Area characteristics.

The calculation solves area characteristics of common profiles and mass characteristics of solids created by drawing or rotation of the profile.

The application enables:

  1. Creation of a profile in a simple graphic editor.
  2. Calculation of area characteristics (Ix, Sx) for main axes passing through the centre of gravity.
  3. Calculation of area characteristics (Ix, Sx) for randomly turned axes passing through the centre of gravity, including a graph.
  4. Calculation of characteristics for the turned axes which pass through a random point.
  5. Calculation of volume and mass characteristics (V, m, Im) of solids created by drawing or rotation of the profile.

Control, structure and syntax of calculations.

Information on the syntax and control of the calculation can be found in the document "Control, structure and syntax of calculations".

Supplements - This calculation:

Controlling this calculation is slightly different from that of common calculations. The calculation sheet is divided into three parts. The upper (fixed) part includes command buttons, parameters setting the graphic editing area and a scale. A description of drawing commands is given below. The central part, which looks like graph paper, is meant for drawing the profile. The lower part includes results and graphs of the values.

Information on the project.

Information on the purpose, use and control of the paragraph "Information on the project" can be found in the document  "Information on the project".

Process of calculation.

Despite the fact that calculating characteristics is a simple task, a brief description of the procedure is given here.

  1. Set units (in, mm) and the scale of the drawing (as large as possible with regards to the intended size of the profile).
  2. Create (draw) the desired profile in the drawing area [1.0].
  3. Press the button "Initiate calculation"
  4. The calculated parameters of the profile can be found in paragraph [2.0, 3.0].
  5. The parameters of volume solids can be found in paragraph [4.0].

Drawing of the profile.

Drawing area. [1.0]

The area for drawing the profile consists of 220 rows x 220 columns with small spacing. Cells in this area are colorless. The profile is created so that the selected cells are filled with the selected color. The X axis is horizontal axis, the Y axis is the vertical axis. The point 0,0 is situated in the lower left hand corner of the drawing area.

Creation of the shape.

Command buttons are situated in the upper part of the sheet. After pressing the button, the selected shape is drawn into the selected area in the drawing field.

Procedure:

  1. Select a rectangular area in the drawing area [1.0] (the same procedure as a selection of more cells in Excel).
  2. Press the command button. The area will be filled with the selected shape.

Example: Drawing a filled elliptic arch

Step 1 - Selection of an area

Step 2 - Pressing the button for filling of the selected area with an arch

Result

Note: As long as the cursor is in the editing graphic area, its standard Excel shape (thick cross) is changed to an arrow.

Description of commands from left to right.

Recovery of the grid.

In case you use the standard Excel command for copying cells with creation of the profile, the millimetre grid can be damaged. This command restores the standard grid in the drawing area.

The color of the shape.

In the list box, select the color which will be used for drawing the shape.

Hint: In case you select the first item - "colorless", a hole of the respective shape will be created. In this way you can also create e.g. an inner rounding or chamfer.

Rectangle.

The command fills the selected rectangle area with the preset color.

Blank rectangle.

The command removes color from the selected area.

Hint: If needed, use this command to clean up the whole drawing area.

Circle.

The command fills the selected square (rectangle) area with a circle (ellipse).

Blank circle.

The command removes the circle (ellipse) shape from the selected square (rectangle) area.

Chamfer.

The command fills the selected area with a triangle shape - chamfer. The triangle is turned according to one of the four buttons.

Rounding.

The command fills the selected area with a quadrant - rounding. The quadrant is turned according to one of the four buttons.

Mirror image.

The command creates a mirror image of the selected area in the defined direction. The mirror image is created along the respective margin of the selective rectangle.

Calculation of the profile.

The button initiates calculation of characteristics.

Calculation and accuracy.

Calculation

The calculation is executed in the VBA module. All cells of the editing area are processed consecutively and if a cell is colored, it is included in the calculation. Standard formulas and procedures with calculations (Integration of the area, calculations of Ix, Ixy, Ip, Sx, gravity centre, area, etc.) are used as available in specialized literature.

The calculation can be initiated using the button "Calculation of the profile" in the upper right hand corner of the calculation sheet.

Accuracy of calculation.

When creating oblique lines and curves, these are replaced by a "polygonal line". This affects the accuracy of the calculation. It is generally applicable for profiles containing oblique lines: The greater the size of the profile compared with the size of the drawing area, the smaller the error in the calculation. Try to set the scale of the grid so that the calculated profile fills as large an area of the drawing field as possible.

Note: With a reasonable size of profile, the calculation error varies in a range from 1 to 2%; this is acceptable for the absolute majority of tasks.

Characteristics to axes going through the centre of gravity. [2]

2.1 The area of the profile.

2.2 Co-ordinates of the centre of gravity.

These are given in actual co-ordinates, measured from the point 0,0. The small button at the end of the row ">" places the cursor (selects a cell) in the centre of gravity.

2.3 Moment of inertia.

The moment of inertia Ix (Iy) is related to the axis which goes through the centre of gravity and is parallel with the X (Y) axis.

2.4 Polar moment of inertia.

It is related to the centre of gravity.

2.5 Product of inertia.

It is related to the centre of gravity.

2.6, 2.7 Minimum / Maximum moment of inertia.

It is the minimum / maximum value Ix to the axis going through the centre of gravity and forming an angle with the X axis.

2.9, 2.10 Bending modulus of section.

It is an important quantity which is decisive in finding of the maximum bending stress acting in a beam with the given profile. The value Sx is calculated according to the formula Sx=Ix/a, where a is the distance between the straight line going through the centre of gravity and the position of the extreme fibre.

Two values for the axis X and two values for the axis Y are given here. The value with the sign "+" Sx+, (Sy+) is related to the upper (right) extreme fibre. The value with the sign "-" is related to the lower (left) extreme fibre.

Graph Ix,Sx.

The graph shows dependences of the "Moment of inertia" and "Bending modulus of section" to the bending angle of the profile around the centre of gravity. The bending angle is drawn on the horizontal axis. The vertical axis on the left includes values Ix, the vertical axis on the right includes values Sx+(extreme upper fibre) and Sx-(extreme lower fibre).

Characteristics to shifted and turned axes. [3]

In this paragraph you can calculate area characteristics of the profile to shifted and turned axes. Marking according to the illustration.

3.1 A shift of axes related to the centre of gravity.

Enter shifts of the axes of which area characteristics you wish to find.

3.2 Axis rotation angle.

Enter an axis rotation angle.

3.3, 3.4, 3.5 Moments of inertia.

These are related to X' and Y' axes (or to the point O' resp.) turned by the angle.

Parameters of volume solids. [4]

In this paragraph you can find volume parameters of the solids which will be created by drawing or rotation of the created profile.

4.1 Density

Enter the density of the material of the beam (or the solid of rotation resp.). Some common materials are given in the list box to the right of the input field.

4.2 The solid created by drawing the profile - beam.

4.3 The length of the beam.

Enter the desired length of the beam created by extraction of the drawn profile in the direction of the axis Z.

4.4 The volume of the beam.

4.5 The weight of the beam.

4.6 The mass moment of inertia to the Z axis.

It is the moment of inertia of the beam to the axis Z. The Z axis is perpendicular to the area of the section and goes through the profile's centre of gravity.

4.7 The solid created by rotation of the profile around the X axis

4.8 The volume of the solid of revolution.

4.9 The weight of the solid of revolution.

4.10 The moment of inertia of the profile to the X axis.

4.11 The mass moment of inertia (axis X).

It is a mass moment of inertia of the solid of rotation (created by rotation of the profile around the X axis) to the X axis.

Setting calculations, change the language.

Information on setting of calculation parameters and setting of the language can be found in the document "Setting calculations, change the language".

Workbook (calculation) modifications.

General information on how to modify and extend calculation workbooks is mentioned in the document "Workbook (calculation) modifications".

Supplements - This calculation:

Except for possible formatting of numbers, no other modifications to this workbook are recommended.