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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.
Definitions
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:
(1)
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:
(2)
(3)
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.
(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 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) :
(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 cross-section. The polar moment of inertia is
defined as:
(8)
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):
(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.
Parallel-Axis Theorems for Moment of Inertia
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:
(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 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 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 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:
(20)
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:
(21)
Equation (21) represents the parallel-axis 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 (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:
(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 angleformulae:
(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
re-written using the double angleas:
(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 angleis
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 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:
(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 re-written 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 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:
(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 counter-clockwise
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)
Mohrs 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 pointsand
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 Mohrs circle is depicted in Figure 7.
Figure 7 Morhs Circle Representation
Note: Practically the Mohrs 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 Mohrs circle;
(c)
The distance is the radius of the circle. Using the radius
and the position of the center
the Mohrs 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
Z, the reflection of the point Z in reference to axis
, has to be constructed. The lines ZP1 and ZP2 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.
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 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:
=0
=0
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 Mohrs 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 Mohrs Circle
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