## A

The FE mathematical idealization of a MoM member. The model is one-dimensional in x. The two end joints are the site of end quantities joint forces and displacements, that interconnect members. The internal quantities characterize the stresses and deformations in the member. Figure 6.2. The FE mathematical idealization of a MoM member. The model is one-dimensional in x. The two end joints are the site of end quantities joint forces and displacements, that interconnect members. The...

## E [ ux Uyi UX Uxn Uyn T

Displacement Interpolation The displacement field u(e) (x, y) over the element is interpolated from the node displacements. We shall assume that the same interpolation functions are used for both displacement components.3 Thus (x, y) J2 Nje)(x, y) Uxi, uy(x, y) N e)(x, y) Uyi, where Nje x, y) are the element shape functions. In matrix form where Nje x, y) are the element shape functions. In matrix form ' N e) 0 N(e) 0 N(e) 0 0 N(e) 0 N(e) 0 N(e) _ The minimum conditions on N(e (x, y)...

## Exercise

A C 25 Derive the 4 x 4 global element stiffness matrix of a prismatic spar element in a two dimensional Cartesian system x, y . Start from the local stiffness (6.10) and proceed as in 3.2.1 and 3.2.2 for the bar element. Since K in (6.10) is 2 x 2, T is 2 x 4. Show that T consists of rows 2 and 4 of the matrix of (3.2). Figure E6.1. Bar element in 3D for Exercise 6.4. Figure E6.1. Bar element in 3D for Exercise 6.4.

## Info

A generic element can be defined, independently of the original circle, by the segment that connects two nodes i and j. The relevant element property length Lj, can be computed in the generic element independently of the others, a property called local support in the FEM. Finally, the desired property the polygon perimeter, is obtained by reconnecting n elements and adding up their length the corresponding steps in the FEM being assembly and solution, respectively. There is of course nothing...

## N n n nNe

2 The proof may be found in texts on variational methods in mechanics, e.g., H. L. Langhaar, Energy Methods in Applied Mechanics, McGraw-Hill, 1960. This is the most readable old fashioned treatment of the energy principles of structural mechanics, with a beautiful treatment of virtual work. Out of print but used copies may be found from web sites. 3 See for example, I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, 1963. Using the fundamental lemma of variational calculus,4...

## Table Of Contents

Mathematical 13.2.3. Assumptions of Classical Beam Theory 13-4 13.3. THE CLASSICAL BEAM THEORY 13-5 13.3.1. The Neutral 13.3.2. Element Coordinate Systems 13-5 13.3.3. Kinematics 13.3.4. Loading 13-6 13.3.5. Support 13.3.6. Strains, Stresses and Bending Moments 13-6 13.4. TOTAL POTENTIAL ENERGY FUNCTIONAL 13-7 13.5. BEAM FINITE ELEMENTS 13-8 13.5.1. Finite Element Trial Functions 13-9 13.5.2. Shape Functions 13-9 13.6. THE FINITE ELEMENT EQUATIONS 13-10 13.6.1....

## Uxl Uy Uy

Whereas the known applied forces are When solving the stiffness equations by hand, the simplest way to account for support conditions is to remove equations associated with known joint displacements from the master system. To apply (3.21) we have to remove equations 1, 2 and 4. This can be systematically accomplished by deleting or striking out rows and columns number 1, 2 and 4 from K and the corresponding components from f and u. The reduced three-equation system is

## N uw

The Tonti diagram for the governing equations of the Bernoulli-Euler beam model. Figure 13.6. The Tonti diagram for the governing equations of the Bernoulli-Euler beam model. where as usual Uand Ware the internal and external energies, respectively. As previously explained, in the Bernoulli-Euler model U includes only the bending energy U f aedV Mk dx f EIk2 dx f EI vf dx f vEIv dx. J V J 0 Jo Jo Jo The external work W accounts for the applied transverse force The three functionals...

## Idealization

The idealization process for a simple structure. The physical system, here a roof truss, is directly idealized by the mathematical model a pin-jointed bar assembly. For this particular structure, the idealization coalesces with the discrete model. Figure 1.6. The idealization process for a simple structure. The physical system, here a roof truss, is directly idealized by the mathematical model a pin-jointed bar assembly. For this particular structure, the idealization coalesces with...

## Remark

If there is an applied distributed moment m x per unit of beam length, the external energy 13.8 must be augmented with a fQ m x O x dx term. This is further elaborated in Exercises 13.4 and 13.5. Such kind of distributed loading is uncommon in practice although in framework analysis occassionally the need arises for treating a concentrated moment C between nodes. Figure 13.7. The two-node plane beam element with four degrees of freedom. Figure 13.7. The two-node plane beam element with four...

## Exercises

Computer algebra systems, known by the acronym CAS, are programs designed to perform symbolic and numeric manipulations following the rules of mathematics.1 The development of such programs began in the mid 1960s. The first comprehensive system the granddaddy of them all, called Macsyma an acronym for Project Mac Symbolic Manipulator was developed using the programming language Lisp at MIT's famous Artificial Intelligence Laboratory over the period 1967 to 1980. The number and quality of...

## Distributed Force On Nodes Finite Element

In many thin structures modeled as continuous bodies the appearance of skinny elements is inevitable on account of computational economy reasons. An example is provided by the three-dimensional modeling of layered composites in aerospace and mechanical engineering problems. A physical interface, resulting from example from a change in material, should also be an interelement boundary. That is, elements must not cross interfaces. See Figure 8.3. In two-dimensional FE modeling, if you have a...