CHAPTER
6 Development of the Plane
Stress and Plane Strain
Stiffness Equations
Introduction
In Chapters 2–5, we considered only line elements. Two or more line elements are
connected only at common nodes, forming framed or articulated structures such as
trusses, frames, and grids. Line elements have geometric properties such as cross-
sectional area and moment of inertia associated with their cross sections. However,
only one local coordinate x^ along the length of the element is required to describe a
position along the element (hence, they are called line elements or one-dimensional ele-
ments). Nodal compatibility is then enforced during the formulation of the nodal
equilibrium equations for a line element.
This chapter considers the two-dimensional finite element. Two-dimensional
(planar) elements are defined by three or more nodes in a two-dimensional plane
(that is, x-y). The elements are connected at common nodes and/or along common
edges to form continuous structures such as those shown in Figures 1–3, 1–4, 1–6,
and 6–6(b). Nodal displacement compatibility is then enforced during the formulation
of the nodal equilibrium equations for two-dimensional elements. If proper displace-
ment functions are chosen, compatibility along common edges is also obtained. The
two-dimensional element is extremely important for (1) plane stress analysis, which
includes problems such as plates with holes, fillets, or other changes in geometry that
are loaded in their plane resulting in local stress concentrations, such as illustrated
in Figure 6–1; and (2) plane strain analysis, which includes problems such as a long
underground box culvert subjected to a uniform load acting constantly over its length,
as illustrated in Figure 1–3, a long, cylindrical control rod subjected to a load that re-
mains constant over the rod length (or depth), as illustrated in Figure 1–4, and dams
and pipes subjected to loads that remain constant over their lengths as shown in
Figure 6–2.
We begin this chapter with the development of the stiffness matrix for a basic
two-dimensional or plane finite element, called the constant-strain triangular element.
We consider the constant-strain triangle (CST) stiffness matrix because its derivation
304
, 6.1 Basic Concepts of Plane Stress and Plane Strain d 305
is the simplest among the available two-dimensional elements. The element is called a
CST because it has a constant strain throughout it.
We will derive the CST stiffness matrix by using the principle of minimum
potential energy because the energy formulation is the most feasible for the develop-
ment of the equations for both two- and three-dimensional finite elements.
We will then present a simple, thin-plate plane stress example problem to illus-
trate the assemblage of the plane element stiffness matrices using the direct stiffness
method as presented in Chapter 2. We will present the total solution, including the
stresses within the plate.
d 6.1 Basic Concepts of Plane Stress and Plane Strain d
In this section, we will describe the concepts of plane stress and plane strain. These
concepts are important because the developments in this chapter are directly appli-
cable only to systems assumed to behave in a plane stress or plane strain manner.
Therefore, we will now describe these concepts in detail.
Plane Stress
Plane stress is defined to be a state of stress in which the normal stress and the shear
stresses directed perpendicular to the plane are assumed to be zero. For instance, in
Figures 6–1(a) and 6–1(b), the plates in the x-y plane shown subjected to surface tractions
T (pressure acting on the surface edge or face of a member in units of force/area) in
the plane are under a state of plane stress; that is, the normal stress sz and the shear
stresses txz and tyz are assumed to be zero. Generally, members that are thin (those
with a small z dimension compared to the in-plane x and y dimensions) and whose
loads act only in the x-y plane can be considered to be under plane stress.
Plane Strain
Plane strain is defined to be a state of strain in which the strain normal to the x-y plane
ez and the shear strains gxz and gyz are assumed to be zero. The assumptions of plane
strain are realistic for long bodies (say, in the z direction) with constant cross-sectional
area subjected to loads that act only in the x and/or y directions and do not vary in the
Figure 6–1 Plane stress problems: (a) plate with hole; (b) plate with fillet
,306 d 6 Development of the Plane Stress and Plane Strain Stiffness Equations
Figure 6–2 Plane strain problems: (a) dam subjected to horizontal loading; (b) pipe
subjected to a vertical load
z direction. Some plane strain examples are shown in Figure 6–2 [and in Figures 1–3
(a long underground box culvert) and 1–4 (a hydraulic cylinder rod end)]. In these
examples, only a unit thickness (1 in. or 1 ft) of the structure is considered
because each unit thickness behaves identically (except near the ends). The finite ele-
ment models of the structures in Figure 6–2 consist of appropriately discretized cross
sections in the x-y plane with the loads acting over unit thicknesses in the x and/or y
directions only.
Two-Dimensional State of Stress and Strain
The concept of a two-dimensional state of stress and strain and the stress/strain rela-
tionships for plane stress and plane strain are necessary to understand fully the develop-
ment and applicability of the stiffness matrix for the plane stress/plane strain triangular
element. Therefore, we briefly outline the essential concepts of two-dimensional stress
and strain (see References [1] and [2] and Appendix C for more details on this subject).
First, we illustrate the two-dimensional state of stress using Figure 6–3. The
infinitesimal element with sides dx and dy has normal stresses sx and sy acting in the
x and y directions (here on the vertical and horizontal faces), respectively. The shear
stress txy acts on the x edge (vertical face) in the y direction. The shear stress tyx acts
on the y edge (horizontal face) in the x direction. Moment equilibrium of the element
results in txy being equal in magnitude to tyx . See Appendix C.1 for proof of this
equality. Hence, three independent stresses exist and are represented by the vector
column matrix
8 9
< sx >
> =
fsg ¼ sy ð6:1:1Þ
>
:t > ;
xy
The element equilibrium equations are derived in Appendix C.1.
, 6.1 Basic Concepts of Plane Stress and Plane Strain d 307
Figure 6–3 Two-dimensional state of stress
The stresses given by Eq. (6.1.1) will be expressed in terms of the nodal displace-
ment degrees of freedom. Hence, once the nodal displacements are determined, these
stresses can be evaluated directly.
Recall from strength of materials [2] that the principal stresses, which are the
maximum and minimum normal stresses in the two-dimensional plane, can be
obtained from the following expressions:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sx þ sy sx sy 2 2 ¼s
s1 ¼ þ þ txy max
2 2
ð6:1:2Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sx þ sy sx sy 2 2 ¼s
s2 ¼ þ txy min
2 2
Also, the principal angle yp , which defines the normal whose direction is perpen-
dicular to the plane on which the maximum or minimum principal stress acts, is
defined by
2txy
tan 2yp ¼ ð6:1:3Þ
sx sy
Figure 6–4 shows the principal stresses s1 and s2 and the angle yp . Recall (as Figure
6–4 indicates) that the shear stress is zero on the planes having principal (maximum
and minimum) normal stresses.
In Figure 6–5, we show an infinitesimal element used to represent the gen-
eral two-dimensional state of strain at some point in a structure. The element is
shown to be displaced by amounts u and v in the x and y directions at point A, and
to displace or extend an additional (incremental) amount ðqu=qxÞ dx along line AB,
and ðqv=qyÞ dy along line AC in the x and y directions, respectively. Furthermore,
observing lines AB and AC, we see that point B moves upward an amount
ðqv=qxÞ dx with respect to A, and point C moves to the right an amount ðqu=qyÞ dy
with respect to A.
6 Development of the Plane
Stress and Plane Strain
Stiffness Equations
Introduction
In Chapters 2–5, we considered only line elements. Two or more line elements are
connected only at common nodes, forming framed or articulated structures such as
trusses, frames, and grids. Line elements have geometric properties such as cross-
sectional area and moment of inertia associated with their cross sections. However,
only one local coordinate x^ along the length of the element is required to describe a
position along the element (hence, they are called line elements or one-dimensional ele-
ments). Nodal compatibility is then enforced during the formulation of the nodal
equilibrium equations for a line element.
This chapter considers the two-dimensional finite element. Two-dimensional
(planar) elements are defined by three or more nodes in a two-dimensional plane
(that is, x-y). The elements are connected at common nodes and/or along common
edges to form continuous structures such as those shown in Figures 1–3, 1–4, 1–6,
and 6–6(b). Nodal displacement compatibility is then enforced during the formulation
of the nodal equilibrium equations for two-dimensional elements. If proper displace-
ment functions are chosen, compatibility along common edges is also obtained. The
two-dimensional element is extremely important for (1) plane stress analysis, which
includes problems such as plates with holes, fillets, or other changes in geometry that
are loaded in their plane resulting in local stress concentrations, such as illustrated
in Figure 6–1; and (2) plane strain analysis, which includes problems such as a long
underground box culvert subjected to a uniform load acting constantly over its length,
as illustrated in Figure 1–3, a long, cylindrical control rod subjected to a load that re-
mains constant over the rod length (or depth), as illustrated in Figure 1–4, and dams
and pipes subjected to loads that remain constant over their lengths as shown in
Figure 6–2.
We begin this chapter with the development of the stiffness matrix for a basic
two-dimensional or plane finite element, called the constant-strain triangular element.
We consider the constant-strain triangle (CST) stiffness matrix because its derivation
304
, 6.1 Basic Concepts of Plane Stress and Plane Strain d 305
is the simplest among the available two-dimensional elements. The element is called a
CST because it has a constant strain throughout it.
We will derive the CST stiffness matrix by using the principle of minimum
potential energy because the energy formulation is the most feasible for the develop-
ment of the equations for both two- and three-dimensional finite elements.
We will then present a simple, thin-plate plane stress example problem to illus-
trate the assemblage of the plane element stiffness matrices using the direct stiffness
method as presented in Chapter 2. We will present the total solution, including the
stresses within the plate.
d 6.1 Basic Concepts of Plane Stress and Plane Strain d
In this section, we will describe the concepts of plane stress and plane strain. These
concepts are important because the developments in this chapter are directly appli-
cable only to systems assumed to behave in a plane stress or plane strain manner.
Therefore, we will now describe these concepts in detail.
Plane Stress
Plane stress is defined to be a state of stress in which the normal stress and the shear
stresses directed perpendicular to the plane are assumed to be zero. For instance, in
Figures 6–1(a) and 6–1(b), the plates in the x-y plane shown subjected to surface tractions
T (pressure acting on the surface edge or face of a member in units of force/area) in
the plane are under a state of plane stress; that is, the normal stress sz and the shear
stresses txz and tyz are assumed to be zero. Generally, members that are thin (those
with a small z dimension compared to the in-plane x and y dimensions) and whose
loads act only in the x-y plane can be considered to be under plane stress.
Plane Strain
Plane strain is defined to be a state of strain in which the strain normal to the x-y plane
ez and the shear strains gxz and gyz are assumed to be zero. The assumptions of plane
strain are realistic for long bodies (say, in the z direction) with constant cross-sectional
area subjected to loads that act only in the x and/or y directions and do not vary in the
Figure 6–1 Plane stress problems: (a) plate with hole; (b) plate with fillet
,306 d 6 Development of the Plane Stress and Plane Strain Stiffness Equations
Figure 6–2 Plane strain problems: (a) dam subjected to horizontal loading; (b) pipe
subjected to a vertical load
z direction. Some plane strain examples are shown in Figure 6–2 [and in Figures 1–3
(a long underground box culvert) and 1–4 (a hydraulic cylinder rod end)]. In these
examples, only a unit thickness (1 in. or 1 ft) of the structure is considered
because each unit thickness behaves identically (except near the ends). The finite ele-
ment models of the structures in Figure 6–2 consist of appropriately discretized cross
sections in the x-y plane with the loads acting over unit thicknesses in the x and/or y
directions only.
Two-Dimensional State of Stress and Strain
The concept of a two-dimensional state of stress and strain and the stress/strain rela-
tionships for plane stress and plane strain are necessary to understand fully the develop-
ment and applicability of the stiffness matrix for the plane stress/plane strain triangular
element. Therefore, we briefly outline the essential concepts of two-dimensional stress
and strain (see References [1] and [2] and Appendix C for more details on this subject).
First, we illustrate the two-dimensional state of stress using Figure 6–3. The
infinitesimal element with sides dx and dy has normal stresses sx and sy acting in the
x and y directions (here on the vertical and horizontal faces), respectively. The shear
stress txy acts on the x edge (vertical face) in the y direction. The shear stress tyx acts
on the y edge (horizontal face) in the x direction. Moment equilibrium of the element
results in txy being equal in magnitude to tyx . See Appendix C.1 for proof of this
equality. Hence, three independent stresses exist and are represented by the vector
column matrix
8 9
< sx >
> =
fsg ¼ sy ð6:1:1Þ
>
:t > ;
xy
The element equilibrium equations are derived in Appendix C.1.
, 6.1 Basic Concepts of Plane Stress and Plane Strain d 307
Figure 6–3 Two-dimensional state of stress
The stresses given by Eq. (6.1.1) will be expressed in terms of the nodal displace-
ment degrees of freedom. Hence, once the nodal displacements are determined, these
stresses can be evaluated directly.
Recall from strength of materials [2] that the principal stresses, which are the
maximum and minimum normal stresses in the two-dimensional plane, can be
obtained from the following expressions:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sx þ sy sx sy 2 2 ¼s
s1 ¼ þ þ txy max
2 2
ð6:1:2Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sx þ sy sx sy 2 2 ¼s
s2 ¼ þ txy min
2 2
Also, the principal angle yp , which defines the normal whose direction is perpen-
dicular to the plane on which the maximum or minimum principal stress acts, is
defined by
2txy
tan 2yp ¼ ð6:1:3Þ
sx sy
Figure 6–4 shows the principal stresses s1 and s2 and the angle yp . Recall (as Figure
6–4 indicates) that the shear stress is zero on the planes having principal (maximum
and minimum) normal stresses.
In Figure 6–5, we show an infinitesimal element used to represent the gen-
eral two-dimensional state of strain at some point in a structure. The element is
shown to be displaced by amounts u and v in the x and y directions at point A, and
to displace or extend an additional (incremental) amount ðqu=qxÞ dx along line AB,
and ðqv=qyÞ dy along line AC in the x and y directions, respectively. Furthermore,
observing lines AB and AC, we see that point B moves upward an amount
ðqv=qxÞ dx with respect to A, and point C moves to the right an amount ðqu=qyÞ dy
with respect to A.
