Chapter: Mechanical - Strength of Materials - Deflection of Beams

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Moment area method

Theorems of Area-Moment Method: Theorem I: The change in slope between the tangents drawn to the elastic curve at any two points A and B is equal to the product of 1/EI multiplied by the area of the moment diagram between these two points.


Moment area method

 

Theorems of Area-Moment Method

 

Theorem I

 

The change in slope between the tangents drawn to the elastic curve at any two points A and B is equal to the product of 1/EI multiplied by the area of the moment diagram between these two points.

 

Theorem II

 

The deviation of any point B relative to the tangent drawn to the elastic curve at any other point A, in a direction perpendicular to the original position of the beam, is equal to the product of 1/EI multiplied by the moment of an area about B of that part of the moment diagram between points A and B.

 

 

 

 

Rules of Sign

 

1.     The deviation at any point is positive if the point lies above the tangent, negative if the point is below the tangent.

 

2.     Measured from left tangent, if ? is counterclockwise, the change of slope is positive, negative if ? is clockwise.

 

 

Columns –End conditions

 

Columns -end conditions

 

What is a Column or Strut?

 

Any machine member, subjected to the axial compressive loading is called a strut and the vertical strut is called column

 


 

The columns are generally categorized in two types: short columns and long columns. The one with length less than eight times the diameter (or approximate diameter) is called short column and the one with length more than thirty times the diameter (or approximate diameter) is called long column.

 

Ideally, the columns should fail by crushing or compressive  stress and it normally happens for the short columns, however, the long columns, most of the times,  failure occurs by buckling.

 

Euler’s                       Buckling   Formula

 

To get the correct results, this formula should only be applied for the long columns. The buckling load calculated by the Euler formula is given by:

 

Fbe = (C*?2*E*I)/

 

Equivalent length of a column

 

Strength Of Columns

 

A stick of timber, a bar of  iron, etc., when used to sustain end loads which act lengthwise of the pieces, are called  columns, posts, or struts if they are so long that they would bend before breaking. When they are so short that they would not bend before breaking, they are called short blocks, and their compressive  strengths are computed by means of equation 1. The strengths of columns cannot, however, be so simply determined, and we now proceed to explain the  method of computing them.

 

77. End  Conditions. The strength of a column depends in part on the way in which its ends bear, or are joined to other parts of a  structure, that is, on its " end conditions." There are practically but three kinds of end conditions, namely:

 

1.  "Hinge" or "pin" ends,

 

2. " Flat" or " square " ends, and

 

3. "Fixed" ends.

 

(1)             When a column is fastened to its  support at one end by means of a pin about which the column could rotate if the other end were free, it is said to be "hinged" or "pinned" at the former end. Bridge posts or columns are often hinged at the ends.

 

(2)             A column either end of which is flat and perpendicular to its axis and bears on other parts of the structure at that  surface, is said to be "flat" or " square" at that end.

 

(3)             Columns are sometimes riveted near their ends directly to other parts of the structure and do not bear directly on their ends; such are called " fixed ended." A column which bears on its flat ends is often fastened near the ends to other parts of the structure, and such an end is also said to be " fixed." The fixing of an end of a column stiffens and therefore strengthens it more or less, but the strength of a column with fixed ends is computed as though its ends were flat. Accordingly we have, so far as strength is concerned, the following classes of columns:

 

78. Classes of Columns. (1) Both ends hinged or pinned; (2) one end hinged and one flat; (3) both ends flat.

 

Other things being the same, columns of these three classes are unequal in strength. Columns of the  first class are the weakest, and those of the third class are the strongest.


Fig. 46.

 

70. Cross=sections of Columns. Wooden columns are usually  solid, square, rectangular, or round in section; but sometimes they are "built up" hollow.  Cast-iron columns are practically always made hollow, and rectangular or round in section. Steel columns are made of single rolled shapes - angles, zees,  channels, etc.; but the larger ones are usually "built up" of several shapes. Fig. 46, a, for  example, represents a cross-section of a "Z-bar" column; and Fig. 46, b, that of a "channel" column.

 

80. Radius of Gyration. There is a quantity appearing in almost all formulas for the strength of columns, which is called "radius of gyration." It depends on the form and extent of the cross-section of the column, and may be defined as follows:

 

The radius of gyration of any  plane figure (as the section of a column) with respect to any line, is such a length that the square of this length multiplied by the area of the figure equals the  moment of inertia of the figure with respect to the given line.

 

Thus, if A denotes the area of a figure; I, its moment of inertia with respect to some line; and r, the radius: of gyration with respect to that line; then


(9)

 

In the column formulas, the radius of gyration always refers to an axis through the center of gravity of the cross-section, and usually to that axis with respect to which the radius of gyration (and moment of inertia) is least. (For an exception, see example 3. Art. 83.) Hence the radius of gyration in this  connection is often called for brevity the "least radius of gyration," or simply the "least radius."

 

The moment of inertia of the square with respect to the axis is 1/12 a4- Since A = a2, then, by formula 9 above,


 

2. Prove that the value of the radius of gyration given for the hollow square in Table A, page 54, is correct.

 

The value of the moment of inertia of the square with respect to the axis is 1/12 (a4 - a1 4). Since A = a2 - a12,


 

 


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