The center of gravity of a rock (or any other three dimensional object) can be found by hanging it from a string. The line of action of the string will always pass through the center of gravity of the rock. The precise location of the center of gravity could be determined if one would tie the string around the rock a number of times and note each time the line of action of the string. Since a rock is a three dimensional object, the point of intersection would most likely lie somewhere within the rock and out of sight.
The centroid of a two dimensional surface (such as the cross-section of a structural shape) is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape. The planar surface (or figure) may represent an actual area (like a tributary floor area or the cross-section of a beam) or a figurative diagram (like a load or a bending moment diagram). It is often useful for the centroid of the area to be determined in either case.
Symmetry can be very useful to help determine the location of the centroid of an area. If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry. This means that if it were required to balance the area (or body or section) in a horizontal position by placing a pencil or edge underneath it, the pencil would be best laid directly under the line of symmetry.
If a body (or area or section) has two (or more) lines of symmetry, the centroid must lie somewhere along each of the lines. Thus, the centroid is at the point where the lines intersect. This means that if it were required to balance the area (or body or section) in a horizontal position by placing a nail underneath it, the point of the nail would best be placed directly below the point where the lines of symmetry meet. This might seem obvious, but the concept of the centroid is very important to understand both graphically and numerically. The position of the center of gravity for some simple shapes is easily determined by inspection. One knows that the centroid of a circle is at its center and that of a square is at the intersection of two lines drawn connecting the midpoints of the parallel sides. The circle has an infinite number of lines of symmetry and the square has four. (Two were described above - what are the other two lines of symmetry?)
The centroid of a section is not always within the area or material of the section. Hollow pipes, L shaped and some irregular shaped sections all have thir centroid located outside of the material of the section. This is not a problem since the centroid is really only used as a reference point from which one measures distances. The exact location of the centroid can be determined as described above, with graphic statics, or numerically.
The centroid of any area can be found by taking moments of identifiable areas (such as rectangles or triangles) about any axis. This is done in the same way that the center of gravity can be found by taking moments of weights. The moment of an large area about any axis is equal to the algebraic sum of the moments of its component areas. This is expressed by the following equation:
The Moment of Inertia (I) is a term used to describe the capacity of a cross-section to resist bending. It is always considered with respect to a reference axis such as X-X or Y-Y. It is a mathematical property of a section concerned with a surface area and how that area is distributed about the reference axis. The reference axis is usually a centroidal axis.
The moment of inertia is also known as the Second Moment of the Area and is expressed mathematically as:
In which:
Ixx = the moment of inertia around the x axis
A = the area of the plane of the object
y = the distance between the centroid of the object and the x axis
The Moment of Inertia is an important value which is used to determine the state of stress in a section, to calculate the resistance to buckling, and to determine the amount of deflection in a beam. For example, if a designer is given a certain set of constraints on a structural problem (i.e. loads, spans and end conditions) a "required" value of the moment of inertia can be determined. Then, any structural element which has at least that specific moment of inertia will be able to be utilized in the design. Another example could be in the inverse were true: a specific element is given in a design. Then the load bearing capacity of the element could be determined.
Let us look at two boards to intuitively determine which will deflect more and why. If two boards with actual dimensions of 2 inches by 8 inches were laid side by side - one on the two inch side and the other on the eight inch side, the board which is supported on its 2" edge is considerably stiffer than that supported along its 8" edge. Both boards have the same cross-sectional area, but the area is distributed differently about the horizontal centroidal axis.
Calculus is ordinarily used to find the moment of inertia of an irregular section. However, a simple formula has been derived for a rectangular section which will be the most important section for this course.
The importance of the distribution of the area around its centroidal axis becomes clear when comparing the values of the moment of inertia of a number of typical beam configurations. All of the members shown below are 2" x 6"; in cross section, equal in length and equally loaded.
BUILT-UP SECTIONS
It is often advantageous to combine a number of smaller members in order to create a beam or column of greater strength. The moment of inertia of such a built-up section is found by adding the moments of inertia of the component parts. This can be done<
B> if and only if the moments of inertia of each component area are taken about a common axis and if and only if the resulting section acts as one unit.
Built-Up Sections
TRANSFER FORMULA
There are many built-up sections in which the component parts are not symmetrically distributed about the centroidal axis. The easiest way to determine the moment of inertia of such a section is to find the moment of inertia of the component parts about their own centroidal axis and then apply the transfer formula. The transfer formula transfers the moment of inertia of a section or area from its own centroidal axis to another parallel axis. It is known from calculus to be:
Asymmetric Cross Sections
Bridge Girder Section