The Analysis of the Impact Response of Elastic Thin-Walled Beams of Open Section Via the Ray Method

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1 The Analysis of the Impact Response of Elastic Thin-Walled Beams of Open Section Via the Ray Method YURY ROSSIKHIN and MARINA SHITIKOVA Voronezh State University of Architecture and Civil Engineering Research Center on Dynamics of Solids and Structures 0-letija Oktjabrja Street 84, Voronezh RUSSIA Abstract: The problem on the normal impact of an elastic rod with a rounded end upon an elastic Timoshenko arbitrary cross section thin-walled beam of open section is considered. The process of impact is accompanied by the dynamic flexure and torsion of the beam, resulting in the propagation of plane flexural-warping and torsionalshear waves of strong discontinuity along the beam axis. Behind the wave fronts upto the boundaries of the contact region, the solution is constructed in terms of one-term ray expansions. During the impact the rod moves under the action of the contact force which is determined due to the Hertz s theory, while the contact region moves under the attraction of the contact force, as well as the twisting and bending-torsional moments and transverse forces, which are applied to the lateral surfaces of the contact region. The procedure proposed allows one to obtain rather simple relationship for estimating the maximal magnitude of the contact force, which can be very useful in engineering applications. Key Words: Ray method, thin-walled beam of open section, transient waves, normal impact Introduction It seems likely that Crook ] pioneered the application of the wave approach in the theory of impact when considering the longitudinal impact of an elastic sphere against the end of a thin elastic bar. As this takes place, the deformation of the bar s material in the contact region was considered through the use of the Hertz s contact theory; but in the vicinity of the contact region, it was taken into account using one-term ray expansions constructed behind the longitudinal wave front. The problem was reduced to the solution of the nonlinear integro-differential equation in the contact force, whose numerical integration allowed the author to determine the time dependence of the contact force and the dynamic stress in the bar. The same approach was used by Rossikhin and Shitikova,3] for investigating the transverse impact of an elastic bar and sphere upon an Uflyand-Mindlin plate 4,5]. The material local bearing dependence of the force has been defined on a basis of quasi-static analysis; however, in this problem, a major portion of energy transformed into energy of the nonstationary transverse shear wave, behind which front, upto the boundary of the contact region, the values to be found were constructed in terms of one-term or multipleterm ray expansions. The ray expansions employed allowed to consider reflected waves as well, if these latter had had time to return at the point of the impact prior to the completion of the colliding process. The conditions of matching of the desired values in the contact region and its vicinity, which were to be fulfilled on the boundary of the contact region, permitted to obtain the closed system of equations for determining all characteristics of the shock interaction. The problem of the response of rods, beams, plates and shells to low velocity impact with the emphasis on the wave theories of shock interaction has been reviewed by Rossikhin and Shitikova in 6]. These theories are based on the fact that at the moment of impact transient waves (surfaces of strong discontinuity) are generated within the contact domain, which further propagate along the thin bodies and thereby influence the process of the shock interaction. The desired functions behind the strong discontinuity surfaces are found in terms of one-term, two-term or multiple-term ray expansions, the coefficients of which are determined with an accuracy of arbitrary functions from a set of equations describing the dynamic behavior of the thin body. On the contact domain boundary, the ray expansions for the desired functions go over into the truncated power series with respect to time and are matched further with the desired functions within the contact region that are represented by the truncated power series with respect to ISSN: ISBN:

2 time with uncertain coefficients. As a result of such a procedure, it has been possible to determine all characteristics of shock interaction and, among these, to find the time dependence of the contact force and the displacements of the contact region. The procedure proposed in 3] for investigating the transverse impact upon a plate has been generalized to the case of the shock interaction of an elastic Timoshenko thin-walled beam of open section with an elastic sphere 7]. It has been revealed that the impact upon a thin-walled beam has its own special features. irst, the transverse deformation in the contact region of colliding bodies may be so large that can result in the origination of longitudinal shock waves. Second, the deflection of the beam in the place of contact may be so large that one is led to consider the projection of the membrane contractive (tensile) forces onto the normal to the beam s median surface in the place of contact. In the present paper, this approach is generalized for the analysis of the thin-walled open section beam response to the impact by a thin long elastic rod with a rounded end. The Engineering Theory of Thin- Walled Rods of Open Section Thin-walled beams of open section are extensively used as structural components in different structures in civil, mechanical and aeronautical engineering fields. These structures have to resist dynamic loads such as wind, traffic and earthquake loadings, so that the understanding of the dynamic behavior of the structures becomes increasingly important. Ship hulls are also can be modelled as thin-walled girders during investigation of hydroelastic response of large container ships in waves. The classical engineering theory of thin-walled uniform open cross-section straight beams as well as horizontally curved ones was developed by Vlasov 8] in the early 60-s without due account for rotational inertia and transverse shear deformations 9]. The Vlasov theory is the generalization of the Bernoulli- Navier law to the thin-walled open section beams by including the sectorial warping of the section into account by the law of sectorial ares, providing that the first derivative of the torsion angle with respect to the longitudinal axis serves as a measure of the warping of the section. Thus, this theory results in the four differential equations of free vibrations of a thinwalled beam with an open inflexible section contour of arbitrary shape. or the case of a straight beam, the first second-order equation determines, independently of the other three and together with the initial and boundary conditions, the longitudinal vibrations of the beam. The remaining three fourth-order differential equations form a symmetrical system which, together with the initial and boundary conditions determines the transverse flexural-torsional vibrations of the beam (see page 388 in 8]). In the case of a curved beam, all four equations are coupled. However, Vlasov s equations are inappropriate for use in the problems dealing with the transient wave propagation. Many researchers have tried to modify the Vlasov theory for dynamic analysis of elastic isotropic thinwalled beams with uniform cross-section by including into consideration the rotary inertia and transverse shear deformations 0 6]. One of the first successful attempts in the field were made in 974 by Korbut and Lazarev ], who generalized the classical engineering theory of thinwalled beams of open cross-section to the case of dynamic response of such beams with due account for rotational inertia and transverse shear deformations. The set of equations derived in ] using the Reissner s variational principle describes the dynamic behavior of a straight beam of the Timoshenko type and has the following form: the equations of motion ρi x Ḃ x M x,z + Q yω = 0 ρi y Ḃ y M y,z Q xω = 0 () ρi ω Ψ B,z Q xy = 0 ρ v z N,z = 0 ρ v x + ρa y Φ Q xω,z = 0 ρ v y ρa x Φ Q yω,z = 0 () ρi p Φ + ρay v x ρa x v y (Q xy + H),z = 0 the generalized Hook s law Ṁ x = EI x B x,z, Ṁ y = EI y B y,z (3) Ḃ = EI ω Ψ,z, Ṅ = E v z,z µ(v x,z B y ) = k y Q xω + k xy Q yω + k yω Q xy µ(v y,z + B x ) = k xy Q xω + k x Q yω + k xω Q xy µ(φ, z Ψ) = k yω Q xω + k xω Q yω + k ω Q xy Ḣ = µi k Φ, z (4) where ρ is the beam s material density, is the crosssection area, ω is the sectorial coordinate, I x and I y are centroidal moments of inertia, I ω is the sectorial moment of inertia, I p is the polar moment of inertia about the flexure center A, I k is the moment of inertia ISSN: ISBN:

3 due to pure torsion, a x and a y are the coordinates of the flexural center, E and µ are the Young s and shear moduli, respectively, B x = β x, B y = β y, Φ = ϕ, β x, β y and ϕ are the angles of rotation of the cross section about x-, y- and z-axes, respectively, Ψ = ψ, ψ is the warping function, v x, v y, v z are the velocities of displacements of the flexural center along the central principal axes x and y and the longitudinal z-axis, respectively, M x and M y are the bending moments, B is the bimoment, N is the longitudinal (membrane) force, Q xω and Q yω are the transverse forces, H is the moment of pure torsion, Q xy is the bending-torsional moment from the axial shear forces acting at a tangent to the contour of the cross section about the flexural center, overdots denote the time derivatives, and the index z after a point defines the derivative with respect to the z-coordinate. In ()-(4), k x, k y, k ω, k xω, k yω, and k xy are the cross-sectional geometrical characteristics which take shears into consideration: k x = Ix k y = Iy k ω = k xω = k yω = k xy = Sx δs d Sy δs Sω Iω δs I x I ω I y I ω I x I y d d S x S ω d (5) δ s S y S ω d δ s S x S y δs d where S x, S y, and S ω are the axial and sectorial static moments of the intercepted part of the cross section, and δ s is the width of the web of the beam.. Velocities of the transient waves propagating in the thin-walled beam of open section The set of equations ()-(4) governs three transient shear waves which propagate with the velocities depending on the geometrical characteristics of the thinwalled beam (5). Really, suppose that the wave surface of strong discontinuity, which can be interpreted as the limiting layer of the width h, propagates along the beam. Inside this layer the desired field Z changes monotonically and continuously from the magnitude Z + to the magnitude Z. If we write ()-(4) inside the layer, apply the condition of compatibility Ż = GZ, z +δz/δt, (6) where G is the normal velocity of the limiting layer and δ/δt is the δ -derivative 7], and further integrate the resulting equations over the layer s width from h/ to h/, then letting h 0 and introducing the notation Z] = Z + Z, where the signs + and refer to the magnitudes of the field Z calculated ahead and behind the wave front, respectively, as a result, we find 7] ρi x GB x ] M x ] = 0 ρi y GB y ] M y ] = 0 (7) ρi ω GΨ] B] = 0 ρ Gv z ] N] = 0 ρ Gv x ] ρ Ga y Φ] Q xω ] = 0 ρ Gv y ] + ρ Ga x Φ] Q yω ] = 0 (8) ρi p GΦ] ρ Ga y v x ] + ρ Ga x v y ] Q xy ] H] = 0 GM x ] = EI x B x ] GM y ] = EI y B y ] (9) GB] = EI ω Ψ] GN] = E v z ] µv x ] = Gk y Q xω ] Gk xy Q yω ] Gk yω Q xy ] µv y ] = Gk xy Q xω ] Gk x Q yω ] Gk xω Q xy ] µφ] = Gk yω Q xω ] Gk xω Q yω ] Gk ω Q xy ] GH] = µi k Φ] (0) Eliminating the values M x ], M y ], B] and N] from (7) and (9), we obtain the velocity of the longitudinal-flexural-warping wave G 4 = Eρ, () on which B x ] 0, B y ] 0, Ψ] 0, and v z ] 0, while v x ] = v y ] = Φ] = 0. Eliminating the values Q xω ], Q yω ], Q xy ], and H] from (8) and (0 ), we arrive at the system of three linear homogeneous equations: a ij v j ] = 0 (i, j =,, 3), () j= where v ] = v x ], v ] = v y ], v 3 ] = Φ], a = ρ G (k y + a y k yω ) µ a = ρ G (k xy a x k yω ) a 3 = ρ G (a y k y a x k xy ) + k yω (ρi p G µi k ) ISSN: ISBN:

4 a = ρ G (k xy + a y k xω ) a = ρ G (k x a x k xω ) µ a 3 = ρ G (a y k xy a x k x ) + k xω (ρi p G µi k ) a 3 = ρ G (k yω + a y k ω ) a 3 = ρ G (k xω a x k ω ) a 33 = ρ G (a y k yω a x k xω ) + k ω (ρi p G µi k ) µ Setting determinant of the set of equations () equal to zero a ij = 0, (3) we are led to the cubic equation governing the velocities G, G, and G 3 of three twisting-shear waves, on which only the values v x ], v y ] and Φ] are nonzero such that where v x ] = γφ], v y ] = δφ], (4) γ = a 3a a 3 a a a a a, δ = a 3a a 3 a a a a a. 3 The Response of a Thin-Walled Beam of Open Section to the Normal Impact of a Rod Let us consider the normal impact of an elastic thin rod of circular cross section upon a lateral surface of a thin-walled elastic beam of open section (ig. ), the dynamic behavior of which is described by system ()-(4). At the moment of impact, the velocity of the impacting rod is equal to V 0, and the longitudinal shock wave begins to propagate along the rod with the velocity G 0 = E 0 ρ 0, where E 0 is its elastic modulus, and ρ 0 is its density. Behind the wave front the stress σ and velocity v fields can be represented using the ray series 8] σ = v = V 0 k=0 k=0 k! k! k ] ( σ t k t n ) k (5) G 0 k ] ( υ t k t n ) k (6) G 0 where n is the coordinate directed along the rod s axis with the origin in the place of contact (ig. ). Considering that the discontinuities in the elastic rod remain constant during the process of the wave igure : Scheme of shock interaction propagation and utilizing the condition of compatibility, we have k+ ] u n t k = G 0 k+ ] u t k+ = G 0 k ] v t k (7) where u is the displacement. With due account of (7) the Hook s law on the wave surface can be rewritten as k σ t k ] = ρ 0 G 0 k v t k Substituting (8) in (5) yields σ = ρ 0 G 0 k=0 k! ] (8) k ] ( v t k t n ) k (9) G 0 Comparison of relationships (9) and (6) gives σ = ρg 0 ( V0 v ) (0) When n = 0, expression (0) takes the form σ cont = ρg 0 (V 0 v ν ) () where σ cont = σ n=0 is the contact stress, and v ν = v n=0 is the normal velocity of the beam s points within the contact domain. ormula () allows one to find the contact force P = πr 0ρ 0 G 0 (V 0 v ν ) () where r 0 is the radius of the rod s cross section. However, the contact force can be determined not only via (), but using the Hertz s law as well P = kα 3 / (3) ISSN: ISBN:

5 where α is the value governing the local bearing of the target s material during the process of its contact interaction with the impactor. If suppose that the end of the rod is rounded with the radius of R, while the lateral surface of the thin-walled beam is flat in the place of contact, then k = 4 / R 3π (k + k ), k = ( ν0)/ πe0, k = ( ν )/ πe, where ν 0 and ν are Poisson s ratios for the impactor and the target, respectively. Eliminating the force P from () and (3), we are led to the equation for determining the value α (t) v ν + k πr 0 ρ 0G 0 α 3 / = V 0 (4) In order to express the velocity v ν in terms of α, let us analyze the wave processes occurring in the thinwalled beam of open section. At the moment of impact, three plane shock shear waves propagating with the velocities G, G, and G 3, which are found from (3), are generated in the beam, as well as the longitudinal wave of acceleration. Since the contours of the beam s cross sections remain rigid during the process of impact, then all sections involving by the contact domain form a layer which moves as rigid whole. Let us name it as a contact layer. If we neglect the inertia forces due to the smallness of this layer, then the equations describing its motion take the form Q xω + P sin β (s) = 0 (5) Q yω + P cos β (s) = 0 (6) (Q xy + H) + P e(s) = 0 (7) where β(s) is the angle between the x axis and the tangent to the contour at the point M with the s coordinate, and e(s) is the length of the perpendicular erected from the flexural center to the rod s axis. The values Q xω, Q yω, and Q xy + H entering in (5)-(7) are calculated as follows: behind the wave fronts of three plane shear waves upto the boundary planes of the contact layer, the ray series can be constructed 8]. If we restrict ourselves only by the first terms, then it is possible to find them from (0). Considering (4), we obtain the following relationships for the values Q xω, Q yω, and Q xy + H: Q xω = L i Φ i (8) i= Q yω = M i Φ i (9) i= (Q xy + H) = d i Φ i (30) i= where L i = ρ G i (γ i + a y ), M i = ρ G i (δ i a x ), and d i = ρ G i (γ i a y δ i a x ) + ρi p G i. rom hereafter the sign...] indicating the discontinuity in the corresponding value is omitted for the ease of presentation. Substituting (8)-(30) and (3) in (5)-(7), we have L i Φ i = kα 3 / sin β (3) where = = i= M i Φ i = kα 3 / cos β (3) i= n d i Φ i = kα 3 / e (33) i= Solving the set (3)-(33), we find L L L 3 M M M 3 d d d 3 L sin β L 3 M cos β M 3 d e d 3 Φ i = kα 3 / i (34), =, 3 = Let us rewrite the relationship for v ν sin β L L 3 cos β M M 3 e d d 3 L L sin β M M cos β d d e v ν = α v x sin β (s) + v y cos β (s) + e (s) Φ (35) with due account for (4) and then consider (34) in (36) v ν = α + l i Φ i (36) i= v ν = α + k α 3 / l i i (37) i= where l i = δ i cos β γ i sin β + e. Substituting (37) in (4), we obtain the equation for defining α α + κα 3 / = V 0 (38) ISSN: ISBN:

6 where ( κ = k πr0 ρ + ) l i i 0G 0 i= The maximum deformation α max is reached at α = 0 and, due to (38), is equal to α max = ( ) / V0 3 κ (39) Substitution of (39) in Eq. (3) gives us the maximal contact force 4 Conclusion P max = kv 0 κ. (40) The problem on the normal impact of an elastic thin rod with a rounded end upon an elastic Timoshenko arbitrary cross section thin-walled beam of open section has been considered. The process of impact is accompanied by the dynamic flexure and torsion of the beam, resulting in the propagation of plane flexuralwarping and torsional-shear waves of strong discontinuity along the beam axis. Behind the wave fronts upto the boundaries of the contact region (the beam part with the contact spot), the solution is constructed in terms of one-term ray expansions. During the impact the rod moves under the action of the contact force which is determined due to the Hertz s theory, while the contact region moves under the attraction of the contact force, as well as the twisting and bendingtorsional moments and transverse forces, which are applied to the lateral surfaces of the contact region. The procedure proposed allows one to obtain rather simple relationships for estimating the maximal magnitude of the contact force and the contact duration, which can be very useful in engineering applications. Acknowledgements: The research described in this publication was made possible in part by Grant No.../50 from the Russian Ministry of High Education. References: ] A.W. Crook, A Study of Some Impacts Between Metal Bodies by Piezoelectric Method, Proc. Royal Soc. A, 95, pp ] Yu.A. Rossikhin and M.V. Shitikova, About Shock Interaction of Elastic Bodies with Pseudo Isotropic Uflyand-Mindlin Plates, Proc. Int. Symp. on Impact Engineering, Sendai, Japan, 99, pp ] Yu.A. Rossikhin and M.V. Shitikova, A Ray Method of Solving Problems Connected with a Shock Interaction, Acta Mech. 0, 994, pp ] Ya.S. Uflyand, Waves Propagation under Transverse Vibrations of Bars and Plates (in Russian), Prikl. Mat. Mekh., 948, pp ] R.D. Mindlin, High requency Vibrations of Crystal Plates, Quart. J. Appl. Math. 9, 96, pp ] Yu.A. Rossikhin and M.V. Shitikova, Transient Response of Thin Bodies Subjected to Impact: Wave Approach, Shock Vibr. Digest 39, 007, pp ] Yu.A. Rossikhin and M.V. Shitikova, The Impact of a Sphere on a Timoshenko Thin-Walled Beam of Open Section with due Account for Middle Surface Extension, ASME J. Pressure Vessel Tech., 999, pp ] V.Z. Vlasov, Thin-Walled Elastic Beams (in Russian), Gostekhizdat, Moscow 956 (Engl. transl.: 96, Nat. Sci. ound., Washington) 9] A. Gjelsvik, Theory of Thin Walled Bars, Wiley, New York 98 0] W.K. Tso, Coupled Vibrations of Thin-Walled Elastic Bars, ASCE J. Eng. Mech. Div. 9, 965, pp ] H.R. Aggarwal and E.T. Cranch, A Theory of Torsional and Coupled Bending Torsional Waves in Thin-Walled Open Section Beams, ASME J. Appl. Mech. 34, 967, pp ] B.A. Korbut and V.I. Lazarev, Equations of lexural-torsional Waves in Thin-Walled Bars of Open Cross Section, Int. Appl. Mech. 0, 974, pp ] R.E.D. Bishop and W.G. Price, Coupled Bending and Twisting of a Timoshenko Beam, J. Sound Vibr. 50, 977, pp ] P. Muller, Torsional-lexural Waves in Thin- Walled Open Beams, J. Sound Vibr. 87, 983, pp ]. Laudiero and M. Savoia, The shear strain influence on the dynamics of thin-walled beams, Thin-Walled Structures, 99, pp ] D. Capuani, M. Savoia and. Laudiero, A Generalization of the Timoshenko Beam Model for Coupled Vibration Analysis of Thin-Walled Beams, Earthq. Eng. Struct. Dyn., 99, pp ] T.Y. Thomas, Plastic low and racture in Solids, Academic Press 96 8] J.D. Achenbach and D. P. Reddy, Note on Wave Propagation in Linearly Viscoelastic Media, ZAMP 8, 967, pp ISSN: ISBN: