Fast approximations of dynamic stability boundaries of slender curved structures

Transcription

1 Fast approximations of dynamic stability boundaries of slender curved structures Yang Zhou a, Ilinca Stanciulescu a,, Thomas Eason b, Michael Spottswood b a Rice University, Department of Civil and Environmental Engineering, 61 Main Street, Houston, TX, 775, U.S.A b Air Force Research Laboratory, Structural Sciences Center, 279 D. Street, WPAFB, OH, U.S.A. Abstract Curved beams and panels can often be found as structural components in aerospace, mechanical and civil engineering systems. When curved structures are subjected to dynamic loads, they are susceptible to dynamic instabilities especially dynamic snap-through buckling. The identification of the dynamic stability boundary that separate the non-snap and post-snap responses is hence necessary for the safe design of such structures, but typically requires extensive transient simulations that may lead to high computation cost. This paper proposes a scaling approach that reveals the similarities between dynamic snap-through boundaries of different structures. Such identified features can be directly used for fast approximations of dynamic stability boundaries of slender curved structures when their geometric parameters or boundary conditions are varied. The scaled dynamic stability boundaries of half-sine arches, parabolic arches and cylindrical panels are studied. Keywords: Dynamic snap-through, Dynamic stability boundary, Scaling approach, Curved structures Corresponding author. Tel.: ; fax: address: ilinca.s@rice.edu (Ilinca Stanciulescu) Preprint submitted to International Journal of Nonlinear Mechanics March 29, 217

2 1. Introduction Slender curved structures such as arches and shells have been widely used in many engineering applications. One type of dynamic instabilities frequently occurring in these slender curved structures is the dynamic snapthrough buckling (sudden jump between remote configurations), accompanied by large amplitude vibrations and rapid curvature reversals (Fig. 1). Such process leads to large and frequent stress reversals that significantly exacerbate fatigue failures [1 4]. Therefore, it is of great importance to identify the dynamic stability boundaries that separate small amplitude non-snap from large amplitude post-snap responses. F Figure 1: Snap-through buckling. Dynamic snap-through is a highly nonlinear problem that usually cannot be solved analytically. Therefore, numerical integration approaches have been used to investigate the dynamic snap-through of curved beams and panels. A great portion of these efforts focus on structures subjected to impulse loads, suddenly applied step loads, or ramp loads [5 16]. These studies examined the effects of geometric imperfections [5, 13], boundary conditions [5], physical damping [6, 1, 14], asymmetric responses [6 8], different initial arch shapes [13], asymmetric perturbations [14], the spatial distributions of an applied load [9, 11], load duration [5, 12], and loading rates [15, 16] on dynamic snap-through. Numerical integration methods have also been used to study the dynamic snap-through of structures under harmonic loads. Plaut 2

3 and Hsieh [17] investigated the effects of a two-frequency harmonic excitation on the dynamic snap-through of curved beams. Chandra et al. proposed a numerical framework [2] and a combined experimental-computational framework [3] to study the dynamic snap-through of harmonically excited shallow arches. Gonçalves and Prado [18] analyzed the influences of non-linear modal interactions on the dynamic snap-through of cylindrical shells. In addition, a considerable amount of research was directed at investigations of dynamic parametric instability of structures under axial harmonic loads [19 23], which is a different type of dynamic instability and does not make the object of our current study. Unlike impulse loads or suddenly applied step loads, the dynamic stability boundary of a structure under harmonic loads consists of dynamic snap-through buckling load amplitudes for different excitation frequencies. When numerical integration methods are used to obtain the stability boundary of a structure under harmonic loads, a number of transient simulations need to be performed for each excitation frequency and the total number of transient simulations required for all excitation frequencies can be large (several hundred or thousand transient simulations depending on the range of interest of the parameters and on the desired resolution). In the design of curved structures, it is necessary to obtain a new stability boundary each time when the design configuration is modified. However, the direct identification of the stability boundary of the structure with modified properties requires another costly set of computations. This brute force approach to obtaining the dynamic stability boundaries is inefficient and can significantly delay the design process. To avoid the high computational cost, researchers have developed energy approaches to study the dynamic snap-through of shallow arches without numerically solving the non-linear differential equations of motion. Hoff and Bruce [24] proposed an energy method in which the condition that the contour of the total potential energy passes through the system s saddle point identifies the initiation of dynamic snap-through. Hsu [25 28] rigorously 3

4 derived a sufficient condition for avoiding dynamic snap-through and used this theory to study the effects of initial thrust, elastic supports and spatial distributions of the applied force on the dynamic stability of curved beams. Simitses [29] presented an approach based on the conservation of total energy, which can be used to obtain sufficient conditions for stability and instability. More recently, Chen et al. [3 35] introduced an energy approach to obtain sufficient conditions to avoid dynamic snap-through of shallow arches. They argued that if the total energy transferred to the arch is smaller than the minimum energy barrier between separated stable equilibrium positions, the arch cannot snap dynamically. Pi and Bradford [36 4] used the approach of energy conservation to derive the lower and upper dynamic buckling boundaries for shallow arches with different boundary conditions and loading types. Kounadis and co-workers [41 47] introduced energy based geometric methods to study the dynamic buckling of discrete systems, approaches that can be extended to continuous systems. Other related work using energy approaches can be found in [48, 49]. Although these approaches do not require integrating the non-linear differential equations of motion, many lead to very conservative results [9, 13, 5] and have difficulties incorporating physical damping effects. Moreover, these methods are currently only applicable to impulse loads and suddenly applied step loads (not applicable to harmonic excitations and more complex dynamic loads) since such methods require that the total potential energy of the dynamic system doesn t vary with time. In this paper, we aim to provide a procedure to quickly estimate the dynamic snap-through boundaries of slender curved structures subjected to harmonic loads. A scaling approach, based on normalizing dynamic forcing amplitudes by static buckling loads and dynamic forcing frequencies by the linear natural frequencies of first symmetric modes, is proposed to characterize the similarities between different dynamic snap-through boundaries of structures with varied properties. It is found that the snap-through boundaries after scaling are alike in shape and closely overlap for certain range 4

5 of forcing parameters. With this finding, only one dynamic snap-through boundary needs to be obtained by performing transient simulations and other dynamic stability boundaries of the structures with modified geometric parameters or boundary conditions can be directly estimated without running additional costly transient simulations, thus significantly reducing the computational cost. To the authors best knowledge, this scaling approach and fast estimates of dynamic stability boundaries have not been proposed in the literature. The proposed scaling approach is shown to be effective for initially symmetric arches and panels for which the linear natural frequency of their first symmetric modes is either the lowest frequency or only slightly larger than the lowest one. 2. The scaling approach In this section, we briefly introduce the proposed scaling approach that identifies the important features of dynamic snap-through boundaries. Forcing amplitudes Displacement x o Forcing frequencies (a) Snap-through boundary Time (b) Pre-snap and post-snap responses Figure 2: A snap-through boundary and two displacement response In static analysis, the singularity of the structural stiffness can be used to identify instabilities, including potential for snap-through. However, a rigorous mathematical definition of dynamic snap-through is hard to obtain [51, 52]. Despite the lack of a rigorous definition, the sudden increase 5

6 of the amplitude of a post-snap response from that of a pre-snap response (Fig. 2) is typically accepted as criterion to identify dynamic snap-through [52 55]. Fig. 2a shows a dynamic snap-through boundary that separates small-amplitude pre-snap and large amplitude post-snap responses. The dramatic change of the amplitudes can be clearly seen in Fig. 2b. Utilizing this inherent feature, Chandra et al. [3] proposed a global criterion to define dynamic snap-through. It first selects a series of displacement thresholds and then calculates the ratios of the cases that snap to the total number of numerical simulations for all thresholds (Fig. 3). A range of flat or slowly varying ratio region is expected to exist in this representation, and any threshold in this region can be used as dynamic snap-through condition. In this paper, this criterion is adopted to select the threshold and then identify the dynamic snap-through boundary. Snap cases/total cases Threshold Figure 3: A criterion for dynamic snap-through. Based on our observations, different snap-through boundaries have very similar shapes in the space of the forcing parameters. Therefore, a well-chosen scaling of the snap-through boundaries may reveal a unique relationship between these instability boundaries, which may be further used to quickly estimate instability boundaries. Fig. 4 shows a schematic representation of 6

7 λ/λcr scaled snap-through boundary f / f sym1 Figure 4: A schematic diagram of the scaled snap-through boundary. one scaled snap-through boundary in the space of forcing parameters when a structure is subjected to a harmonic excitation λ sin(2πft). Here λ represents the dynamic forcing amplitude and λ cr denotes the static buckling load of the structure; f represents the excitation forcing frequency and f sym1 is the linear natural frequency of the first symmetric mode. In the scaled representation, the dynamic forcing amplitudes and frequencies are normalized: λ/λ cr and f/f sym1. The static buckling load (limit or bifurcation load) is a good choice for the scaling of dynamic forcing amplitudes since a structure s static behavior provides some insightful information about its dynamic responses. The ratio of the dynamic to static buckling load should be below 1 when the excitation frequency is between and the linear frequency of the fundamental resonance mode. For the scaling of dynamic forcing frequencies, the linear natural frequency of the first symmetric mode is a reasonable candidate for the systems whose response is dominated by this mode. We find that the structures whose static buckling loads are equal to or a little smaller than the limit-point loads often satisfy such condition. When a structure loses stability at a limit point, the first symmetric mode is the system s fundamental resonance mode and dominates the deformations. For structures that snap asymmetrically at a 7

8 bifurcation point but for which the bifurcation load is close to the limit-point load, the linear frequency of the first symmetric mode is also used to normalize the dynamic forcing frequencies because the dominant deformation mode of the asymmetric post-snap oscillations in this case is still the first symmetric mode. When a structure s static buckling load is significantly smaller than the limit-point loads, the proposed scaling approach becomes less effective since the first asymmetric mode and other higher modes now significantly contribute to the deformations. 3. Numerical experiments The proposed scaling approach is applied to half-sine arches, parabolic arches and cylindrical panels subjected to dynamic loads that vary sinusoidally in time. A mass-proportional damping (C = ηm), commonly used in the simulations of structural dynamics [56], is adopted to model physical damping. The value of η is determined by assuming the damping ratio of the first symmetric mode equal to.2% for each case Shallow arches To generate more data to verify the similarities of various snap-through boundaries, a lower-order model (LOM) is used to analyze pinned-pinned shallow arches that are under a concentrated load at the midspan (Fig. 5). The arches are assumed to be elastic, isotropic and homogeneous. A large deformation Euler-Bernoulli beam theory is used to model the geometric nonlinearity. In Fig. 5, ρ, E, A, I, H and L are the density, Young s modulus, area and moment of inertia of the cross section, rise and horizontal span of the beam. The LOM is obtained from simple modifications of the derivations in [57] and a brief introduction of this LOM is included in the Appendix. Convergence studies and verification with finite element analysis (FEA) demonstrate that the LOM with 1 sine-terms gives accurate results for both static and dynamic responses. 8

9 y x ρ,e,a,i L F H Figure 5: A shallow arch under a point load at its midspan. For half-sine and parabolic shallow arches with non-dimensional rise h, the initial dimensionless configuration can be written as h sin ξ the half-sine arch, u (ξ) = n 32h i=1 sin(2i 1)ξ the parabolic arch. π 3 (2i 1) 3 where the dimensionless height h is equal to the physical rise (H) divided by the radius of the gyration (r) of the cross section which is proportional to the thickness [57]. Therefore, increasing the value of the non-dimensional rise is equivalent to either increasing the physical rise or decreasing the physical thickness. Before performing transient analyses, it is necessary to obtain the static buckling loads and the linear natural frequencies of the first symmetric mode since they will be used to scale the dynamic snap-through boundaries. Figs. 6 and 7 show the equilibrium paths of half-sine and parabolic arches respectively when the initial non-dimensional rise is varied. λ is the non-dimensional external loading parameter and u mid is the vertical dimensionless displacement at the midspan of the arch. The solid and dashed lines represent the primary and bifurcated equilibria obtained using the LOM, while the markers denote the solution obtained from FEA. It should be noted that an initially 9

10 5 4 3 Λ 2 1 L 1a u mid L 1b (a) h= L 1a 5 4 Λ L 1b u mid (b) h= Λ 3 L 1a B 1a B 1b u mid (c) h=4.27 L 1b Λ L 1a B 1a B 1b u mid (d) h=4.53 L 1b Λ B 1a L 1a u mid (e) h=5.2 B 1b L 1b Λ B 1a L 1a u mid (f) h=5.6 Figure 6: Equilibrium paths of half-sine arches with different non-dimensional rises h. Primary path from LOM, Bifurcated path from LOM, Results from FEA. L 1a and L 1b represent limit points, while B 1a and B 1b denote bifurcation points. L 1b B 1b symmetric structure can have bifurcated secondary equilibrium states due to the existence of strong nonlinearity [2, 58]. It can be seen that the results from the LOM match those from the FEA for both types of arches. The material properties and geometric parameters of these arches used in the FEA are shown in Tables 1-4. All finite element simulations are performed in the Finite Element Analysis Program (FEAP) [59], a research code that includes most commonly used finite element algorithms. The large-deformation Euler- Bernoulli beam element [59] is used for the spatial discretization of the arch. The arc-length method is used to retrieve the primary equilibrium paths, and the numerical procedure introduced in [58] is used to obtain the bifurcated equilibrium paths. Statically, a structure loses stability at the first encountered critical point 1

11 5 4 3 Λ 2 1 L 1a u mid L 1b (a) h=2.63 u mid (d) h= L 6 1a 5 4 Λ L 1b u mid (b) h= L 1a B 1a 2 B 1a L 1a 1 1 Λ 5 Λ 1 L B 5 1b 1b L 1b B 1b u mid (e) h= Λ 3 L 1a B 1a L 1b B 1b Λ 1 B 1a u mid (c) h=4.37 L 1a u mid L 1b (f) h=6.17 B 1b Figure 7: Equilibrium paths of parabolic arches with different dimensionless rises h. Primary path from LOM, Bifurcated path from LOM, Results from FEA. L 1a and L 1b represent limit points, while B 1a and B 1b denote bifurcation points. Table 1: Material properties of shallow arches used in FEA Parameters Values Poisson s ratio.28 Density (kg/m 3 ) 78 Young s Modulus (MPa) 27 when the applied load is increased. The critical points, where the structure s stiffness is singular, can be further differentiated as limit points (locations with zero load variation or horizontal tangent to an equilibrium path) and bifurcation points (locations where the primary and bifurcated path intersect on an equilibrium path). In nonlinear buckling analyses, an initially symmetric structure can buckle symmetrically at a limit point or asymmetrically 11

12 Table 2: Unchanged dimensions of shallow arches used in FEA Parameters Values Thickness (mm) 1. Width (mm) 12. Horizontal span (mm) 6 Table 3: Rises and rise-to-span ratios of half-sine arches used in FEA H(mm) H/L (1 3 ) Table 4: Rises and rise-to-span ratios of parabolic arches used in FEA H(mm) H/L (1 3 ) at a bifurcation point depending on the system s properties such as its rise and thickness. Detailed discussions about critical points and symmetric or asymmetric buckling can be found in [58]. Comparing Figs. 6 and 7, we can see that these half-sine and parabolic arches have similar static buckling and post-buckling behaviors. The equilibrium states of parabolic arches shown in Fig. 7 can also be classified into three different groups. In the first group (Figs. 6a, 6b, 7a and 7b), one pair of limit points (L 1a and L 1b ) exist on the equilibrium path. The load of the first reached limit-point L 1a is the static buckling load and will be used to scale dynamic forcing amplitudes. Figs. 6c, 6d, 7c, and 7d show that the equilibrium states have one pair of limit points (L 1a and L 1b ) and bifurcation points (B 1a and B 1b ). The arches in this group also display limit-point buckling since the limit point L 1a is still the first reached critical point on the equilibrium path. The corresponding limit-point buckling load of L 1a will be used for the normalization of dynamic forcing amplitudes. The last group (Figs. 6e, 6f, 7e, and 7f) includes cases for which the first bifurcation point B 1a occurs before the first limit point 12

13 L 1a. The arches in this group buckle asymmetrically at the first encountered bifurcation point B 1a and the load of this critical point B 1a will be the one adopted in the scaling of dynamic forcing amplitudes..8.8 Linear natural frequencies f asym1 f sym h 5 6 (a) Half-sine arches Linear natural frequencies f asym1 f sym h 5 6 (b) Parabolic arches Figure 8: The dimensionless linear natural frequencies of the first symmetric and asymmetric modes of half-sine and parabolic arches for different non-dimensional rises. Frequencies of symmetric mode from LOM, Frequencies of asymmetric mode from LOM, Solutions from FEA. The non-dimensional linear natural frequencies of the first symmetric and asymmetric (f sym1 and f asym1 respectively) modes of half-sine and parabolic arches are shown in Fig. 8 for different dimensionless rises. The solid and dashed lines represent the frequencies calculated from the LOM respectively. The square markers denote the results obtained from the FEA for the arches whose equilibrium states are shown in Figs. 6 and 7. The solutions from the LOM match closely with those from the FEA. Like the equilibrium states, half-sine and parabolic arches also have similar varying patterns in the linear natural frequencies as the non-dimensional rise is changed. The frequency of the symmetric mode f sym1 significantly increases, but the frequency of the asymmetric mode f asym1 barely varies, when the rise is increased. For relatively small rises for which the structures display limit-point buckling, the linear natural frequencies of the symmetric mode f sym1 are well below those of 13

14 the asymmetric mode f asym1. When the arches with relatively high rise buckle asymmetrically at a bifurcation point, the frequencies of the first asymmetric mode f asym1 are close to the frequencies of the first symmetric mode. The linear natural frequencies of the first symmetric mode are still used to scale the dynamic forcing frequencies. The reason is that the primary deformation mode of the post-snap response is still the lowest symmetric mode even for the cases whose frequencies of the asymmetric mode are slightly smaller than those of the symmetric mode. This will be demonstrated later by the comparison of the mode coefficients of asymmetric post-snap responses. The discussions of the structures with significantly higher rises will be presented at the end of this paper LOM FEA 25 2 LOM FEA u mid 1 5 u mid τ (a) Half-sine arch (λ = 1, f =.45) τ (b) Parabolic arch (λ = 12, f =.52) Figure 9: Two post-snap transient responses of the half-sine with h=5.6 and parabolic arch with h=6.17. In the transient analysis, we are not interested in the cases where the dynamic snap-through buckling loads are larger than the static buckling loads since the static buckling governs the instability behavior for these cases. Therefore, the excitation forcing amplitude λ is varied from to the static buckling load and the excitation forcing frequency f is varied from to a value whose dynamic buckling load is close to the static buckling load, which we find is about 1.4 times of the linear natural frequency of the first symmet- 14

15 1 α 1 -β 1 α 2 -β 2 1 α 1 -β 1 α 2 -β 2 Mode coefficient τ (a) Half-sine arch (λ = 1, f =.45) Mode coefficient τ (b) Parabolic arch (λ = 12, f =.52) Figure 1: Mode coefficient comparison of two post-snap responses shown in Fig. 9. α 1 and β 1 represent the first symmetric mode coefficient of the initial and deformed configurations respectively while α 2 and β 2 denote the first asymmetric mode coefficient of the initial and deformed configurations respectively. ric mode f sym1. When the excitation forcing frequency is larger than 1.4f sym1, the dynamic buckling loads are usually larger than the static buckling loads, and these cases are less important and outside the interest of the current work. Fig. 9 shows two post-snap steady state responses of the half-sine arch with h=5.6 and the parabolic arch with h=6.17. An excellent match between the solutions calculated from LOM (solid line) and FEA (square markers) can be observed. Newmark s method [6], a widely used time integration method in structural dynamics, is used here to numerically integrate the differential equations. Two steady-state periodic responses are chosen for this quantitative comparison because chaotic responses from LOM and FEA will inevitably have quantitative differences in their long-term behavior. Note the half-sine and parabolic arches with these two relatively high nondimensional rises dynamically snap-through asymmetrically since they both display asymmetric bifurcation buckling in their static behavior (Fig. 6f and Fig. 7f). This can be further confirmed by the contribution of asymmetric modes to the post-snap responses illustrated by non-zero values of the first asymmetric mode coefficient of the displacement α 2 β 2 (dashed lines), which 15

16 together with the first symmetric mode coefficient of the displacement α 1 β 1 (solid lines) are shown in Fig. 1. The amplitude of the first symmetric mode coefficient of the displacement (α 1 - β 1 ) is much larger than that of the first asymmetric mode coefficient of the displacement (α 2 - β 2 ), indicating that the first symmetric mode is still the dominant deformation mode even when these arches dynamically snap-through asymmetrically. Therefore, the linear natural frequency of the first symmetric mode f sym1 is used for scaling even for arches with relatively high rises that snap-through asymmetrically and whose lowest mode is asymmetric λ /λ cr.6 h= h=3.33 h= h=4.53 h=5.2 h= f/f sym1 (a) Half-sine arches λ/λ cr.6.4 h=2.63 h=3.31 h= h=4.75 h=5.65 h= f /f sym (b) Parabolic arches Figure 11: The scaled snap-through boundaries of half-sine and parabolic arches. The close match between LOM and FEA for the static and dynamic snap-through responses gives us confidence in this LOM of shallow arches. Using the LOM and the scaling approach, that is, normalizing the excitation forcing amplitude λ by static buckling load λ cr and the excitation forcing frequency f by the linear frequency of the first symmetric mode f sym1, we obtain the scaled snap-through boundaries of the half-sine arches (Fig. 11a) and parabolic arches (Fig. 11b) for different non-dimensional rises h. For each stability boundary, 56 different transient simulations are performed. Although these half-sine or parabolic arches have different static buckling loads and linear frequencies of their first symmetric mode, the scaled dynamic stability 16

17 boundaries of either type of arch have very similar V shapes and are almost on top of each other for excitation frequencies between and 1.4. Therefore, when it is necessary to identify multiple dynamic snap-through boundaries of shallow arches with varied properties such as rises or thicknesses (increasing the non-dimensional rise h is equivalent to either increasing the physical rise or decreasing the physical thickness), we only need to calculate one dynamic snap-through boundary by performing transient simulations and other snap-through boundaries can be approximately obtained without the need to perform additional large numbers of transient simulations. Only the static buckling loads and the linear frequencies of the first symmetric modes need to be known, but they can be easily obtained with much less computation cost than performing large sets of transient simulations. In the scaled stability boundaries the lowest dynamic buckling loads are only about.15 or.2, indicating a significant reduction compared with the static stability limits and demonstrating the importance of identifying dynamic buckling loads. The lowest dynamic buckling loads occur for scaled forcing frequencies f/f sym1 [.85,.9], indicating the existence of a softening non-linearity with respect to the first symmetric mode, which is expected for initially curved beams. In addition, when the excitation frequency is very close to, the scaled dynamic buckling loads (λ/λ cr ) of all arches are approximately 1, which is also as expected because the dynamic effect in this situation is very small and the dynamic buckling loads should be close to the static buckling loads. Fig. 12 shows all scaled dynamic snap-through boundaries of half-sine and parabolic arches. It demonstrates that the scaled dynamic stability boundaries of arbitrary shapes of shallow arches are similar, suggesting that the dynamic snap-through boundary of a half-sine arch can be used to directly estimate that of arches of other shapes. 17

18 1.8 λ/λ cr Half-sine arches Parabolic arches f/f sym1 Figure 12: A comparison of the scaled stability boundaries of shallow arches Cylindrical panels In this section, the scaling approach is applied to cylindrical panels (Fig. 13), another commonly used curved structure. The panel considered here is subjected to a uniformly distributed load in the vertical (z) direction. Its two longitudinal straight edges are pinned-pinned while its two circumferential edges are free of constraints. The material and geometric parameters are chosen to match those of a popular benchmark example [58, 61 7], except for the rise (H) and thickness (t) which will be varied in the following studies. Only FEA is used here for the static and dynamic buckling analyses of cylindrical panels because a LOM that correctly captures the snap-through buckling is more difficult to obtain for curved panels than for shallow arches. The locking-free 27-node solid element [58] is used for the spatial discretization of the panels. In the static analysis, the arc-length method is used to identify the primary equilibrium paths and the numerical procedure proposed in [58] is used to capture the secondary paths. In the transient analysis, the HHT-α method is used for the time integration so that it can efficiently damps out the spurious high frequency components that may be induced by the spatial discretization in FEA [71]. Fig. 14 shows the equilibrium paths of the cylindrical panels when the rise (H) is increased but the thickness (t) is held constant at 6.35 mm; Fig. 15 shows the equilibrium paths when the thickness (t) is decreased but the rise 18

19 circumferential direction t longitudinal direction b P C H P L a y E= MPa 3 ρ=119 kg/m υ=.3 a=58 mm b=58 mm z x Figure 13: The cylindrical panel Distributed load [Pa] L 1a L 1b Vertical displacement [mm] (a) H=5.8 mm Distributed load [Pa] 1 5 L 1a B 1a B 1b L 1b Vertical displacement [mm] (b) H=6.86 mm Distributed load [Pa] L 1a B 1a B 2a B 2b B 1b L 1b Vertical displacement [mm] (c) H=7.75 mm Distributed load [Pa] B L 1a 1a B 2a B 3a B 1b B 3b B 2b L 1b Vertical displacement [mm] (d) H=9.78 mm Figure 14: Equilibrium paths obtained from FEA for cylindrical panels with a thickness t=6.35mm but different rises H. 19

20 (H) is kept constant as 5.8 mm. The solid line represents the primary equilibrium states and different types of markers denote the bifurcated equilibrium states, which are obtained from FEA. It can be found that increasing the rise of the cylindrical panels has the same effect on their buckling and post-buckling behavior as decreasing the thickness of these panels. Similar to shallow arches, cylindrical panels have one pair of limit points (L 1a and L 1b ) for a relatively small rise or large thickness (Fig. 14a and 15a). When the rise of the panel is increased or the thickness of the panel is decreased, the first bifurcation point (B 1a ) initially appears after the first limit point (L 1a ) and then moves before the first limit point, indicating that the loss of stability changes from the limit-point buckling to the bifurcation buckling. Unlike shallow arches, cylindrical panels have two additional bifurcated branches due to the presence of multiple anti-symmetries. The bifurcated buckling mode shapes of all these branches are illustrated in Fig. 14d and 15d. The static buckling and post-buckling behavior of this type of cylindrical panel subjected to a concentrated load was extensively discussed in [58]. As just discussed, the equilibrium states of these cylindrical panels have three asymmetric modes that control the bifurcated branches. Therefore, the linear natural frequencies of the first symmetric and three lowest asymmetric modes are calculated by FEA and shown in Fig. 16a and Fig. 16b respectively for different rises and thicknesses. The shapes of these three asymmetric modes are also displayed in these two plots. The linear natural frequencies of the first symmetric and asymmetric modes greatly increase when the rise of the panel becomes larger, but slightly decrease when the thickness of the panel decreases. In contrast, the linear frequencies of the second and third asymmetric modes barely change when the rise of the panel becomes larger, but significantly decrease when the thickness of the panel becomes smaller (due to the variation of the mass of the structure). In the dynamic analysis, a harmonic load λ sin(2πft) is applied to the panels. Using FEA and the same scaling approach as the one applied to shallow arches, we obtain the scaled snap-through boundaries of cylindrical 2

21 Distributed load [Pa] L 1a L 1b Vertical displacement [mm] (a) t=6.35 mm Distributed load [Pa] L 1a B 1a B 1b L 1b Vertical displacement [mm] (b) t=4.15 mm Distributed load [Pa] L 1a B 1a B 2a B 2b B 1b L 1b Vertical displacement [mm] (c) t=4.8 mm Distributed load [Pa] B L 1a 1a B 2a B 3a B 1b B 3b B 2b L 1b Vertical displacement [mm] (d) t=3.3 mm Figure 15: Equilibrium paths obtained from FEA when the thickness of the panel is varied but its rise H=5.8mm. panels when their rise is increased or thickness is decreased, which are shown in Fig. 17a and Fig. 17b respectively. Here, λ is the forcing amplitude, λ cr is the static buckling load, f is the excitation frequency, and f sym1 is the linear natural frequency of the first symmetric mode. Similar to shallow arches, the snap-through boundaries of cylindrical panels are also exhibiting a zigzagged V shape profile and closely match in their scaled representation in the range of dynamic forcing frequencies where the dynamic buckling loads are equal or smaller than the static buckling loads. Once again, the similarities and overlap identified by the scaling approach for snap-through boundaries of 21

22 Linear natural frequencies [Hz] f sym1 t=6.35 mm H [mm] 9 1 Linear natural frequencies [Hz] t [mm] (a) Increasing the rise H (b) Decreasing the thickness t f sym1 H=5.8 mm 3 Figure 16: The linear natural frequencies, obtained from FEA, of the first symmetric and three lowest asymmetric modes of cylindrical panels, when increasing the rise or decreasing the thickness of cylindrical panels. Symmetric mode, Asymmetric mode. 1 t=6.35 mm 1 H=5.8 mm.8.8 λ/λcr.6.4 H=5.8 mm.2 H=6.86 mm H=7.75 mm H=9.78 mm f/f sym1 λ/λ cr.6.4 t=6.35 mm.2 t=4.8 mm t=4.15 mm t=3.3 mm f/f sym1 (a) Different rises (b) Different thicknesses Figure 17: The scaled snap-through boundaries, obtained from FEA, of cylindrical panels when their rise is increased or thickness is decrease. cylindrical panels with different geometry, suggest that only one dynamic stability boundary is necessary and other boundaries can be directly approximated based on the information of the structure s static buckling loads and linear natural frequencies of the first symmetric modes, which is more ef- 22

23 ficient than the brute force approach of conducting additional large sets of dynamic simulations Other boundary conditions As pointed out in [58, 72], the snap-through buckling of slender curved structures is also very sensitive to boundary conditions. All previous sections in this paper focus on using the scaling approach to reveal the similarities of dynamic stability boundaries when certain critical geometric parameters (e.g., the rise and thickness) are changed but only a pinned-pinned boundary condition was considered for all cases discussed so far. In what follows, we will investigate the applicability of the scaling approach on the dynamic snap-through boundaries when varying the boundary conditions. Fig. 18 shows the comparison of the scaled snap-through boundaries of two half-sine arches with the same properties but different boundary conditions: pinned-pinned and clamped-clamped on the two ends. Similar to the cases studied with different geometric properties, the scaled dynamic snapthrough boundaries of curved structures with different boundary conditions also approximate each other well, indicating that the proposed scaling approach is also applicable to the cases where boundary conditions are changed. 1.8 λ/λ cr Pinned-Pinned Clamped-Clamped f/f sym1 Figure 18: The comparison of scaled snap-through boundaries of the half-sine arches (h=5.6) with pinned-pinned and clamped-clamped boundary conditions. 23

24 3.4. Comparison between arches and panels The comparison of scaled snap-through boundaries is extended to different types of curved structures (e.g., arches versus panels). Fig. 19 shows the scaled dynamic snap-through boundaries of all previously discussed halfsine arches, parabolic arches and cylindrical panels (2 different cases). It is found that the scaled dynamic stability boundaries of all initially curved beams and cylindrical panels analyzed are very similar in shapes and values. Consequently, the dynamic snap-through boundaries of cylindrical panels can be approximately obtained from one dynamic snap-through boundary of a shallow arch rather than computing dynamic snap-through boundaries of cylindrical panels. This can be much more efficient because the computational cost of calculating one dynamic stability boundary of a shallow arch by performing transient simulations is usually much lower than that of a cylindrical panel. 1.8 λ/λ cr Half-sine arches Parabolic arches Cylindrical panels f / f sym Figure 19: The scaled snap-through boundaries of shallow arches and cylindrical panels. It has been shown that the scaling approach works well for the snapthrough boundaries of curved beams and cylindrical panels when their rises or thicknesses are changed in the range of values described. We will further discuss the effectiveness of the scaling approach when varying these geometric parameters outside these ranges. Fig. 2a shows the scaled dynamic stability boundaries of the shallow arches and cylindrical panels when they 24

25 display limit-point buckling, that is P cr = P lim. Fig. 2b shows the stability boundaries of all previously examined cases in which.95 P cr /P lim 1. Fig. 2c shows the stability boundaries of all previously discussed cases and the half-sine arches with three higher non-dimensional rises (h=6.3, 7. and 8.) and cylindrical panels with two higher rises (H=11.81 mm and 12.7 mm) corresponding to.74 P cr /P lim 1. λ/λ cr f/ f sym1 (a) P cr P lim = 1 λ/λ cr f/ f sym1 (b).95 P cr P lim 1 λ/λ cr f/ f sym1 (c).74 P cr P lim 1 Figure 2: The scaled snap-through boundaries of curved beams and panels for different P cr ranges of. P lim Natural frequencies h (a) Half-sine arch Natural frequencies [Hz] H [mm] (b) Cylindrical panel Figure 21: The linear natural frequencies of half-sine arches and cylindrical panels. We can see that the proposed scaling approach is very effective for arches and panels that display limit-point buckling (Fig. 2a). These scaled dynamic 25

26 stability boundaries are almost identical for the excitation frequencies on the left side of the V shapes and closely overlap for the excitation frequencies on the right side of the V shapes. When the first bifurcation point appears before the first limit-point but it is very close to it, the scaled snap-through boundaries become narrowly banded but are still reasonably close (Fig. 2b). We conjecture that the scaling approach works effectively for these cases because the linear natural frequencies of the first symmetric mode are smaller or only slightly larger than those of the asymmetric mode (Fig. 8 and 16) and the dominatnt deformation mode of post-snap responses is still the first symmetric mode. When the first bifurcation point moves further away from the first limit point, the linear natural frequencies of the first symmetric mode become much larger than those of the asymmetric mode whose shape is similar to the static bifurcation buckling mode shape (Fig. 21) and this asymmetric mode greatly contributes to the deformations of post-snap responses. Therefore, the scaled stability boundaries are more widely banded (Fig. 2c) and the approximation of the boundaries using this scaling approach gradually loses accuracy. 4. Discussions When numerical integration methods are used to obtain different dynamic snap-through boundaries of symmetric slender curved structures subjected to harmonic loads, extensive parametric studies with large numbers of transient simulations are usually necessary for each design solution examined. The very large computational costs associated with these studies make them impractical for complex geometries. To overcome these difficulties, we propose to use scaled representations of the boundaries by normalizing the dynamic forcing amplitudes by the static buckling loads and the dynamic forcing frequencies by the linear natural frequencies of the first symmetric mode. These representations identify similarities of different dynamic stability boundaries suggesting an alternate (and much faster) method of approximating them. The approximation requires that the equilibrium paths with the correspond- 26

27 ing critical points and the lowest linear natural frequencies are identified, a much faster computation than a single transient simulation. The range of excitation frequencies of interest is from to 1.4 times the linear natural frequencies of the first symmetric modes, which we find is the interval where the dynamic buckling loads are approximately less than the static buckling loads. We find that the scaling is very effective for curved structures whose post snap-through response is dominated by the first symmetric mode. In practice, the best approximations are obtained for shallow arches or panels (small rises), or larger thickness, geometries for which limit loads are either the lowest critical loads or at most 5-1% larger than the lowest critical critical loads, and the lowest linear natural frequency corresponds to a symmetric mode or is very close to the first symmetric mode. While similarities between the boundaries are still found outside these ranges, the quality of the approximation degrades gradually with the decrease in Pcr P lim, i.e.,when limit-point loads and the linear natural frequencies of the first symmetric mode are significantly larger than the static buckling loads and the lowest linear frequencies respectively. From the three plots in Figure 2a-c one can also observe that for frequencies larger than that corresponding to the lowest dynamic critical load (on the right side of the V shaped boundary) the overlap of various boundaries is a little less accurate than in the range of smaller frequencies. A possible explanation is the fact that more chaotic responses exist for higher forcing frequencies as it was previously shown in [2]. The examples presented in this paper also demonstrate that the scaling can be applied (1) for quite varied geometries: arches of various shapes and panels, with different rise, thickness and material properties, provided the critical loads and natural frequencies satisfied the conditions mentioned above, (2) for distributed or concentrated loads (provided they are symmetrically applied and harmonic), (3) for various boundary conditions (clamped, pinned, etc.) Extension of this approximation to handle cases with behavior 27

28 dominated by asymmetric modes necessitates further investigation. It should also be noted that initial conditions can influence the dynamic snap-through boundaries. For example, initial conditions on the instability boundaries may cause the structure dynamically snap-through with smaller external force due to the large kinetic energy these initial conditions may introduce to the system. However, we focus on studying the effects of forcing parameters on the instability boundaries in the current paper. Therefore, for all cases considered in the current paper, the systems are all excited from the initial rest condition (zero displacement and velocity). A systematic investigation of the effect of initial conditions on dynamic snap-through boundaries is out of the scope of the current paper. Acknowledgements The authors greatly appreciate the financial support by AFOSR under the grant no. FA and by DOD under the High Performance Computing Modernization Program (HPCMP) grant GS4T9DBC17. Appendix Following [57], the equation of motion of a shallow arch under a harmonic load at the mid-span can be written as ρay,tt + c y,t + EI(y y ),xxxx P y,xx = Q δ (x L/2) sin(ω t) (1) Under the assumption of mass-proportional damping, the damping coefficient can be written as c = η ρa. Using (u, u ) = 1 r (y, y ), ξ = π L x, E π 2 r ρ τ = ρ L t, η = L 2 2 E π 2 r η, p = P L 2 π 2 EI, q = Q L 3 π 3 EIr, δ(ξ) = L π δ (x), and ρ L 2 ω = E π 2 r ω, Eq (1) can be non-dimensionalized into the following forms: u,ττ + ηu,τ + (u u ),ξξξξ pu,ξξ = qδ(ξ π/2) sin(ωτ) (2) 28

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