dynamic response of beams and plates to impact loading condition

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Dynamic Response of Beams and Plates to Impact loading Condition Project-I (CE47005) report submitted to Indian Institute of Technology, Kharagpur in partial fulfillment for the award of the degree of Bachelor of Technology (Hons.) in Civil Engineering by Abhik Bhattacherjee (11CE10002) Under the supervision of Prof. L.S.Ramachandra Department of Civil Engineering Indian Institute of Technology, Kharagpur Spring Semester, 2015February 26, 2015 Dynamic Response of Beams and Plates to Impact loading Condition 2 | P a g e DECLARATION I certify that (a) The work contained in this report has been done by me under the guidance of my supervisor. (b) The work has not been submitted to any other Institute for any degree or diploma. (c) I have conformed to the norms and guidelines given in the Ethical Code of Conduct of the Institute. (d) Whenever I have used materials (data, theoretical analysis, figures, and text) from other sources, I have given due credit to them by citing them in the text of the thesis and giving their details in the references. Further, I have taken permission from the copyright owners of the sources, whenever necessary. Date:(Abhik Bhattacherjee) Place: Kharagpur (11CE10002) Dynamic Response of Beams and Plates to Impact loading Condition 3 | P a g e DEPARTMENT OF CIVIL ENGINEERING INDIAN INSTITUTE OF THECHNOLOGY, KHARAGPUR KHARAGPUR-721302, INDIA CERTIFICATE This is to certify that the project report entitled “ Dynamic Response of Beams and Plates to Impact loading Condition ” submitted by Abhik Bhattacherjee (Roll No. 10CE10002) to Indian Institute of Technology, Kharagpur towards partial fulfillment of requirements for the award of degree of Bachelor of Technology(Hons.) in Civil Engineering is a record of bonafide work carried out by him/her under my/our supervision and guidance during Autumn Semester 2014-15. Prof. __ ________ Date: Department of Civil Engineering Place: KharagpurIndian Institute of Technology Kharagpur, India Dynamic Response of Beams and Plates to Impact loading Condition 4 | P a g e Table of contents Sl. No. Subject Page Index 1 Introduction 5 - 6 2 Scope and Objective 7 3 Literature Survey 8 - 9 4 Finite element modelling 10 - 13 5 Result and Discussion 14 - 21 6 Conclution 22 7 Works to be carried out 23 8 Reference 24 Dynamic Response of Beams and Plates to Impact loading Condition 5 | P a g e 1 Introduction The threats to the public security and public premises are on rise because of increasing terrorism and violence. Safety of individual is matter of concern, hence there is need to develop impact resistant solutions for convenience stores, detention facilities, hospitals, government offices, schools and banks, public installations, military applications and sensitive premises. Generally, constructions in these remises can be made strong by using solution with first line of defence. There are various protection levels in NIJ and UL standards as mentioned below: Sr. No Protection level Test Round Test bullet Equivalent UL 752 Ratings 1 IIA 1 9 mm FMJ RN Level 1 2 0.40 S&W FMJ 2 II 1 9 mm FMJ RN Level 2 2 0.357 Magnum JSP 3 IIIA 1 0.357 SIG FMJ Level 3 2 0.44 Magnum SJHP 4 III 1 7.62 mm NATO FMJ Level 8 5 IV 1 0.30 Caliber M2 AP Level 4 NIJ level III plate provides the resistance against the 7.62 soft core AK47 ammunition. Dynamic Response of Beams and Plates to Impact loading Condition 6 | P a g e When a projectile strikes an object during high velocity due to the complexity of deformation of material at high strain rates During the high velocity penetration that takes place where a projectile strikes a target, many significant phenomena appear to be experimentally intractable due to the complexity of materials deforming at extremely high strain rates. To better understand the physics of the penetration process during high velocity impact, finite element analysis methods have proven to be an invaluable diagnostic and design tool. In general, very good agreement between FEA experiments and theory can be obtained but care must be taken in the definition of the problem from the numerical and material modelling standpoints. In addition to defining an adequate finite element mesh, an important aspect of conducting successful penetration simulations is the use of adequate material failure models. Generally, for the dynamic analysis a simplified single-degree-of-freedom system is considered to simulate the dynamic responses of the steel plate under the impact load of bullet, by which the maximum distortion and time-history of acceleration can be investigated. The dynamic loading characteristics such as bullet speed and its weight need to be ensured prior to analysis. Dynamic Response of Beams and Plates to Impact loading Condition 7 | P a g e 2 Scope and Objective The objective of the present work is to study the deformation characteristics of steel beam and plate subjected to projectile impact having both translation and rotation. To realize the above mentioned object the scope of the work has been determined. Step 1. A convergence study is done to find out the time step of integration and the mesh size for finite element modelling Step 2. Using the above parameters deformation characteristics of a simply supported, clamped and cantilever beam and clamped plate has been analyzed The deflection of the beam and along with contact force and contact area has been plotted over time for different cases. Step 3. Once the Stress and Plasticity pattern is known next step will be an attempt to find out the velocity of the plastic hinge Step 4. The plastic hinge will move to the end of the beam section and revart back. It is also necessary to find the time when the wave has rebound. Step 5. An attempt to corelate the relation between plastic hinge velocity and rebound time for different impact velocity and angle of incident Dynamic Response of Beams and Plates to Impact loading Condition 8 | P a g e 3 Literature Survey Introduction : A brief review of literature dealing with the dynamic response of beams and plates is presented.This project attempts to find out the impact behavior of steel structure.The study of extremely high velocity impacts is still very new. An early result is due to Newton; the impact depth of any projectile is the depth that a projectile will reach before stopping in a medium; in Newtonian mechanics, a projectile stops when it has transferred its momentum to an equal mass of the medium. If the indenter and medium have similar density this happens at an impact depth equal to the length of the indenter. For this simple result to be valid, the arresting medium is considered to have no integral shear strength. Note that even though the projectile has stopped, the momentum is still transferred, and in the real world fragmentation and similar effects can occur. For short range target shooting on ranges up to 50 meters, aerodynamics is relatively unimportant and velocities are low. As long as the bullet is balanced so it does not tumble, the aerodynamics are unimportant. So for modelling such a system the knowing impact velocity and endurance of the indenter is quite sufficient. 1.Sun and Huang (1975) modelled a higher order finite elementmodel to evaluate impact situation for beam element. Non elasticdeformation has also been considered in that model. The efficiency Dynamic Response of Beams and Plates to Impact loading Condition 9 | P a g e of the model is good enough to estimate the transient response of the beam under impact loading condition. 2.Sankar and Sun (1985) added the transverse property to the Sunand Huang model. They also proposed the a numerical model moreefficient than the existing one which includes the nonlineariteration. The effect of time increments has also been discussed. 3.Schonbergl et al (1987) considered the indenter as rigid andsmooth. The result is analysed for simply supported beams andplates 4.Vaziri et al (1996) broadens the scope of the predecessor andinclude plate structure. Thin plate andvon Karman kinetics hasbeen combined to formulate the governing equation. Hertziancontact law and coarse mesh accuracy is an important factor in thisresearch. 5.Ahmed et al. (2001) introduced elasto plastic analysis for beamssubjected to low velocity impact. Newmark’s integration andLagrangian formulation has been used respectively for nonlinearequation of motion and large displacement finite elementformulation with small strain. 6.Olsson (2003) generalized it farther by incorporating linear contactlaw along with Hertzian contact law. He also include Bending orshear along with indentation to standardise it further. 7.Khalili et al. (2011) used ABAQUS finite element so that it canserve as a benchmark problem in low velocity impact analysis. Dynamic Response of Beams and Plates to Impact loading Condition 10 | P a g e 4 Finite Element Modelling Introduction : Existing analytical model is quite ineffective when the question arises for evaluating the dynamic response of a structure behaving as plastic. So numerical analysis is required to approximate the transient response of such structures. In this work ABAQUS has been used as the numerical solver. Basic assumption of motion : Some assumption has to be considered like, 1. The indenter is rigid and it has a sharp pointed tip 2. The effect of damping between bodies is neglected. Stiffness Matrix Algorithm For a single element, the local degree of freedom is explained by q = [q1, q2, q3, q4]T = [w1, ?1, w2, ?2]T The shape function for interpolating w on an element are defined in terms of ? on -1 to 1. The hermite shape functioncxan be used to write w in the form ? ? 3 2 1 2 1 1 2 4 dw w N w N N w N dw d? d? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? Where Ni are Hermite shape function and are defined as, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 1 2 2 2 3 4 1 1 N 1 2 N 1 1 4 4 1 1 N 1 2 N 1 1 4 4 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Dynamic Response of Beams and Plates to Impact loading Condition 11 | P a g e The coordinates transform by the relationship, 1 2 1 2 1 2 1 1 x x x x x x x 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? Since le = x1-x2 is the length of the element, and dx =0.5 le d?, so e dw l dw d 2 dx ? ? The element strain energy is given by 2 2 e f 2 e 1 w U E I dx 2 x ? ? ? ? ? ? ? ? ? ? So 2 2 2 2 2 e e e 2 e T T e f e e w 4 N q Bq x l 4 3 1 3 l 3 1 3 l B , , , l 2 2 2 2 2 2 1 1 U E I qB Bq dx q k q 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Where the stiffness matrix ke is e e 2 2 e f e e e e 3 e e e 2 2 e e e e 12 6l 12 6l E I 6l 4l 6l 2l k l 12 6l 12 6l 6l 2l 6l 4l ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? Convergence study A convergence study has been done for three different time steps i.e. 0.0275ms, 0.0375ms, 0.0485ms shows that the using of 0.0375 ms gives close Dynamic Response of Beams and Plates to Impact loading Condition 12 | P a g e enough data to maintain the approximately correct result in case of clamped beam. For the simply supported beam a beam the following design parameters has been used Indenter Dimension 10 mm diameter Density 7966 Elastic 2.158 X 1011 Poisson 0.3 Target Dimension 153.5 mm X 10 mm X 10 mm Density 7966 Elastic 2.158 X 1011 Plastic 2.002 X 108 0 2.46 X 108 0.02374 2.94 X 108 0.04784 3.74 X 108 0.09436 4.37 X 108 0.1388 4.8 X 108 0.1814 Poisson 0.3 Johnson Cook Damage d1 d2 d3 d4 Reference Strain rate 5.625 0.3 -7.2 -0.0123 1 For the clamped beam Indenter Dimension 0.025 m diameter Density 7801 (Weight 0.5 kg) Dynamic Response of Beams and Plates to Impact loading Condition 13 | P a g e Elastic 199 X 109 Poisson 0.3 Target Dimension 240 mm X 12.7 mm X 4 mm Density 7801 Elastic 2.158 X 1011 Plastic 2.075 X 108 0 7.775 X 108 1 Poisson 0.3 Johnson Cook Damage A B n C Reference Strain rate 217MPa 234MPa 0.643 0.076 1 Dynamic Response of Beams and Plates to Impact loading Condition 14 | P a g e 5 Results and Discussions Simply Supported beam (Finite Element Modelled) Velocity of the indenter : 10 mm/s Deflected Beam (Colour coded only) Deflection of the beam at the point of contact due to impact by the solid sphere Dynamic Response of Beams and Plates to Impact loading Condition 15 | P a g e Contact Force of the beam at the point of contact due to impact by the solid sphere This graph has been obtained using hard contact i.e. contact stiffness is infinite whereas using contact stiffness as 5.536 X 1010 the contant force data as shown: Dynamic Response of Beams and Plates to Impact loading Condition 16 | P a g e The mass of the imdentor is 32.26 g and the acceleration is g = 9.81 m/s2. So total force = 0.316N in case of static loading. Clamped Beam (Finite Element Model) Deflected Beam (There two different kind of loading has been applied on this beam : vertical and inclined) In case of vertical impact the deformation spectrum at the point of impact: Dynamic Response of Beams and Plates to Impact loading Condition 17 | P a g e And the equivalent plastic strain at the point of impact : The deformation spectrum at the point of impact: Dynamic Response of Beams and Plates to Impact loading Condition 18 | P a g e Total Stress generated : Dynamic Response of Beams and Plates to Impact loading Condition 19 | P a g e Model Validation : Model Validation has been done using the similar parameter given in the Paper 'Dynamic behaviour of elastic plastic free free beams subjected to impulse loading' by T X Yu and J L Yang As shown in the graph the velocity profile is similar to that of given in the paper and also the velocty is 293m/s whereas it has been calculated as 307m/s Also another important observation has been made in a referenced paper 'Elasto-plastic response of free free beams subjected to impact loads' by T. U. Ahmed and L.S. Ramachandra that there has been 5points where plastic hinge formation occured which has been recreated in the current study. The plastic hinges formed is clear from the below mentioned snapshot of the stress pattern. Dynamic Response of Beams and Plates to Impact loading Condition 20 | P a g e For Simply supported beam the result : Sl.No. Contact Stiffness Contact Force Contact Time Remarks 1 Infinite 7.375N 145 X 10-6 s Layered Solid Beam Element 2 5.536 X 1010 5.294 N 163 X 10-6 s Solid Beam Element For, the clamped beam the result : Sl. No. Angle of Indentation Mass of the indenter Velocity of the indenter Maximum Deflection Maximum Plastic Strain Maximum Stress 1 900 0.5 kg 15 m/s 2.32 X 10-2 1.947 X 10-2 1.494 X 109 2 450 1.5 X 10-2 1.386 X 10-2 1.1 X 109 3 300 1.046 X 10-2 6.316 X 10-3 6.836 X 108 Dynamic Response of Beams and Plates to Impact loading Condition 21 | P a g e For Clamped beam the result : Dynamic Response of Beams and Plates to Impact loading Condition 22 | P a g e 6 Conclusion In the present study the study of elasto-plastic analysis of beam under low velocity impact. The results predicted by the analysis is not always very close to the available literature. The main reasons behind this errors are 1.A simplified model has been utilised in the present analysis 2.Geometric nonlinearity has not been considered Some important feature came out of the analysis 1.The contact time is reduced and the contact force is increased for anincrease in impact velocity 2.The impact algorithm is time step dependent. Small changes in the timestep significantly changes the result.Some observation related to plastic hinge velocity and rebound time Dynamic Response of Beams and Plates to Impact loading Condition 23 | P a g e 7 Works to be done on the next part 1.The further analysis of clamped beam is required to be done with angular rotation included 2.The ballistic limit of the plate should be investigated Dynamic Response of Beams and Plates to Impact loading Condition 24 | P a g e 8 Reference 1.Harsoor R. K. (2009) Influence of notch on the response of mild stillbeam subjected to low velocity impact, IIT kharagpur 2.Achintya Das (2013) Response of beams and plates to low velocityimpact, IIT kharagpur 3.Godrej and HTC (2012) Simulation of bullet impact on bullet resistantstill plates 4.Vedantam, K.; Bajaj, D.; Brar, N. S.; Hill, S. (2006) Johnson - CookStrength Models for Mild and DP 590 Steels 5.Ahmed T. U (2001) Ph. D Thesis on Elasto-plastic response of beamssubject to low velocity impact : numerical and experimental studies. IITkharagpur 6.Khalili S. M. R. (2011) Finite Element Modelling of low velocity impact onlaminated composite plates and cylinder shells Composite Structures,93, 1363 – 1375 7.Olsson R. (2003) Closed form prediction of peak load and delaminationonset under small mass impact. Composite structures, 59, 341 – 349 8.Sun C. T. and Hung C. T. (1985) Transverse impact problems by higherorder beam finite element, Computers and Structures, 5, 297 – 303 9.Sankar B. V. and Sun C. T. (1985) An efficient numerical algorithm fortransverse impact problems, computers and structures 20(6), 1009 –1012 10. Vaziri R., Quan X., Olson M. D. (1996) Impact analysis of laminated composite plates and shells by super finite elements. International journal of solids structure 18(7 - 8) 765,782


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