One-dimensional Bars/Springs

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Introduction

A large number of practical problems are governed by the one-dimensional boundary value problem (1D BVP) of the following form:

$frac{d}{dx}(k(x)frac{du(x)}{dx})+p(x)u(x)+q(x)=0;; : x_0<x<x_L$

Since the differential equation is of second order, for a unique solution, there must be at least two specified boundary conditions.

Concepts and Formulas

Governing differential equation (ODE):

The general governing differential equation for axial deformation of bars/springs is:

$frac{d}{dx}(EA(x)frac{du}{dx})+q(x)=0;; : 0<x<L$

where A(x) is the area of cross section and can vary along the length; E is the elastic modulus; and q(x) is the applied distributed load in the axial direction.

Types of boundary conditions

The boundary conditions of the following forms may be specified at one or more points:

1- Essential boundary conditions (EBC):

$u=specified$

2- Natural boundary conditions (NBC):

$EAfrac{du}{dx}=F=specified$

Explicit equations for a two-node linear element:

The interpolation functions are simple linear functions of x. Furthermore, if we assume that k,p, and q are constant over an element, then it is easy to carry out integrations and write explicit formulas for element equations:

the position of element nodes:

$[x_1, x_1+L]$

Interpolation functions (shape functions):

$N^T=(frac{x_1+L-x}{L} , -frac{x_1-x}{L})$

and consequently derivatives of shape functions:

$B^T=frac{dN^T}{dx}=(frac{-1}{L} , frac{1}{L})$

Thus, the element stiffness matrix is:

$K=frac{EA}{L}egin{bmatrix} 1 & -1 -1 & 1 end{bmatrix}$

and the explicit equations for a linear element:

$frac{EA}{L}egin{bmatrix} 1 & -1 -1 & 1 end{bmatrix}egin{bmatrix} u_1 u_2 end{bmatrix}=egin{bmatrix} F_1 F_2 end{bmatrix}$

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