# Analysis of Elastic Frames

DOI: 10.4018/978-1-4666-6379-4.ch004

## Abstract

The formulation of a mathematical model that describes some physical phenomenon is just a first step; a second one, equally important, is the development of numerical procedures that transform this model into a potentially predictive tool with practical engineering applications; no computer software can be developed without robust numerical algorithms. This chapter describes some of these procedures in the case of elastic frames. First, it considers the direct stiffness method that permits the analysis of linear elastic and quasi static structures (Section 4.1); then, the procedure is extended to the more complex cases of nonlinear structures (Section 4.2) and dynamic loading (Section 4.3).
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## 4.1 The Direct Stiffness Method

### 4.1.1 Stiffness Matrix of a Frame and Matrix of Total Forces

Consider the particular case of a structure under quasi-static loading; assume that nonlinear terms can be neglected. Then, the problem is formulated as follows:

• 1.

Compute: The free displacements, reaction forces, deformations and stresses.

• 2.

With the Following Data: the initial configuration of the structure, the restricted displacements, the external forces corresponding to the free displacements and those applied over the elements; finally, the material and cross-section properties (E, ν, I, A).

• 3.

Such that they verify:

• a.

The Kinematic Equation:

• b.

The Equilibrium Equation: (4.1.1)

• c.

The Constitutive Equation:

The problem can now be resolved as follows; the combination of Equations (4.1.1) leads to:

(4.1.2) where

; (4.1.3)
(4.1.4)

The matrix is called stiffness matrix of the structure; is denoted expanded stiffness matrix of element b. The elasticity matrix used for the computation of the stiffness matrix of the element b can be based on the Euler-Bernoulli beam theory (Equation 3.3.11), if the element is slender, or on the Timoshenko theory (Equation 3.4.11) in all cases.

The term is called matrix of total forces. It is decomposed into two terms; the first one, , includes external forces applied directly on the nodes; the second matrix, , depends exclusively on the external forces distributed over the element. The terms and were defined in the previous chapter (section 3.2.3).

Notice that (4.1.2) is again an equilibrium equation but this time it is expressed in function of displacements instead of stresses; the left hand term gives the internal forces, the right hand one includes the external forces.

The resolution of the linear equation (4.1.2), by any conventional method, gives the values of the unknown displacements and reaction forces. Then, element deformations can be computed by the kinematic equation (4.1.1a), finally the element stresses are calculated by the constitutive law (4.1.1c); (see examples 4.4.1 and 4.4.2).

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