Numerical simulation of warhead transportation

G.Martynenko ,K.Avramov a,,d,*,V.Martynenko ,M.Chernoryvko a,,A.Tonkonozhenko ,V.Kozharin

a Podgorny Institute for Mechanical Engineering,National Academy of Science of Ukraine,Department of Vibrations,2/10 Pozharskogo St.,61046 Kharkiv,Ukraine

b National Technical University“Kharkiv Polytechnic Institute”,2 Kyrpychova Str,61002 Kharkiv,Ukraine

c Yangel Yuzhnoye State Design Office,Dnipro,Ukraine

d Kharkiv National University of Radio Electronics,Department of Technical Systems,Kharkov,Ukraine

Keywords:Cartridge warhead Land operation Finite element simulation Operability condition

ABSTRACT The new numerical approach for analysis of the warhead transportations is suggested.This approach allows to control the warhead operability before its experimental analysis.The approach is implemented by the adequate models for the software ANSYS.Analysis of the loads at land operations and transportations of the warhead by natural roads,water and aviation allows to obtain the maximal values of loads,which are used in numerical simulations of the warhead.These loads give an opportunity to analyze the operability and the fatigue strength of the cartridge warhead.The numerical simulations of the attachments of the warhead combat elements are performed on the basis of the suggested method.The data of the numerical simulations verifies the operability of the fastener system of the warhead combat elements.

1.Introduction

Warhead is very important element of missile.Therefore,a lot of investigations performed devoted to this unit.Fast and accurate estimation of explosion fragmentations is treated in the paper[1].A projectile system is proposed to improve efficiency and effectiveness of damage done by anti-tank weapon system on its target by designing a ballistic projectile that can split into multiple warheads and engage a target at the same time[2].Natural fragmentation of the warheads that detonates causes the casing of the warhead to split into various sized fragments through shear or radial fractures depending on the toughness,density,and grain size of the material.The fragmentation behavior of radially expanding steel rings cut from a 25 mm warhead by using an in house smooth particle hydrodynamic is examined in the paper[3].The warhead consisting of carbon composite casings and high explosive is treated in the paper[4].In order to evaluate its blasting damage effect on concrete target,the three types of charges were researched by means of experiment,which are bare charge,charge with carbon composite material shell and charge with steel shell.The tuneable effects concept is used to achieve selectable blast and fragmentation output,to enable one charge to be used in different scenarios requiring different levels of blast and fragmentation lethality[5].The paper[6]presents a design for a protective bulkhead in the form of a multi-layered composite structure,which consists of a front plate,front aerogel felt,anti-penetration layer,back aerogel felt,and back plate.Hemispherical-nosed warheads were manufactured and then used to simulate the warhead of semi-armorpiercing anti-ship missiles.Numerical simulation of a missile warhead dynamic fracture is treated in Ref.[7].The parameters of the warhead are chosen in order to a fracture takes place in the structure specified area.The methodology of calculation of the fragments dispersion and determination of the hit locations is treated in Ref.[8].Based on the developed methodology,vulnerability assessment and survivability analysis of a generic aircraft is performed for different predefined approach angles of the missile threat.

In some novel warheads,the explosive charge is not compactly filled within the cylindrical casing but has a hollow core instead.However,most existing formulas can only predict the fragment velocity of warheads fully filled with explosive charge.In the paper[9],a new formula was proposed to predict the initial fragment velocity of novel warheads with hollow core.Numerical simulations of the warheads fragmentation are treated in Ref.[10].The applicability of the Johnson-Cook strength and fracture model is evaluated by comparing the fracture behavior of an expanding steel casing of a warhead with experimental data.The strength and ductility at high strain rates of the warhead are treated in Ref.[11].The experimental results of the ring fragmentation are compared with the numerical simulations using the hydrocodes.New advanced numerical computer model enabling accurate simulation of fragmentation parameters of shells has been developed and validated in Ref.[12].The use of sintered metal compositions for the shells of explosive warhead is treated in Ref.[13].The analysis is based on the simulations with different numerical approaches and the models for the launch and the impact phase.The spatial dispersion of fragment generator warhead is analyzed experimentally in Ref.[14].This paper presents the explanation of the fragment dispersion phenomenon using one dimensional shock theory.The experimental results[15]show,that the acceleration process of the warhead can be divided into two distinct phases:initial acceleration under shock waves and further acceleration under the load of detonation products.The simulation was developed to predict damage from closely spaced tumbling rods in Ref.[16].This simulation predicts the synergistic effects from any collateral damage against submunition and bomblet payloads.The response of buried shelters to blast loadings due to conventional weapon detonation has been investigated using the finite element method in Ref.[17,18].The finite element analysis was carried out using a commercial finite element software package,ABAQUS.The validity of finite element model was established by comparisons with existing empirical results.

The implementation of the warhead battle task is treated mainly in the above-mentioned papers.It seems,the warhead transportation is not treated.Analysis of warhead transportation is very important matter,as unsuccessful transportation can led to loss of the warhead operability.The approach for the simulation of the warhead transportation is suggested in the present paper.This approach is implemented in finite element software ANSYS.

2.Problem formulation

As follows from the review of the modern scientific journals,the warhead operation is treated mainly.The transportation of the warhead is not considered.The methodology of the warhead transportation simulation is suggested in this paper.The warhead is installed on the missile[7].Such warhead is transported by broken country,water and air in wide ranges of the temperatures.The suggested methodology is based on the finite element calculations.

As follows from the request for the proposal,which is produced by the design office of this warhead,the numerical simulations of the warhead transportations replace the bulk of the expensive experimental investigations.Therefore,the systematic experimental analysis is not carried out.

On the other hand,the authors of this paper have experience of the simulation of this warhead operation,which is treated in the paper[7].In this research,the authors compare the results of the numerical modeling with the experimental data.

The sketch of the cassette warhead is shown on Fig.1.The warhead has two stages.Every stage consists of the set of the combat elements(CE)(Fig.1).The bolted joints(Fig.2a)are used to attach CEs of the first(internal)stage.The CEs are pressed to locating blocks with preset tightness.The tension bands(Fig.2b)are used to attach CEs of the second(external)stage.The band tension assembles the warhead by means of doweled joints(Fig.3).Friction forces between contact surfaces prevent the slip of the CEs.

The missile flight is the basic mode,which is investigated traditionally.On this stage,the starting mechanism is actuated.Then the CEs fly away and the part of earth surface is covered by CEs.Such operation of this cartridge warhead(Fig.1)is treated in the previous paper of the same authors[7].

The approach for the simulations of the transportation of the cartridge warhead is suggested in the present paper.These loads,which act on the warhead,describe the land operation of the warhead;water and aircraft transportation.The list of the operation conditions leads to huge full-scale experimental tests or numerical calculations.The finite element simulations of such complex branched structure lead to huge numbers of finite elements and contact surfaces.Plastic states of the structure material are accounted close to contact surface.Such analysis leads to huge number of the computer simulations.Therefore,the approach,which is reduced the number of the numerical calculations,is suggested in the present paper.If less number of calculations is carried out,then more accurate models with huge number of degrees-of-freedom,contact surface and nonlinearities are derived.This increases the accuracy of the results.

CEs operability is verified for different conditions of transportations in the following way.First of all,the possibility of CEs slip is examined by numerical simulations.In the second place,the conditions of static and fatigue strength of the warhead under the action of quasi-static and dynamic loads are examined.Such loads are observed at the warhead transportations by broken country,water and air.The examination of the fatigue strength is performed for maximal values of the dynamic loads.Note,that the dynamic loads are accounted together with the pretension,which originates due to the first stage bolts tightening and the doweled joints of the second stage tension band(Fig.2b).

The loads acting on the warhead are analyzed for the transportations on broken country,water and air.The following dangerous loads are determined:

1)the thermal loads,which act during water transportation and storage;

2)the operation loads and the vibratory loads,which act on the warhead at discharge,crane overcharge and transportation by railway,aircraft and water.These overloads are preset by meansquare values of the vibrations accelerations;

3)vibratory loads at car,railway,aviation and water transportations,which are preset by the values of the acceleration amplitudes.

The loads,which are applied on the warhead to analyze its operability and the fatigue strength,are determined.The types of transportations and the loads,which are applied to the warhead,are systematized in Table 1.The types of transportations are shown in the first column of Table 1.The vibrations accelerations amplitudes in longitudinal,vertical and lateral directionsnx,ny,nzare shown in the second,third and fourth columns of Table 1.The considered values of the warhead temperatureTare shown in the fifth column.Three temperature conditions are used for the numerical analysis.The calculations are carried out with three values of temperatures:-40,+20 and+50°C.The dependence of the mechanical parameters on the temperature is accounted.

All types of the calculations,which are presented in Table 1,are treated in this paper.The first case is consistent with the thermal strength stationary analysis.The static tightening and the thermal loads are accounted in the calculations.In order to simulate the second and the third types of the warhead transportations(Table 1),the overloads in different directions are applied.The second case,which corresponds to the action of the longitudinal overloads,is the most dangerous for the warhead operability,as the CEs slip can be observed.The conclusions about the warhead operability are reduced to the reliability of CEs attachments in the bonds and the clamps.The thermal strength calculations with examination of the holding elements strength are carried out for the second and the third cases(Table 1).

Fig.1.Sketches of cartridge warhead.

Fig.2.Junctions of structure elements:a).bolted joints of first stage;b).doweled joints of the second stage tension band.

Fig.3.Symmetrical part of the cartridge warhead.

The dynamical strength calculations with stationary values of the temperatures are carried out to analyze the fourth and the fifth cases(Table 1).Two types of the calculations are performed.The harmonic analysis of the structure with account of the inertial loads is carried out for the first type of calculations.Vibration analysis of the warhead under the action of multiharmonic loads is related to the second type of the calculations.As a result of this analysis,the multiharmonic response is obtained.Then the fatigue strength is analyzed numerically.

The following phenomena are accounted in these calculations.In the first place,the tension loads and the temperature field generate the static stress-strain state.The vibrational loads act on the structure besides the static loads.Therefore,the asymmetrical fatigue cycles are considered.In the second place,if the eigenfrequencies of the structure do not belong to the frequency range of the vibrational loads,then the resonance vibrations are not observed.In this case,the amplitudes of the vibrations are close to the static stress-strain state.Then the dynamical analysis is replaced by the quasi-static calculations.In the third place,if the dynamic load with the preset frequency does not cause to highcycle fatigue(the number of cycles is less 107),then the dynamic analysis can be reduced to the quasi-static one.

On the preliminary stage of the analysis,the adequate simplified model of the contact interactions between the bodies of the warhead is obtained to transform the comprehensive model into the symmetric one,which contains 1/8 part of the structure(Fig.3).The comprehensive model is used to perform the structure analysis under the action of the inertial loads,which has the following three projections of the vibrations accelerations:nx(longitudinal),ny(lateral),nz(horizontal).If only longitudinal vibration acceleration nxacts on the structure,the 1/8 part of the structure with the contact interactions is analyzed.

Table 1The most dangerous operational loads.

3.Comprehensive model with contact interactions between the bodies

3.1.Model of 1/8 part of the structure

Analysis of the structure 1/8 part is impossible to calculate stress-strain states of the warhead with the vibrations accelerations,which contains nx,ny,nzcomponents.In this case,the complete geometrical model of the structure is used for the numerical analysis.Moreover,the threaded connections with the pretension are accounted besides the contact interactions.

If only longitudinal load acts on the structure,it is possible to simulate the stress-strain state by using 1/8 part of the warhead.The model of the warhead 1/8 part is shown on Fig.4.This model contains 18 nonlinear contacts interactions and 4 threaded connections.It is produced by ANSYS Workbench.

Fig.4.The warhead model with contact interactions.

The objective of the calculations is obtainment of the adequate model for the warhead stress-strain state.Preliminary calculations are performed for 1/8 part of the warhead.The contact interactions statuses are shown on Fig.5a.The pretension loads of the bolted and the threaded joints are shown on Fig.5b.More than 2 million degrees-of-freedom are used in the model of 1/8 part of the warhead.This allows to obtain the relative error of the stresses calculations about 0.2%.Fig.6a shows the status of the contact interactions after tightening.The longitudinal displacements field of the warhead under the action of gravity is shown on Fig.6b.The values of the displacements verify,that the friction forces hold CEs.These forces originate due to the pretension.

The nonlinear analysis of the static stress-strain state of the warhead under the action of the pretension loads is carried out.The results are shown on Fig.7.The fields of the displacements and the von Mises equivalent stresses are shown on Fig.7a and b,respectively.

The following types of the bodies contacts interactions are used in the simulations.“Bonded”presents linear contact interaction,which describes the total bonding of bodies.“No Separation”presents linear contact,which describe free motions of the contact surfaces of two bodies without separation.“Frictionless”describes nonlinear contact interaction of two bodies without friction.“Frictional”presents nonlinear contact of two bodies accounted friction force.The mechanical properties of the material are accounted by using the following options.“Elastic”describes linear material with preset elastic properties;“Plastic”describes nonlinear plasto-elastic material.The calculations of the total model of the warhead under the action of the pretension loads are performed to choice the combination of the calculation parameters,which are treated above.The calculations with the following combinations of the options are performed:1)“Plastic”and“Frictional”;2)“Elastic”and“Frictional”;3)“Plastic”and“Bonded”;4)“Elastic”and“Bonded”;5)“Plastic”,“Bonded”and“No Separation”;6)“Elastic”,“Bonded”and“No Separation”.

The following conclusions are made using the calculations results.Plastic behavior of the warhead material is observed near the clamp.Thus,the plasto-elastic material properties are accounted to calculate the total warhead stress-strain state.The slipping in the“Bonded”contacts interactions must be accounted to describe adequately stresses fields in the bands and the clamps.“Frictional”options describe adequately bodies contact interactions using the nonlinear model.The use of“Frictional”contact results in the significantly increase of the computation time due to iterative calculations.The contact model“No Separation”is taken into account slipping of the bodies.This model is linear.Moreover,the use of this contact model does not result in the significant errors of the stress-strain state close to the band contacts.The contact model“No Separation”is used to simulate the band contacts.

3.2.Verification of the finite element model

As the finite element method is approximate,the obtained solution needs error estimation.The verification of the model of the structure 1/8 part is performed in several stages.

The choice of the appropriate finite elements is performed on the first stage.3D finite elements in the form of hexahedrons or tetrahedrons are used to mesh the warhead.The ordered finite element mesh is used for the regular domain.As follows from the theory of the finite element method[19],this discretization provides small error.

The calculations to obtain the optimal finite element dimensions are performed on the second stage.More than ten calculations of the structure under the action of the static loads were performed.The average dimensions of the finite elements are changed in these calculations.The local maxima of the stresses are analyzed for different dimensions of the finite elements.As a result of the analysis,the finite elements mesh,the error of which does not exceed 1÷3%,is obtained.

Fig.5.Sketches of warhead part:a).status of contact interactions up to loads application;b).pretension loads.

Fig.6.Sketches of the structure parts:a).status of contact interactions after pretension;b).axial displacements of warhead under the action of the gravity.

The error,which is caused by the discretization,is estimated on the third stage.More than five calculations of the structure with prestress and the bolt tightening are performed.The errors are estimated by the comparison of the nodal(averaged)stresses and elemental(unaveraged)stresses in the most loaded regions.For example,the averaged and unaveraged von Mises equivalent stresses in the holding elements of the structure under the action of the pretension are equal to 493и500 MPa,respectively.These stresses are equal to 496 and 510 MPa in the band.Then the relative errors are equal to 1.4%and 2.8%,respectively.For the rest cases,the relative errors are within 3%.This is permissible result,which indicates the good quality of the finite element discretization.

3.3.Total model of the structure under the action of nonaxisymmetric loads

The total model of the warhead is shown on Fig.8a.The tightening of the threated connections is accounted.The contacts between the band and the CEs and between the clamp and the CEs are described by the options“No Separation”.The rest contacts are described by the“Bonded”type.The finite-element mesh of the total structure is shown on Fig.8b.The number of the finite elements is eight times as large,than the number of finite elements in the warhead 1/8 part.

Fig.7.The calculations results of the warhead 1/8 part:a).displacements field in mm;b).von Mises equivalent stresses in MPa.

Fig.8.Total model of warhead:a).model with preload;b).finite-element mesh of the warhead.

4.Numerical simulation of warhead land operation

4.1.Stress-strain state of structure under the action of temperature loads

The effect of the temperature on the stress-strain state of the warhead and its operability is analyzed in this section.By operability we shall basically mean absence of CEs slip.The model of the warhead 1/8 part(Fig.4)is used to simulate the slip of the CEs.

The thermoelastic problem is solved for the first case(Table 1).Then it is assumed that the temperature of the warhead is constant and it equals to-40°C,+20°C and+50°C.The preloads of the threaded connections are accounted in the model.The plastic behavior of the material is not accounted.The symmetry of the structure is used for the boundary conditions.The fields of the displacements and the von Mises equivalent stresses are calculated by the software ANSYS.Strength analysis is performed by comparison of the von Mises equivalent stresses with the limiting values.The fields of displacements and the von Mises equivalent stresses are shown on Figs.9-13.

The operability criterion at the negative temperature is impossibility of the CEs slip due to body contraction and decrease of the preloads at the bolted connections and the doweled joints.The thermal stresses in the warhead are shown in Table 2.The elements of the warhead are presented in the first column.The middle von Mises equivalent stresses at different warhead temperatures are shown in the rest columns.The von Mises equivalent stresses are in excess of the yield stress.The dependences of the plasto-elastic material properties on the temperature are accounted.If the temperature is increased,the equivalent stresses in the bands and the pins are increased and the equivalent stresses in the bolts and the clamps are decreased.The additional calculations are performed to account the dependences of the plasto-elastic material properties on temperature.The equivalent stresses exceed the yield stress only in local zone under the clamp washer of the first stage CEs.This leads to the local bearing of the material.Moreover,this does not effect on the strength of the bracket.If the temperature is increased up to+50°C,the stresses in the bands,the pins of CEs of the second stage are increased.Note,that the stresses are decreased in others brackets.

Fig.9.The displacements in mm with the following preset constant temperature:a)+20 °C;b)-40 °C;с)+50 °C.

Fig.10.Radial displacements in mm at the following temperatures:a)T=+20 °C;b)T=-40 °C;c)T=+50 °C.

Fig.11.The von Mises equivalent stresses(MPa)in the second stage at the following temperatures:a)T=+20 °C;b)T=-40 °C;T=+50 °C.

Fig.12.The von Mises equivalent stresses(MPa)in the first stage at the following temperatures:a)Т=+20 °C;b)Т=-40 °C;Т=+50 °C.

If the temperature is decreased up to-40°C,the stresses in the bolts and in the clamps of the first stage CEs are increased.The values of the friction forces are enough to keep CEs without slip by means of the clamps and the bands.The maximal von Mises equivalent stresses do not exceed the yield stress.

4.2.Operability of structure under the action of longitudinal inertial loads

The model of the warhead 1/8 part(Fig.4)is used to analyze the operability of the structure under the action of the axial inertial loads.The warhead has constant temperature.Internal cylindrical column is added into the warhead model(Fig.14).The obtained model corresponds to the second case(Table 1).The thermoelastic stresses of the warhead under the action of the inertial forces are analyzed at the following temperatures:-40°C,+20°C;+50°C.The tension of the threated connection and the plasto-elastic material properties are accounted in this model.The values of the temperature are assumed constant in the whole structure.Symmetry of the structure is used as the boundary conditions.The fields of the displacements and the von Mises equivalent stresses are obtained numerically.The operability of the structure is checked during the calculations.The absence of the CEs slip is used as the operability condition.

Fig.13.Von Mises equivalent stresses(MPa)in the joint at the following temperatures:a)T=+20 °C;b)T=-40 °C;T=+50 °C.

Fig.14.Symmetrical finite-element model.

Table 2The dependence of the middle von Mises equivalent stresses on temperature.

The longitudinal displacements of the warhead under the action of the longitudinal inertial forces with the acceleration 2 g are shown on Figs.15-17.The calculations results of the warhead under the action of the inertial forces downwards and upwards are shown on Figs.15-17 a and b,respectively.These Figures show the numerical results for different values of the temperature+20,-40 and+50°C.The fields of the von Mises equivalent stresses in the elements of the warhead are shown on Figs.18-20.The stresses in the clamps of the first stage CEs and the bands are shown on the upper parts of these figures.The possibility of CEs slip is determined by using these stresses.The data of the stress calculations under the action of the axial loads downwards and upwards are shown on these figures with notations(a)and(b),respectively.

As follows from the numerical analysis,the stresses in the pins of the couples“the first stage cradle-the first stage CE”,“the first stage CE-the second stage cradle”,“the second stage cradle-the second stage CE”do not exceed the yield stress.If the temperature of the warhead is decreased up to-40°C(Figs.16 and 19),the minimum value of the safety factor is equal to 1.8.The decrease of the temperature up to-40°C is very dangerous owing to the thermal compression.This thermal compression results in tension decrease in the holding elements(clamps,bolts,bands,locks).The principal loads are transferred into the pins between the CEs and the cradles.In this case,the maximal von Mises stresses(500 MPa)take place in the pin between the first stage cradle and the first stage CEs.These stresses in the structure are consistent with the safety factor near 2.

Fig.15.The fields of the axial displacements(in mm)of the warhead at the temperature+20 °C under the action of the following longitudinal forces:a)downwards;b)upwards.

Fig.16.The longitudinal displacements of the warhead at the temperature-40 °C under the action of the following longitudinal forces:a)upwards;b)downwards.

The most dangerous state of the warhead is uniform heating up to the temperature+50°C(Figs.17 and 20).In this case,the minimal safety factor is equal to 1.2,which is explained by decrease of the admissible stress and increase of the temperature.At the rest temperatures,the safety factors in all pins are in exceed of 3.This corroborates the operability of the structure.

The maximal variations of the stresses in the warhead under the action of the maximal inertial longitudinal loads,which are summed up with the stresses from tightening and temperature,add up to 5%.Therefore,the stresses in the warhead under the action of the inertial longitudinal forces contribute insignificantly to the stressed state of the structure.

Fig.17.Longitudinal displacements of the warhead at the temperature+50 °C under the action of the following longitudinal forces:a)upwards;b)downwards.

4.3.Calculations of warhead loads at transportation

The static and quasi-static loads acting on the structure are treated above.The loads acting on the warhead are treated for the third,the fourth and the fifth cases(Table 1)in this subsection.These loads act on the warhead at automobile,railway,aircraft and water transportations.These loads are set by the longitudinal,vertical and lateral amplitudes of accelerations.

The solution of the dynamical problem is transformed into the equivalent quasi-static formulation only after analysis of the eigenfrequencies and the eigenmodes of the warhead vibrations[20-23].The data of the total structure modal analysis(Fig.8)accounting prestresses due to the bolts and the doweled joints tightening are shown in the form of the eigenfrequencies spectrum(Fig.21).The values of the eigenfrequencies in Hz versus their numbers are shown on this Figure.The structure eigenmodes are presented on Figs.22-25.The nonlinear contacts interactions are replaced by the linear contacts“Bonded”in modal analysis.Thus,the calculations with the combination“Elastic”&“Bonded”&“No Separation”are performed.The eigenfrequencies are analyzed in the range 0÷1200 Hz.The upper range limit is equal to the maximal frequency of the dynamic load.

The first eigenmode(Fig.22a)is mainly torsional.The second eigenmode(Fig.22b)is flexural with one nodal diameter and the third eigenmode(Fig.23a)is shear.The vibrations of the bands are observed on the eigenmodes No.4-7(Figs.23b,24a,24b,25a).The circumferential vibrations of the CEs are observed on the eigenmode No.8(Fig.25b).

The eigenmodes,which are shown on Figs.22-25,are orthogonal to the loads acting on the warhead.Therefore,these eigenmodes are not excited by the loads,which are observed at the land operation.Thus,if the frequencies of the excitation are close to the eigenfrequencies,then the dynamical system analysis is reduced to the quasi-static one.In this case,the static loads coincide with the harmonic loads amplitudes.

The others eigenmodes have eigenfrequencies,which do not coincide with the frequencies of the excitations.Mainly,the frequencies of the dynamic loads are less than the first eigenfrequency.Therefore,the dynamic calculations can be replaced by the quasi-static ones for the cases 3,4 and 5(Table 1).The use of the quasi-static models gives great advantages.It is possible to use the total model with account of the pretension and nonlinearity.The total finite-elements model of the warhead(Fig.8b)has 1524295 nodes and 629836 tetrahedral finite elements of the second order with 10 nodes and hexagonal elements of the second order with 20 nodes.Moreover,128 contacts(64“Bonded”contacts;48“No Separation”contacts and 16“Frictional”contacts)are accounted in this model.The loads are applied sequentially in four stages:1).bolt tightening of the first stage CEs;2).bend tensioning of the second stage(Fig.8a);3).temperature loading;4).the overloads are applied.

4.4.Analysis of operability at land operation and aircraft transportation

The third case(Table 1)is devoted for strength analysis of the warhead under the action of three dimensional loads,which take place at the land operation and the aircraft transportation.The temperatures fields are taken into account.The strength of the structure with the constant temperature under the action of the maximal three dimensional overload is solved in the quasi-static formulation.The total warhead model(Fig.8)is used.The detailed description of this model is presented in Section 4.3.The symmetry of the warhead is accounted in the boundary conditions.As a result of the calculations,the parameters of the stress-strain state are obtained.The Von Mises equivalent stresses are compared with their limiting values to analyze the structure strength.

The structure under the action of the maximal quasi-static inertial forces with the lateral component is analyzed.These inertial forces are obtained from the analysis of the warhead crane transfer and its transportation by broken country,water and railway.Fig.26a and b and c show the results of the warhead stress-strain state calculations at three different values of the temperatures+20°C,+50°C and-40°C.

Fig.18.Von Mises equivalent stresses(MPa)in the warhead at temperature+20 °C under the action of the following longitudinal forces:a)upwards;b)downwards.

The operability analysis of the warhead under the action of the inertial forces(the case 3 of Table 1)results in the following conclusion.The stresses in the pins of the couples“the first stage cradle-the first stage CE”,“the first stage CE-the second stage cradle”,“the second stage cradle-the second stage CE”do not exceed the yield stress.The yield stresses depend on the temperature.As the difference between stresses in the structural components,which are treated in Section 4.2 and in this section,do not exceed 7-8%,all conclusions about the strength,which are made in Section 4.2,are true for the considered cases.

4.5.Analysis of warhead operability at broken country and aircraft transportations

The fourth case(Table 1)is devoted to the stress calculations and the fatigue strength analysis of the warhead at different values of temperature with the vibrational accelerations,which occur during warhead transportation at the broken country and aircraft transportations.As follows from the eigenfrequencies and the eigenmodes analysis,the dynamical calculations for the fourth and fifth cases(Table 1)can be replaced by the quasi-static analysis.Such calculations are performed to determine the mean values and the amplitudes of the stresses.In this case,the constant component of the Fourier series is observed in the fatigue cycle of the structure.The conditions of the fatigue strength are verified.

The warhead under the action of the quasi-static inertial forces is analyzed in order to study the fourth case from Table 1.The vector of the inertial force has lateral component.The maximal excitation frequency is equal to 40 Hz.This excitation frequency is significantly less,than the first excited eigenfrequency.Therefore,the calculations of the vibratory amplitudes are performed far away from the resonance by the quasi-static calculations with the loads,which are equal to the amplitude values of the dynamic loads.

For the fifth case from Table 1,the excitation frequencies of the aircraft transportation have wide range 10÷2000 Hz.As the spectrum of the eigenfrequencies is dense,the resonance phenomenon is possible.The maximal amplitudes values of three components of the inertial forces are equal to(1.2÷1.3)g for the excitation frequency range 500÷2000 Hz.In such high frequencies,the resonances are possible with insignificant increase of the vibratory amplitudes.As the material damping and the dry friction damping in the contact pairs decrease significantly the amplitudes of the vibrations on the high frequencies.In the fifth case from Table 1,the fatigue strength of the structure is estimated using the data of the quasi-static analysis of the structure.The static loads are assumed to be equal to the amplitudes of the dynamic loads.

Fig.19.Von Mises equivalent stresses(MPa)in the warhead at temperature-40 °C under the action of the following longitudinal forces:a)upwards;b)downwards.

The complete model(Fig.8)is used for the fourth and the fifth cases.The symmetry of the warhead is used for the boundary conditions.The stress-strain state of the warhead is analyzed numerically.The von Mises equivalent stresses are compared with the limiting values to verify the warhead strength.The fatigue strength is estimated using the fatigue asymmetric cycle of the equivalent stresses in the fourth and fifth cases(Table 1).The mean stressesσmare obtained as a result of the action of the pretension and the temperature field.The difference between the mean stresses and the stresses,which are obtained for the cases 4 and 5(Table 1),are considered as fatigue amplitudes values of the stressesσa.

The condition of fatigue strength is the following:

whereσmandσaare mean and amplitude values of stresses;σRis fatigue point;R=σmin/σmax.The symmetric fatigue cycle satisfy the following equation:R=-1.

The dependence of the fatigue point on the mean stress is approximated by the phenomenological equation[24-26].The Goodman form of this equation is the following[24]:

Fig.20.Von Mises equivalent stresses(MPa)in the warhead at temperature+50 °C under the action of the following longitudinal forces:a)upwards;b)downwards.

Fig.21.The eigenfrequencies spectrum of the warhead.

The approximation of the phenomenological equation in the Gerber form is[24]:

The Goodman model is used for brittle material and the Gerber model is used for the plastic material.As the warhead is fabricated from the brittle material,Eq.(2)is used.

Figs.27 and 28 show the results of the quasi-static analysis of the structure under the action of the loads,which corresponds to the fourth and fifth cases(Table 1).The stresses in the warhead under the action of the tightening of the bolts and the doweled joints,the temperature fields and the quasi-static loads are shown on Figs.27 and 28.In this case,the mean stressesσmare consistent with the ones,which are shown on Figs.11-13.The differences between the total stresses(Figs.27 and 28)and mean stresses(Fig.11)are calculated to obtain the amplitudes of the stresses.These stresses are used in the conditions of the fatigue strength(1)in order to obtain the conclusion about the warhead operability for the fourth and fifth cases from Table 1.We come to the conclusion,that the warhead is operable.

Fig.22.The eigenmodes,which conform to the following eigenfrequencies:a)No.1-244 Hz;b)No.2-433 Hz.

Fig.23.The eigenmodes,which conform to the following eigenfrequencies:a)No.3-574 Hz;b)No.4-657 Hz.

Fig.24.The eigenmodes,which conform to the following eigenfrequencies:a)No.5-696 Hz;b)No.6-730 Hz.

5.Discussions and global analysis

The data of the above considered calculations of the warhead under the action of the static loads for the cases 1-3(Table 1)gives the complete understanding of the structure behavior at the transportation.The equivalent stresses in the warhead at different temperatures are determined by the numerical simulations.These results are used to verify the strength conditions by comparison the equivalent stresses with the conventional yield strengthσ0.2.The most dangerous from the viewpoint of strength is the temperature-40°C.In this case,the safety factor is minimal for the doweled joint of the tension band.It is equal to 1.14.In the most cases,which are treated here,the safety factor belongs to the range 1.4÷2.8.Thus,the strength condition is satisfied.The exceeding of the permissible stresses is observed only in the aluminum clamp.Such maximal stresses are equal to 190,162,161 MPa.Note,that the permissible stresses are 166, 162, 160 MPa at temperatures-40,+20,+50°C,respectively.This phenomenon is local.It results in plastic state only under the washer of the bolts.This does not effect on the conclusion about the structure strength.The equivalent stresses in the welded joints are close to the limiting values.

Fig.25.The eigenmodes,which conform to the following eigenfrequencies:a)No.7-783 Hz;b)No.8-902 Hz.

Fig.26.Von Mises equivalent stresses(MPa)in the warhead at the following temperature values:a)+20 °C;b)+50 °C;c)-40 °C.

Fig.27.Equivalent stresses(MPa)for different values of temperature:a)+20 °C;b)+50 °C;c)-40 °C.

Fig.28.Equivalent stresses(MPa)for different values of temperature:a)+20 °C;b)+50 °C;c)-40 °C.

The following conclusion about the operability and the structure strength are made on the basis of the calculations of the warhead transportation under the action of the dynamical loads for the cases 4-5(Table 1).The maximal values of the equivalent stressesσmaxfor each structural element are obtained.Using these values and the mean stressesσm,the amplitudes fatigue stressesσaare calculated.The mean stresses originate due to the action of the tightening and the temperature(case 1,Table 1).The fatigue points σRare calculated according to formula(2)for every value of mean stress.They have acceptable values for asymmetric fatigue cycle(1).

The maximal equivalent stressesσmaxdoes not exceed the yield stressσ0.2.Then,if the dynamical loads act on the warhead and the number of the fatigue cycle is equal to N=107,high-cycle fatigue is observed.As follows from the comparison of the amplitude of the fatigue stressσawith the fatigue pointsσR,the condition of the fatigue stress is satisfied for all structural elements.The safety factor belongs to the range[1.6,20].Even if the dynamic factor is taken 1.5,then the condition of the safety strength is satisfied.The pin of the first stage CEs and the welding joints of the first stage cradle are exceptions.The maximal equivalent stressesσmaxof these elements are exceed the yield strengthσ0.2and the amplitudes of the fatigue stressesσaare exceed the fatigue pointσR.Exceeding of the permissible stress in the pin has local nature.It leads to plastic state under the bolt washer.This does not effect on the general strength of the structure.

6.Conclusion

The analysis of the structure operability under the action of the dynamical and temperature loads,which occur due to the warhead transportation,is necessary for the design of the CE fastener system.The terms and costs of the experimental analysis are considerable,if full-scale experimental analysis is performed.Therefore,the numerical simulations of the structure behavior are very important for the design.

The approach for the computational modeling of the warhead transportation is suggested in the present paper.It allows to simulate numerically the warhead operability.This approach consists of two stages.

The first stage includes analysis of the operational loads and the temperature conditions for determination the maximal loads.The analysis of all possible loads acting on the warhead at the land operation and the aircraft transportation is performed to determine the most dangerous loads.This allows to analyze the warhead operability.

The numerical simulations of the warhead transportation are performed in the quasi-static formulation on the second stage of the analysis.The conclusion about the structure operability is made on the basis of the numerical simulations of the fastener system.Analysis of the fastener system incorporates investigations of the stress-strain state,which is induced the structure assembly,temperature fields and transportation loads.Analysis of operability and the fatigue stresses of the warhead fastener system are performed by calculations of the mean stresses and the amplitude fatigue stresses.Accuracy of the results is verified by the validity of the structure model,which accounts contact elements,nonlinear material behavior and the dependence of the mechanical characteristics on the temperature.The data of the numerical simulations verifies the operability of the CEs fastener system during the structure transportation.

Declaration of competing interest

The authors declare,that there are not conflict of interests.