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DRAFT - 1
CHAPTER 3
Truss Element
3.1 Introduction
The single most important concept in understanding FEA, is the basic understanding of
various finite elements that we employ in an analysis. Elements are used for representing
a real engineering structure, and therefore, their selection must be a true representation of
geometry and mechanical properties of the structure. Any deviation from either the
geometry or the mechanical properties would yield erroneous results.
The elements used in commercial codes can be classified in two basic categories:
1.
Discrete elements
: These elements have a well defined deflection equation that can
be found in an engineering handbook, such as, Truss and Beam/Frame elements. The
geometry of these elements is simple, and in general, mesh refinement does not give
better results. Discrete elements have a very limited application; bulk of the FEA
application relies on the Continuous-structure elements.
2.
Continuous-structure Elements:
Continuous-structure elements do not have a well
define deflection or interpolation function, it is developed and approximated by using
the theory of elasticity. In general, a continuous-structure element can have any
geometric shape, unlike a truss or beam element. The geometry is represented by
either a 2-D or a 3-D solid element – the continuous- structure elements. Since
elements in this category can have any shape, it is very effective in calculation of
stresses at a sharp curve or geometry, i.e., evaluation of stress concentrations. Since
discrete elements cannot be used for this purpose, continuous structural elements are
extremely useful for finding stress concentration points in structures.
Element
Figure 3.1 A Discrete element Structure
Node
DRAFT –1
Chapter 3 Truss Element
As explained earlier, for analyzing an engineering structure, we divide the structure into
small sections and represent them by appropriate elements. Nodes always define
geometry of the structure and elements are generated when the applicable nodes are
connected. Results are always obtained for node points – and not for elements - which are
then interpolated to provide values for the corresponding elements.
For a static structure, all nodes must satisfy the equilibrium conditions and the continuity
of displacement, translation and rotation.
In the following sections, we will get familiar with characteristics of the basic finite
elements.
3.2 Structures & Elements
Most 3-D structures can be analyzed using 2-D elements, which require relatively less
computing time than the 3-D solid elements. Therefore, in FEA, 2-D elements are the
most widely used elements. However, there are cases where we must use 3-D solid
elements. In general, elements used in FEA can be classified as:
- Trusses
- Beams
- Plates
- Shells
- Plane
solids
- Axisymmetric
solids
- 3-D
solids
Since Truss element is a very simple and discrete element, let us look at its properties and
application first.
3.3 Truss
Elements
The characteristics of a truss element can be summarized as follows:
Truss is a slender member (length is much larger than the cross-section).
It is a two-force member i.e. it can only support an axial load and cannot support a
bending load. Members are joined by pins (no translation at the constrained node,
but free to rotate in any direction).
The cross-sectional dimensions and elastic properties of each member are
constant along its length.
The element may interconnect in a 2-D or 3-D configuration in space.
The element is mechanically equivalent to a spring, since it has no stiffness
against applied loads except those acting along the axis of the member.
ME 273 Lecture Notes © by R. B. Agarwal
3-2
DRAFT –1
Chapter 3 Truss Element
However, unlike a spring element, a truss element can be oriented in any direction
in a plane, and the same element is capable of tension as well as compression.
j
i
Figure 3.2 A Truss Element
3.3.1 Stress – Strain relation
:
As stated earlier, all deflections in FEA are evaluated at the nodes. The stress and strain
in an element is calculated by interpolation of deflection values shared by nodes of the
element. Since the deflection equation of the element is clearly defined, calculation of
stress and strain is rather simple matter. When a load F is applied on a truss member, the
strain at a point is found by the following relationship.
x
L
or,
ε
=
δ
L/L
L
+
δ
L
dx
du
=
ε
Figure 3.3 Truss member in Tension
where,
ε
= strain at a point
u = axial displacement of any point along the length L
By hook’s law,
ε
σ
E
=
Where, E = young’s modulus or modulus of elasticity.
From the above relationship, and the relation,
ME 273 Lecture Notes © by R. B. Agarwal
3-3
DRAFT –1
Chapter 3 Truss Element
F = A
σ
the deflection,
δ
L, can be found as
δ
L
=
FL/AE
(3.1)
Where,
F = Applied load
A
=
Cross-section
area
L = Length of the element
3.3.2 Treatment of Loads in FEA
For a truss element, loads can be applied on a node only. If loads are distributed on a
structure, they must be converted to the equivalent loads that can be applied at nodes.
Loads can be applied in any direction at the node, however, the element can resist only
the axial component, and the component perpendicular to the axis, merely causes free
rotation at the joint.
3.3.3 Finite Element Equation of a Truss Structure
In this section, we will derive the finite element equation of a truss structure. The
procedure presented here is the basis for all FEA analyses formulations, wherever h-
element are used.
Analogues to the previous chapter, we will use the direct or equilibrium method for
generating the finite element equations. Assembly procedure for obtaining the global
matrix will remain the same.
In FEA, when we find deflections at nodes, the deflections are measured with respect to
a global coordinate system, which is a fixed frame of reference. Displacements of
individual nodes with respect to a fixed coordinate system are desirable in order to see
the overall deformed structural shape. However, these deflection values are not
convenient in the calculation of stress and strain in an element. Global coordinate system
is good for predicting the overall deflections in the structure, but not for finding
deflection, strain, and stress in an element. For this, it’s much easier to use a local
coordinate system. We will derive a general equation, which relates local and global
coordinates.
In Figure 3.4, the global coordinates x-y can give us the overall deflections measured
with respect to the fixed coordinate system. These deflections are useful for finding the
final shape or clearance with the surroundings of the structure. However, if we wish to
find the strain in some element, say, member 2-7 in figure 3.4, it will be easier if we
know the deflections of node 2 and 3, in the y’ direction. Thus, calculation of strain
value is much easier when the local deflection values are known, and will be time-
ME 273 Lecture Notes © by R. B. Agarwal
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DRAFT –1
Chapter 3 Truss Element
consuming if we have to work with the x and y values of deflection at these nodes.
Therefore, we need to establish a trigonometric relationship between the local and global
coordinate systems. In Figure 3.4, xy coordinates are global, where as, x’y’ are local
coordinates for element 4-7
y
y’
6 7 x’ 8
Node
Element
1
2 3 4 5 x
Fixed-Frame
Origin
Figure 3.4. Local and Global Coordinates
3.3.4 Relationship Between Local and Global Deflections
Let us consider the truss member, shown in Figure 3.5. The element is inclined at an
angle
θ
, in a counter clockwise direction. The local deflections are
δ
1
and
δ
2
. The global
deflections are: u
1
, u
2
, u
3
, and u
4
. We wish to establish a relationship between these
deflections in terms of the given trigonometric relations.
u
4
δ
2
, R
2
u
2
2
u
3
1
θ
u
1
R
1
,
δ
1
Figure 3.5 Local and Global Deflections
ME 273 Lecture Notes © by R. B. Agarwal
3-5