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Finite Element Method universal resource
Force-Displacement Relationship for
Single Degree of Freedom (SDOF) Linear Spring
Degrees of freedom is
defined as the number of independent coordinates necessary to
specify the configuration of a system. The figure below shows a
translational spring which only needs one coordinate to specify
its position. The displacement u defines the horizontal movement
(translation) of point P due to a horizontal external force f
applied at that point. Therefore, the spring system shown has a
single (one) degree of freedom.
Single Degree of Freedom Spring
In a three-dimensional problem there
are six degrees of freedom. Three are translational, i.e., x, y
and z, and three are rotational, i.e., about x, y, z axes.
Multi-degree of freedom system that arise in FEM will be
discussed later.
The SDOF spring shown above is a
simple setting for introducing the force-displacement equation
that arises in FEM. A spring is a mechanical element
which exerts a force when deformed. When a force f is
incrementally applied as 
f, one can measure the corresponding displacement.
The figure below shows the force-displacement curve for the SDOF
spring. Since the force-displacement curve is linear, we have a linear
spring.
Force-Displacement Curve for a Linear Spring
The force-displacement relationship
for a single degree of freedom linear spring is
f = K u
where
f - External Force (units : force
- lb or Newton)
u - Displacement (units : length - in. or millimeter)
K - Stiffness or Spring Rate (units : force/length - lb/in or
Newton/millimeter)
The linear force-displacement
relationship f = K u is commonly found in FEM.
For a linear
system, the stiffness K is defined as the force necessary to
produce a unit displacement; or the ratio of force to
displacement. The constant K is often referred to as the
spring constant for a linear spring. For a linear system, K
depends on the material properties and geometry of the problem.
This will be discussed at a later time.
In a given practical application the
external force f and stiffness (spring constant) K are known
and the displacement u is unknown. Solving for the
unknown displacement yields
u = f / K = K-1
f
where K-1 denotes the
inverse of the stiffness. In FEM we solve for displacements
associated with the degrees of freedom.
It is very important in FEM that we
pay close attention to the sign convention for the force and
displacement. The sign conventions for force and displacement are
as follows:
In FEM the convention is to always
assume f and u are in the same direction, and positive coordinate
direction (in this case positive Cartesian coordinate x).
The stiffness K is always positive (K > 0)
for any physically possible linear application. The sign
of K can be determined by considering the three cases when K is
positive, zero and negative. We will also consider if the
solution is physically possible for all three cases.
Case #1: Positive K
Consider the positive force acting to the right
as shown below, where K is positive. Therefore, from u = K-1
f, we know that the displacement is also to the right in the
positive x-direction. Thus, a force acting in the positive
x-direction would correspond with a displacement in the same
direction. The stiffness K is always positive (K > 0) for a
physical linear system.
f = K u
Case #2: Zero K
If K = 0, the displacement u would generate no
resisting force f, which implies that the structure is unstable
as shown below. In other words, f = K u = 0.
Case #3: Negative K
If K < 0, the force f and its corresponding
displacement u would be oppositely directed as found from the
force-displacement relationship, u=-K< 0. The system below is
physically unreasonable for a linear system.
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Copyright 1996. All Rights Reserved. Finite Element Method universal resource (FEMur).