The shear buckling and postbuckling behavior of laterally pressured curved panels

Curved panels are widely used in different structures from fuselage of planes to curved bridge girders. An accurate understanding of buckling and postbuckling behavior of curved panels under different loadings is essential for efficient structural design. The shear buckling and postbuckling behavior of laterally pressured thin curved panels under gradually increasing in – plane shear forces is investigated. The magnitude of the lateral forces, the radius of curvature and the aspect ratio of panels are considered in the parametric studies. A classic theoretical formulation of curved panels buckling load is reexamined and compared to experimental results. The results showed that inward pressure eliminates the snap – through phenomenon and the softening stage in the response of shallow curved panels. However, the buckling characteristics are not significantly affected in the moderately curved panels under small pressures. In addition, the magnitude of inward pressures that would affect the shear buckling and postbuckling behavior of panels depends on their radius of curvature. The ultimate shear capacity of a highly curved panel is considerably reduced due to the increasing presence of inward pressures. The failure mode of 1 Corresponding author


INTRODUCTION
Curved panels are widely utilized as load-bearing elements in various naval, aerospace and civil engineering structures. Examples of such structures are web plates of curved I -girders, wings and fuselage of planes, ships and boats, missiles, water storage tanks and pressure vessels. These elements are of great interest, where light weight and high inplane load-bearing capacity is required. In addition to the aforementioned properties, the reserved postbuckling strength of these elements opens extra possibilities for the engineering design. This paper aims to investigate the buckling and postbuckling behavior of curved panels under the combined action of in-plane shear force and lateral normal pressure, commonly found in the above mentioned structures.
There have been some previous researches on the stability of curved panels under pure in-plane shear forces as discussed by Amani et al. [1]. The early works on the theoretical aspects were conducted by Donnell [2]. In a majority of the theoretical treatments of the buckling of cylindrical shells, three simultaneous partial differential equations were used to express the relationship between the components of the shell's median-surface axial, circumferential and radial displacements [3]. Leggett proposed a new set of equations to predict the critical shear stress of a long, slightly bowed sheet [4]. The panels were assumed to be sufficiently long, allowing the short edge boundary conditions to be neglected. The problem with the Donnell's equation is that it only includes the out-of -plane displacements, whereas the boundary conditions on the radial and circumferential boundaries cannot be imposed directly.
Batdorf proposed a method to find the critical shear stress of curved panels by the use of the Donnell's equation [5]. The method presumes that the radial and circumferential boundaries are simply supported. Some curves were computed for the panels under pure shear loading and the results were compared to those obtained by Leggett. Domb presented a nonlinear modeling technique to predict the buckling of simply supported curved panels under the combined action of shear and compression loads [6]. A set of interaction buckling curves was generated to include the effects of initial imperfections on the combined buckling characteristics.
In general, the buckling and postbuckling behavior of an ideal curved panel under pure in-plane shear force can be divided into the following stages: a. Linear growth of the in-plane shear force up to the critical shear load, b. Snap reduction of the shear force immediately after buckling, c. Further growth of the shear force after reaching a local minimum limit, d. Reaching the maximum load -bearing capacity, e. Reduction of the shear force due to the material's softening.
The effect of normal lateral pressure on the buckling behavior of curved panels was further considered during the World War II. The researchers at the National Advisory Committee for Aeronautics (NACA) found that on an airplane in level flight, the upper surface of the wing is subjected to an outward-acting pressure due to the difference in the internal and external pressures [7]. The effect of normal pressure on the stability of thinwalled circular cylinders under torsion was investigated experimentally and theoretically by Rafel and Sandlin [8]. They found that the critical shear stress was appreciably raised by the outward pressure. A semi-empirical interaction formula was derived to predict the critical shear stress. The results of the formula were in good accordance with the experimental results.
However, the postbuckling behavior of panels was not considered in that study.
Hopkins and Brown theoretically investigated the stability of long and bowed curved panels under the combined action of normal pressure and shear forces [9]. Their results were in good accordance with the NACA's experimental results. They also considered the influence of boundary conditions on the critical shear stress of the panels and concluded that the difference between the buckling stress of simply supported and clamped edges decreases rapidly with the curvature and pressure, thus making the indeterminacy of practical edge conditions less important.
Aghajari et al. numerically and experimentally investigated the buckling and postbuckling behavior of thin -walled cylindrical steel shells with varying thickness subjected to uniform external pressure [10]. The results showed that when there is a low variation in the wall thickness, the buckling mode expands over the whole length of the panel; while when there is a high variation in the wall thickness, only the thinner parts are involved.
Golzan and Showkati conducted experimental and FE studies to investigate the elastic and plastic responses and asymmetric imperfection sensitivity of truncated conical shells and shallow conical caps under the action of external uniform pressure [11]. Chen et al. described a new method of determining the critical buckling resistance of cylindrical shells with stepped walls exposed to uniform external pressure, and demonstrated the manner in which the method can be used to assess the buckling behavior in a wide range pattern of the wall thickness changes [12].
The buckling and postbuckling behavior of structural elements often involves material and geometric nonlinearities. In slender elements, the load-displacement path shows a limit point, where the curve becomes horizontal or vertical, followed by a snap -through or snapback. General static analysis procedures like the standard load -control and displacementcontrol methods encounter problems in snap -through and snap -back points, respectively. The Riks method [13,14] or the dynamic approaches should be applied to deal with such problems. Kobayashi et al. [15] used a validated static stabilizing method to simulate the successive path -jumping in the deep postbuckling behavioral region of the cylindrical shells under axial compression.
In this paper, the Donnell's equation of equilibrium for curved panels under combined action of normal pressure and shear force was solved by use of the Mathematica software.
The calculated critical shear stress coefficients were compared against the NACA's experimental results. Next, the shear buckling and postbuckling behavior of the perfect curved panels with various curvatures subjected to pure shear forces were investigated via four nonlinear FE analysis methods (i.e., the general static, the arc length, the dynamic explicit, and the dynamic implicit). Results of these analyses were then compared and their advantages and shortcomings were discussed. Subsequently, a set of parametric studies on the laterally pressured curved panels with various curvatures and aspect ratios under gradually increasing in-plane shear was conducted. Both inward and outward pressures were included in these investigations, and the results are presented and discussed.

CRITICAL SHEAR STRESS OF PRESSURED CURVED PANELS
In this section, the modified form of the Donnell's equation of equilibrium was solved for curved panels subjected to uniform lateral outward pressure and uniform shear stress along their four edges and compared against the NACA's experimental results [7]. The Donnell's equation is a partial differential equation for the radial displacement of panel median surface (!), which takes into account the effects of the axial ("), and circumferential (#) displacements. Here, a modified form of the Donnell's equation (Eq. (2)), which was proposed by Batdorf is solved by means of the Galerkin Method [5]. This form of the Donnell's equation is equivalent to its initial form, but has certain advantages in physical interpretations and in ease of solution for different boundary conditions. In the solution of the Donnell's equation, $ % represents the critical shear stress coefficient, and & is a measure of panel curvature given by: in which * is the radius of curvature, + is the panel's thickness, and / is the Poisson's ratio.
Accordingly, the modified form of the Donnell's equation for pressured panels under shear forces is: where 9 represents the applied lateral pressure. Eq. (2) should be solved to find the critical shear stress coefficient ($ % ) as below: where < % is the critical shear stress, and : is the flexural stiffness of the panel.
According to the Galerkin's method, Eq. (2) is solved by expanding the unknown function of ! in terms of a suitable set of functions, each satisfying the simply supported boundary conditions. Curved panels either have long straight sides (= = > ( ≥ 1.0 ⁄ ), or long circumferential sides (= ≤ 1) (see Fig. 1). The radial displacement (!) for simply supported curved panels with axial sides longer than circumferential sides is written in the form of Eq. (4).
where L and K are the number of buckling half-waves in the axial and circumferential directions respectively. According to the Galerkin's method, the coefficients of > DE should be chosen to satisfy the following equation: By substituting Eq. (4) into (5), a set of homogeneous linear algebraic equations is obtained as follows: By assigning different values for L and K, two sets of equations, which are associated with the symmetric and anti -symmetric buckling modes, are established. Critical shear stress of panels with particular aspect ratios (=) and curvature parameters (&) is found via minimizing the value of $ % for which the two sets of equations have a non-vanishing solution.
The above mentioned solution procedure was implemented by use of the Mathematica software [16]. The comparison between this solution and available NACA's experimental results is given in Fig. 2. A very good agreement between the theoretical estimates and experimental results for $ % was observed for both panels under the whole range of applied pressure forces. Though theoretical equations given in this section seem to provide a very reasonable estimate of shear buckling load of curved panels, they do not give any information regarding their postbuckling capacity, which is the subject of next section.

NONLINEAR POSTBUCKLING ANALYSIS METHODOLOGY
Finite elements has proven to be a robust tool for analysis of wide range of engineering problems including buckling and postbuckling of plates and shells and is considered in this study for nonlinear investigation of postbuckling behavior of shear curved panels [17][18][19][20][21][22][23][24]. Details of implemented finite element model is discussed in this section.
Nonlinear FE analyses were carried out using ABAQUS 6.9 package. A 6-degree of freedom, 4-node quadrilateral doubly curved general-purpose shell element (S4R) was used for modeling, which is appropriate for both thin and thick shells. 33.33×33.33 mm mesh size was chosen per recommendations of another past study [25].
The panels were assumed to be made of an elastic-perfectly plastic structural steel were restrained to simulate simply supported boundary conditions. Also, the translational degrees of freedom at one corner node of the panels were restricted to prevent any solid displacements.
The implemented nonlinear analyses were divided into two successive steps. In the first step, the normal acting inward or outward pressure was exerted to the surface of the panel, and analyses were conducted by the use of a general load-control static method. In the second step, in-plane shear forces were applied to the edges of pressured panel; and shear buckling and postbuckling responses was monitored. Low frequency dynamic effects were removed using viscous damping [26,27].
The buckling and postbuckling responses of curved panels frequently involve structural instabilities where the load-displacement path shows a negative stiffness and panels releases strain energy to remain in equilibrium. In such cases, the response of a structural element involves limit-points (maximum and/or minimum loads) rather than bifurcation points, showing both snap-through and snap-back characteristics. Standard static load-control and displacement-control methods are not able to track such behavior and other nonlinear schemes must be used which will be discussed subsequently.

FE analysis method
Various dynamic and static approaches are available for nonlinear analysis of unstable structures [28]. A standard static solution procedure involves load or displacement increments coupled with full or modified Newton -Raphson iterations. When applying such techniques, severe difficulties can be encountered in the vicinity of limit points, where the loaddisplacement response becomes horizontal (or vertical). Thus, any attempt to find the true, complete response of a structure with such behavior, including shear curved panel, via standard methods would either fail, or miss some portion of the response and other numerical methods such as Arc-length must be used.
The Arc -length (Riks) method is intended to enable the solution algorithms to pass the limit points. This method works when the applied loading is proportional; i.e., where the load magnitude is defined by a single scalar parameter. The main essence of the Arc-length method is that the load magnitudes become additional unknowns in the formulations. This approach gives solution regardless of whether the response is stable or unstable [29].
Taking inertia effects into account, dynamic approaches are also applicable to find the load-displacement response of structural elements with unstable response. Although dynamic procedures do not fail in the vicinity of limit points, the true load-displacement response of panels during unstable phases cannot be traced. implicit method, the explicit procedure is conditionally stable. The stability limit for an explicit operator is that the maximum time increment must be less than the critical value of the smallest transition time for a dilatational wave to cross any element in the mesh.
Applicability of the explicit method is limited to the analysis of short transient problems. If this method is used for quasi -static problems, the inertia effects must be small enough to be neglected [30].
In order to compare the mentioned available analytical methodologies, the shear buckling and postbuckling responses of two curved panels were studied via the four

Validation of FEM procedure
In order to validate the nonlinear FE analysis procedure, a set of analyses was The panels were loaded and analyzed in two subsequent steps similar to the actual tests. In the first step, outward pressures were applied to the surface of the panel and a nonlinear general static analysis was carried out. Subsequently, in the second step, shear forces were uniformly applied to the edges and a nonlinear Riks analysis was conducted.
The results are presented in Figs. 4 and 5. Fig. 4 shows the load-displacement response of three studied panels under various lateral pressure loads which was used to estimate their buckling shear stress. These values are compared to those predicted by Donnell equation and also NACA test results in Fig. 5. As it can be seen in Fig. 5

RESULTS AND DISCUSSION
The influence of the lateral pressures on the shear buckling and postbuckling behavior of curved panels is discussed in this section. Four curved panels with various curvature parameters of & = 5, 15, 50 and 150, aspect dimensions of 1000LL × 1000 LL and wall thickness of 3 mm were considered. The influence of both uniform and hydrostatic outward and inward pressures ranging from 1 kPa to 10 kPa was studied. For panels subjected hydrostatic pressure, the loads were linearly distributed from zero to a maximum specified magnitude. Lateral pressures and in-plane shear loads were applied in two subsequent stages; and in each stage, a full nonlinear FE analysis was performed.

Inward pressures
The shear buckling and postbuckling behavior of curved panels in the presence of the inward lateral pressures is presented in Figs. 6 to 12. According to Figs. 6 and 7, the panels which had very low (& = 5) and low (& = 15) curvatures did not exhibit any buckling bifurcation points due to the presence of the inward pressures. As a result, the subsequent post-critical load reductions disappeared and panels became more stable at the buckling load levels. Therefore, a further growth of the shear force was observed beyond the theoretical load levels. In addition, there was no considerable reduction in the ultimate load-bearing capacities. The initial deflections, which are induced by the inward pressure, disturb the membrane action of the panels; thus, no strain energy was conserved before the buckling and the snap-through point was eliminated from the response.
In Fig. 8 the load-displacement response of the moderately curved panel (& = 50) is given. It is observed that for small inward pressures (1 kPa), the panel behaved similar to the case of pure shear; exhibiting snap-through unstable buckling, with a considerable reduction in the buckling load. In contrast, when the inward pressure was increased to 5 kPa or more, the behavior of the moderately curved panel became stable similar to the low curved panels. Fig. 9 shows the out-of-plane deflection contours of the two panels under pure shear loads and a combination of shear and 5kPa inward pressure. Accordingly, the buckling mode shape was changed from two half-waves in the case of the pure shear to one half-wave in the presence of inward pressure.  Fig. 11 compares the buckling mode shapes of the panel in three mentioned loading conditions. As it can be seen, the buckling mode shape of the pressured panels was similar to the case of pure shear. In order to investigate the influence of the aspect ratio on the stability behavior of curved panels, the results of twelve FE analyses under the two different loading conditions (pure shear loads vs. combined shear and hydrostatic inward pressures) are compared in Figs. [13][14][15]. It is observed that the ultimate capacity was considerably decreased with the aspect ratio regardless of the curvature parameter. However, the unstable buckling was clearly evident in all the panels with & = 50, 150.

Outward pressures
The results for panels subjected to outward lateral pressures are given in Figs

Conclusions
The shear buckling and postbuckling behavior of thin curved steel panels subjected to gradually increasing in-plane shear forces in the presence of inward and outward normal pressures were investigated. The main concluding remarks based on the considered cases in this study are as follows: • Amongst the various available nonlinear FE analysis methods, the Arc length Riks method can more realistically trace the complete load-displacement response of curved panels. Both the stable and unstable buckling phenomena, as well as all strain hardening and softening phases, are well predicted by the use of Riks method.
• Small amounts of inward pressures eliminate the snap-through shear buckling and the post unstable buckling softening stages in shallow curved panels. The shear capacity continued to increase after the theoretical critical loads. In moderately curved panels, however, small amounts of pressures did not alter their buckling behavior.
• The magnitude of inward pressures required to influence the shear buckling and postbuckling behavior of curved panels, depends on their curvature. Panels with greater curvature parameter require higher pressures. The buckling mode shapes of curved panels are also considerably affected by the amount of pressures.
• The ultimate shear capacity of highly curved panels is considerably reduced by the increasing presence of inward pressures. The failure modes of such panels are associated with the occurrence of unstable buckling, and the amount of the released strain energy prevents hardening effects.
• Increasing the aspect ratio of curved panels, considerably decreases their ultimate shear capacity. This is regardless of the radius of curvature, although signs of unstable buckling are clearly evident in moderate to highly curved panels.
• In general, outward pressures increase, and inward pressures decrease the shear buckling capacity of curved panels; the magnitude of such changes increase with the curvature parameter. Inward pressures can remove the bifurcation points, and make panels behave like imperfect flat panels. Outward pressures make the buckling phenomenon to be more stable.

General view
In-plane shear forces Outward uniform pressure Inward uniform pressure Outward hydrostatic pressure Outward hydrostatic pressure