The General Beam Analysis module performs a finite element analysis of a general beam with multiple sections and supports subject to transverse loads.  The module calculates static displacements, rotations, shears, moments, and stresses; and natural frequencies and mode shapes.

The beam problem is defined using multiple nodes and elements as shown in Figure 1.  The beam can be split into as many as 19 discrete elements of varying lengths, and with independent boundary conditions, cross-sectional properties, and materials. Results are output in both text and graphic form.

Figure 1.  Beam model with multiple sections

The General Beam Analysis module is based on a simplified two-dimensional analytical model in which the beam is assumed to be slender and straight and all loads and deflections act in the transverse direction in a single plane. The element x-axis runs left to right along the axis of the beam. The y-axis is perpendicular to the beam axis, as shown in Figure 1.

Displacements and Slopes
The sign convention for calculated displacements and slopes is shown in Figure 2.

Figure 2.  Sign convention for nodal displacements and slopes

External Loads And Reactions
The sign convention for applied loads and calculated reactions is shown in Figure 3.

Figure 3.  Sign convention for applied loads and calculated reactions

Element Internal Shears and Moments
The internal shears and moments for a beam element are derived using the force balance and sign conventions shown in Figure 4.  The deflected shape of the beam with positive internal shears and moments is shown in Figure 5.

Figure 4.  Beam deflected shapes for positive internal moments and shears

Figure 5.  Beam deflected shapes for positive internal moments and shears

The General Beam Analysis title panel shown in Figure 6 provides the primary interface for specifying the solution type, navigating between input and output panels, viewing the beam model, and running an analysis. The title panel is partitioned into 3 groups:

• Analysis Input Group -
  Contains the basic controls for defining global analysis parameters, editing the beam model, and post-processing results.
  
• Graphics Options Input Group -
  Contains controls for customizing the display of the analytical model.
  
• Beam Design Display-
  Contains a graphical view of the analytical model, including beam geometry, boundary conditions, and applied loads.

Figure 6. General Beam Analysis title panel

Analysis Input Group

The Analysis input group contains the basic controls for specifying the solution type, defining global analysis parameters, and navigating to other input and output panels.

Solution Type
Two solution types are available: 

• Static Analysis  - Computes the displacements, rotations, shears, moments, and stresses of a beam subject to transverse loads and moments.
• Natural Frequency Analysis  - Computes the natural frequencies and mode shapes of the beam.

Edit Beam
Press the Edit Beam button to display the input table defining the beam analytical model. The input table allows you to specify beam geometry, section properties, materials, boundary conditions, and external loads.  More details are provided in the Beam Analytical Model section.

View Beam
Press the View Beam button at any time to open a new Beam Animation window. The Beam Animation window displays a three-dimensional graphical representation of the beam model. If analytical results are available, calculated beam deflections can be viewed and animated.

Axial Force
Select the Axial Force checkbox to specify an axial force F to be applied to each end of the beam.  The beam sign convention defined by the module specifies a positive axial load in tension, as shown in Figure 3. A tensile (positive) force acts to stiffen a beam in the transverse direction.  A compressive (negative) force reduces the transverse stiffness.  An axial force can be specified in both a static analysis and a natural frequency analysis.

Gravity Load
Select the Gravity Load checkbox to apply an accelerative force g in the transverse direction.  The module sign convention defines downward gravity load as positive, as shown in Figure 3.

Accelerative forces are calculated internally by applying a concentrated force -mg to all nodal masses m, and applying a linearly distributed force with end points (force per unit length) defined by

to all elements with material density r, left-end crossectional area A1, and right-end crossectional area A2.

A gravity load can only be applied in a static analysis.

Graphics Options Input Group

The Graphics Options input group contains controls for customizing the graphical display of the analytical model in the title panel.

Beam Properties And Materials

Height
Specifies the geometric or material property that will be represented by the height of the rectangles shown in the Properties And Materials beam schematic (see the Beam Design Display section below). The height of each rectangle will reflect the relative magnitude of the specified property value for the corresponding beam element (i.e., the beam element represented by the rectangle).

Plot
Click the Plot button to generate a plot of the beam property selected in the Height drop-down box. The plot will show the specified property value versus the longitudinal beam coordinate X for all elements in the beam.

Color
Specifies the geometric or material property that will be represented by the color of the rectangles shown in the Properties And Materials beam schematic (see the Beam Design Display section below). The colors associated with the minimum and maximum property values can be specified using the palette buttons to the right of the list. Intermediate property values will be represented using an interpolated color range. The color of each rectangle will reflect the relative magnitude of the specified property value for the corresponding beam element (i.e., the beam element represented by the rectangle).

Loads And Boundary Conditions

Color
To change the color scheme used in the Loads And Boundary Conditions schematic (see the Beam Design Display section below), select the graphical element that you want to modify from the drop-down list and then click on the palette button to choose a desired color.

Beam Design Display

The Beam Design section of the title panel, shown in Figure 7, provides a graphical depiction of the analytical model. The display contains two beam schematics, stacked vertically. The upper schematic, labeled Properties And Materials, shows the beam's cross-sectional and/or material properties. The lower schematic, labeled Loads And Boundary Conditions, displays applied loads and enforced boundary conditions. Both the upper and lower beam schematics are described in more detail in the sections that follow.

Figure 7. Beam Design section of the title panel

Properties And Materials
The Properties And Materials schematic contains a series of contiguous rectangles of varying dimensions and colors that represent relative geometric and/or material properties of the elements comprising beam. The position and length of each rectangle corresponds to the location and length of its associated beam element. The height and color of each rectangle represent specified section or material properties as defined in the Graphics Options input group.

For example, click on the Height drop-down list in the Graphics Options input group and select Moment Of Inertia. The height of each rectangle will reflect the relative magnitude of the corresponding beam element's cross-sectional moment of inertia property. Note that due to display limitations, the height of the rectangles may not be exactly proportional to the property value. The Properties And Materials schematic is primarily intended to give a general indication of relative element property values.

Loads And Boundary Conditions
The Loads and Boundary Conditions schematic displays the loads applied to the beam and the enforced boundary conditions. The screen colors of the various loads and boundary conditions can be changed to suit user preferences in the Graphics Options input group.

The beam problem is defined by entering beam geometry, section properties, materials, boundary conditions, and external loads into the input table shown in Figure 8.  The beam model consists of nodes which are connected by beam elements. Each element may have a different cross section and/or material. By definition, element 1 connects nodes 1 and 2 and is defined using section and material properties from row 1.

Up to 20 rows of data can be entered in the spreadsheet, allowing the beam problem to be broken into 19 discrete elements using a maximum of 20 node points. Section and material properties entered in row 20 are not used.

Figure 8. Beam definition spreadsheet input

Input Table Basics
The input table looks and operates very similar to Microsoft Excel. To move between cells in the table, either click the mouse button or use the arrow keys. When you move to a cell, it becomes the active cell and is highlighted. The data that the cell contains is shown in the large input area at the top of the spreadsheet. Only the first 8 characters of the data is shown in cell.

To enter new data in the active cell, simply type the data and press the enter key (the new data will also be stored if the cell is made inactive by a mouse click or arrow key). To edit existing data in a cell, press the F2 function key or double click on the cell. To cancel edits, press the escape key (Esc).

Input Fields
The beam model input table contains 13 columns of data. Each column defines a specific input parameter for the node or element associated with each data row. Use two dots (..) to indicate that the data from the prior row is to be used in current row also. A minus sign (-) is interpreted as a blank field.

A detailed description of the input requirements for all data columns is provided below.

The General Beam Analysis module's Static Solution requires that the beam be constrained in at least 2 degrees-of-freedom (DOF). For example, a simply supported beam is constrained in 2 translational DOF, one at each end of the beam. A cantilevered beam is constrained in one translational DOF and one rotational DOF, at one end of the beam.

The following data are calculated in a static analysis:

• Displacements
• Slopes
• Reaction Forces and Moments
• Internal Shears, Moments and Bending Stresses

Results Plot

The module generates plots of calculated deflections, slopes, moments, and shears along the length of the beam. New results are plotted on top of data from previous analyses, and all plot data is retained until explicitly cleared. A typical static analysis results plot is shown in Figure 11.

Figure 11. Plot of beam displacements, slopes, shears, and moments

Text Output

The General Beam Analysis module generates a standard text-based ETBX results report. The results report documents the input model and tabulates the calculated data. A typical static analysis results report is shown in Figure 12.

Figure 12. Results report from a static solution

Beam Graphics and Animation

The General Beam Analysis module provides an animation utility for visualization of the beam analysis. The animation utility, shown in Figure 13, has the capability to display and animate calculated beam deflections in three-dimensional space.

Click the View Beam button on the module's title panel to open a new animation window. The current beam model will be displayed in the main view. If an analysis has been run and analytical results are available, click the Deformed radio button to see the deflected shape.

Note that an animation window only contains a 'snapshot' of the beam model (and any calculated results) at the time the user presses the View Beam button. The graphics displayed in the animation window are NOT updated to reflect changes to the beam model or subsequent analysis runs. To get an updated animation, open a new animation window by clicking the View Beam button again. Don't forget to close the previous animation window if it is no longer needed.

Similarly, any changes to the beam model will invalidate previous results calculations, and the previous results data will not be available to a new animation window when the View Beam button is pressed. Click the Calculate button to calculate new results data for the current beam model, and then click View Beam to open a new animation window with the results.

Figure 13. Beam results animation

The General Beam Analysis module's Natural Frequency Solution calculates the natural frequencies and mode shapes of the beam. The natural frequency solution does not require that the beam be constrained to ground; the solution will calculate the frequencies and modes shapes of a free-free beam.

Results Plot

The module generates plots of calculated mode shapes. New results are added to existing data from previous analyses, and all plotted mode shapes are retained until explicitly cleared. A typical natural frequency analysis results plot is shown in Figure 14.

Figure 14. Plot of calculated mode shapes

Text Output

The natural frequency solution generates a similar results report to the static solution. The results report documents the input model and tabulates the calculated natural frequencies and mode shape data. A typical modal analysis results report is shown in Figure 15.

Figure 15. Results report showing tabulated natural frequencies

Mode Shape Animation

The General Beam Analysis module's animation utility provides the capability to display and animate calculated mode shapes in three-dimensional space.

After running a natural frequency analysis, click the View Beam button on the module's title panel to open a new animation window. The current beam model will be displayed in the main view. Choose a mode shape to display from the Results Case list and click the Deformed radio button to see the deflected shape. A typical mode shape animation is shown in Figure 16.

Note that an animation window only contains a 'snapshot' of the beam model (and any calculated results) at the time the user presses the View Beam button. The graphics displayed in the animation window are NOT updated to reflect changes to the beam model or subsequent analysis runs. To get an updated animation, open a new animation window by clicking the View Beam button again. Don't forget to close the previous animation window if it is no longer needed.

Similarly, any changes to the beam model will invalidate previous results calculations, and the previous results data will not be available to a new animation window when the View Beam button is pressed. Click the Calculate button to calculate new results data for the current beam model, and then click View Beam to open a new animation window with the results.

Figure 16. Beam mode shape animation