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Frame structures are widely used in engineering practice. They are likely to lose their stability before damage. As an indicator of load-carrying capability, the first critical load plays a crucial role in the design of such structures. In this paper, a new method of identifying this critical load is presented, based on the governing equations in rate form. With the presented method, a great deal of well-developed numerical methods for ordinary differential equations can be used. As accurate structural tangent stiffness matrices are essential to stability analysis, the method to obtain them systematically is discussed. To improve the computational efficiency of nonlinear stability analysis in large-scale frame structures, the corotational substructure elements are formulated as well to reduce the dimension of the governing equations. Four examples are studied to illustrate the validity and efficiency of the presented method.

Frame structures are widely used in engineering practice and are progressively optimized towards sparsity and slenderness. These frame structures often show strong nonlinear behaviors after carrying heavy loads. Instability is one of the causal factors for damages and injuries, which results in permanent disability. In general, the more slender such structures become, the more likely they are to lose stability and even result in damage. To improve the stable load-carrying capacity, a simplified and efficient computational method for determining the first critical loads is required for the structural design, application, and safety of these structures.

In the stability analysis of structures, the load on the structure is usually assumed to be varied in a proportional manner; that is, it can be represented by a single load-control parameter. In this case, identifying critical loads requires the solution of the equilibrium equations under the variation of the load-control parameter and the detection of critical points where the tangent stiffness matrix becomes singular. Generally, the critical points are detected by two main processes called direct and indirect methods [

In the direct method, the equilibrium equations are augmented with a criticality condition [

Both of the direct and indirect methods need the tangent stiffness matrix which is not easy to obtain when the effect of geometric nonlinearity is considered. Such difficulties can be avoided by the Dynamic Relaxation (DR) method, in which the static system is treated as a fictitious vibration system through adding fictitious inertia and damping forces to the static equilibrium equations [

A proper and efficient finite element model is necessary for describing the geometrical nonlinear behaviors of the frame structures. Three kinematic descriptions are used for the finite element analysis of nonlinear structural mechanics: the total Lagrangian formulations, the updated Lagrangian formulations [

However, when the corotational formulation is adopted for modeling the large-scale frame structures, all components in these structures should be treated as corotational elements, which will eventually lead to a set of high-dimension nonlinear equilibrium equations to be solved. To reduce the dimension of nonlinear equilibrium equations, the large-scale structure can be divided into several substructures [

The paper is organized as follows. In Section

In general, the nonlinear finite element analysis of structures leads to a set of nonlinear algebraic equilibrium equations, which yields

Governed by (

Theoretically, (

As an indicator of load-carrying capacity, the critical load associated with the first limit or bifurcation point on the equilibrium path plays a crucial role in the structural design. In this paper, we focus on the method for identifying such critical loads and suggest using (

From a mathematical point of view, following the equilibrium paths is the same thing as describing the history of the solutions varying with a parameter, which can be achieved better by the differential equations with the same solutions [

Another benefit of such a transformation is the facilitation for identifying the critical loads. As is well known, at limit points on the equilibrium path, the derivatives of displacement with respect to the load-control parameter are infinite, as shown in Figure

(a) Limit points on the equilibrium path; (b) bifurcation point on the equilibrium path.

In this case, if the right side of (

In engineering practice, when the point in question is to find the critical load, the types of critical points are less important, and we can identify the critical load by monitoring the rapid changes in the derivatives of displacement

It is worth noting that there is always the danger of omitting the local buckling when finite element methods are applied in the critical load analysis. This is chiefly because any finite element model of structures includes the displacement of only part of the points in the structure. Assuming some new points between the element nodes are added in the model as shown in Figure

Local buckling of a frame.

In a frame structure, the shorter a beam element is, the greater the required force for the local buckling would be. So the elements should be divided fine enough in order to prevent the local buckling force from being smaller than the critical load.

From the above discussion, it can be seen that the tangent stiffness matrix

In general, the maximum load sustained by a frame structure can be determined by two specific loads. The first one, called the failure load, is the load that can make some components broken. The second one, called the critical buckling load, is the load that can turn the structure from stable to unstable states while structural strains remain small. The latter is more important to structural design. If this load is less than the failure load, the structure would be not optimal. Consequently, we can assume that the critical buckling load is mainly caused by geometric nonlinearity. In this case, the corotational formulation is more desirable among many methods of representing the influence of geometric nonlinearity in virtue of its acceptance of abundant linear elements [

The corotational formulation in geometrical nonlinear problems is a well-known technique. Its most important advantage is that it leads to artificial separation of the geometric nonlinearity, and a linear strain definition in the local coordinate system is used [

A local coordinate system of a substructure element can be established with the positions of three noncollinear points, denoted by I, II, and III, respectively, as shown in Figure

Local coordinate systems for (a) a substructure element and (b) a beam element.

Its origin is placed at point I and its base vectors are defined by

From (

The local coordinate system defined in this manner is element-independent in the sense that it applies to substructure elements, so long as the displacement relative to the local coordinate system is small enough.

In the local coordinate system, a node

Corotational kinematic descriptions for (a) a substructure element and (b) a beam element.

In accordance with (

Another way to describe the orientation matrix

From (

When the relative displacement and rotations of element nodes are all small enough in a substructure, we can attach a local coordinate system to this substructure and treat the whole substructure as a single generalized element. In this case, (

The angular velocity of a rotation can be defined by the following equation:

According to (

According to (

In the case that the substructure is actually a beam element, the third point in the definition of the local coordinate system is located on the left end-section of the beam, so

Moreover, the relationships described by (

As rigid motions play no role in internal forces, the virtual power of internal forces for a substructure

In general, the external forces of substructures are composed of surface and body forces. Concentrated forces are regarded as the specific surface forces, and in most cases, body forces are gravity forces. Their contributions to the virtual power of external forces, denoted by

By means of (

Some insights can be revealed by the characteristics of the nodes in a substructure. The nodes in a substructure

Three groups of nodes in the substructure

Local displacement and rotational vectors of the nodes in the substructure

In expanded form, (

Substituting (

In the case that the substructure

Substituting (

Formulations of the tangent stiffness matrices can be obtained by taking the derivatives of the generalized internal nodal forces and torques in (

By the definitions of

In accordance with (

In this section, four examples are studied to illustrate the validity and efficiency of the proposed method. Except for Section

The first example is a slender beam clamped at its bottom and subjected to an axial force

(a) Axial compression beam and (b) its model with

According to Euler’s formula of fixed-free beam under axial compression, the critical load of this simple structure is given by

Load-displacement curves of the nodes on the beam when the beam is modeled by different numbers of elements.

The critical load can be detected by the maximum norm of the displacement derivatives with respect to axial force

The comparison between the first Euler buckling load and the critical loads obtained from presented method.

Euler buckling load | The number of elements along its axial direction | ||||
---|---|---|---|---|---|

2 | 4 | 8 | 16 | ||

Critical load (kN) | 139.953 | 147.316 | 141.791 | 140.434 | 140.096 |

Relative error (%) | — | 5.26 | 1.31 | 0.34 | 0.10 |

It can be seen that, with the increasing number of divided elements, the relative error becomes smaller, and once the number of elements exceeds four, the relative error

However, Euler buckling load is not the analytical solution, because the effect of axial displacement is ignored. In this example, the axial displacement of the beam top is near

This example also indicates that the presented method can identify the critical load with satisfying accuracy and the relative errors are decreased with the number of elements increasing.

The star-shaped shallow dome has been studied by a number of researchers as a benchmark problem due to its obvious geometrical nonlinear characteristics before collapse [

Geometry of 24-member star-shaped shallow dome.

By using

Load-displacement curves of the node

In this example, the derivative

The comparison of the first critical loads from the presented method and Meek and Xue [

Meek and Xue [ |
The number of elements per member | ||||
---|---|---|---|---|---|

2 | 4 | 8 | 16 | ||

Critical load (N) | 561.4 | 626.1 | 582.9 | 571.9 | 569.3 |

Relative error (%) | — | 11.52 | 3.83 | 1.87 | 1.41 |

This example indicates that the presented method can also adapt to the space frame with limit point, and the relative errors are decreased with the number of elements increasing.

As shown in Figure

Frame with a subsidiary beam.

A small perturbation moment of −1 N·m is applied at the middle point of its top line, as shown in Figure

It can be easily predicated that the subsidiary beam

Local buckling of the frame.

When the subsidiary beam

Load-displacement curve of

It is predicted by this model that the critical load is 3514.966 kN, and 1372.245 kN is carried by the subsidiary beam

When all beams are divided into twenty elements, the corresponding equilibrium paths of

Load-displacement curves of

The results of this example imply that each component in a structure has to be modeled by enough elements to capture the possible local buckling and identify the critical load precisely.

The forth example is a slender space frame structure clamped at its bottom and carrying axial load ^{3} kg/m^{3}. The gravity acceleration is 9.807 m/s^{2}, which is parallel to the negative

(a) Slender space frame structure carrying an axial load; (b) geometric parameters of the periodic insert section.

Along the longitudinal direction, each insert section is modeled as one corotational substructure element in the presented method to reduce the dimension of the governing equations, as shown in Figure

(a) Corotational substructure element of an insert section; (b) substructure model of the slender frame structure.

For all boundary nodes displacement values, the absolute values of all components of the vector

Under the first critical load, contour plot of the displacement vector sum of all nodal solutions is shown in Figure

Contour plots displacement vector sum of nodal solution of the slender frame structure.

The structure is studied with ANSYS software, and Beam188 element is used for modeling each member of the frame structure in the ANSYS model. NLGEOM command in ANSYS software is selected to consider geometrical nonlinear behavior of the structure, and the number of loading substeps is chosen as 45. Load-displacement curves of the point

Load-displacement curves of the point

In the ANSYS model, 1680 nonlinear equations are solved to identify the first critical load. Only 264 nonlinear equations are required by using the corotational substructure element formulation in the presented method. The corotational substructure element formulation improves the numerical efficiency by means of reducing the degrees of freedom. Meanwhile, the accuracy of the presented procedure is acceptable while comparing with the ANSYS results. The unknowns in the governing equations are greatly reduced with the proposed method.

As an indicator of load-carrying capability, the first critical load plays a crucial role in the structural design. This critical load can be identified by the path following method presented in this paper on the basis of governing equations in rate form. In doing so, the well-developed numerical methods for ordinary differential equations can be used to facilitate the structural stability analysis. Considering the geometrical nonlinear effects of large-scale frame structures, a corotational formulation for substructure elements is presented, by which the number of the governing equations can be greatly reduced. The accurate structural tangent stiffness matrices required in the method can be obtained systematically through the proposed element-independent corotational formulation. The presented method can be extended to the fields of mechanical and civil engineering, and so forth, such as the slender lattice boom and tower and jib structures of cranes and derricks.

Base vectors of the global coordinate system

Global position vectors of three noncollinear points I, II, and III for defining the substructure local coordinate system

Base vectors of the substructure local coordinate system

3 × 3 identity and zero matrices

Rotational matrix and angular velocity of the substructure local coordinate system with respect to the global coordinate system

Local and global initial position vectors of a node

Local and global current position vectors of a node

Local and global translational displacement vectors of a node

Local and global rotational vectors of a node

Local and global rotational matrix of a node

Local and global angular velocities of a node

A substructure in the frame system

Group of the three nodes I, II, and III for defining the local coordinate system of substructure

Group of the interface nodes of substructure

Group of the boundary nodes of substructure

Group of the interior nodes of substructure

Group of all nodes of substructure

Gravity acceleration in the global coordinate system

Stiffness and gravity influence coefficient matrices of substructure

Condensation stiffness and gravity influence coefficient matrices of substructure

Generalized internal forces and torques in the local coordinate system of substructure

Generalized internal forces and torques in the global coordinate system of substructure

Position of the mass center in the local coordinate system

Mass density of the frame structures.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the China National Science Foundation under Grant no. 11372057.