Structural analysis by matrix methods for forces not at the coordinates

In this report a critical study is made of the flexibility and stiffness methods for the static calculation of beam like structures. I found that the computation speed and the accuracy of both methods are essentially determined by the number of elements in the structure and the degree of redundancy. Several computer trials have shown that the flexibility method is the most economic and the most accurate for the calculation of structures with a small degree of redundancy. The stiffness method is the most efficient for the calculation of highly redundant structures. Practical values are given enabling one to choose the optimal computation method in each case.

Statically indeterminate structures: by Jack R. Benjamin. 350 pages, diagrams, 6 × 9 in. New York, McGraw-Hill Book Co., Inc., 1959. Price, $11.00

Vehicle‐bridge interaction is the core for a variety of applications, including vehicle vibration, bridge vibration, bridge structural health monitoring, weight‐in‐motion, bridge condition inspection, and load rating. These applications give rise to a great interest in pursuing a high‐efficiency method that can tackle intensive computation in the context of vehicle‐bridge interaction. This paper studies the accuracy and efficiency of discretizing the beam in space as lumped masses using the flexibility method and as finite elements using the stiffness method. Computational complexity analysis is carried out along with a numerical case study to compare the accuracy and efficiency of both methods against the analytical solutions. It is found that both methods result in a similar level of accuracy, but the flexibility method overperforms the stiffness method in terms of computational efficiency. This high efficiency algorithm and corresponding discretization schema are applied to study the dynamics of vehicle‐bridge interaction. A system of coupled equations is solved directly for a simply supported single‐span bridge and a four‐degree‐of‐freedom vehicle modeling. Pavement roughness significantly influences dynamic load coefficient, suggesting preventative maintenance or timely maintenance of pavement surface on a bridge, to reduce pavement roughness, is of significant importance for bridge’s longevity and life‐cycle cost benefit. For class A and B level pavement roughness, the dynamic load coefficient is simulated within 2.0, compatible with specifications of AASHTO standard, Australian standard, and Switzerland standard. However, the Chinese code underestimates the dynamic load coefficient for a bridge with a fundamental frequency of around 4 Hz. The proposed method is applicable to different types of bridges as well as train‐bridge interaction.

The development of the electronic digital computer and the application of matrix algebra has made it possible for the structural engineer to analyze complex or highly redundant structures which previously had been prohibitive to analyze by the hand method of calculation. Papers recently written have utilized either the “flexibility” or “stiffness” matrix approach. It is the purpose of this paper to point out the common origin of these methods in energy and to discuss procedures for calculating the total energy of a structure. The concept of the “total” structure will be considered. Applications include analysis of a truss with elastic supports and a continuous beam.

In this study, a mixed-based approach that combines the classical stiffness and flexibility methods is developed to analyze the rotating dynamic behavior of thin-walled, tapered composite blades with built-in twist angles. The analysis model includes the effects of elastic couplings, shell wall thickness, transverse shear couplings, torsion warping, and constrained warping. The resulting theory describes the beam kinematics in terms of the axial, flap and lag bending, flap and lag shear, torsion, and torsion-warping deformations. The equations of motion for the beam are obtained by using Hamilton's principle. The analysis is validated against experimental test data and detailed finite element analysis results for pretwisted and tapered beams with different material distributions and cross-section shapes. Good correlation is achieved for various configuration cases considered in this study. The influence of pretwist, taper and fiber orientation angles on the structural dynamic response of rotating composite beams are investigated.

On-machine measurement is crucial in industrial production because it allows the measurement of the surface properties of products without having to remove them from the machine tool; however, measurement results are superimposed with the motion error caused by the table traverse. Therefore, eliminating motion errors is this field's main objective. This paper discusses measuring a surface profile using an extension of the displacement method to eliminate motion errors. The displacement method can separate the surface profile and motion error of the stages in measurement results; however, it introduces a specific cumulative error. The tolerance of the accumulated error was first investigated by simulating a one-axis measurement. The displacement method was extended to a two-axis stage to measure the surface profile and correct the stage tilt to each axis. Moreover, carbon steel finished using a face mill was measured in the experiments, and a coordinate measuring machine was used to compare the results with those of the extended displacement method. It was found that the extended displacement method was able to remove the motion error. Applying the displacement method resulted in an improvement of approximately 88.9%. Error analyses were also formulated and evaluated for sensor drift, reference piece measurement, motion error, table rotation, and thermal expansion, and the percentage of these errors in the total was clarified. The results showed that the motion error and table rotation for the y-axis of the prototype experimental apparatus were larger than the others. These errors are discussed to analyze each error and listed in the error budget. The proposed extended displacement method can eliminate motion errors despite the simplicity of the measurement principle and can be easily integrated into machine tools and other tables.

Earlier work on the flexibility analysis of skeletal structures via a combinatorial approach is extended to give a procedure for enforcing compatibility of the nodal displacements of a finite element idealization, which here is called the flexibility method. The procedure relies upon a cycle basis of minimal measure of an associated graph, and upon a distinction between ‘infilling’ and other simple cycles; it also relies upon the enforce­ment of local conditions of compatibility along inter-element boundaries. The paper is intended to give a clear physical description of the procedures involved. A plane strain and a viscous flow example are given.

The direct stiffness method is used to analyze the structural response of the cross section of an arteriole which is both heterogeneous and nonlinear in terms of structural response. A typical segment of the cross section is represented by an assembly of triangular elements. The stiffness matrix and strain-displacement relationships are then derived in terms of polar coordinates for a triangular plate element. Viscoelasticity is included in this structural analysis by using a piecewise linear approach. Stress-time and displacement-time data are presented for two typical cross sections of the arteriole. These results reveal special properties of a heterogeneous structure which is undergoing viscoelastic deformations.

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

The effects of prestressing forces on a reinforced concrete frame are shown in the paper. The structure is a statically indeterminate frame with two fixed supports. The load is due to tendon prestressing with Nk = 600 kN and due to self-weight of columns and beam. The columns height is 6.00 m and the beam span is 20.00 m. Effects of prestressing can be replaced by an external equivalent load. Calculations are performed by the force method, by the direct stiffness method and using the 'Tower' software. The rotational restraints are released at fixed supports and the restraint moments are calculated by the force method. The direct stiffness method uses the members' stiffness relations for computing member forces and displacements in structures. The unknown nodal rotations and translations are obtained by solving a system of equations. The considered tendon profile is without eccentricity above the columns. 'Tower 7' is software for static and dynamic structural analysis and for design of concrete elements. It enables overall structural analysis of plane and three-dimensional structures. Automated, integrated tools allow fast and reliable design or the verification of the results obtained by standard methods. Version 7 is a professional tool with high performance. Three different calculation methods have given close results, with minor differences.

This paper presents a method to define two-dimensional (2D) and three-dimensional (3D) elastic-stiffness matrices for nonprismatic elements (tapered or haunched), based on traditional beam theory and the flexibility method. The proposed formulation includes deformations and the shape of the cross section but neglects warping deformations. Although more rigorous formulations for tapered elements have already been addressed, the proposed procedure is presented so its direct application or implementation in computer programs for structural analysis is straightforward. The procedure is compared against the design tables of the Portland Cement Association (PCA). It is demonstrated that the PCA tables are obsolete for today's state-of-the-knowledge on nonprismatic members because they can lead to significant errors. A new set of design aids for most common cross sections used in building structures have been developed to substitute the PCA handbook of frame constants. Closed-form solutions for linearly tapered elements of rectangular, square, and circular cross sections are provided. Finally, it is demonstrated that the stiffness factors for nonprismatic elements depend on the span-to-depth ratio of the element ( L /h). To the writer's knowledge, no one has proved this fact before.

Metamaterials with thermal contraction and auxetic effect can effectively prevent irreversible damage caused by thermal stress and impact in engineering applications, playing a crucial role in aerospace and other fields. Therefore, we propose a novel double-negative metamaterial structure by combining triangular arrowhead structures and triangular thermal metamaterial structures, which can achieve thermal expansion coefficient over a wide range from -167×10-6 to 162×10-6 K-1, and Poisson's ratios range from -1.7 to 1.95. Furthermore, we study the impact of the geometric parameters of the designed structures on the coefficient of thermal expansion (CTE), elastic modulus, and Poisson's ratio of the metamaterial by using the stiffness matrix method (SMM). Results show that changing the included angle of the re-entrant triangle of the structure can change the sign of thermal expansion coefficient and Poisson's ratio. This paper provides new ideas and methods for designing mechanical metamaterials with anisotropic adjustable thermal expansion coefficients and Poisson's ratios.

Fiber optic sensors have the capability to simultaneously measure multi-scale data (e.g. strain, strain gradients, and integrated strains) for the purpose of structural health monitoring of structures. Presented here is a technique that can detect and localize damage in a structure suitable for the fusion of such multi-scale data. The flexibility method has been previously proposed for damage localization using changes in natural frequencies and mode shapes. However, such changes are difficult to apply in a real world application due to the extreme accuracy required for the input excitation frequencies. In addition, computer modeling of the undamaged structure to compare with the damaged structure can lead to significant errors due to imperfections. The flexibility method for solving indeterminate structures using static conditions presented here avoids these difficulties. The method follows a procedure of applying a known load to determine the flexibility matrices of the structure in pre- and post- damaged states. The introduction of fiber optic strain and displacement measurements permits the calculation of these matrices from the static loading conditions. These matrices are subtracted and the resulting null space calculated. When reapplied to the original system this null space locates the damage. To demonstrate the validity of this process numerical models are created for a simple truss structure and a plate instrumented with several fiber optic sensors. The examples show that damage can be located in both cases.

A beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamic analysis of reinforced concrete (RC) structural element. Stiffness matrix of this element is obtained by using rigidity method. A solution technique that included nonlinear dynamic substructure procedure is developed for dynamic analyses of RC frames. A predicted-corrected form of the Bossak-α method is applied for dynamic integration scheme. A comparison of experimental data of a RC column element with numerical results, obtained from proposed solution technique, is studied for verification the numerical solutions. Furthermore, nonlinear cyclic analysis results of a portal reinforced concrete frame are achieved for comparing the proposed solution technique with Fibre element, based on flexibility method. However, seismic damage analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumped/distributed mass and load. Damage region, propagation, and intensities according to both approaches are researched.