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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 
The definitions of the beam and crosssection were specified in the previous lectures. Some geometrical characteristics of the crosssection, such as the area and moments of inertia, have a central role in the theoretical development of Mechanics of Materials and are the main subject of this lecture.
Definitions
The beam crosssection is a plane area bounded by a closed curve _{}. For mathematical convenience the Cartesian plane coordinate system _{}, as illustrated in Figure 1, is defined.
Figure 1 CrossSection Area
The total area of the crosssection is calculated as:
_{} (1)
The integral contained in equation (1) defines the summation of the differential areas_{}over the two defining variables _{} and_{}. The area is characterized by units of length squared [L^{2}], with the symbol [L] representing length.
The first moments of the area _{} about the coordinate system axes _{}and _{}are called the static moments. These are defined as:
_{} (2)
_{} (3)
The units of the static moments are [L^{3}].
The geometric center of the crosssection is called the centroid. Equations (4) and (5) are used to calculate the plane position _{}of the centroid_{}. The notation is shown in Figure 2.
_{} (4)
_{} (5)
Figure 2 Centroid Location
Note: Analyzing the integrals contained in the equations (4) and (5), the following important conclusions regarding the position of the centroid _{} may be drawn:
(a) if the crosssection area possesses one axis of symmetry, the centroid _{} lies on that axis;
(b) if the crosssection area possesses two axes of symmetry, the centroid _{} is located at their intersection;
(c) if the crosssection area is symmetric about a point, the centroid _{}is located at the location of that point.
These cases are illustrated in Figure
Figure 3 Types of Symmetry for Plane Area
(a) One Symmetry Axis, (b) Two Symmetry Axes and (c) Point Symmetry
The second moments of the crosssection area _{} about the coordinate system axes _{}and _{}are called the moments of inertia and are defined as (see Figure 4 for notation) :
_{} (6)
_{} (7)
Figure 4 Second Moments of Inertia Notation
The summation of the moment of inertia _{} and _{}is called the polar moment of inertia and represents the second moment of inertia about the axis _{}normal to the crosssection. The polar moment of inertia is defined as:
_{} (8)
The unit of the second moments of inertia is [L^{4}].
Note: The second moments of inertia are always positive values.
Another important geometric property is the product of inertia of the area _{} also called in the Romanian technical literature the centrifugal moment of inertia. The definition of the centrifugal moment of inertia is given in equation (9):
_{} (9)
The unit of the product of inertia is [L^{4}].
Note: Contrary to the moments of inertia which are always positive values, the product of inertia moment of inertia may have either a positive or negative value. If the area _{} has an axis of symmetry the product moment of inertia _{}.calculated for a coordinate system including that axis is zero.
ParallelAxis Theorems for Moment of Inertia
The above described moments of inertia are usually calculated relative to a coordinate system _{} anchored at the crosssection centroid _{}. A new translated coordinate system_{}, with axes _{} and _{} parallel to the centroidal axes _{} and _{}, respectively, is defined in Figure 5. The correspondence between the moments of inertia relative to this new coordinate system _{} and those calculated with respect to the centroidal coordinate system _{} is studied in this section.
Figure 5 ParallelAxis Theorems Notation
The position _{} of an arbitrary point located on the crosssection area _{} expressed relative to the _{} coordinate system is written as:
_{} (10)
_{} (11)
where the distances _{}and _{} are the horizontal and vertical distances, respectively, between the two coordinate systems considered.
The moments of inertia about axes _{}and_{}, _{} and_{}, are expressed in the equations (6) and (7), respectively. Substitution of equations (10) and (11) into equations (6) and (7) yields the following new expressions for the moments of inertia:
_{} (12)
_{} (13)
Note that the integrals _{} and _{} become zero when the static moments are calculated relatively to the centroidal axes.
The moments of inertia calculated about centroid axes are expressed as:
_{} (14)
_{} (15)
Substituting equations (14) and (15) into equations (12) and (13) final expressions of the moments of inertial calculated about the axes of the translate system _{} are obtained as:
_{} (16)
_{} (17)
Equations (16) and (17) are called the parallelaxis theorem for moments of inertia.
Note: Examination of equations (16) and (17) shows that the minimum values for the moments of inertia are obtained when the axes _{}and_{}are coincident with the centroidal axes _{} and _{}, respectively.
A similar approach can be used for the case of the polar moment of inertia _{}defined by equation (8). Substituting equations (10) and (11) into equation (8), the polar moment of inertia about point _{}is obtained as:
_{} (18)
Grouping the terms in the equation (18), the final expression may be written as:
_{} (19)
where _{} and _{}
Equation (19) represents the parallelaxis theorem for the polar moment of inertia.
The parallelaxis theorem for the product of inertia is derived in a similar manner to that for the moments of inertia. By substitution of equations (10) and (11) into equation (9), the following expression is obtained:
_{} (20)
Since the coordinate system _{} passes through the centroid of the crosssection the integrals representing the static moments are zero and consequently, the equation (20) reduces to:
_{} (21)
Equation (21) represents the parallelaxis theorem for the product of inertia.
Moment of Inertia about Inclined Axes
Consider Figure 6 where a new coordinate system _{} is shown rotated by an angle _{} from the position of the original coordinate system_{}. The rotation angle _{} is positive when increasing in the trigonometric positive sense (counterclockwise). This convention corresponds to the righthand rule previously used.
Figure 6 Axes Rotation Notation
The position _{}of a current point located in the crosssection _{} relative to the coordinate system _{}can be written as:
_{} (22)
_{} (23)
Accordingly, with the definition equations (6) and (7) the moments of inertia in the rotated coordinate system _{}follow as:
_{} (24)
_{} (25)
After algebraic manipulations the moments of inertia _{} and _{} are obtained as:
_{} (26)
_{} (27)
Further, the equations (26) and (27) are expressed in an alternative form by substitution of the double angle_{}formulae:
_{} (28)
_{} (29)
The product of inertia _{}is defined as:
_{} (30)
After the algebraic manipulations, the equation (30) becomes:
_{} (31)
Further more, the equation (31) is rewritten using the double angle_{}as:
_{} (32)
If the derivatives of the moments of inertia shown in equations (28) and (29) are taken relative to the double angle _{}an interesting result is obtained:
_{} (33)
_{} (34)
Note: The first derivative of the moment of inertia relative to the double angle_{}is the product of inertia.
The sum of equations (28) and (29) reveal the following important relationship:
_{} (35)
Note: Equation (35) indicates the invariance of the sum of the moments of inertia with the rotation of the axes.
Principal Moments of Inertia
The moments of inertia _{} and_{}, expressed by equations (28) and (29), are functions of the angle _{} of the rotated coordinate system_{}. The extreme values (the maximum and minimum) of the moments of inertia _{} and_{} are called principal moments of inertia. The corresponding values of the rotation angle _{} describe the principal axes of inertia. The principal axes of inertia passing through the centroid of the crosssection area are called centroidal principal axes of inertia.
As known from Calculus, the condition for a real function to have an extreme point (a maximum or minimum) is that the first derivative of the function be equal to zero at that point. For the case of the moments of inertia_{}and_{}, using equation (33) and (34), the condition of extreme, is written as:
_{} (36)
From equation (36) the value of the angle _{} corresponding to the principal directions is obtained as:
_{} (37)
From the trigonometry, it is known that equation (37) has two solutions, _{} and _{}, related as shown in the equation (38):
_{} (38)
Consequently, it is concluded that the principle directions are perpendicular to each other:
_{} (39)
Using the following trigonometric identities
_{} (40)
_{} (41
and substituting them into equations (33) and (34) the final expressions for the principle moments of inertia are obtained as:
_{} (42)
_{} (43)
Note: The invariance of the sum of the moments of inertia is also preserved for the case of the principal moments of inertia. By addition of equations (42) and (43) the invariance is proven:
_{} (44)
To identify which of the two angles, _{} or_{}, corresponds to the maximum moment of inertia _{} the second derivative of the function_{}, shown in equation (28), is used. The condition for the point to be a maximum is:
_{} (45)
The expression (45) is rewritten as:
_{} (46)
After the trigonometric manipulations and the usage of the equation (37) the inequality (46) became:
_{} (47)
The condition for the inequality (47) to hold is:
_{} (48)
Note: Here, in order for inequality (48) to hold true, the sign of product of inertia _{}must be opposite to that of the tangent of the angle _{}.
Practically, the angle corresponding to the direction of the maximum moment of inertia is obtained by successively assigning to angle _{}the values _{} and _{} and identifying which angle verifies the inequality (48).
Maximum Product Moment of Inertia
Consider the angle of the principal directions_{}established and the original coordinate system_{} shown in Figure 6 rotated such that the _{}and _{}axes align with the principal directions. Then, the following expressions hold:
_{} (49)
_{} (50)
_{} (51)
Substituting equations (49) through (51) into the equations (28), (29) and (32) the expression for the moments of inertia as functions of the principal moments of inertia are obtained:
_{} (52)
_{} (53)
_{} (54)
From equation (54), it is easy to see that the maximum value for the product of inertia is obtained when:
_{} (55)
Then,
_{} (56)
The maximum value of the product of inertia is obtained for an angle of rotation _{} measured in the counterclockwise direction from the position of the principal axes is expressed in equation (57).
_{} (57)
Substituting the principal moments of inertia given by equations (43) and (44) and (42) into equation (57) a new expression for the _{} is obtained:
_{} (58)
The corresponding moments of inertia are obtained by replacing _{} in equations (52) and (53):
_{} (59)
Mohr’s Circle Representation of the Moments of Inertia
A very interesting and useful relationship, shown in equation (60), is obtained by manipulating the equations (28) and (32) in the following manner: (a) the equation (28) is rearranged by moving in the left hand side the term _{} and then squaring both sided of the equation, (b) the equation (332) is squared and (c) adding together the previous obtained equations
_{} (60)
The following notation is employed in the implementation of the equation (60):
_{} (61)
_{} (62)
_{} (63)
_{} (64)
Substitution of equations (61) through (64) into the equation (60) yields a new form for equation (60)
_{} (65)
Geometrically, equation (65) represents the equation of a circle located in the _{}plane. The circle has center _{} located at _{}and radius_{}.
The coordinates of the intersection points, _{}and_{}, between the circle and the horizontal axis_{}, are obtained by solving the algebraic system composed of equation (65) and the equation of the axis_{}:
_{} (66)
_{} (67)
Substituting equations (63) and (64) into equations (66) and (67) the position of the intersection points_{}and_{} are expressed as shown in equations (68) and (69) and are identified as the principal moments of inertia.
_{} (68)
_{} (69)
The graphical representation of the Mohr’s circle is depicted in Figure 7.
Figure 7 Morh’s Circle Representation
Note: Practically the Mohr’s circle is constructed using the following steps:
(a) The coordinates system _{} is drawn as shown in Figure 2.7. The horizontal axis _{} represents the moments of inertia, while the vertical axis _{} represents the product of inertia (note that the positive axis _{} is drawn upwards). The drawing should be done roughly to scale. The representation considers that the following conditions are met:_{} , _{}, _{}and _{};
(b) Using the calculated values of the moments of inertia_{}and_{} and the product of inertia_{} two points noted as _{} and _{} are placed on the drawing. The line _{} intersects the horizontal axis in point _{} which represents the center of the Mohr’s circle;
(c) The distance _{}is 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 moments of inertia;
(d) The absolute value of the _{} can be calculated from the graph;
(e) The angle of the principal direction 1 is the angle measured in the counterclockwise direction between lines CY and CP1. To obtain the position of the two principal directions corresponding to the crosssection system _{} an additional point Z’, the reflection of the point Z in reference to axis _{}, has to be constructed. The lines Z’P1 and Z’P2 represent the principal direction 1 (associated with the maximum moment of inertia) and 2 (associated with the minimum moment of inertia), respectively. The two directions can then be transcribed on the crosssection sketch.
Radii of Gyration
The square root of the ratio of the moment of inertia to the area is called the radius of gyration and has the unit of [L].
The radii of gyration relative to the original coordinate system _{}are calculated as:
_{} (70)
_{} (71)
For the case of the principal moments of inertia, the corresponding radii of gyration are:
_{} (72)
_{} (73)
Examples
To clarify the theoretical aspects and the formulae derived in this lecture, two examples are presented.
8.1 Rectangle CrossSection
A rectangular crosssection is shown in Figure 8. The rectangle is characterized by two symmetry axes and consequently, the centroid _{} is located at their intersection. The coordinate system used is the centroidal coordinate system shown in the Figure 8.
Figure 8 Rectangular CrossSection
The following crosssectional characteristics are calculated using the formulae previously developed:
_{}
_{}=0
_{}=0
_{}
_{}
_{}
_{}
_{}
It is shown thus, that for a rectangular crosssection the centroidal coordinate system represents the principal axes of inertia_{}.
If the moment of inertia about the axis coinciding with the lower edge of the rectangle is required, using the notation shown in Figure 8(b), the parallelaxis theorem for moments of inertia is employed:
_{}
8.2 Composite CrossSection
The Lshaped crosssection illustrated in Figure 9(a) is proposed for investigation. The vertical and horizontal legs have a height of _{} and_{}, respectively, while thickness _{} is uniform for the entire figure. The Lshaped crosssection can be decomposed into two rectangular areas, _{} and _{}, representing the areasof the individual legs of the crosssection. The distances of the centroids,_{}and_{}, of the two rectangular areas are positioned relative to the coordinates system _{} without any difficulty as depicted in Figure 9(b).
Figure 9 LShaped CrossSection
The individual area of each leg and total area of the Lshaped crosssection are calculated as:
_{}
_{}
_{}
The position of the crosssection centroid _{} is obtained:
_{}
_{}
_{}
_{}
A new coordinate system aligned with the system _{}and with the origin at the centroid _{}of the entire crosssection is established as _{}. The following calculations are performed with reference to this centroidal coordinate system. The moments of inertia about the centroidal coordinate axes are calculated as:
_{}
_{}
_{}
The principal moments of inertia are obtained as:
_{}
_{}
The angle of the principal direction of inertia is calculated as:
_{}
_{}
_{}
Using the test contained in the equation (48) to determine if the rotation angle_{}is the angle of the principal direction results in the following:
_{}
Consequently, the angle _{}is the angle of the direction of the minimum moment of inertia and_{}, while the complementary angle _{}represents the direction of the maximum moment of inertia.
The angles _{}and _{}are illustrated in Figure 10.
Figure 10 LShaped CrossSection Principal Directions
The radii of gyrations are obtained as:
_{}
_{}
_{}
_{}
The construction of the Mohr’s circle is conducted as explained in Section 6. Using the moments and the product of inertia calculated above the following values are determined:
_{}
_{}
_{}
_{}
Figure 11 LShaped CrossSection Mohr’s Circle
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