Metal matrix composites findings metal matrix composites mmcs usually consist of a lowdensity metal, such as aluminum or magnesium, reinforced with particulate or fibers of a ceramic material, such as silicon carbide or graphite. Nptel syllabus advanced structural analysis video course course outline this course mainly deals with matrix analysis of structures. The next step is to add the stiffness matrices for the elements to create a matrix for the entire structure. As a general rule, the construction of the master mass matrix m largely parallels of the master stiffness matrix k. Now once we have the element stiffness matrices for all the members, then you have to assemble them, all the matrices to get the global stiffness matrix of the. Bar and beam elements stiffness matrix a formal approach we derive the same stiffness matrix for the bar using a formal approach which can be applied to many other more complicated situations. Structural analysis iv chapter 4 matrix stiffness method 9 dr. Force method for analysis of indeterminate structures. Each row and column of every element stiffness matrix can be associated with a global degree of freedom.
Unfortunately is used for both the stiffness matrix and the coordinate transformation matrix. Relates forces at the element nodes to displacements of those nodes. A summary of classical lamination theory defining the laminate a laminate is an organized stack of unidirectional composite plies unidirectional meaning the plies have a single fiber direction rather than a weave pattern. Assemble member stiffness matrices to obtain the global stiffness matrix. Stiffness method for frame structures for frame problems with possibly inclined beam elements, the stiffness method can be used to solve the problem by transforming element stiffness matrices from the local to global coordinates. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Mod01 lec03 introduction to finite element method youtube. The slopedeflection and moment distribution methods were extensively used for many years before the computer era.
Computation of shape functions for constant strain triangle properties of the shape functions computation of straindisplacement matrix computation of element stiffness matrix computation of nodal loads due to body forces. Laminated composite plates david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029 february 10, 2000 introduction. State of the element is entirely defined by a filament a curved line. The matlab toolbox of this chapter is developed according to the stiffness method. Matrix method stiffness method for structural analysis. Skew roller support beam analysis using the stiffness method. Write the global loaddisplacement relation for the beam. Or but s 3 the surface over which convection occurs now equal to the cross. Note that in addition to the usual bending terms, we will also have to account for axial effects. In this method, a truss is first decomposed into a number of slender members with their ends interconnected at nodes.
Compared with unreinforced metals, mmcs offer higher specific strength and stiffness. Use of matrix method definition of stiffness method definition of flexibility method. Advanced structural analysis online course video lectures. Chapter 3 formulation of fem for twodimensional problems.
Introduction force method and slopedeflection method can be used, with hand calculation, for solving the indeterminate structures when the degree of static or kinematical indeterminacy is small. The convection at the free end of an element gives us. A more efficient method involves the assembly of the individual element stiffness matrices. Force displacement relations stiffness matrix all displacement methods follow the above general procedure. For each member, a stiffness matrix relating the nodal displacements to the nodal forces of the member is derived. Construct the stiffness equation by superposing the member end forces based on the equilibrium equations.
Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. To demonstrate the solution of an axisymmetric pressure vessel using the stiffness method. The elastic forces are related to displacement through stiffness matrix. The stiffness matrix can be obtained from stiffness influence coefficient kij, which is the force required along dof i due to a unit displacement at dof j and zero displacement at all other dofs. To derive the axisymmetric element stiffness matrix, body force, and surface traction equations. Finite element analysis for fixture stiffness by yi zheng a ph. In the last two lessons, the analysis of continuous beam by direct stiffness matrix method is discussed. The direct stiffness method throughout chapters 2, 3, 4, and 5 we look at deriving the stiffness matrix for line elements. Calculate reactions and other quantities as needed. Theory and analysis of structures 475 principle of superposition the principle states that ifthe st ructural behavior is linearly elastic, the forces acting on a structure may be separated or divided into any convenient fashion and the structure analyzed for the separate cases. Stiffness methods for systematic analysis of structures. Assemble member stiffness matrices to obtain the global stiffness matrix for a beam.
The b matrix straindisplacement corresponding to this element is we will denote the columns of the b matrix as computation of the terms in the stiffness matrix of 2d elements recap. Review of strain energy methods and introduction to. Sap2000 stiffness matrix extract the sap2000 stiffness matrix for a twostory frame and compare it to the manual solution. This application is directly applicable to the solution of pinconnected truss problems. Force method for analysis of indeterminate structures number of unknown reactions or internal forces number of equilibrium equations note. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials. A summary of classical lamination theory william a.
For element 1, we divide the outside by 15 and multiply each element of the matrix by 15. Pe281 finite element method course notes summarized by tara laforce stanford, ca 23rd may 2006 1 derivation of the method in order to derive the fundamental concepts of fem we will start by looking at an extremely simple ode and approximate it using fem. Now we can write the elemental stiffness matrix integral given in equation 3. To demonstrate how to determine the stiffness matrix and stresses for a. Strain energy and matrix methods of structural analysis 3 1. Assemble member stiffness matrices to obtain the global stiffness matrix of the plane frame. Ce316 structural matrix analysis 2 member truss using ms excel this is a supplementary lecture video for numerical solutions to ce problems ce316 and structural matrix analysis ce504. Devdas menon, department of civil engineering, iit madras for more details on nptel visit. Chapter 2 introduction to the stiffness displacement. Development of truss equations stiffness matrix for a bar element consider the derivation of the stiffness matrix for the linearelastic, constant crosssectional area prismatic bar element show below. Constant strain triangle cst recommendations for use. Introduction to the finite element method fem lecture 1.
In this video, we look at an indeterminate beam and decide to solve for the reactions using the stiffness method. To derive the constantstrain triangle cst element stiffness matrix and equations. An introduction to fem and stiffness method is given in this lecture. Mod04 lec24 matrix analysis of structures with axial. Smaller deflections for similar members redundancy in load carrying capacity redistribution increased stability advantages. This chapter provides a quickintroduction to standard methods for computing this matrix. It is assumed in the analysis that the supports are. Chapter 4 metal matrix composites princeton university.
Caprani linpro linpro is very useful as a study aid for this topic. Write down global load vector for the beam problem. Derive the element stiffness matrix and equationsdefine the stiffness matrix for an element and then consider the derivation of the stiffness matrix for a linearelastic spring element. Assembly of member stiffness matrices, imposition of boundary conditions, and the problem of inclined supports were discussed. The materialsproperty matrix with all of the qs is known as the stiffness matrix. Most structures in the real world are statically indeterminate.
Devdas menon, department of civil engineering, iit madras. Analyse plane truss by the direct stiffness matrix. Indeterminate structures direct stiffness method 1. Mass matrices for individual elements are formed in local coordinates, trans. The goal of this site is presently to help the undergraduates pursuing b. Assemble the element equations to obtain the global or total equations and introduce boundary. Yimingkevin rong, advisor, professor of mechanical engineering. Write the global loaddisplacement relation for the. In this chapter, direct stiffness method which is also called the. The displacement method follows essentially the same steps for both statically determinate and indeterminate structures. Dissertation submitted to the faculty of the worcester polytechnic institute in partial fulfillment of the requirements for the degree of doctor of philiosophy in manufacturing engineering by april 2005 approved. In the displacement method of analysis, primary unknowns are joint.