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TABLE OF CONTENT
Configuration
Start from scratch
Arcs
Fillets
Shear area
DXF picture file, scale, font size
Examples - description
Examples - files
Calculation formulas
Torsional properties (alfa)
Inertia calculates properties of any section which may be constructed from straight lines, arcs and circles. It's purpose is to supply mechanical designer with properties of any section for which no ready made formula may exists. Examples are given for the following sections:
All above figures were created by Inertia. Inertia calculates: center of gravity, area, all moduluses of inertia and all section moduluses. The section may be rotated about center of gravity (= COG.) and new set of moduluses of inertia and section moduluses is calculated for a given angle of rotation. Torsional constant and torsional modulus is calculated for solid, open or hollow sections. For the purpose of calculations, it is assumed that coordinate x is horizontal, coordinate y is vertical.
Inertia mimics potential planimeter and its input is based on a polygon. Needle of the planimeter has to be moved in clockwise fashion along the perimeter of the section to get positive results and in counter-clockwise fashion to get negative results. Therefore for the positive output, the polygon vertices must be numbered and entered in a clockwise sequence. Counter-clockwise sequence gives negative results. This way, it is possible to deduct a hole. For example, see the hollow section below.
Straight lines between two polygon vertices may be replaced by small or large arc, sharp corners may be filleted to a given radius.
Lines of the polygon are drawn to follow the movement of the planimeter needle, that is from vertice (n) to vertice (n + 1) e.g. from vertice 1 to 2, from 2 to 3 etc. The last line is drawn automatically from the last vertice to starting vertice (similar to command 'Close' in AutoCAD).
Any line between two vertices may be replace by arc. There are four options for an arc to be drawn based on arc radius and location of two points only. It's center may lay below or above the line and the arc may be large or small, see picture below. Arc is drawn from the point (n) to point (n + 1) as would be the line which is replaced by the arc.
Arcs are actually built from line segments. For this purpose, arc angle is divided into smaller fragments, and the arc is build in steps. Default maximum angle of step is 10 degrees, or arc is build from minimum 5 segments . The more line segments is used to represent the arc, the more accurate are the results. Default figures are reasonable compromise between complexity of the model and accuracy. They give about -0.5% error for a full circle. Maximum number of steps for any arc is 800. If you exceed this figure, the program shall stop execution.
Position of the center is indicated by the sign of the arc radius. Negative radius shall place the center above line which connects the arc terminal points 'n' and 'n + 1', positive radius place the center below the line.
Large or small arc is determined by the sign of the step. Positive step builds small arc, negative step builds large arc. With small arc, positive radius would bulge (convex) the positive polygon, while negative radius would excavate it (concave). With negative polygon it is opposite. It may be therefore said, that positive arcs tend to add area, while negative tend to decrease area of a polygon. Note further that the large arcs may reverse the course of polygon vertice numbering, that is from clockwise to counter-clockwise. If this is the case, you have to adjust the polygon numbering accordingly, to get positive movement along the arc, see 'Pin' example.
Fillet is used to round sharp corners. Circle of the fillet is tangent to two lines and therefore only the value of the radius must be specified. It is always built from five line segments. If the radius of the fillet is smaller than the first line of an arc, arcs may be filleted too, see example Segment.
Purpose of menu 'Configuration' is to make and record path to Inertia files. Go to directory from which you want to call Inertia and specify path to: editor (e.g. Notepad, Winword etc. - if you specify Inertia.exe as editor, Inertia would use its own build-in editor), path to Browser (e.g. Internet Explorer - Iexplore.exe, Netscape Navigator etc.), path to this file (Inertia.htm - Manual)), path to Dxfgen.txt file which is used in DXF generation process , and finally path to DXF viewer (e.g. MS Word, Excel, Powerpoint, AutoCAD etc.). This configuration is stored in hidden file Inertia.ini and you do not have to configure Inertia the next time you use it.
To enter new section into Inertia, click on File/Start from scratch and enter: number of polygons and number of full circles which form the section. Do not forget that holes must be specified with negative diameter to be deducted from the polygon. If a straight line between two vertices of a polygon has to be replaced by arc, enter number of arc definitions - an arc needs only one definition no matter how many times it is used in a polygon or polygons.
If specific torsional properties of the section are required, follow the input form and directions in torsional properties chapter.
Further you have to enter number of vertices for each polygon. Just prior to clicking on O.K. button, click and select the units in red letters located just above the button.
Inertia creates input table and you have to supply the numbers. Then save the file and using Commands menu, Run Inertia. After that you may create DXF file to review geometry of the section and, if required, do additional editing etc. Note that to get section centre of gravity, you have to Run Inertia prior generating DXF picture.
To visualize the section, program creates picture file which may be viewed by array of other programs like AutoCAD, Microsoft Word, Excel, Powerpoint etc. For Inertia, the most practical is MS Word. Create DXF file and run menu 'Insert, Picture, From File...', and insert the picture into a document. Then 'Insert, File...' and insert input/output file, and document for your report is ready. If your installation of MS Word do not insert DXF files, the DXF converter was not installed. Find your MS Office CD ROM and install it. It is an excellent tool for engineers. DXF is an text file and following Autocad DXF specifications you may create any picture using only your text editor!
Inertia is using 'Dxfgen.txt' file to create DXF file of particular section. Note that when creating file from the scratch, the unit of length may be specified - see the red letters on the input form, just above the O.K. button. Click on these letters and units shall alternate from 'inch' to 'foot' to 'mm' to 'cm'. The picture shall be created to show the section in scale 1:1 using units of length as specified. If letter size paper is not large enough to show the section in full scale, then DXF converter shall automatically scale the picture to fit the paper size. For the best results, this should be avoided; specify the scale when creating DXF picture. Scale down '2' (1/2 = 0.5) shall create picture of half size (e.g. 2 x 2 angle shall be shown as 1 x 1 angle), scale down '0.5' (1/0.5 = 2) shall create double size section (e.g. 2 x 2 angle shall be shown as 4 x 4 angle) etc.
Font size is always in inches and shall not be scaled, unless the scaling is done automatically by the DXF converter - situation to be avoided, see above. Default font size is 0.2 inch = 5 mm high. If the picture of the section is appropriately scaled, the font size usually does not require to be changed. For font size 0, vertice numbers shall not appear, center of gravity 'COG' shall be shown. For small font size e.g. 0.01 or 0.001 inch, both vertice numbers and COG shall not be shown - actually they are there but they are so small that they are invisible.
Inertia calculates cross sectional area Az only. To calculate tension or compression use sectional area Az, e.g. for the joist shown here Az = 0.5 * 2.5 * 2 + 0.25 * 3 = 3.25 and tension or compression = Fz/Az. Cross sectional area Ax and Ay which are used to calculate shear has to be estimated separately. Shear area Ax and Ay is the sectional area which resists shear force Fx and Fy. It depends on the shape of the section. e.g. for a flat bar Ax = Ay = 2/3 Az, for a circular section Ax = Ay = 3/4 Az. For sections like joist, channel etc. the shear areas may be given in the tables. If not, you may estimate Ax and Ay as area of rectangular shapes which have longer side parallel with the given direction. E.g. for joist with vertical web as shown, the Ax = area of top and bottom flanges Ax = 0.5 * 2.5 * 2 = 2.5 and shear = Fx/Ax where Fx is force parallel with x coordinate - in Inertia it is horizontal coordinate. Ay = area of the web Ay = 0.25 * 3 = 0.75 and similarly shear = Fy/Ay. For any rectangular shape (web, flange etc.), the maximum shear is located in the middle of the longer side. Shear stress in web may be used to calculate size of the weld connecting the beam web and flange.
Note that the examples are in MS Word format. The document was created by Inserting the Picture From File and inserting the Inertia output/input file. If you want to re-create an examples, delete the picture and copy the text into a new document. Save document in ASCII text format (MS-DOS text format). Then you can open the file using Inertia program and run it.
Description and explanations for the following examples: turbine Blade, Pin, Rebar and Segment. For full file input and output including picture as created by Inertia see Example files.
Example of the input for the turbine Blade section. Input is in millimeters. To scale DXF picture down or up, Inertia provides its own scaling ability. Recommended scale for this input is 0.1 (1:0.1=10) and change the size of letters from default of 0.2 inch to 0.1 inch. To be in control of the picture it is better to scale the picture using Inertia scaling functions. The best results are obtained when the size is scaled to the size, which would fit letter size paper. If you leave the scaling to DXF viewer (it shall do it automatically), the letters are scaled too and they may become very small.
In this example number of: polygons = 1, arcs = 4, circles = 0, rectangular = 0, HS's = 0, vertices = 9.
Example of the Pin. The pin is not closed by weld to form a pipe, therefore it is an open section. Such pin torsional properties far differ from those of pipe. Note that the polygon is very small, it is actually only the closing gap 0.2" wide and 0.1" high (= thickness of plate). Note as well, that the large arc is indicated by negative step angle and that the arcs dominates the picture. This is the only example where negative step was used. To get positive movement (clockwise) along the arc, the rectangular actually has to be numbered counter-clockwise.
In this example number of: polygons = 1, arcs = 2, circles = 0, rectangulars = 0, HS's = 0, vertices = 4.
Rebar section is typical section for which a ready made formula would be difficult to find. Note that only one arc is specified, as the same arc is used in all cases.
In this example number of: polygons = 1, arcs = 1, circles = 0, rectangulars = 0, HS's = 0, vertices = 4. Note the calculations of torsional properties for solid section.
The last example shows simple round bar Segment. This example is remarkable by the fact that we are making polygon from two vertices. As we all know, the lowest polygon is a triangle with three vertices. Yet because we use only one segment as straight line and the other line is replaced by arc, the missing vertices are supplied by the arc.
In this example number of: polygons = 1, arcs = 1, circles = 0, rectangulars = 0, HS's = 0, vertices = 2.
Note: It is possible to make half circle from both lines (that is from line 1 - 2 and 2 - 1) and then deduct the circular hole. This would show you the difference between real circle and the polygon which is created by program to represent a circle.
Reference: Felix Wojciechowski, Rockford Ill., Machine Design, January 22, 1976 p. 105, 106
The method divides a cross section into series of trapezoids (or rectangles), then adds or subtracts the properties of the elemental areas to find the composite properties of the total area. This technique replaces integration by summation of finite elements, and applies mainly to areas bounded by straight lines. But because curves can be approximated by straight-line segments, the method can be used on any shape plane cross section.
Two rules must be observed while using this technique:
1. The area of the cross section must be entirely within the positive quadrant, that is the coordinates of the vertices must be always positive.
2. The sides of the cross section must be traversed clockwise. The direction should be reversed for inside boundaries.
Basic Equations:
Plane cross sections are made up of connected straight lines (curved boundaries can be approximated by straight line segments). The area DA under a particular segment can be calculated from the basic rules of plane geometry as
DA = (xn+1 - xn)(yn+1 + yn)/2
where xn and yn - coordinates of the line starting point and xn+1 and yn+1 - coordinates of the line end point.
The static moment of DA (defined by trapezoid ABEG in the fig.) about the x-axis DMx can be considered to consists of contribution from rectangle ABCD and triangle FCE minus the contribution from triangle GFD. Using the areas and centers of gravity (known from basic formulas) gives DMx as
DMAx = [(xn+1 - xn)/8][(yn+1 + yn)2 + (yn+1 - yn)2/8]
Similarly the moment of inertia DIx of DA about the x-axis consists of contribution from rectangle ABCD and triangle FCE minus the contribution from triangle GFD. Known formulas for inertia and the parallel axis theorem gives
DIAx = [(xn+1 - xn)(yn+1 + yn)/24][(yn+1 + yn)2 + (yn+1 - yn)2]
To find the total area A, static moment MAx and moment of inertia Ix of a plain cross section, then equations are first solved for each line segment. Then the individual results are summed that is,
A = SDA, MAx = SDMAx, Ix = SDIx
Once A and MAx are known the position of the center of gravity y' can be calculated from
y' = MAx/A
Also the moment of inertia about the axis through y' can be found from
Ix' = Ix - A.y'2
Torsional properties are calculated using a special input.
Reference: Roark & Young, Formulas for Stress and Strain.
Three types of sections are recognized: open, solid and hollow. Input for open and hollow sections are independent - no polygon has to be entered, while solid may be only calculated if a polygon was entered too. Typical open section is joist, channel, angle etc. The examples of Blade, Pin and Segment are open sections. Open section has to be idealized as collection of rectangulars.
Program is using formula K = alfa*F/(3+(4F/A*U2)) where F = sum(U*t3), U = length of section of thickness t, A = sectional area. Section modulus is calculated from formulas:
c1 = (p Di2/4)2/(4 A)2,
c2 = Di/(2rc),
c = Di[1 + 0.15(c1 - c2)]/(1 + c1),
Q = K/c
where Di is diameter of maximum inscribed circle and rc curvature of the section boundary at the two points where the inscribed circle touches the boundary, e.g. see I beam section. Concave radius is positive, convex radius (as in picture) is negative. For straight line, input rc = 0. Required input is: thickness and length of partial rectangulars, maximum diameter of inscribed circle Di, radius r1 and r2 of the curvature of the section boundary at the two points where the inscribed circle touches the boundary. Shape correction coefficient alfa to be input as 1.0 for angle, 1.1 for channel, Z and T section, 1.2 for cross and 1.3 for joist or I beam. If in doubts, use default value of alfa, which is 1.0 (conservative).
Solid section has a dominant inscribed circle, e.g. in example Rebar. Required input is described in the input table. It is maximum diameter of inscribed circle and radius of the curvature of the section boundary at the point where the inscribed circle touches the boundary. Program is using formula K = A4/(40*Jz), where A is section area and Jz is polar moment of inertia. Section modulus is calculated from Q = K/c, where c is calculated as described in open section. Required input is: maximum diameter of inscribed circle Di, radius r1 and r2 of the curvature of the section boundary at the two points where the inscribed circle touches the boundary. Concave radius is positive, convex radius is negative. For straight line, input rc = 0
Hollow section must be a closed section. Pin example (see top of the manual) is not a hollow section as the gap is not closed by weld to form a pipe. Hollow section torsional properties are calculated using K = 4A2/(sum(U/t)) and Q = 2tmin*A where A is area enclosed by the hollow section wall centerline, U is length of hollow section wall of thickness t, tmin is minimal wall thickness.