_{} (3)

_{} (4)

There are instances when some of the stress tensor
components vanish and the general three-dimensional stress tensor _{}degenerates into a tensor characterized by only three
independent components. This condition is called **plane state of stress. **For the present theoretical development, it
is assumed that all stress components pertinent to the planes having normal
vector parallel to the _{} axis are zero:

_{} (5)

_{} (6)

_{} (7)

This type of plane stress tensor is illustrated in Figure 2.

Figure 2 Plane Stress Tensor

Additionally, the remaining non-zero stress tensor
components contained in equation (1) are assumed to be independent of the
variable_{}.

_{} (8)

Consequently, the plane stress tensor _{} can be represented in
plane_{} as depicted in Figure 3:

Figure 3 Plane
State of Stress

The plane stress tensor_{}is defined by three non-zero components:

_{} (9)

Several of the conditions encountered in previous lectures,
including the study of the axial and torsional deformation and pure and
non-uniform bending, are characterized by different states of plane stress. In
reality, the plane stress tensor is a direct result of the assumptions imposed
on the deformation.

Examples of plane stress tensors, such as uni-axial, pure
shear and bi-axial, are illustrated in Figure 4.

Figure 4 Examples of Plane State
of Stress

(a) Uni-Axial, (b) Pure Shear and (c) Bi-Axial

**2 Plane Stress Transformation Equations**

__ __

Suppose that the components of the plane stress tensor_{}, as expressed by equation (9), are defined at any point _{} of the vertical plane_{}. The variation of stress components when the reference
system attached to point _{} is rotated with a
counterclockwise angle _{}, as illustrated in Figure 5, is the subject of this section.

Figure 5 Representation of the Plane State
of Stress

(a) Normal Planes and (b) Rotated Planes

The transformation relations between stresses _{}, _{} and_{}pertinent to a rotated plane and the stresses_{}, _{} and _{} are obtained by
writing the equilibrium equations for the infinitesimal triangular element depicted
in Figure 6. The inclined plane is defined by its positive normal _{}which is rotated counterclockwise with an angle_{} from the horizontal direction _{}.

Figure 6 Equilibrium of the Infinitesimal
Triangular Element

(a) Stresses and (b) Forces

Using the notation shown in Figure 6.b the equilibrium
equations are written as:

_{}

_{} (10)

_{}

_{} (11)

From Figure 6 the following
geometrical relations can be derived:

_{} (12)

_{} (13)

Substituting equations (12) and (13)
into equations (10) and (11) and using the shear stress duality principle the
following expressions are obtained for normal_{} and shear_{}stresses:

_{} (14)

_{} (15)

Equations (14) and (15) can be re-written using the trigonometric
relations between the angle_{} and the double angle_{}:

_{} (16)

_{} (17)

Equation (16) and (17) are called the **plane stress transformation equations.**

In general, two faces are needed to express the plane stress
tensor around a point _{}. Consequently, the formulae (16) and (17) are applied twice:
first, considering the rotation angle _{} and, secondly, for the
complementary angle_{}. The notation is illustrated in Figure 7. The rotation
angles _{}and _{}are related as:

_{} (18)

The relation between the double angles necessary in equations
(16) and (17) is obtained as:

_{} (19)

Consequently, the following trigonometric relations can be
established:

_{} (20)

_{} (21)

Figure 7 Stresses on Orthogonal Rotated Faces

The stresses on two orthogonal rotated faces are expressed
as:

_{} (22)

_{} (23)

_{} (24)

_{} (25)

If equations (22) and (24) are summed, the invariance of the
summation of normal stresses is established:

_{} (26)

__ __

The maximum and minimum normal stresses are called **principal stresses **and mathematically
represent the extreme values of the normal stress function _{}. The extreme values are obtained by imposing the condition that
the first derivative of the normal stress _{} relative to the
rotation angle _{} is zero:

_{} (27)

The explicit expression for equation (27) is obtained by differentiating
equation (16):

_{} (28)

Dividing by 2*_{}the trigonometric equation (28) is transformed into equation
(29) relating the tangent of twice the principal directions angle, _{}, to the stresses ion the orthogonal planes _{} and _{}:

_{} (29)

The angle _{}represents the angle for which the normal stress_{} reaches its extreme value. The geometrical illustration of
the equation (29) is presented in Figure 8.a where the distance_{}is calculated as:

_{} (30)

Figure 8 Geometrical Representation of Equation (29)

Solution of the trigonometric equation (29) yields two
solutions, _{} and_{}, where the two angles are related as:

_{} (31)

From equation (31), the orthogonality of the two principal
directions is established:

_{} (32)

To calculate the values of the normal stress _{}corresponding to the angles _{} and _{} it is necessary to
evaluate the trigonometric functions _{}and _{} contained in equation
(16). Using the notation shown in Figure 8.a and the trigonometric relations (20)
and (21) these functions can be expressed as:

_{} (33)

_{} (34)

_{} (35)

_{} (36)

Substituting first the trigonometric expressions (33) and (34)
into equation (16) and then (35) and (36) the **principal stresses** _{} and _{}are obtained:

_{} (37)

_{} (38)

The average normal stress is calculated as:

_{} (39)

The principal stresses are schematically depicted in Figure 9.

Figure 9 Principal Stresses and Directions

The right-hand expression of equation (28) being identical with
the expression (17), representing the shear stress_{}, allows equation (28) to be re-written as:

_{} (40)

Equation (40) indicates that *the principal normal stresses are obtained for a rotated plane where
the shear stress is zero*.

The invariance of the sum of the normal stresses is again
shown to be valid for the case of the principal stresses. Summation of
equations (37) and (38) yields:

_{} (41)

To identify which of the two angles, _{} or_{}, corresponds to the maximum principal stress _{} the second derivative
of the function _{} relative to the
rotation angle _{} is employed. The
condition for the point to be a maximum is:

_{} (42)

The condition (42) is explicitly written as:

_{} (43)

The inequality (43) can be manipulated and cast in a new
form:

_{} (44)

If the _{}is expanded the following trigonometric expression is established:

_{} (45)

Substituting equations (29) and (45), representing the _{} and _{}, into the inequality (44) the following expression is
obtained:

_{} (46)

The condition for the inequality (46) to hold true is:

_{} (47)

__Note:__ It is important to note that for the inequality (47) to hold true,
the signs of the tangent of the angle _{} and shear stress_{} must be identical.

The angle corresponding to the direction of the maximum normal
stress can also be obtained by successively assigning to the angle _{} in equation (16) the
values _{} and _{} and observing which
angle produces the maximum principal stress.

__ __

The maximum shear stresses are determined in a similar
manner as the principal stresses. The extreme condition for the shear stress
function_{}contained in equation (17) is written as:

_{} (48)

The explicit format of equation (48) is obtained by
differentiating the expression (17):

_{} (49)

Dividing by _{} the trigonometric
equation (49), the tangent of the principal directions angle _{} is obtained:

_{} (50)

The angle _{}represents the angle for which the shear stress_{} reaches its extreme value. Equation (50) is illustrated in
Figure

Figure 10 Angular Relation between _{}and _{}

Solving the trigonometric equation (50), two solutions _{} and_{}are obtained. They are related as:

_{} (51)

Dividing equation (50) by two (2), the orthogonality of the two
angles_{}and_{} is obtained:

_{} (52)

Equation (50) indicates that a relation between the angles _{}and _{}can be established. With this intent, equation (50) is recast
into a new format as follows:

_{} (53)

First, multiplying by the terms in the denominator, equation
(53) becomes:

_{} (54)

Then, equation (54) is simplified as:

_{} (55)

Therefore,

_{} (56)

The relationship between angles _{} and _{} is calculated from
equation (56):

_{} (57)

Still, equation (57) does not indicate how to identify the
direction of the maximum shear stress. By examination of Figure 10, the
following angular relations can be established:

_{} (58)

The relationship between the angles of the maxim principal
and shear stress directions, _{} and _{}, is obtained from (58) as:

_{} (59)

Again using the notation shown in Figure 10, the following
trigonometric relations are obtained:

_{} (60)

_{} (61)

_{} (62)

_{} (63)

Successively substituting the two groups of expressions, (60)
and (61), and, (62) and (63), into equation (17) the maximum and minimum shear
stresses are calculated as:

_{} (64)

_{} (65)

The normal stresses corresponding to the maximum and minimum
shear stresses are calculated by substituting the expressions (60) through (63)
into equation (16):

_{} (66)

_{} (67)

The maximum and minimum shear stresses and the corresponding
normal stresses are illustrated in Figure 11.

Figure 11 Relationships between Principal Planes and Maximum
Shear Stress Planes

__Note:__ From Figure 12.b it can be concluded that, in contrast, to the
principal planes which are free of shear stress, the planes on which the shear
stress achieves extreme values are not necessarily free of normal stresses.

**5 Mohr’s Circle for Plane Stresses**

__ __

Mohr’s circle is a graphical construction reflecting the
variation of the plane state of stress around a particular point, including
information pertinent to the principal and maximum shear stresses.

From equations (16) and (17), first squared and then summed,
the following relation is obtained:

_{} (68)

Equation (68) represents the equation of a circle of radius_{} defined in the _{} plane. The center of
the circle is located at point_{}. The values of circle radius_{}and average normal stress_{}are calculated employing equations (30) and (39), respectively.

Intersecting the equation of the circle (68) with the
horizontal axis _{} the intersection points
_{} and _{}are obtained:

_{} (69)

_{} (70)

It can be concluded that these intersection points represent
the principal stresses _{}and_{}.

The Mohr’s circle for plane stress condition is drawn
relative to a Cartesian system with the abscissa and the ordinate axis
representing the normal stresses _{} and the shear stress_{}, respectively. The following sign convention is employed as
illustrated in Figure 12:

(a) the positive shear stress_{}axis is downward**;**

(b) ** **the positive angle is measured
counterclockwise**;**

(c) the shear stress on a face plots as positive shear if tends to rotate
the face counterclockwise.

Figure 12 Mohr’s Circle Notation

__Note:__ The positive direction of the vertical axis, representing the
shear stress_{}, pointing downward (sign convention (a)) is elected in order
to be able to enforce the positive measurement of the angle (sign convention
(b)). Examining the notations shown in Figure 12 clarifies that all the angles
are measured from the line _{}(_{}) in the anticlockwise direction.

Morh’s circle for plane stress is constructed in the following
steps:

(a) The coordinate system is drawn as shown in Figure 12. The horizontal
axis represents the normal stress _{}, while the vertical axis represents the shear stress _{}. To gain full advantage of the graphical benefits of the
method it is necessary that the drawing to be made on scale. However, the
method is also helpful as a conceptual tool in combination with the governing
equations wherein it may be drawn more roughly. The representation considers
that the following conditions are met:_{} and _{};

(b) Using the calculated values of the normal stresses_{}and_{} and the shear stress_{} two points noted as_{}and_{} are placed on the drawing. The line _{} intersects the
horizontal axis at point _{} which represents the
center of the Mohr’s circle;

(c) The distance _{}represents the radius of the circle. Using the radius _{}and the position of the center_{} the Mohr’s circle is constructed. The intersection points, _{}and_{}_{,} between the circle and the horizontal axis
represent the maximum and the minimum principal stresses;

(d) The value of the _{} can be calculated from
the graph. The angle _{} is identified on the
graph by the _{} angle and is measured
from the _{}to the principal directions line in the counterclockwise
direction;

(e) The lines_{} and_{}represent the principal direction1 (associated with the
maximum principal stress) and 2 (associated with the minimum principal stress),
respectively.

Every point on the Mohr’s circle corresponds to a pair of
stresses _{} and _{} on a particular face. To
emphasize the face involved the point is labeled identically with the face
where it belongs. For example, the face_{}, _{}and _{}are represented on the Mohr’s circle by the points_{}, _{}and_{}. To reinforce the shear stress sign convention (c) two icons
indicating the rotation sense induced by the shear stress are shown in Figure 12.
The angle measured from _{} to the radius line _{}in the counterclockwise direction is equal to twice the angle
of the plane rotation _{}.

__Note:__ The points _{}and_{} represent the case of orthogonal planes having normals
parallel to axes _{}and_{}, respectively. The line _{} corresponds to angle _{}. The double angle _{} of the maximum
principal direction is measured from the line _{} to the line_{}. By these conventions, the double angle sense is established
as being positive in the counterclockwise direction. The angle _{} is the angle _{} and has the same
direction as the double angle _{}. The angle associated with the minimum direction _{} is perpendicular to
the angle_{}. The stresses corresponding to a plane rotated with an angle
_{} are obtained by
placing on the circle the radius_{} located by measuring in the counterclockwise direction an
angle of _{} from the line_{}. The corresponding stresses _{}and_{} are a function of the location of the point_{} position in the _{}coordinate system and may be obtained by scaling them from
the figure or by the use of the analytical equations (30), (39) and (68). The
opposite point _{}represents the stresses on the orthogonal rotated face.

*In comparison with the
technique used to show the principal axes in the *_{} representation
(Figures 7 through 11) the plot obtained from the Mohr’s circle appears to be
misleading. The cause is that the Mohr’s circle is drawn in the _{} coordinate system. In
the _{}representation the principal directions are correctly plotted
by artificially rotating the principal directions obtained from the Mohr’s
circle around the point _{}with an angle _{} measured in the
counterclockwise direction.

**6 Principal Stresses Distribution in Beams**

__ __

One of the most important applications of the plane state of
stress theory described above is found in the study of variation of the
stresses in beams under non-uniform bending. Recall from Lecture 7 that under
some imposed kinematic assumptions, a beam subjected to transversal loading is
in a state of plane stress. With the exception of some areas (around the
supports or the application points of concentrated loads) the beam theory characterizes
the existence of only two types of stresses: normal stress _{}and shear stress_{}. The normal stress_{} is calculated using Navier’s formula expressed by equation (71),
while the shear stress _{}is obtained employing Jurawski’s formula contained in
equation (72):

_{} (71)

_{} (72)

The notation used in the formulae (71) and (72) is explained
in Lecture 7 and is not repeated herein.

The plane stress tensor previously expressed in equation (9)
is written for the case of the beam in nonuniform bending as:

_{} (73)

The entire theoretical development described in the previous
sections can be without restriction applied to the study of the particular
plane stress tensor (73). Consequently, the variation of the stresses around
any point in a beam subjected to nonuniform bending can be calculated. Figure 13
represents an example of the application of plane stress theory for the case of
a simply supported beam.

In the example, the beam has a rectangular cross-section and
is subjected to a single concentrated force _{} located at the
mid-span. It is evident that the ratios of the beam dimensions and the loading
do not violate any of the assumptions related to the applications of the formulae
(71) and (72). The shear force diagram _{}and the bending diagram_{}, where the axis identification indices were dropped for
clarity, are plotted.

Figure 13 Simple Supported Beam

The geometrical characteristics of the rectangular
cross-section involved in the evaluation of the formulae (71) and (72) are:

_{} (74)

_{} (75)

_{} (76)

For the left half of the beam _{}the shear force _{}and the bending moment _{} are expressed as:

_{} (77)

_{} (78)

Substituting equations (74) through (78) into equations (71)
and (72), the expressions for normal and shear stresses are obtained as:

_{} (79)

_{} (80)

__Note:__ The minus (-) sign appearing in formula (81) has been inserted in
order to comply with the shear sign convention (c).

To obtain an illustrative variation of the principal
stresses, the rectangular domain of the beam is divided by superimposing a
rectangular mesh. For the case under study, the mesh has five spaces in the longitudinal
_{} direction and eight
spaces in the vertical _{} direction. Using
Mathcad programming capabilities the principal stresses and corresponding
angular directions can be easily calculated for every point of the mesh. The
principal stresses calculated for two cross-sections_{}and _{} and all nine points
vertically describing the cross-sections are contained in Table 1. The ratios _{}and _{}are tabulated instead of the _{}and_{}, where_{}.

A review of the results presented in Table 1 shows that at
the extreme fibers the principal stresses correspond with the normal stresses
and reach the maximum values. At the extreme fiber locations the shear stress _{}is zero. The situation is different for the case of
wide-flange beams where both _{}and _{}have significant values at the junction between the web and
the flange.

__Table 1__

Today, with the help of modern computer codes, the formulae
involved in the calculation of the principal stresses and directions can be
computed using a very refined mesh. The graph containing the curves tangent to
the principal directions in every point of the mesh is called the **stress trajectory**. Two sets of curves
are drawn and they are orthogonal at every point. The stress trajectory graph
pertinent to the simply supported beam investigated above is pictured in Figure
14. A typical example of practical usage of the stress trajectory curves is the
placement of the reinforcement in reinforced concrete beam. Because the stress
trajectory graph does not give any indication about the magnitude of the
principal stresses another type of graph is also used. This is called a **stress contour plot **and contains curves
of equal principal stress magnitudes. The commercial codes employed today can
provide these plots.

Figure 14 Stress Trajectory Plot

__ __

The theoretical formulation derived above is used to
investigate the following practical case:

_{}_{}

_{}_{}

_{}_{}

The corresponding stress tensor is written as:

_{}

The state of stress for the case above is shown in Figure 13.

Figure 13 Example
Plane State
of Stress

The following values illustrated in Figure 13 are calculated
as:

_{}

_{}

_{}

_{}

_{}

Figure 14 Geometrical Relations

From equation (47) it is established that the angle related
to the maximum principal direction must have a negative tangent. Consequently,
the angles of the principal direction are:

_{}

_{}

The principal stresses, shown in Figure 15, are obtained as:

_{}

_{}

Figure 15 Principal Stresses

The angle of the maximum shear stresses is calculated as:

_{}

The maximum shear stresses are calculated as:

_{}

_{}

The normal stresses acting on the maximum shear planes are
calculated as:

_{}

_{}

Figure 16 Maximum Shear Stresses

The maximum shear stresses and the corresponding normal
stresses are illustrated Figure 16

Morh’s circle pertinent to the problem is illustrated in
Figure 17.

Figure 17 Mohr’s Circle

__Note:__ The points _{} and _{} represent the case of
orthogonal planes having the normal directions rotated with angles of _{}and _{}, respectively, from the _{} axis. Successively substituting
the above angular values in equations (16) and (18) the following stresses pertinent
to points _{} and _{} are obtained:

·
for _{}

_{}

_{}

·
for _{}

_{}

_{}