FEA Unit 2 notes

1. Learning/ Resource Material
UNIT – 2 (Spring Element and Spring Assembly)
Development of a spring stiffness matrix, Development of spring assembly by direct
equilibrium method, Work/Energy method, Methods of weighted residuals- Galerkin's method,
and Variational method, Development of force-displacement relation, Strain-displacement
relation, Calculation of elemental reaction force and nodal displacement.
DEVELOPMENT OF A SPRING STIFFNESS MATRIX
Definition of the Stiffness Matrix
Familiarity with the stiffness matrix is essential to understanding the stiffness method.
We define the stiffness matrix as follows: For an element, a stiffness matrix 𝑘
̂ is a matrix such
that 𝑓
̂ = 𝑘
̂ 𝑑
̂, where 𝑘
̂ relates local-coordinate (𝑥
̂, 𝑦
̂, 𝑧̂)nodal displacements 𝑑
̂ to local forces 𝑓
̂ of
a single element. (Throughout this text, the underline notation denotes a matrix, and the ^ symbol
denotes quantities referred to a local-coordinate system set up to be convenient for the element as
shown in Fig 2.1.)
Fig 2.1: Local (𝑥
̂, 𝑦
̂, 𝑧̂) and global (x, y, z) coordinate systems
For a continuous medium or structure comprising a series of elements, a stiffness matrix
K relates global-coordinate (x, y, z) nodal displacements d to global forces F of the whole
medium or structure. (Lowercase letters such as x, y, and z without the ^ symbol denote global-
coordinate variables.)
Derivation of the Stiffness Matrix for a Spring Element
Using the direct equilibrium approach, we will now derive the stiffness matrix for a one-
dimensional linear spring—that is, a spring that obeys Hooke’s law and resists forces only in the
direction of the spring. Consider the linear spring element shown in Fig 2.2. Reference points 1
and 2 are located at the ends of the element. These reference points are called the nodes of the
spring element. The local nodal forces are 𝑓1𝑥
̂ and 𝑓2𝑥
̂ for the spring element associated with the

2. local axis 𝑥
̂. The local axis acts in the direction of the spring so that we can directly measure
displacements and forces along the spring. The local nodal displacements are 𝑑1𝑥
̂ and 𝑑2𝑥
̂ for the
spring element. These nodal displacements are called the degrees of freedom at each node.
Positive directions for the forces and displacements at each node are taken in the positive 𝑥
̂
direction as shown from node 1 to node 2 in the figure. The symbol k is called the spring
constant or stiffness of the spring.
Fig 2.2: Linear spring element with positive nodal displacement and force conventions
We now want to develop a relationship between nodal forces and nodal displacements for
a spring element. This relationship will be the stiffness matrix. Therefore, we want to relate the
nodal force matrix to the nodal displacement matrix as follows:
(2.1)
where the element stiffness coefficients 𝑘𝑖𝑗 of the 𝑘
̂ matrix in Eq. (2.1) are to be determined. 𝑘𝑖𝑗
represent the force 𝐹𝑖 in the ith degree of freedom due to a unit displacement dj in the jth degree
of freedom while all other displacements are zero. That is, when we let dj = 1 and dk = 0 for k ≠
j, force Fi = 𝑘𝑖𝑗.
Our approach is to derive various element stiffness matrices and then to illustrate how to
solve engineering problems with the elements, step 1 now involves only selecting the element
type.
Step 1 Select the Element Type
Consider the linear spring element (which can be an element in a system of springs)
subjected to resulting nodal tensile forces T (which may result from the action of adjacent
springs) directed along the spring axial direction 𝑥
̂ as shown in Fig 2.3, so as to be in
equilibrium. The local 𝑥
̂ axis is directed from node 1 to node 2. We represent the spring by
labeling nodes at each end and by labeling the element number. The original distance between
nodes before deformation is denoted by L. The material property (spring constant) of the element
is k.

3. Fig 2.3: Linear spring subjected to tensile forces
Step 2 Select a Displacement Function
We must choose in advance the mathematical function to represent the deformed shape of
the spring element under loading. Because it is difficult, if not impossible at times, to obtain a
closed form or exact solution, we assume a solution shape or distribution of displacement within
the element by using an appropriate mathematical function. The most common functions used
are polynomials.
Because the spring element resists axial loading only with the local degrees of freedom
for the element being displacements 𝑑1𝑥
̂ and 𝑑2𝑥
̂ along the 𝑥
̂ direction, we choose a
displacement function 𝑢
̂ to represent the axial displacement throughout the element. Here a
linear displacement variation along the 𝑥
̂ axis of the spring is assumed [Fig 2.4(b)], because a
linear function with specified endpoints has a unique path.
Therefore,
𝑢
̂ = 𝑎1 + 𝑎2𝑥
̂ (2.2)
Fig 2.4: (a) Spring element showing plots of (b) displacement function 𝑢
̂ and shape functions
(c) N1 and (d) N2 over domain of element

4. In general, the total number of coefficients a is equal to the total number of degrees of
freedom associated with the element. Here the total number of degrees of freedom is two—an
axial displacement at each of the two nodes of the element.
In matrix form, Eq. (2.2) becomes
(2.3)
We now want to express u^ as a function of the nodal displacements 𝑑1𝑥
̂ and 𝑑2𝑥
̂ as this
will allow us to apply the physical boundary conditions on nodal displacements directly as
indicated in Step 3 and to then relate the nodal displacements to the nodal forces in Step 4. We
achieve this by evaluating 𝑢
̂ at each node and solving for a1 and a2 from Eq. (2.2) as follows:
(2.4)
(2.5)
or, solving Eq. (2.5) for a2,
(2.6)
Upon substituting Eqs. (2.4) and (2.6) into Eq. (2.2), we have
(2.7)
In matrix form, we express Eq. (2.2.7) as
(2.8)
(2.9)
(2.10)
are called the shape functions because the Ni’s express the shape of the assumed
displacement function over the domain (x^ coordinate) of the element when the ith element
degree of freedom has unit value and all other degrees of freedom are zero.

5. In this case, N1 and N2 are linear functions that have the properties that N1 = 1 at node 1
and N1 = 0 at node 2, whereas N2 = 1 at node 2 and N2 = 0 at node 1. See Fig 2.4(c) and (d) for
plots of these shape functions over the domain of the spring element. Also, N1 + N2 = 1 for any
axial coordinate along the bar. In addition, the Ni’s are often called interpolation functions
because we are interpolating to find the value of a function between given nodal values. The
interpolation function may be different from the actual function except at the endpoints or nodes,
where the interpolation function and actual function must be equal to specified nodal values.
(Follow Book for this topic: Logan – Page no. 33 to 45)
(Follow Book for topic Weighted Residue: DV Hutton – Page no. 131 to 135)