Download the video from iTunes U or the Internet Archive. Study Guide PDF. The following content is provided under a Creative Commons license. In these lectures I would like to give you an introduction to the linear analysis of solids and structures. You are probably well aware that the finite element method is now widely used for analysis of structural engineering problems.

The method is used in civil, aeronautical, mechanical, ocean, mining, nuclear, biomechanical, and other engineering disciplines. Since the first applications two decades ago of the finite element method we now see applications in linear, nonlinear, static, and dynamic analysis. However, in this set of lectures, I would like to discuss with you only the linear, static, and dynamic analysis of problems. The finite element method is used today in various computer programs.

And its use is very significant. My objective in this set of lectures is to introduce to you the finite element methods or some of the finite element methods that are used for linear analysis of solids and structures.

And here we understand linear to mean that we're talking about infinitesimally small displacements and that we are using a linear elastic material law. In other words, Hooke's law applies. We will consider, in this set of lectures, the formulation of the finite element equilibrium equations, the calculation of finite element matrices of the matrices that arise in the equilibrium equations.

We will be talking about the methods for solution of the governing equations in static and dynamic analysis. And we will talk about actual computer implementations. I will emphasize modern and effective techniques and their practical usage. The emphasis, in this set of lectures, is given to physical explanations of the methods, techniques that we are using rather than mathematical derivations.

However, this program is also very effectively employed for linear analysis. The nonlinear analysis being then a next step in the usage of the program. These few lectures really represent a very brief and compact introduction to the field of finite element analysis.

We will go very rapidly through some or the basic concepts, practical applications, and so on. We shall follow quite closely, however, certain sections in my book entitled Finite Element Procedures in Engineering Analysis to be published by Prentice Hall. And I will be referring in the study guide of this set of lectures extensively to this book to the specific sections that we're considering in the lectures in this book.

The finite element solution process can be described as given on this viewgraph. You can see here that we talk about a physical problem. We want to analyze an actual physical problem. And our first step, of course, is to establish a finite element model of that physical problem. Then, in the next step, we solve that model.

And then we have to interpret the results. Because the interpretation of the results depends very much on how we established the finite element model, what kind of model we used, and so on.

And in establishing the finite element model, we have to be aware of what kinds of elements, techniques, and so on are available to us. Well, therefore, I will be talking, in the set of lectures, about these three steps basically here for different kinds of physical problems. Once we have interpreted the results we might go back from down here to there to revise or refine our model and go through this process again until we feel that our model has been an adequate one for the solution of the physical problem of interest.

Let me give you or show you some models that have been used in actual structural analysis. You might have seen similar models in textbooks, in publications already.As one of the methods of structural analysisthe direct stiffness methodalso known as the matrix stiffness methodis particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures.

The direct stiffness method is the most common implementation of the finite element method FEM. In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematicscompiled into a single matrix equation which governs the behaviour of the entire idealized structure. The direct stiffness method forms the basis for most commercial and free source finite element software.

The direct stiffness method originated in the field of aerospace. Researchers looked at various approaches for analysis of complex airplane frames. These included elasticity theoryenergy principles in structural mechanicsflexibility method and matrix stiffness method. It was through analysis of these methods that the direct stiffness method emerged as an efficient method ideally suited for computer implementation.

Between and A. Collar and W. Duncan published the first papers with the representation and terminology for matrix systems that are used today.

## Direct stiffness method

Aeroelastic research continued through World War II but publication restrictions from to make this work difficult to trace. The second major breakthrough in matrix structural analysis occurred through and when professor John H. Argyris systemized the concept of assembling elemental components of a structure into a system of equations.

Finally, on Nov. For a system with many members interconnected at points called nodes, the members' stiffness relations such as Eq. The system stiffness matrix K is square since the vectors R and r have the same size. Once the supports' constraints are accounted for in 2the nodal displacements are found by solving the system of linear equations 2symbolically:. Subsequently, the members' characteristic forces may be found from Eq. It is common to have Eq. The method is then known as the direct stiffness method.

The advantages and disadvantages of the matrix stiffness method are compared and discussed in the flexibility method article. The first step when using the direct stiffness method is to identify the individual elements which make up the structure. Once the elements are identified, the structure is disconnected at the nodes, the points which connect the different elements together.

Each element is then analyzed individually to develop member stiffness equations.Need help working from home with your Bentley software? We're here to help - click here. Fundamentals of Analysis and Design for Stability.

The recording of this webinar is available at: Fundamentals of Analysis and Design for Stability. The slides from the presentation can be downloaded from here:. Webinar Description. The process of designing for strength and for deflection and drift are generally well understood, but the issue of Stability is less understood and often ignored.

What can potentially cause a structure to be unstable? How can those instabilities be discerned and addressed in the design of a structure?

In this webinar the fundamentals of Stability, including P-delta, out-of-plumbness, and member imperfections were presented, with an explanation of how Stability or a lack thereof can impact both the member design forces and the structural drift and deflection.

Strategies and techniques for addressing stability issues through 1 st - and 2 nd -order analysis, amplification of design forces, and reduction of member capacity are presented. It is shown how RAM Frame and other software can be used to consider and satisfy these stability demands. The following questions were submitted during the presentation of the webinar. Questions and Answers:. Why is it necessary to perform a 2 nd -order analysis?

**Civil engineering / Stiffness method structural analysis - Concepts & Basics**

We use a 1 st -order analysis for most common buildings. Historically it was difficult to include P-delta and the various 2 nd -order effects in the analysis of a structure that was often done by hand or charts.

When the equations for calculating member capacity were established in the various specifications they were calibrated i. The ability to perform more sophisticated analyses has become more accessible through computer automation, so therefore the capacity equations can be made more precise. Recognizing this, the AISC specification committee developed more refined capacity equations based on the assumption that a more sophisticated analysis would be performed to calculate the member forces.

This resulted in the development of the requirements for the Direct Analysis Method, which describes in detail what needs to be done with the analysis so that the capacity equations are valid. This means that the capacity equations of Chapters D through K are only valid if you have performed an acceptable analysis.

This is true for all modern steel and concrete design specifications, not just AISC What quick method you do recommend to determine if P-delta is critical or not? Or should we go straight to performing a P-delta analysis? It is calculated for each story, and is a function of the total vertical load, the story height, the story drift, and the horizontal force that produced that drift.

This can be calculated by performing a 1 st -order analysis. If the value of B 2 is 1. For example, if B2 is 1. Similarly, Appendix 8 gives a method for determining the impact of small P-delta P-dwith a B 1 factor. ASCE Section Although appropriate for other structures, I recommend that this not be used to determine that a P-delta analysis is not necessary for a steel structure; AISC requires that P-delta be considered, and as explained above, the AISC member capacity equations are not valid unless P-delta has been accounted for.

Having said all of this, I recommend that you always include P-delta in your analysis and avoid the effort required to determine if it is significant or not.

### Flexibility method

Most software readily accommodates this. For analysis of steel structures as per AISCshould both P- D and P- d effects be considered simultaneously or can any one be done individually?In structural engineeringthe flexibility methodalso called the method of consistent deformationsis the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices also has the name the matrix force method due to its use of member forces as the primary unknowns.

Flexibility is the inverse of stiffness. For example, consider a spring that has Q and q as, respectively, its force and deformation:. For a system composed of many members interconnected at points called nodes, the members' flexibility relations can be put together into a single matrix equation, dropping the superscript m:.

Unlike the matrix stiffness methodwhere the members' stiffness relations can be readily integrated via nodal equilibrium and compatibility conditions, the present flexibility form of equation 2 poses serious difficulty.

To resolve this difficulty, first we make use of the nodal equilibrium equations in order to reduce the number of independent unknown member forces.

The nodal equilibrium equation for the system has the form:. In the case of determinate systems, matrix b is square and the solution for Q can be found immediately from 3 provided that the system is stable. The vector X is the so-called vector of redundant forces and I is the degree of statical indeterminacy of the system.

We usually choose jkWith suitable choices of redundant forces, the equation system 3 augmented by 4 can now be solved to obtain:. That is, using the unit dummy force method :. Equation 7b can be solved for Xand the member forces are next found from 5 while the nodal displacements can be found by. While the choice of redundant forces in 4 appears to be arbitrary and troublesome for automatic computation, this objection can be overcome by proceeding from 3 directly to 5 using a modified Gauss-Jordan elimination process.

This is a robust procedure that automatically selects a good set of redundant forces to ensure numerical stability. It is apparent from the above process that the matrix stiffness method is easier to comprehend and to implement for automatic computation. It is also easier to extend for advanced applications such as non-linear analysis, stability, vibrations, etc. For these reasons, the matrix stiffness method is the method of choice for use in general purpose structural analysis software packages.

On the other hand, for linear systems with a low degree of statical indeterminacy, the flexibility method has the advantage of being computationally less intensive. This advantage, however, is a moot point as personal computers are widely available and more powerful.

The main redeeming factor in learning this method nowadays is its educational value in imparting the concepts of equilibrium and compatibility in addition to its historical value. In contrast, the procedure of the direct stiffness method is so mechanical that it risks being used without much understanding of the structural behaviors.

The upper arguments were valid up to the late s. However, recent advances in numerical computing have shown a comeback of the force method, especially in the case of nonlinear systems.

New frameworks have been developed that allow "exact" formulations irrespectively of the type or nature of the system nonlinearities. The main advantages of the flexibility method is that the result error is independent of the discretization of the model and that it is indeed a very fast method. For instance, the elastic-plastic solution of a continuous beam using the force method requires only 4 beam elements whereas a commercial "stiffness based" FEM code requires elements in order to give results with the same accuracy.

To conclude, one can say that in the case where the solution of the problem requires recursive evaluations of the force field like in the case of structural optimization or system identificationthe efficiency of the flexibility method is indisputable.Account Options Sign in.

Top charts. New releases. Add to Wishlist. It is the most useful App for last minute preparations for civil engineering students. Some of the topics Covered in the ebook app are: 1. Degrees of Freedom and Indeterminacy 2. Member Stiffness Matrix 4. Coordinates Transformation 5. Displacement Transformation 6. Assembly of Structure Stiffness Matrix 7. Calculation of Member Forces 8. Treatment of Internal Loads 9.

Treatment of Pins Temperature Effects Temperature Gradient Elastic and Plastic Behavior of Steel Elastic - Plastic Behavior Fully Plastic Section Plastic Hinge Comparison of Linear Elastic and Plastic Designs Limit States Design Overview of Design Codes for Plastic Design General Elastoplastic Analysis of Structures Shear Force Concept of Yield Surface Yield Surface and Plastic Flow Rule Derivation of General Elastoplastic Stiffness Matrices Elastoplastic Stiffness Matrices for Sections Stiffness Matrix and Elastoplastic Analysis Use of Computers for Elastoplastic Analysis Effect of Force Interaction on Plastic Collapse Distributed Loads in Elastoplastic Analysis Theorems of Plasticity Continuous Beams and Frames Application to Portal Frames Calculation of Member Forces at Collapse Introduction of Limit Analysis by Linear Programming Limit Analysis Theorems as Constrained Plastic Rotation Capacity As one of the methods of structural analysisthe direct stiffness methodalso known as the matrix stiffness methodis particularly suited for computer-automated analysis of complex structures including the statically indeterminate type.

It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method FEM.

In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematicscompiled into a single matrix equation which governs the behaviour of the entire idealized structure. The direct stiffness method forms the basis for most commercial and free source finite element software. The direct stiffness method originated in the field of aerospace.

Researchers looked at various approaches for analysis of complex airplane frames. These included elasticity theoryenergy principles in structural mechanicsflexibility method and matrix stiffness method. It was through analysis of these methods that the direct stiffness method emerged as an efficient method ideally suited for computer implementation. Between and A. Collar and W. Duncan published the first papers with the representation and terminology for matrix systems that are used today.

Aeroelastic research continued through World War II but publication restrictions from to make this work difficult to trace. The second major breakthrough in matrix structural analysis occurred through and when professor John H. Argyris systemized the concept of assembling elemental components of a structure into a system of equations. Finally, on Nov. For a system with many members interconnected at points called nodes, the members' stiffness relations such as Eq.

The system stiffness matrix K is square since the vectors R and r have the same size. Once the supports' constraints are accounted for in 2the nodal displacements are found by solving the system of linear equations 2symbolically:. Subsequently, the members' characteristic forces may be found from Eq. It is common to have Eq. The method is then known as the direct stiffness method. The advantages and disadvantages of the matrix stiffness method are compared and discussed in the flexibility method article.We use cookies to give you the best possible experience.

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Bestselling Series. Harry Potter. Popular Features. Home Learning. Categories: Engineering: General Mechanical Engineering. Notify me. The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer.

The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems. Table of contents 1. Introduction to Matrix Notation. Role of the Computer.

General Steps of the Finite Element Method. Applications of the Finite Element Method. Advantages of the Finite Element Method. Computer Programs for the Finite Element Method. Derivation of the Stiffness Matrix for a Spring Element. Example of a Spring Assemblage. Boundary Conditions. Selecting Approximation Functions for Displacements. Transformation of Vectors in Two Dimensions. Computation of Stress for a Bar in the x-y Plane. Solution of a Plane Truss. Use of Symmetry in Structure.

Inclined, or Skewed, Supports. Example of Assemblage of Beam Stiffness Matrices. Distribution Loading. Beam Element with Nodal Hinge. Rigid Plane Frame Examples. Inclined or Skewed Supports - Frame Element.

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