Model of a simple supported beam
This tutorial will show you how to create one dimensional model of a simple supported beam shown on fig. 1 with a loading force. This model allows you to calculate nonlinear statical and dynamical behaviour of the beam, including stability problems and large deflections.
|Fig. 1 Scheme of a simple supported beam|
FyDiK model of such beam will be composed from three types of FyDiK objects: Mass Points, Translational Springs and Rotational Springs. In addition for every spring must be defined a Spring Function which describes its behaviour.
Mass Points are objects carrying mass of the beam. Initial and boundary conditions, loads, etc. are applied on them. The position and the mass of a Mass Point is given by a division of the beam. We can divide the beam equidistantly into ten parts. Each part will have length 1 meter and mass 24 kg (800 x 0.03 x 1). Mass Points will be located at the ends of the parts, see fig. 2. Therefore each Mass Point will have mass 24 kg, except two Mass Points at the ends of the beam, which will be half-size (12 kg).
|Fig. 2 Division of the beam and location of the Mass Points|
Launch the FyDiK application and use menu Object--New--Mass Point, the dialog New Mass Point will be shown. Set the mass 12 kg for the first Mass Point and choose true for restraint x and y. Then use Apply button. The second Mass Point will have mass 24 kg, initial coordinate x0 = 1 meter and all restraints false. The third Mass Point will have initial coordinate x0 = 2 meters, and so on. After creation of all eleven Mass Points open View--View Panel and use the All button. The working plane should look like on fig. 3.
|Fig. 3 Working plane after creation of Mass Points|
If some Mass Points are not created correctly, select them by picking or by box and use Object--Change menu. The Change Mass Point dialog will be shown. Use it to change attributes of the selected objects.
Before creation of springs we need to create Spring Functions for them. Created Spring Functions will be shown in Spring Function organizer from menu View. Spring Function for a Translational Spring will be linear with stiffness EA/dL, where dL is the length of the Translational Spring. Rotational Springs will have linear Spring Function with stiffness EI/dL.
Use menu Object--New--Spring Function--Linear to create new Linear Spring Function, the New Spring Function dialog will be shown. Set name as EA for the first Spring Function and choose relative true. The choice true means that stiffness will be automatically divided by length of the spring. Enter stiffness k = 0.36e9 N (12 x 0.03 x 109) and use button Apply. The second Spring Function will have name EI and stiffness k = 0.012e9 Nm2 (12 x 0.001 x 109).
Translational Springs can be created using menu Object--New--Translational Spring, the New Translational Spring dialog will be shown. First pick in the Spring Function organizer the EA Spring Function. Then pick in the working plane the first Mass Point, second Mass Point and use button Apply. You should see created Translational Spring as a line between Mass Points with directional mark.
Second Translational Spring will be successfully created after picking second Mass Point and third Mass Point. Create next Translational Springs the same way. After creation, the working plane should look like on fig. 4.
|Fig. 4 Working plane after creation of Translational Springs|
Translational Springs can be selected by picking and by box, but the box selects Mass Points too. For selecting Translational Springs only, use dialog Visibility Panel from menu View. There you can disable visibility of the Mass Points.
Create Rotational Springs using menu Object--New--Rotational Spring, the New Rotational Spring dialog will be shown. First pick in the Spring Function organizer the EI Spring Function. Then pick in the working plane the first Translational Spring, second Translational Spring and use button Apply. Create next Rotational Springs the same way. After creation, the working plane should look like on fig. 5.
|Fig. 5 Working plane after creation of Rotational Springs|
Forces can be created using menu Object--New--Force, the New Force dialog will be shown. First enter the size 100 000 N, the angle 270 and choose angle unit as degrees. Finally pick the Mass Point on the midspan (MP6) and press button Apply. Now the working plane should look like on fig. 6.
|Fig. 6 Working plane after creation of loading Force|
The model is finished, we can try to start the simulation. Open dialog Control Panel from menu Simulation and push Solve button. If the solution is not stable, carefully decrease the step and press Restart button.
In the Control Panel you can also change used numerical method, speed of the simulation and the number of drawed frames per second. The field status serves for indication of your CPU capabilities. If it is green, CPU is fast enought, else decrease the speed of the simulation.
Every FyDiK object has option 'save state' which is useful for monitoring of its time series. If you set up this option as true then a state file will be created and the object state will be automatically saved. Period of saving is given by numerical step and by fps parameter. Both can be specified in the Control Panel. State files are overwritten by reopening the model data files.
On fig. 6 the resulting time serie of the y coordinate of the midspan Mass Point (negative deflection) is shown. It converges to deflection 0.177 m which corresponds well with exact linear solution 0.174 m (without taking account of the shear deformations).
|Fig. 7 Graph of time serie of the y coordinate of the midspan Mass Point|
- Increase the size of the loading force and observe large deflections.
- Try to verify the critical force for the typical buckling load.
- Change the model to a continuous beam and/or a cantilever beam.
Resulting data files.