Vibration analysis of plates with arbitrary stiffener arrangements using a semi-analytical approach

320 V International Conference on Computational Methods in Marine Engineering MARINE 2013 B. Brinkmann and P. Wriggers (Eds) VIBRATION ANALYSIS OF PLATES WITH ARBITRARY STIFFENER ARRANGEMENTS USING A SEMI-ANALYTICAL APPROACH LARS BRUBAK∗,a,b, JOSTEIN HELLESLANDaAND OLE J. HAREIDEb aMechanics Division, Department of Mathematics, University of Oslo, 0316 Oslo, Norway bSection for Ship Structures and Concepts, Technical Advisory for Ship and Offshore Structures, Det Norske Veritas, 1322 Høvik, Norway ∗e-mail: [email protected] - Web pages: www.dnv.com and www.math.uio.no Key words: Stiffened plates, Arbitrary stiffener orientations, Vibration analysis, Semianalytical method, Rayleigh-Ritz method Abstract. Vibration of plates with arbitrarily oriented stiffeners is studied. The plate may be subjected to in-plane loads and the plate boundaries may be simply supported or rotationally restained. The main objective is to present and validate an approximate, semi-analytical computational model for such plates subjected to in-plane loading. The formulations derived are implemented in a Fortran computer code, and numerical results are obtained for a variety of plate and stiffener geometries. The model may handle complex plate geometries, by using inclined stiffeners to enclose irregular plate shapes. The method allows for a very efficient analysis. Relatively high numerical accuracy is achieved with low computational efforts. 1 INTRODUCTION Computationally efficient, semi-analytical methods are becoming more common as an alternative to finite element analyses (FEA) and explicit design formulas. The amount of published literature on semi-analytical methods for analysis of stiffened plates is growing. In a review paper by Liew, Xiang and Kitipornchai [1], most of the literature on vibration of stiffened plates till the year 1994 has been summarized. This literature study goes back to the well known study of vibration by Rayleigh [2] in 1877 and Ritz [3] in 1909. The latter approach is known as the Rayleigh-Ritz or Ritz method where a series of admissible trial functions are used. This approach is used in the present paper. Semi-analytical methods such as the Rayleigh-Ritz approach and other mesh-less methods have been used to investigate many different aspect of vibration. For example, the 1 Vibration analysis of in-plane loaded plates with arbitrary stiffener arrangements using a semi-analytical approach 321 Lars Brubak, Jostein Hellesland and Ole J. Hareide Longitudinal stiffener (continuous) Longitudinal girder Transverse girder Sniped stiffener Longitudinal stiffener (a)(b) Figure 1: Examples of stiffened plates: (a) Stiffened plate enclosed by longitudinal and transverse girders, and (b) girder stiffened by sniped stiffeners. Rayleigh-Ritz approach has been used to analyse free vibration of unstiffened plates with various boundary conditions [4, 5, 6], and vibration of plates subjected to in-plane loads [7, 8, 9]. Semi-analytical approaches have also been developed to investigate various aspects of vibration on stiffened plates [10, 11, 12, 13, 14]. In a research work by Xu, Du and Li [15], vibration of irregularly stiffened plates without in-plane loads is studied. The semi-analytical methods mentioned above are restricted to irregularly stiffened plates without in-plane preloads, or to unstiffened or regularly stiffened plates. In the present work, the main objective has been to develop a computationally efficient semianalytical model for eigenfrequency computations of stiffened plates subjected to in-plane prestress and with arbitrarily oriented stiffeners. Analyses by the present model can be performed for plates with simply supported, clamped of partially clamped boundary conditions, or combinations of these. The model may also handle interior supports, along lines with arbitrary orientations and lengths. By using inclined stiffeners or strong translational springs to enclose triangular, trapezoidal and other plate shapes, the present model may handle complex plate geometries. 2 PLATE DEFINITION AND BOUNDARY CONDITIONS Typical engineering applications for stiffened plates are illustrated in Fig. 1(a) where a plate is enclosed by the strong longitudinal and transverse girders, and in Fig. 1(b) where the plate web is supported by a strong girder flange. Girder stiffeners may be be oriented horizontally or vertically. The stiffeners may be sniped at their ends (such as in Fig. 1(b)), and will then, unlike continuous stiffeners such as in Fig. 1(a), not be subjected to external axial loading (in the stiffener direction). Sniped stiffeners may also be used in conjunction with cases where a rather non-regular stiffener arrangement is required, such as for instance in the stern and in the bow of a ship hull. In order to model such cases, the plate defined in Fig. 2 is considered. It may be subjected to in-plane shear stress and linear varying in-plane compression or tension stress. It may have none, one or more stiffeners, and the stiffener orientations may be 2 322 Lars Brubak, Jostein Hellesland and Ole J. Hareide Sx(y) S2 x S1 x Sxy Sxy Sxy Sxy S1 y S2 y S1 y S2 y Sy(x)Sy(x) Sx(y) S2 x S1 x (x1,y 1) (x2,y 2) L b x y t hw tf bf tw Stiffener (b)(a) Figure 2: (a) Stiffened plate subjected to in-plane shear stress and in-plane, linear varying applied compression or tension stress, and (b) cross-section of an eccentric stiffener. arbitrary. The stiffeners may have different cross-section profiles, and may be eccentric, as in Fig. 2(b), or symmetric about the middle plane of the plate. The stiffeners are modeled as simple beams. A boundary (plate edge) or a part of a boundarymaybe simply supported, clamped or something in between. 3 MATERIAL LAW AND KINEMATIC RELATIONSHIPS The usual plane stress assumption for thin isotropic plates is adopted. The well known Hooke’s law for this case is defined by σx=E 1−ν2(ǫx+νǫy) (1) σy=E 1−ν2(ǫy+νǫx) (2) τxy =E 2(1 + ν)γxy =Gγxy (3) where σx,σyand τxy are the in-plane stresses, and ǫx,ǫyand γxy the in-plane strains, defined positive in tension, and the material coefficients Eand νare Young’s modulus and Poisson’s ratio, respectively. The total strain can be divided into a membrane strain (ǫm) and a bending strain (ǫb) and given by ǫx=ǫm x+ǫb x=ǫm x−zw,xx (4) ǫy=ǫm y+ǫb y=ǫm y−zw,yy (5) γxy =γm xy +γb xy =γm xy −2zw,xy (6) where wis the out-of-plane displacement in the z-direction (positive downwards in Fig. 2). The conventional notation w,xy for ∂2w/∂x∂y, etc., is adopted. The bending strain distribution complies with Kirchhoff’s assumption [16]. 3 323 Lars Brubak, Jostein Hellesland and Ole J. Hareide 4 VIBRATION ANALYSIS – EIGENVALUES The eigenfrequencies of a perfect, stiffened plate are computed using the well known Rayleigh-Ritz method. The assumed displacement field, which satisfies the boundary conditions of a simply supported plate, is given by w(x, y)= M  i=1 N  j=1 aijsin(πix L)sin(πjy b) (7) where aij are amplitudes, Lthe plate length and bthe plate width. By assuming harmonic vibrations, the usual eigenvalue problem [17] is (KM ijkl +Λ preKG ijkl −ω2Mijkl)akl = 0 (8) where KM ijkl =∂2U ∂aij ∂akl ,Λ preKG ijkl =∂2T ∂aij∂akl and ω2Mijkl =∂2Hmax ∂aij ∂akl (9) Here, ωdenotes the natural circular eigenfrequencies and akl the corresponding eigenvectors. The desired prestress level is obtained by multiplying some initial (reference) applied in-plane stress by a load factor Λpre. In the eigenvalue problem, Mis the mass matrix, KMthe material stiffness matrix and KGis the geometrical stiffness matrix for the predefined in-plane stresses. These matrices are, as seen, expressed by U,Tand Hmax, which are the strain energy, potential energy due to external loads and the maximum kinetic energy, respectively. Each energy contribution is described in more detail below. In the common matrix notation, the eigenvalue problem above can be written (KM+Λ preKG−ω2M)a=0(10) In an analysis of a clamped plate, it would be more appropriate to assume a displacement field defined with a series of cosine functions. However, although each component in a series of sine functions represents a simply supported condition, added together they are nearly able to describe a clamped, or partially restrained, condition at a negligible distance from the support. The sine curve assumption is therefore able to handle plates with various boundary conditions along the edges. The elastic strain energy contribution from bending of the plate is given by Ub plate =D 2b 0L 0(w,xx +w,yy)2−2(1 −ν)(w,xxw,yy −w2 ,xy)dx dy (11) where D=Et3/12(1 −ν2) is the plate bending stiffness and tis the plate thickness. By substituting the assumed displacement field, an analytical solution of this integral may be derived. More details can be found in Brubak [18] and Brubak, Hellesland and Steen [19]. The membrane strain energy of the plate and the stiffeners (below) is not included 4 324 Lars Brubak, Jostein Hellesland and Ole J. Hareide as it does not affect computed eigenvalues, since small deflection theory is used. This will have some consequences that will be discussed in the results sections. The curvature of the stiffeners is equal to the curvature in the plate along the stiffeners. Thus, the bending strain energy due to an arbitrarily oriented stiffener, with length Ls and cross-section area As, can be given by Ub stiff =EIe 2L4 sLs L2 xw,xx +2LxLyw,xy+L2 yw,yy 2 dLs(12) where (x1,y 1) and (x2,y 2) are the coordinates of the stiffener ends, Lx=(x2−x1), Ly=(y2−y1) and Ie=As (z−zc)2dAs+tb ez2 c(13) is an effective moment of inertia about the axis of bending. Here, zcis the distance from the middle plane of the plate to the centroidal axis (through the centre of area) of a crosssection consisting of the stiffener and an effective plate width be. The effective moment of inertia Iereflects the fact that eccentric stiffeners tend to “lift” the axis of bending. For a symmetric stiffener, zc= 0. This value also represents a reasonable simplification in many cases also for eccentric stiffeners. The strain energy integral in Eq. (12) may be solved analytically or by numerical integration. More details can be found in Brubak [18] and Brubak, Hellesland and Steen [19]. The potential energy of the external, in-plane prestress due to plate bending is given by T=−Λpre L 0b 0 t 2Sx0(y)w2 ,x +Sy0(x)w2 ,y −2Sxy0w,xw,ydy dx (14) where Sx0(y), Sy0(x) and Sxy0are the initial (reference) stresses and Λpre the load factor for the prestress. More details can be found in Brubak [18] and Brubak, Hellesland and Steen [19]. The stiffeners in the example computations, presented below, are sniped. As a consequence, the in-plane stresses are applied at the midplane of the plate. Since the stiffeners will try to resist the corresponding strains, eccentricities (bending) will be introduced. However, such eccentricity effects are not accounted for in the model. In line with the sniped stiffener assumption, Eq. (14) does not include any contribution from stiffeners. It would be reasonably straightforward to extend Eq. (14) to also include end loaded (continuous) stiffeners. However, in cases with local plate buckling, which are of most practical interest, the stiffeners will remain nearly straight and only contribute negligibly to T. Then it makes little difference whether the stiffeners are sniped or continuous. More details on how to include the energy contribution for end loaded stiffeners can be found in Brubak and Hellesland [20]. 5 325 Lars Brubak, Jostein Hellesland and Ole J. Hareide L b x y Stiffener t hw tf bf tw n s vw vf w,n (b)(a) Figure 3: (a) The velocity of the centroid of the web and flange due lateral displacement of the stiffener and (b) the stiffener orientation. Both the displacements and the rotations along an arbitrary oriented line with length Smay be restrained by applying translational and rotational springs, respectively. The strain energy due to these springs can be written Uspring =1 2S (krw2 ,n +ktw2)dS (15) Here, w,n is the derivative of wnormal to the line, and ktand krare the stiffness of the translational springs and the rotational springs, respectively. More details can be found in Brubak [18] and Brubak, Hellesland and Steen [19]. Eq. 15 can also be applied for rotational springs at plate boundaries. The maximum kinetic energy of the plate is given by Hmax plate =ω2ρt 2b 0L 0 w2dx dy (16) where ρis the density of the material, and the maximum kinetic energy of the vertical movement of the stiffener is given by Hmax v,stiff =ω2mst 2Ls w2dLs(17) where mst =ρ(hwtw+bftf) is the mass per unit length of the stiffener. The kinetic energy of the rotational movement of the stiffener can be expressed by Hr,stiff =1 2Ls (mwv2 w+mfv2 f)dLs(18) where mw=ρhwtwis the mass per unit length of the web and mf=ρbftfis the mass per unit length of the flange. As shown in Fig. 3(a), vwand vfis the lateral velocity of 6 326 Lars Brubak, Jostein Hellesland and Ole J. Hareide the centroid of the web and the flange of the stiffener, respectively. By using geometrical considerations, these velocities can be written as vw= (0.5hw+0.5t)∂w,n ∂t and vf= (0.5t+hw+0.5tf)∂w,n ∂t (19) where w,n is the derivative of wnormal to the stiffener orientation as illustrated in Fig. 3(b). By substituting Eq. (19) into Eq. (18), the expression for the maximum kinetic energy of the rotational movement of the stiffener can be written as Hmax r,stiff =ω2¯mrot 2L2 sLs L2 yw,x +2LxLyw,xw,y+L2 xw,y dLs(20) where ¯mrot =mw(0.5hw+0.5t)+mf(0.5t+hw+0.5tf) (21) The strain energy integral in Eq. (20) may be solved analytically or by numerical integration. The latter is used in the present work. When all the expressions for the potential energy above are computed, the eigenvalue problem of Eq. (8) can be established and solved. 5 VALIDATION PREMISES The present model was incorporated into a Fortran computer code and computed results have been compared with finite element analyses (FEA) using ABAQUS [21] for a variety of plate and stiffener dimensions, and in-plane prestress loads. Eigenfrequency results, presented as natural eigenfrequencies f=ω/2π[Hz], are verified by comparisons with FEA. Results, presented in subsequent sections, are limited to simply supported plates with sniped stiffeners. The finite element model, based on shell elements (shell elements S4R both for plate and stiffeners), is supported in the out-of-plane direction along the edges of the plate, and the edges are forced to remain straight during deformation. The plate is also supported in the in-plane directions, just enough to prevent rigid body motions. Further, the ends of the stiffeners are completely free and not loaded (sniped). The adopted elastic material properties in each computation are Young’s modulus E= 208000 MPa and Poisson’s ratio ν=0.3. In the present model, 225 degrees of freedom (15x15) are used in all the cases. Comparable convergence studies carried out previously [19] have shown that this choice of degrees of freedom may overestimate the eigenfrequency predictions by, typically, about 1-2 %. In comparisons, the number of degrees of freedom used in the FEA analysis is typically about 20000, which is believed to be a sufficiently large number to ensure satisfactory results. 7 327 Lars Brubak, Jostein Hellesland and Ole J. Hareide Sx[MPa] (a) (b) Figure 4: (a) Eigenfrequencies for a regularly stiffened plate subjected to uniaxial prestress Sxin the stiffener direction and (b) the eigenmode of the plate for free vibration. 6 REGULARLY STIFFENED PLATE WITH IN-PLANE PRESTRESS Stiffened plates subjected to in-plane prestress have been analysed. A typical case is shown in Fig. 4(a), where the eigenfrequencies for a plate is computed for various magnitudes of prestress in the stiffener direction for both compression and tension. The plate (L/b/t = 2000/2000/20mm) is simply supported and is provided with one regular, sniped stiffener with a flatbar section (hw/tw= 100/12mm). The agreement between the model and FE results is very good for external prestress smaller than about 120 MPa (Fig. 4(a)). This corresponds to about 63% of the elastic buckling stress, which has been found to be Sx,cr = 189 MPa in this case. Beyond this level, it can be seen that the model frequency results continues to decrease toward the elastic buckling stress (at the intersection with the Sx-axis) while the FEA results reaches a minimum value and then starts increasing for increasing prestress values. The reason for these differences is that the stiffness matrix in the FEA is computed using large deflection theory while the model is based on small deflection theory (in that the energy contribution from membrane strains does not affect the eigenvalues). In large deflection theory, membrane stresses are redistributed from the interior of the plate fields to the stiffer parts, which are at the edges. This leads to a reduction in membrane compression stresses at the interior of the plate, which causes a larger eigenfrequency for the vibrations computed by FEA. It is worthwhile noting that this leads to solutions also for prestress values exceeding the classical elastic buckling stress. For plates without eccentricities (i.e., due to sniped stiffeners or imperfections), the membrane stress redistribution will start taking place at the classical buckling stress. With increasing eccentricities, the transition from the descending to the ascending portion of the frequencyprestress curve (FEA, Fig. 4(a)) becomes increasingly gradual. In the semi-analytical 8 328 Lars Brubak, Jostein Hellesland and Ole J. Hareide LSySy 2b 3 b 3 L b t hwtw Plate 1 1200 2400 12 200 12 Plate 2 1600 2400 12 200 12 y xStiffeners (a) (b) Figure 5:(a) Plate definitions and (b) eigenmode of Plate 1 subjected to prestress Sy= 50 MPa. model, there is no redistribution of compression membrane stresses. Due to this, the frequency-prestress curve decreases continuously toward the elastic buckling stress. At and above this stress there is no solutions. In order to extend the model to include large deflection theory [22], it is expected that a similar formulation as in Brubak and Hellesland [23] for computing the post-buckling behaviour can be used. The eigenmode for free vibration of the plate is shown in Fig. 4(b). This is a global mode where the stiffener deflection is one half wave. Similar modes with global deflections are also found when the plate is subjected to a prestress. In its present form, application of the model should be limited to prestress values below about 60% of the classical elastic buckling stress for such cases, associated with global buckling modes and eccentric loads (sniped stiffeners). For local buckling modes such eccentricity effects are not that pronounced and the model will be able to handle prestress values closer to the elastic buckling stress (as shall be seen below). 7 IRREGULARLY STIFFENED PLATES Typical results for two simply supported, irregularly stiffened plates defined in Fig. 5(a) are presented in Fig. 6 . The plates are provided with two inclined, eccentric stiffeners. The rather irregular stiffener locations are chosen such as to provide quite severe test cases for the present model. In this case, with a local eigenmode, the redistribution of compression stresses (discussed above) is less pronounced than above due to smaller eccentricity effects. This is reflected in the FEA results by a rather abrupt change in slope (sharp angle) for prestress values close to the elastic buckling stress. As already explained, the model does not reflect this phenomenon. As seen (Fig. 6), the agreement between the model and the FE results is good for compression prestresses smaller than about 95% of the elastic buckling stress (as given by the intersection by the model results (full line) and the Sy-axis). For tensile prestress 9