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. Spend time on studying, not for searching material to study. Devdas menon, department of civil engineering, iit madras. 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. Note that in addition to the usual bending terms, we will also have to account for axial effects. 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. Smaller deflections for similar members redundancy in load carrying capacity redistribution increased stability advantages. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials.
The materialsproperty matrix with all of the qs is known as the stiffness matrix. Finite element analysis for fixture stiffness by yi zheng a ph. In this video, we look at an indeterminate beam and decide to solve for the reactions using the stiffness method. Strain energy and matrix methods of structural analysis 3 1.
Write the global loaddisplacement relation for 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. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. For each member, a stiffness matrix relating the nodal displacements to the nodal forces of the member is derived. Each row and column of every element stiffness matrix can be associated with a global degree of freedom.
Chapter 10 isoparametric elements learning objectives to illustrate by example how to evaluate the stresses at a given point in a plane quadrilateral element using gaussian quadrature to evaluate the stiffness matrix of the threenoded bar using gaussian quadrature and compare the result to that found by explicit evaluation of the. Skew roller support beam analysis using the stiffness method. 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. In this method, a truss is first decomposed into a number of slender members with their ends interconnected at nodes. 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. Write the global loaddisplacement relation for the beam. Chapter 4 metal matrix composites princeton university. Chapter 3 formulation of fem for twodimensional problems. 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. Devdas menon, department of civil engineering, iit madras for more details on nptel visit. In this chapter, direct stiffness method which is also called the. Force displacement relations stiffness matrix all displacement methods follow the above general procedure.
Mod01 lec03 introduction to finite element method youtube. Assemble member stiffness matrices to obtain the global stiffness matrix. 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. The displacement method follows essentially the same steps for both statically determinate and indeterminate structures. A more efficient method involves the assembly of the individual element stiffness matrices. Mass matrices for individual elements are formed in local coordinates, trans. In the last two lessons, the analysis of continuous beam by direct stiffness matrix method is discussed. Mod04 lec24 matrix analysis of structures with axial.
Structural analysis iv chapter 4 matrix stiffness method 9 dr. This application is directly applicable to the solution of pinconnected truss problems. Unfortunately is used for both the stiffness matrix and the coordinate transformation matrix. Force method for analysis of indeterminate structures number of unknown reactions or internal forces number of equilibrium equations note. Advanced structural analysis online course video lectures. Assemble member stiffness matrices to obtain the global stiffness matrix of the plane frame. Yimingkevin rong, advisor, professor of mechanical engineering.
It is assumed in the analysis that the supports are. Most structures in the real world are statically indeterminate. Compared with unreinforced metals, mmcs offer higher specific strength and stiffness. To derive the axisymmetric element stiffness matrix, body force, and surface traction equations. 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. A summary of classical lamination theory william a.
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. We can facilitate this by creating a common factor for youngs modulus and the length of the elements. 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. Matrix method stiffness method for structural analysis. Laminated composite plates david roylance department of materials science and engineering massachusetts institute of technology cambridge, ma 029 february 10, 2000 introduction. This chapter provides a quickintroduction to standard methods for computing this matrix. Construct the stiffness equation by superposing the member end forces based on the equilibrium equations. Introduction to the finite element method fem lecture 1. Force method for analysis of indeterminate structures. The direct stiffness method throughout chapters 2, 3, 4, and 5 we look at deriving the stiffness matrix for line elements. Now we can write the elemental stiffness matrix integral given in equation 3. Imagine an arbitrary solid body oriented in a cartesian coordinate system. Indeterminate structures direct stiffness method 1.
To demonstrate how to determine the stiffness matrix and stresses for a. In the displacement method of analysis, primary unknowns are joint. 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. As a general rule, the construction of the master mass matrix m largely parallels of the master stiffness matrix k. Nptel syllabus advanced structural analysis video course course outline this course mainly deals with matrix analysis of structures. Class note for structural analysis 2 fall semester, 20 hae sung lee, professor. 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. Chapter 2 introduction to the stiffness displacement. The slopedeflection and moment distribution methods were extensively used for many years before the computer era. The goal of this site is presently to help the undergraduates pursuing b. The convection at the free end of an element gives us.
Easiest way to learn how to analyse indeterminate members by matrix method. An introduction to fem and stiffness method is given in this lecture. For element 1, we divide the outside by 15 and multiply each element of the matrix by 15. Assemble the element equations to obtain the global or total equations and introduce boundary. Review of strain energy methods and introduction to. Analyse plane truss by the direct stiffness matrix. Assemble member stiffness matrices to obtain the global stiffness matrix for a beam. Stiffness methods for systematic analysis of structures. 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. Or but s 3 the surface over which convection occurs now equal to the cross.
Calculate reactions and other quantities as needed. To demonstrate the solution of an axisymmetric pressure vessel using the stiffness method. Caprani linpro linpro is very useful as a study aid for this topic. 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. To derive the constantstrain triangle cst element stiffness matrix and equations. The elastic forces are related to displacement through stiffness matrix. Constant strain triangle cst recommendations for use. 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. Assembly of member stiffness matrices, imposition of boundary conditions, and the problem of inclined supports were discussed.
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