# The Plastic Methods of Structural Analysis (Science Paperbacks)

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Matrix methods of structural analysis. Theory of Matrix Structural Analysis. Matrix Analysis Of Structural Dynamics. Advanced Methods of Structural Analysis.

Matrix Structural Analysis, Second Edition. Matrix Methods in Analysis. Structural Mechanics: Graph and Matrix Methods. Optical Methods of Engineering Analysis. Lucent Library of Science and Technology.

Methods of Matrix Algebra. Handbook of Food Science Technology and Engineering. Methods of matrix algebra. International Encyclopedia of Information and Library Science,. Matrix Analysis of Structures. Once the dimensional requirement for a structure have been defined, it becomes necessary to determine the loads the structure must support.

Structural design, therefore begins with specifying loads that act on the structure. The design loading for a structure is often specified in building codes. There are two types of codes: general building codes and design codes, engineers must satisfy all of the code's requirements in order for the structure to remain reliable. There are two types of loads that structure engineering must encounter in the design.

The first type of loads are dead loads that consist of the weights of the various structural members and the weights of any objects that are permanently attached to the structure.

For example, columns, beams, girders, the floor slab, roofing, walls, windows, plumbing, electrical fixtures, and other miscellaneous attachments. The second type of loads are live loads which vary in their magnitude and location. There are many different types of live loads like building loads, highway bridge loads, railroad bridge loads, impact loads, wind loads, snow loads, earthquake loads, and other natural loads.

To perform an accurate analysis a structural engineer must determine information such as structural loads , geometry , support conditions, and material properties. The results of such an analysis typically include support reactions, stresses and displacements. This information is then compared to criteria that indicate the conditions of failure.

Advanced structural analysis may examine dynamic response , stability and non-linear behavior. There are three approaches to the analysis: the mechanics of materials approach also known as strength of materials , the elasticity theory approach which is actually a special case of the more general field of continuum mechanics , and the finite element approach. The first two make use of analytical formulations which apply mostly simple linear elastic models, leading to closed-form solutions, and can often be solved by hand.

The finite element approach is actually a numerical method for solving differential equations generated by theories of mechanics such as elasticity theory and strength of materials.

## The Plastic Methods of Structural Analysis

However, the finite-element method depends heavily on the processing power of computers and is more applicable to structures of arbitrary size and complexity. Regardless of approach, the formulation is based on the same three fundamental relations: equilibrium , constitutive , and compatibility. The solutions are approximate when any of these relations are only approximately satisfied, or only an approximation of reality.

Each method has noteworthy limitations. The method of mechanics of materials is limited to very simple structural elements under relatively simple loading conditions. The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems. The theory of elasticity allows the solution of structural elements of general geometry under general loading conditions, in principle.

Analytical solution, however, is limited to relatively simple cases. The solution of elasticity problems also requires the solution of a system of partial differential equations, which is considerably more mathematically demanding than the solution of mechanics of materials problems, which require at most the solution of an ordinary differential equation.

## Structural analysis

The finite element method is perhaps the most restrictive and most useful at the same time. This method itself relies upon other structural theories such as the other two discussed here for equations to solve. It does, however, make it generally possible to solve these equations, even with highly complex geometry and loading conditions, with the restriction that there is always some numerical error. Effective and reliable use of this method requires a solid understanding of its limitations.

## The Plastic Methods of Structural Analysis

The simplest of the three methods here discussed, the mechanics of materials method is available for simple structural members subject to specific loadings such as axially loaded bars, prismatic beams in a state of pure bending , and circular shafts subject to torsion. The solutions can under certain conditions be superimposed using the superposition principle to analyze a member undergoing combined loading.

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Solutions for special cases exist for common structures such as thin-walled pressure vessels. For the analysis of entire systems, this approach can be used in conjunction with statics, giving rise to the method of sections and method of joints for truss analysis, moment distribution method for small rigid frames, and portal frame and cantilever method for large rigid frames. Except for moment distribution, which came into use in the s, these methods were developed in their current forms in the second half of the nineteenth century.

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They are still used for small structures and for preliminary design of large structures. The solutions are based on linear isotropic infinitesimal elasticity and Euler—Bernoulli beam theory. In other words, they contain the assumptions among others that the materials in question are elastic, that stress is related linearly to strain, that the material but not the structure behaves identically regardless of direction of the applied load, that all deformations are small, and that beams are long relative to their depth.

As with any simplifying assumption in engineering, the more the model strays from reality, the less useful and more dangerous the result. There are 2 commonly used methods to find the truss element forces, namely the Method of Joints and the Method of Sections. Below is an example that is solved using both of these methods. The first diagram below is the presented problem for which we need to find the truss element forces.

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The second diagram is the loading diagram and contains the reaction forces from the joints. Since there is a pin joint at A, it will have 2 reaction forces. One in the x direction and the other in the y direction. At point B, we have a roller joint and hence we only have 1 reaction force in the y direction. Let us assume these forces to be in their respective positive directions if they are not in the positive directions like we have assumed, then we will get a negative value for them.

Since the system is in static equilibrium, the sum of forces in any direction is zero and the sum of moments about any point is zero. Therefore, the magnitude and direction of the reaction forces can be calculated. This type of method uses the force balance in the x and y directions at each of the joints in the truss structure. Although we have found the forces in each of the truss elements, it is a good practice to verify the results by completing the remaining force balances.