This series includes technical reports prepared by faculty, students and staff who are associated with the John A. Blume Earthquake Engineering Center at Stanford University. While the primary focus of Blume Center is earthquake engineering, many of the reports in this series encompass broader topics in structural engineering and materials, computational mechanics, geomechanics, structural health monitoring, and engineering life-cycle risk assessment. Each report includes acknowledgments of the specific sponsors for the report and underlying research. In addition to providing research support, the Blume Center provides administrative support for maintaining and disseminating the technical reports. For more information about the Blume Center and its activities, see https://blume.stanford.edu.
In analyses of plane or space frames by the stiffness method on a digital computer. it is common practice to include axial as well as flexural and torsional strains. In some cases. however. it is desirable to omit the axial strains while retaining the other two types. for rectangular frames this can be done rather easily, but for frames of general geometry the task is much more complicated. Assigning large axial stiffnesses for these members is an artifice that can be used in certain cases. However, mixing large axial stiffnesses with relatively small flexural and torsional stiffnesses can cause a significant loss of accuracy in the calculations .
Axial constraints can be imposed by specifying zero length change for each member within a frame. Such constraints involve joint translations only. The joint rotations are not involved directly in the constraints equations, and therefore can be eliminated from the set of joint equilibrium equations for both static and dynamic analyses. By using axial constraint conditions. we can solve for some of the unknown (dependent) joint translations in terms of the remaining (independent) translations. The number of dependent translations is equal to the rank of the coefficient matrix for the constraint equations. After the number of dependent translations is found, the equilibrium equations can be reduced further, resulting in a set which can be solved for the independent translations.
Four types of constrained structures can be identified: a) underconstrained b) fully constrained c) overconstrained and d) general. The classification of a frame to a group depends on the rank of the constraint matrix and the connectivity of the structure. For the first two types the rank of the constraint matrix is equal to the number of constrained members. Frames for which the number of constraints is larger than the rank are of either type c or d. A unique solution for the unknown constraint forces can be found for types a and b, which do not contain redundant constraints. However, we can also identify frames of type c or d and determine which members produce the redundant constraints.
For static analysis we can obtain a reduced set of equilibrium equations that involve only the independent joint translations and the axial forces in the constrained members. It is possible to solve directly for the translations from a subset of the equations. Substitution of these values into the remaining equations will produce the constraint forces. Using appropriate back-substitutions, we can find the dependent translations and the joint rotations. Internal actions (other than axial forces) for each member are then calculated using standard procedures of the stiffness method.
For dynamic analysis, we can obtain a reduced set of equations of motion that include inertial forces corresponding to the independent joint translations. These equations are solved for the dynamic response of the independent translations due to various excitations, using the normal-mode method of analysis. Calculation of other quantities proceeds as for static analysis.
Computer programs have been developed for the static and dynamic analyses of plane and space frames. Examples illustrating the use of these programs are also provided. Comparative studies show that omitting the axial strains by the procedures described herein produces results that are more accurate and efficient than those obtained by other means.
Eisenberg, M and Weaver, W. (2013). Static and Dynamic Analysis of Plane and Space Frames. John A. Blume Earthquake Engineering Technical Report 48, Stanford Digital Repository. Available at: http://purl.stanford.edu/jw937tf0340
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