effect of reflection accounted for in the blast loading characteristic strongly influences the dynamic response of considered shell structures. Although the assumed pulses are of the same impulse value, peak overpressure and duration time the triangular distribution (pulse I) is the most dangerous and results in greater values of strain energy and effective stresses. Therefore, the traditional approach of blast pressure time relation seems to
Fig. 10. Relation between deflection and peak overpressure value for pulse ‘‘I’’ in arbitrary point of conical shell, registered in different time of pulse duration.
Fig. 9. Distributions of deformations and effective stresses for p0 ¼ 5 MPa (a, b) and for p0 ¼ 6 MPa (c, d).
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
strain energy [Nm]
0.7 0.6
250
pulse I pulse II pulse III
effective stress [MPa]
0.8
0.5 0.4 0.3 0.2 0.1 0
609
pulse I pulse II pulse III
200 150 100 50 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [ms]
0
0.2
0.4 0.6 time [ms]
0.8
1
Fig. 11. Strain energy and effective stresses time relations for hemispherical shell when p0 ¼ 4.5 MPa.
1.4 strain energy [Nm]
1.2 1
pulse I pulse II pulse III
0.8 0.6 0.4 0.2 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [ms]
time [ms]
Fig. 12. Strain energy and effective stresses time relations for hemispherical shell when p0 ¼ 5.0 MPa.
strain energy [Nm]
5
pulse I pulse II pulse III
4 3 2 1 0
300 effective stress [MPa]
6
pulse I pulse II pulse III
250 200 150 100 50 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time [ms]
0
0.2
0.4 0.6 time [ms]
0.8
1
Fig. 13. Strain energy and effective stresses time relations for hemispherical shell when p0 ¼ 5.5 MPa.
It should be also noted that from two proposed shell structures the conical one can sustain pulses of greater peak overpressure than the hemisphere. At last some general remarks:
In calculations the pulses duration is kept constant and equal to 0.5 ms while the period of fundamental natural vibrations equals approximately 0.1 ms. Therefore, according to the definition given in Section 1 the phenomenon should be
treated as quasi-static. In authors’ opinion this can be true if the pulse shape is rectangular but for other shapes, especially triangular (almost right triangle) the shell behavior is dynamic not static. The assumption of damping neglecting should be verified in further investigations. In this paper it was assumed that for each time step the blast pressure is uniformly distributed on shell surface. Perhaps it would be worth-while effort to determine pressure values depending on time and location on the shell surface.
610
K. Kowal-Michalska et al. / Thin-Walled Structures 49 (2011) 604–610
Fig. 14. Distributions of deformations and effective stresses for p0 ¼4.5 MPa (a, b) p0 ¼5.5 MPa (c, d).
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[5] Le Blanc G, Adoum M, Lapoujade V. External blast load on structures—empirical approach. In: Proceedings of the fifth European LS-DYNA conference, Ulm, Germany; 2006. [6] Remennikov AM, Rose TA. Modelling blast loads on buildings in complex city geometries. Computers and Structures 2005;83:2197–205. [7] Simitses GJ. Dynamic stability of suddenly loaded structures. New York: Springer Verlag; 1990. [8] Simitses GJ. Buckling of moderately thick laminated cylindrical shell: a review. Composites Part B 1996;27 B:581–7. [9] User’s guide ANSYS 11sp1, Ansys, Inc., Houston, USA.