Second-order Stiffness Research Articles

Exact Second-Order Stiffness Matrix of Axial-Loaded Beam–Columns on Winkler Foundation

This paper establishes the exact second-order stiffness matrix of axial-loaded beam–columns on Winkler foundation. This Winkler foundation model has been widely used in engineering applications. The equilibrium analysis of an axial-loaded beam–column on Winkler foundation is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. Two dimensionless factors for the axial force and foundation spring stiffness are defined, respectively. The exact second-order element stiffness matrix of the axial-loaded beam–column on Winkler foundation is then formulated, showing the relationship between the element-end deformations (translation and rotation angle) and the corresponding forces (shear force and bending moment). Such an exact second-order stiffness matrix may be used for the buckling and second-order solutions with allowance for the use of one element per member for the exact solution. The classical buckling problem of axial-loaded beam–columns on Winkler foundation with typical element-end boundary conditions is then analyzed by the developed matrix stiffness method. By comparing with classical buckling criterion expressions of axial-loaded beam–columns on Winkler foundation with various boundary conditions, it is demonstrated that the derived exact second-order stiffness matrix can be used to obtain the exact buckling solutions systematically.

A new mesh-type rail pad with second-order stiffness

In this paper, a novel mesh-type rail pad with second-order stiffness (MTRPSOS) is proposed, which is comprised of a mesh-type rail pad (MTRP) and multiple filled blocks. By adjusting the height of the filled blocks, the stiffness curves of the MTRPSOS exhibit pronounced second-order stiffness (SOS) characteristics. A finite element (FE) model of the MTRPSOS and a dynamic model were established to investigate the SOS characteristics and their impact on various dynamic indices. The FE calculations reveal that both static and dynamic stiffness of the MTRPSOS increase as the height of the filled blocks decreases. Moreover, as the height of the filled blocks increases, stress distribution becomes more uniform. The dynamic calculations demonstrate that the SOS characteristics of the MTRPSOS significantly affect various dynamic indices. As the SOS of the MTRPSOS decreases, rail displacement correspondingly increases, while vibration acceleration, wheel-rail forces, fastener reaction forces, and derailment coefficient all decrease. This indicates that the MTRPSOS offers superior vibration reduction under the SOS conditions compared to the first-order stiffness conditions. Additionally, a comparative study was conducted to assess the vibration reduction effect of the MTRPSOS in contrast to traditional rail pad, and the results show that the MTRPSOS consistently exhibits lower vibration levels under the same operating conditions, underscoring its superior capacity for vibration reduction.

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.

Technical Note: Internal Second-Order Stiffness: A Refined Approach to the Rm Coefficient to Account for the Influence of P-? on P-?

One component of the B2 amplifier method of addressing second-order effects is the RM coefficient, which represents the influence of P-? on P-? effects. This paper presents the background for RM based on LeMessurier’s paper, “A Practical Method of Second-Order Analysis: Part 2—Rigid Frames,” (1977), and makes explicit the simplifications entailed in the AISC Specification for Structural Steel Buildings (AISC, 2016b) formulation for this coefficient. These simplifications, while providing for reliable strength design, can overestimate the P-? effect for typical building applications, especially if applied to drift. A simple formula for RM based on the work of LeMessurier permits a more precise estimate, which can be used as a component of both force and displacement amplifiers presented in this paper. This explicit approach to the RM coefficient provides the basis for clear presentation of the relationships first-order and second-order stiffness (both internal and external), including the distinct effects of P-? and P-? stiffness reductions on equilibrium at the second-order displacement.

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.

A DISCRETIZATION OF TAPERED BEAMS UP TO THE SECOND-ORDER NONLINEARITY

It is quite complicated for the linear differential equation of beam deflection to be expanded as to varying cross section. But, once a tapered beam element is solved, its nodal stiffness relations can be adopted into a general discrete analysis of framed structures. In this paper, the method of separation into rigid displacement and deformation, which has been developed in the geometrically nonlinear analysis, is found to have a fitness for dealing with the varying beam elements; and typical two types of tapered 2-D beams are discretized from their linear solutions into the geometric and second-order stiffness relations.

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.

First- and Second-Order Stiffness Matrices and Load Vector of Beam-Columns with Semirigid Connections

The firstand second-order stiffness matrices and load vector of a prismatic beam-column of double symmetrical cross section with semirigid end connections including the effects of end axial loads (tension or compression) and shear deformations are derived in a classical manner. The derived matrices can be used in the stability, firstand second-order elastic analyses of framed structures with rigid, semirigid, and simple connections. The classical stability functions are utilized in the stiffness matrix and in the load vector. The proposed stiffness matrices can also be utilized in the inelastic analysis of frames whose members suffer from flexural degradation or, on the contrary, stiffening at their end connections. The validity of both matrices is verified against available solutions of stability analysis and nonlinear geometric elastic analysis of framed structures. Three examples are included that demonstrate the effectiveness of the proposed matrices.

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.

Second-order tapered beam-column elements

The second-order finite element formulations of linearly tapered beam-column elements are derived. Different tapering types such as width changing only, height changing only, width and height changing at the same rate and different rates for various section shapes are incorporated into the formulations. The stiffness matrices are formed using the shape functions considering various tapering types. The derived second-order stiffness matrices are the simple addition of the small-displacement stiffness matrices and a few other component matrices. The small-displacement stiffness matrices are derived into the closed form and other component matrices are formed by the Gauss numerical integration method. The standard finite element approach presented in this paper makes the beam-column elements compatible with other finite elements and can be easily incorporated into the existing finite element programs. The application of the developed tapered beam elements is illustrated with an example of a simply supported beam with axial restraints.

Temperature dependences of the third-order elastic constants and acoustic mode vibrational anharmonicity of vitreous silica

Measurements of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components (TOEC) of vitreous silica between 77 and 293 K. In the absence of such data previous practice has been to assume that these TOEC are independent of temperature; however a significant temperature dependence has been observed. With the exception of the smallest C 456 , each of the TOEC is anomalously positive: a feature consistent with the well established aberrant negative values for the hydrostatic pressure derivatives (∂ C ij /α P ) P =0 of the second-order elastic stiffness tensor components (which reveal softening of the long-wave-length acoustic modes under pressure). The TOEC increase as the temperature is reduced showing that the pressure-induced acoustic mode softening becomes enhanced at lower temperatures. The largest TOEC is C 111 which is 46.7 × 10 10 Pa at 293 K and increases to 99.1 × 10 10 Pa at 77 K. The hydrostatic pressure derivatives of the second-order stiffnesses have been calculated from the TOEC; (∂ C 11 /∂ P ) p =0 is much larger than (∂ C 44 /∂ P ) P =0 over the whole temperature range: the longitudinal acoustic mode softens more with pressure than the shear mode. This effect is emphasised by the finding that the acoustic mode Gruneisen parameters are negative with |γ L | > |γ S |. As the temperature is reduced the longitudinal acoustic mode Gruneisen parameter, γ L , increases in magnitude considerably, reaching a value of −5.5 at 77 K. The mean acoustic mode Gruneisen parameter, γ el , increases to a larger negative value (−3.1 at 77 K) than previously suspected on the basis of the assumption of temperature-independent third-order elastic constants: the discrepancy between the low-temperature limits of the thermal, γ th L , and mean acoustic mode, γ el L , Gruneisen parameters is not quite so large as had been thought. The vibrational anharmonicity of the acoustic modes plays an important part in causing the thermal expansion of vitreous silica to be negative at low temperatures.

Pressure and temperature dependence of the elastic properties of synthetic MnO

The adiabatic elastic stiffness constants of synthetic single-crystal MnO were measured in this study using pulse superposition interferometry. Data were obtained up to 1.0 GPa in pressure and over the temperature range 273 to 473 K. As a result, we were able to determine the complete set of second-order stiffness moduli (C ij s ) and their pressure and temperature derivatives, as well as higher-order properties for selected modes. Relevant results for the adiabatic bulk modulus are: Ks=155.1±0.8 GPa; (∂Ks/∂P)T=4.70±0.13; and, (∂Ks/∂T)P= -0.0203±0.0009 GPa/K. Our results for the second-order moduli are generally consistent with the data from previous studies. However, relative to the estimated uncertainties, small and systematic discrepancies appear to characterize the data set. The available evidence indicates that the differences result from microstructural variations (in particular, microcracks and Mn3O4 inclusions) between the synthetic MnO specimens used in different investigations. The pure shear mode C44 exhibits anomalous soft-mode behavior with both temperature (the ambient derivative is positive) and pressure (the ambient derivative is negative). In both cases the C44 data trends appear to primarily reflect the influence of Mn-Mn magnetoelastic interactions associated with the onset of a paramagnetic-antiferromagnetic (PM → AFM) phase transition.