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Bulgara  Ceha slovaca  Croata  Engleza  Estona  Finlandeza  Franceza 
Germana  Italiana  Letona  Lituaniana  Maghiara  Olandeza  Poloneza 
Sarba  Slovena  Spaniola  Suedeza  Turca  Ucraineana 
Introduction
As discussed in Lecture 2, the general threedimensional state of stress is obtained, as was discussed in Lecture 2, by passing a set of three orthogonal planes though a point of the solid body and isolating the infinitesimal volume located around the point. The threedimensional state of stress is illustrated in Figure 1 where, for clarity, only the stresses drawn on the faces with positive normal are represented.
Figure 1 ThreeDimensional State of Stress
Mathematically the threedimensional state of stress is represented by a generalized stress tensor_{}defined by nine distinct stress components:
_{} (1)
Applying the shear stress duality principle the number of independent tensorial components is reduced from nine to six and, consequently, the symmetric shear stress components are related as:
_{} (2)
_{} (3)
_{} (4)
There are instances when some of the stress tensor components vanish and the general threedimensional 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 nonzero 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
The plane stress tensor_{}is defined by three nonzero components:
_{} (9)
Several of the conditions encountered in previous lectures, including the study of the axial and torsional deformation and pure and nonuniform 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 uniaxial, pure shear and biaxial, are illustrated in Figure 4.
Figure 4 Examples of
(a) UniAxial, (b) Pure Shear and (c) BiAxial
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
(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 rewritten 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)
Principal Stresses
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 righthand expression of equation (28) being identical with the expression (17), representing the shear stress_{}, allows equation (28) to be rewritten 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.
Maximum Shear Stresses
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.
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.
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 nonuniform 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 crosssection and is subjected to a single concentrated force _{} located at the midspan. 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 crosssection 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 crosssections_{}and _{} and all nine points vertically describing the crosssections 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 wideflange 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
Example
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
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 _{}
_{}
_{}
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