Second-order Stiffness Matrix Research Articles

Optimal Design of Crossbeam Stiffness Factor in Bridge Towers Using a Reliability-Based Approach

Optimal design of the crossbeam is essential for the economical design of bridge towers as the crossbeam could considerably enhance the lateral stiffnesses of these towers by providing a special bracing for the tower columns. By using a reliability-based approach, this paper studies the optimal design of the crossbeam stiffness factor in bridge towers; this is defined as a dimensionless crossbeam stiffness relative to the tower column stiffness. A novel second-order matrix stiffness method (MSM) is applied to obtain a closed-form solution of the lateral stiffness of the bridge tower. The structural second-order stiffness matrix consists of combinations of the second-order element stiffness matrices and coordinate transformations. Subsequently, a reliability analysis to study the optimal design of the bridge tower is performed by considering the uncertainties arising from the design and construction of the bridge tower. The lateral stiffness of the bridge tower is set as an objective function while the total usage of materials is set as a constraint condition. Then, the influence of the crossbeam stiffness factor on the lateral stiffness of the bridge tower, including the fragility curve and the probabilistic behavior, is examined. Based on the reliability analysis, optimal design recommendations on the crossbeam stiffness of the bridge tower are presented.

Exact Solutions for Torsion and Warping of Axial-Loaded Beam-Columns Based on Matrix Stiffness Method.

The typically-used element torsional stiffness GJ/L (where G is the shear modulus, J the St. Venant torsion constant, and L the element length) may severely underestimate the torsional stiffness of thin-walled nanostructural members, due to neglecting element warping deformations. In order to investigate the exact element torsional stiffness considering warping deformations, this paper presents a matrix stiffness method for the torsion and warping analysis of beam-columns. The equilibrium analysis of an axial-loaded torsion member is conducted, and the torsion-warping problem is solved based on a general solution of the established governing differential equation for the angle of twist. A dimensionless factor is defined to consider the effect of axial force and St. Venant torsion. The exact element stiffness matrix governing the relationship between the element-end torsion/warping deformations (angle and rate of twist) and the corresponding stress resultants (torque and bimoment) is derived based on a matrix formulation. Based on the matrix stiffness method, the exact element torsional stiffness considering the interaction of torsion and warping is derived for three typical element-end warping conditions. Then, the exact element second-order stiffness matrix of three-dimensional beam-columns is further assembled. Some classical torsion-warping problems are analyzed to demonstrate the established matrix stiffness method.

Second-order Stiffness Matrix and Loading Vector of a Tapered Rectangular Timoshenko Beam-column With Semirigid Connections

The second-order stiffness matrix and loading vector of a linearly tapered Timoshenko beam-column with rectangular cross-section, constant width and generalized end conditions subject to a constant axial load is derived. The proposed model includes the simultaneous effects of: (1) axial load (tension or compression) applied at both ends; (2) transverse load; (3) generalized end conditions; (4) bending and shear deformation along the member; (5) shear stress correction factor induced by the tapering of the cross section; and (6) the shear force induced by the applied axial load as the member deflects laterally using two different approaches, those by Engesser and Haringx which are proportional to the total slope (dy/dx) and to the angle of rotation (θ) along the span of the member, respectively. The governing differential equations are solved by the power series method. The concept of fixity factor of tapered Timoshenko beam-columns with semirigid end connections is introduced. The proposed method and corresponding equation are capable of determining the static and stability behavior of elastic framed structures made of tapered beam-columns with semi-rigid connections using a single segment per element. Three comprehensive examples are presented that show the accuracy and validity of the proposed method and the calculated results are compared with those obtained using ABAQUS.

Second-order stiffness matrix and load vector of an imperfect beam-column with generalized end conditions on a two-parameter elastic foundation

The stiffness matrix and load vector for an imperfect Euler–Bernoulli beam-column with generalized end conditions subjected to axial and transverse loads are presented. The proposed method includes the effects of initial imperfections (i.e., out-of-straightness, out-of-plumbness, and axial load eccentricities at both ends), a two-parameter elastic foundation, partially restrained sidesway and rotational semirigid connections at both ends, and transverse and end axial loads (tension or compression) on the stiffness matrix and load vector. The proposed method is capable of solving the second-order response and lateral stability, and capturing the phenomenon of deflection reversals in 2D framed structures by using a single segment per element. The effects of shear deformations and torsion along the member are not included in the present research. Three comprehensive examples are provided to show the effectiveness and validity of the proposed matrix method.

Nonlinear Finite Element Analysis of Steel Frame with Semi-Rigid Connections

In conventional analysis and design, the beam-column connections are divided into two categories: fully rigid joints and perfectly pinned joints. Actually, most beam-column connections exhibit flexibility. Idealized connections may diverge from the real reaction of structures. This article presents a hybrid finite-element model that consists of a beam element with two semi-rigid connections at the ends for studying the effect of the flexibility on the structures. Based on the theory of finite deformation, the first-order and the second-order stiffness matrix and the equilibrium equation of hybrid beam element were got. A nonlinear finite element analysis program was developed to analysis the steel frames with semi-rigid connections. At last, one-bay, two-story semi-rigid planar steel frame was analyzed using the program, and some conclusions about the effect of semi-rigid connections flexibility were obtained.

Matrix method for stability and second-order analysis of Timoshenko beam-column structures with semi-rigid connections

The first- and second-order stiffness and load matrices of a beam-column of symmetric cross section with semi-rigid connections including the effects of end axial loads (tension or compression) and shear deformations along the member are derived in a classical manner. Both matrices can be used in the stability, first- and the second-order elastic analyses of framed structures made of Timoshenko beam-columns with rigid, semi-rigid and simple connections of symmetric cross sections. The “modified” stability approach based on Haringx’s model described by Timoshenko and Gere [1] is utilized in all matrices. The model proposed which is an extension of that presented by Aristizabal-Ochoa [2] captures the models of beams and beam-columns based on the theories of Bernoulli–Euler, Timoshenko, and bending and shear. The closed-form second-order stiffness matrix and load vector derived and presented in this paper find great applications in the stability and second-order analyses of structures made of beam-columns with relatively low shear stiffness such as orthotropic composite polymers (FRP) and multilayer elastomeric bearings commonly used in seismic isolation of buildings. The effects of torsional warping along the members are not included. Analytical studies indicate that the buckling load and the stiffness of framed structures are reduced by the shear deformations along the members. In addition, the phenomenon of buckling under axial tension forces in members with relatively low shear stiffness is captured by the proposed equations. Tension buckling must not be ignored in the stability analysis of beam-columns with shear stiffness GA s of the same order of magnitude as EI/ h 2. The validity of both matrices is verified against available solutions of stability analysis and nonlinear geometric elastic behavior of framed structures with semi-rigid connections using a single segment for each beam and column member without introducing additional degrees of freedom. Four examples are included that demonstrate the simplicity, effectiveness and accuracy of the proposed method and corresponding matrices.

Stability analysis of hyper symmetric skeletal structures using group theory

The main objective of this article is to develop a methodology for the efficient calculation of buckling loads for frame structures having high-order symmetry properties in order to reduce the size of their associated eigenvalue problems. This is achieved by decomposing the second-order stiffness matrix of a symmetric model into submatrices using a representation of its symmetry group, via a step-by-step approach. The physical interpretation of the resulting submatrices is shown as substructures (factors), and the possibility of further decomposition is then investigated for each of the constructed submodels. Due to the similarity in transformation, the constructed submatrices contain the eigenvalues of the main structural matrix. The buckling load of the entire structure is obtained by calculating the buckling loads of its factors. The methods of the present paper provide a mathematical foundation and a logical means to deal with symmetry instead of looking for various boundary conditions to be imposed for symmetric structures, as in the traditional methods. Examples are provided to illustrate the simplicity and efficiency of the present method.

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.

Consideration of out-of-plane buckling in advanced analysis for planar steel frame design

A technique that may be used to augment most in-plane advanced analysis procedures to account for inelastic out-of-plane flexural–torsional buckling in the design of planar steel frames is presented. The linear stability theory is adopted and the finite element analysis method is used to derive the second-order stiffness matrix. The effects of material nonlinearities and geometric imperfections on out-of-plane buckling strength are accounted for by substituting the elastic rigidities ( EI y , EC w and GJ) with their effective values, which are determined from calibrations against specification equations for member strength. Based on a set of attributes found in typical building frames, it is demonstrated that out-of-plane buckling is likely to govern the strength of nonsway frames and may control the strength of sway frames. Thus, the proposed advanced analysis technique is recommended for the design of both types of planar steel frames.

Geometrically Nonlinear Formulations of Beams in Flexible Multibody Dynamics

In this paper, the equations of motion of flexible multibody systems are derived using a nonlinear formulation which retains the second-order terms in the strain-displacement relationship. The strain energy function used in this investigation leads to the definition of three stiffness matrices and a vector of nonlinear elastic forces. The first matrix is the constant conventional stiffness matrix; the second one is the first-order geometric stiffness matrix; and the third is a second-order stiffness matrix. It is demonstrated in this investigation that accurate representation of the axial displacement due to the foreshortening effect requires the use of large number or special axial shape functions if the nonlinear stiffness matrices are used. An alternative solution to this problem, however, is to write the equations of motion in terms of the axial coordinate along the deformed (instead of undeformed) axis. The use of this representation yields a constant stiffness matrix even if higher order terms are retained in the strain energy expression. The numerical results presented in this paper demonstrate that the proposed new approach is nearly as computationally efficient as the linear formulation. Furthermore, the proposed formulation takes into consideration the effect of all the geometric elastic nonlinearities on the bending displacement without the need to include high frequency axial modes of vibration.