SIMILARITY METHODS IN ELASTO-PLASTIC BEAM BENDING

Transcription

1 Similarity methods in elasto-plastic eam ending XIII International Conference on Computational Plasticity Fundamentals and Applications COMPLAS XIII E Oñate, DRJ Owen, D Peric and M Chiumenti (Eds) SIMILARIT METHODS IN ELASTO-PLASTIC BEAM BENDING CHRISTIAN ZEHETNER *, FRANZ HAMMELMÜLLER *, HANS IRSCHIK AND WOLFGANG KUNZE x * Linz Center of Mechatronics (LCM) Altenergerstrasse 69, 44 Linz, Austria christianzehetner@lcmat, we-page: wwwlcmat Institute for Technical Mechanics Johannes Kepler University Altenergerstrasse 69, 44 Linz, Austria hansirschik@jkuat we page: x Salvagnini Maschinenau GmH, Dr Guido Salvagnini-Strasse 1, 4482 Ennsdorf, Austria wolfgangkunze@salvagninigroupcom - we page: wwwsalvagninigroupcom Key words: plastic ending, similarity methods, drop-test Astract In industrial metal forming processes a large numer of parameters has to e considered As is well known, this numer can e reduced y a non-dimensional representation Based on the example of an elasto-plastic cantilever eam, the nondimensional form is derived in the framework of the Bernoulli-Euler theory In the second step, a Finite Element analyis of a drop-test experiment is performed, and the results are presented in non-dimensional form The results illustrate the advantage of the normalization 1 INTRODUCTION In industrial metal forming processes, high versatility is claimed with respect to material types, geometric dimensions and process parameters In order to control and optimize the production process, efficient simulation models are required Usually, metal forming simulation models are very complex and highly non-linear Thus, parameter studies take a large computational effort An efficient and practicale way to reduce the numer of the essential parameters is the application of non-dimensional formulations, a procedure that finds its theoretical foundation in Similarity Methods [1], [2] In the following, plastic ending of a cantilever eam is investigated Especially, the plastic deformations are caused y a drop test as discussed in [1]: A cantilever eam is fixed on a drop mass which is falling down from a predefined height and stopped y an elastic spring, such that the impact causes plastic deformations In [1] experimental results have een presented with two materials and two configurations of geometric dimensions Applying the concept of similarity, ie the Buckingham Pi Theorem, the numer of parameters to descrie 96

2 the process has een reduced y introducing appropriate normalized quantities In the present paper, the non-dimensional representation of the elasto-plastic cantilever eam and the drop-test experiment are investigated in more detail First, in section 2, the ending process is modelled quasi-statically in the framework of Bernoulli-Euler eam theory assuming a uni-axial state of stress and elasto-plastic material ehavior, cf [5] and [6] In the latter references ideal plastic material ehaviour has een assumed In the present contriution, the formulation is extended for exponential stress-strain relation of the Ludwik type With an appropriate normalization of the equilirium equations, a representation is found in terms of the same non-dimensional quantities as suggested in [1] Furthermore, in section 3, we discuss the outcomes of a Finite Element model, which we have implemented for the drop test process, using the software package ABAQUS, version The eam is represented y continuum elements, elasto-plastic material ehavior with exponential stres-strain relation is assumed, and large deformation is taken into account A dynamic analysis is performed in order to simulate the drop-test With this model, parameter studies have een performed, and the results are represented in terms of the non-dimensional form as suggested in [1] With the simulation model, a very detailed analysis of the influences of several geometrical and material properties ecomes possile 2 ELASTO-PLASTIC BENDING OF A CANTILEVER BEAM In the following we consider a cantilever eam with length L as shown in Figure 1 The cross-section of the eam is rectangular with width B and height H The eam is fixed to a moving rigid ase with prescried acceleration a z Figure 1: Accelerated cantilever eam With the density ρ of the eam, the influence of the rigid ody acceleration can e represented y a uniform constant distriuted transversal load with amplitude q =ρ HBa z (1) In the framework of a quasi-static modeling, the moment follows to M= 1 2 ql ( x) 2 (2) The transversal deflection of a point on the eam axis relative to the ase motion is denoted as w The material ehavior is assumed to e elasto-plastic with exponential stress-strain relation of Ludwik's type For a uni-axial state of stress the constitutive relations read σ= Eε for σ σ, σ= aε for σ<σ, (3) 97

3 where σ is the axial stress, ε the axial strain, E oung's modulus, a and specify the exponential stress-strain relation, and σ is the yield stress, following from solving Eq (3), ( / ) 1/( 1) σ = E E a (4) 21 Plastic zones The ending moment is given y M = B σ() z zdz In the elastic range, the distriution of the stress is linear over the cross-section The yield moment corresponds to the state where the yield stress is reached at z=h/2, ie 1 2 M = 6 σ BH In the quasi-static case, the yield load follows from Eq (2) y sustituting x=, q 2 2 = BH L σ 1 3 If the load is higher than the yield load, the domain is divided into elastic and plastic zones as shown in Figure 2 Assuming that the deformations remain small and that the Bernoulli-Euler hypothesis holds, the axial strain reads ε= zw (7) Thus, the stress in the elastic and plastic zones can e expressed y σ=σ z z for z z, σ z σ= a for z< z E z (5) (6) (8) Figure 2: Elastic and plastic zones 98

4 The oundary of the plastic zone follows from equating Eqs (2) and (5), using Eq (8): 2 B zdz a dz q( L x) z z H /2 σ z 1 2 σ + z z = 2 E z From Eq (9) the length x of the plastic zone is otained y sustituting z=h/2 An analytical solution is possile On the other hand side, the height z of the plastic zone is otained as a function of x Due to the nonlinearity of the equation, z has to e solved numerically 22 Curvature From Eqs (3) and (7) follows that σ = Ezw (1) Inserting Eq (1) into Eq (9) yields an equation for the curvature in the elasto-plastic case: z 2 H/2 1 2 ( z ( ) ) 2 2 B Ew z dz + a w z dz = q( L x) Due to the nonlinearity with respect to w,only a numerical solution is possile For the elastic range, x> x, Eq (11) simplifies to the well-known form 32 Normalization q 2 w = 6 ( L x) 3 BH E In order to otain a generalized representation, a normalization of the aove equations has een performed as follows: By introducing the non-dimensional quantities Eq (9) can e rewritten as with x z * H * a * q * σ ξ=, ζ=, H =, a =, q =, σ =, L L L E ab E 1+ ** 2 ** 1 * *2 ** * 1 3 * * 2 2 a ζ (2+ 3 a ) + 3 ah ( a Hζ ) = qa( ξ 1) (2 + ) 2 2 a ** 1 = * a Sustituting ζ = H * /2 and solving Eq (14) yields the normalized extension ξ of the plastic zone On the other hand side ζ is otained as a function of ξ Figure 3 shows the 1 1 (9) (11) (12) (13) (14) (15) 99

5 * * * solutions for ξ as a function of Q = q /q and ζ ( ξ ) for some values of the normalized curvature and deflection Here and in the following the convention is used * Q Introducing * * w w = Lw, w = (16) L 2 * 2 * d w d w w =, w = 2 2 dξ dx Sustituting Eq (13) into Eq (11) yields the equation for the normalized curvature for the elasto-plastic range ( ξ ξ ): (17) 1 * 3 * 2 * * *2 1 * * * * 4w ζ a ζ w ζ 3aH w H = 3qa ξ ( ) ( ) ( ) 2 (18) ξ ζ 6 x Q * = 11 Q * = 12 Q * = 13 Q * = 14 Q * = Q * ξ Figure 3: (a) Extension of the plastic zone, () Boundary of the plastic zone Sustituting Eqs (13) and (16) into Eq (12) yields the normalized curvature for the elastic range ( ξ>ξ ) * * * qa 2 w = 6 ( ξ 1 ) (19) *3 H Solving Eq (18) numerically yields the curvature as a function of ξ as shown in Figure 4a Numerical integration of Eqs (18) and (19) provides the deflection as shown in Figure 4 91

6 -1-2 w * ξ Figure 4: Results for (a) Curvature and () Deflection Figure 5a shows the plastic tip displacement as a function of the rigid ody acceleration a z The rigid-ody acceleration at first yield is denoted as a z, which is otained y inserting Eq (6) into Eq (1) In this dimensional representation, for each configuration of H and L a separate curve is otained However, in the non-dimensional representation in Figure 5, only one curve is otained for the three cases w tip,plast () -2-4 w * tip,plast * 13-6 H=5, L=5 H=1, L=1-3 H=2, L= a / a z z, q * * Figure 5: Plastic tip displacement (a) dimensional and () non-dimensional representation Discussion of the non-dimensional formulation In the dimensional representation with the coordinates x and z, the ending process is specified y eight physical quantities: wa, z, EaH,,,, Lρ, (2) On the other hand, in the dimensionless formulation derived in section 22 with the normalized coordinates ξ= x/ L, ζ= z/ L the numer of essential parameters has een reduced to five non-dimensional quantities w ρah, z a H,,, L a E L (21) 911

7 The results show that for the simple model of an elasto-plastic eam ending process the same normalized quantities have een derived as have een otained y Baker et al [1] using the Buckingham Pi theorem Recall that all the results in this section are ased on the geometrically linear Bernoulli- Euler eam theory with exponential stress-strain relation Thus the results are valid for small deformations Goal of this section has een an analytic derivation of the non-dimensional formulation In the next section, the drop test as presented in [1] is simulated y means of Finite Element computations, considering large deformations The results are expressed in terms of the same non-dimensional form 3 FINITE ELEMENT SIMULATION OF THE DROP-TEST 31 Simulation model Figure 6 shows the drop-test as discussed in [1] A cantilever eam is fixed on a drop mass At the cantilever tip an additional eam with higher thickness is attached The length of the cantilever is L, and H D is the drop height Figure 6: Drop-test For this setup a Finite Element model has een implemented as follows: The drop mass is modelled as a rigid ody, and the eam y three-dimensional Finite Elements A dynamic analysis is performed to model the complete drop-test After falling down and colliding with the spring the cantilever eam deforms due to inertial forces Two kinds of eams are considered with the properties according to Tale 1 Cantilever Additional eam length height width length height Tale 1: Parameters Beam 1 Beam width Material type Steel Aluminium oung's modulus N/m² 268e e1 yield stress N/m² 23e8 115e8 density kg/m Drop mass weight of drop mass kg

8 32 Numerical results Figure 7 shows the deformed configuration after the impact of the drop mass with the spring S, Mises (Avg: 75%) +2175e e e e e e e e e e e e e+2 Z X Figure 7: Deformed configuration after impact with spring The time response of the transversal displacement is shown in Figure 8 The deformation of the cantilever consists of the plastic part with super-imposed elastic virations In Figure 8 the maximum deflection and the plastic part are marked -2 time (sec) deflection () Figure 8: Transversal displacement Figure 9 shows the maximum deflection and the plastic deflection as a function of the drop height H D In this dimensional representation the deflection of the aluminium eam is larger than the deflection of the steel eam On the other hand side, Figure 1 shows the nondimensional deflection w as a function of the non-dimensional drop height H, given * * y H * D ρgh =, σ w w L * = (22) D 913

9 Max deflection (m) Aluminium Steel Drop height (m) Plastic deflection (m) Aluminium Steel Drop height (m) 1 12 Figure 9: (a) Maximum deflection and () plastic deflection Non-dimensional max deflection Aluminium Steel Non-dimensional drop height Non-dimensional plast deflection Aluminium Steel Non-dimensional drop height Figure 1: (a) Maximum deflection, () Plastic deflection In this normalized representation, the non-dimensional deflection is equal for oth eams, as it has een shown in the experimental drop-test in [1] From the investigations in section 2 it follows that one curve is otained for equal relations of the non-dimensional parameters a/e, and H/L In the present case, the ratio H/L is the same for oth eams The exponent and the ratio a/e are differing slightly for aluminium and steel However, this influence is negligile Note that the result for a differing ratio H/L would e another curve in the nondimensional representation 4 CONCLUSIONS A non-dimensional representation of the elasto-plastic ending process has een derived in the framework of Bernoulli-Euler eam theory It has een shown how the numer of parameters can e reduced y a non-dimensional formulation In the second step, a Finite Element simulation has een performed for a drop-test In the non-dimensional representation, the results for the steel and aluminium eam coincide The results coincide with [1], where the non-dimensional quantities have een derived y the Buckingham Pi theorem, and applied to 914

10 an experimental drop test The results show that in complex industrial prolems the computational effort for numerical simulations and parameter studies can e reduced significantly y applying appropriate normalization strategies REFERENCES [1] Baker, WE, Westine, PS, and Dodge, FT Similarity Methods in Engineering Eynamics, Theory and Practice of Scale Modeling, Elsevier (1991) [2] Sedov, LI Similarity and Dimensional Methods in Mechanics, CRC Press (1993) [3] Szaó, I Höhere Technische Mechanik, Springer (21) [4] Ziegler, F Mechanics of Solids and Fluids, Springer (25) [5] u, TX, and Zhang, LC Plastic Bending, Theory and Applications, World Scientific (1996) [6] Natarajan, A and Peddieson, J Simulation of eam plastic forming with variale ending moments, International Journal of Non-Linear Mechanics (211), 46: