The definitions of the beam and cross-section were specified in the previous lectures. Some geometrical characteristics of the cross-section, 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.
The beam cross-section 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 Cross-Section Area
The total area of the cross-section is calculated as:
The integral contained in equation (1) defines the summation of the differential areasover the two defining variables and. The area is characterized by units of length squared [L2], 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:
The units of the static moments are [L3].
The geometric center of the cross-section 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.
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 cross-section area possesses one axis of symmetry, the centroid lies on that axis;
(b) if the cross-section area possesses two axes of symmetry, the centroid is located at their intersection;
(c) if the cross-section 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 cross-section area about the coordinate system axes and are called the moments of inertia and are defined as (see Figure 4 for notation) :
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 cross-section. The polar moment of inertia is defined as:
The unit of the second moments of inertia is [L4].
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):
The unit of the product of inertia is [L4].
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.
The above described moments of inertia are usually calculated relative to a coordinate system anchored at the cross-section 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 Parallel-Axis Theorems Notation
The position of an arbitrary point located on the cross-section area expressed relative to the coordinate system is written as:
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:
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:
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:
Equations (16) and (17) are called the parallel-axis 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 andare 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:
Grouping the terms in the equation (18), the final expression may be written as:
Equation (19) represents the parallel-axis theorem for the polar moment of inertia.
The parallel-axis 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:
Since the coordinate system passes through the centroid of the cross-section the integrals representing the static moments are zero and consequently, the equation (20) reduces to:
Equation (21) represents the parallel-axis theorem for the product of inertia.
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 (counter-clockwise). This convention corresponds to the right-hand rule previously used.
Figure 6 Axes Rotation Notation
The position of a current point located in the cross-section relative to the coordinate system can be written as:
Accordingly, with the definition equations (6) and (7) the moments of inertia in the rotated coordinate system follow as:
After algebraic manipulations the moments of inertia and are obtained as:
Further, the equations (26) and (27) are expressed in an alternative form by substitution of the double angleformulae:
The product of inertia is defined as:
After the algebraic manipulations, the equation (30) becomes:
Further more, the equation (31) is re-written using the double angleas:
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:
Note: The first derivative of the moment of inertia relative to the double angleis the product of inertia.
The sum of equations (28) and (29) reveal the following important relationship:
Note: Equation (35) indicates the invariance of the sum of the moments of inertia with the rotation of the axes.
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 cross-section 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 inertiaand, using equation (33) and (34), the condition of extreme, is written as:
From equation (36) the value of the angle corresponding to the principal directions is obtained as:
From the trigonometry, it is known that equation (37) has two solutions, and , related as shown in the equation (38):
Consequently, it is concluded that the principle directions are perpendicular to each other:
Using the following trigonometric identities
and substituting them into equations (33) and (34) the final expressions for the principle moments of inertia are obtained as:
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:
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:
The expression (45) is re-written as:
After the trigonometric manipulations and the usage of the equation (37) the inequality (46) became:
The condition for the inequality (47) to hold is:
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).
Consider the angle of the principal directionsestablished 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:
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:
From equation (54), it is easy to see that the maximum value for the product of inertia is obtained when:
The maximum value of the product of inertia is obtained for an angle of rotation measured in the counter-clockwise direction from the position of the principal axes is expressed in equation (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:
The corresponding moments of inertia are obtained by replacing in equations (52) and (53):
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
The following notation is employed in the implementation of the equation (60):
Substitution of equations (61) through (64) into the equation (60) yields a new form for equation (60)
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:
Substituting equations (63) and (64) into equations (66) and (67) the position of the intersection pointsand are expressed as shown in equations (68) and (69) and are identified as the principal moments of inertia.
The graphical representation of the Mohrís circle is depicted in Figure 7.
Figure 7 Morhís Circle Representation
Note: Practically the Mohrís 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 inertiaand 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 Mohrís circle;
(c) The distance is the radius of the circle. Using the radius and the position of the center the Mohrís 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 counter-clockwise direction between lines CY and CP1. To obtain the position of the two principal directions corresponding to the cross-section system an additional point Zí, the reflection of the point Z in reference to axis , has to be constructed. The lines ZíP1 and ZíP2 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 cross-section sketch.
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:
For the case of the principal moments of inertia, the corresponding radii of gyration are:
To clarify the theoretical aspects and the formulae derived in this lecture, two examples are presented.
8.1 Rectangle Cross-Section
A rectangular cross-section 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 Cross-Section
The following cross-sectional characteristics are calculated using the formulae previously developed:
It is shown thus, that for a rectangular cross-section 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 parallel-axis theorem for moments of inertia is employed:
8.2 Composite Cross-Section
The L-shaped cross-section 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 L-shaped cross-section can be decomposed into two rectangular areas, and , representing the areasof the individual legs of the cross-section. 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 L-Shaped Cross-Section
The individual area of each leg and total area of the L-shaped cross-section are calculated as:
The position of the cross-section centroid is obtained:
A new coordinate system aligned with the system and with the origin at the centroid of the entire cross-section 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 angleis 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 L-Shaped Cross-Section Principal Directions
The radii of gyrations are obtained as:
The construction of the Mohrís 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 L-Shaped Cross-Section Mohrís Circle
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