1. Summary
Regarding the uniformly distributed load problem of rectangular cantilever beams, different element models were used to analyze the differences in the analysis results.
2. Problem description
The rectangular cross-section cantilever is shown in the picture, the left end is fixed, and the material is steel. The material parameters are: elastic modulus E=2E11Pa, Poisson's ratio u=0.28, density ρ=7800kg/m^3. The geometric parameters are shown in the figure. There is a uniform load acting on the beam.
Three-dimensional solid elements, plane stress elements, and beam elements are used for analysis; the displacement at the right end of the beam and the stress distribution at the left end are compared; the difference between the theoretical calculation value and the finite element calculation result is discussed.
3. Technical Route
This problem belongs to the category of structural analysis, and is realized through software interface operation with the help of ANSYS Mechanical APDL module. Unit system: m, kg, s, N, Pa.
Three-dimensional solid element Brick 8 node 185, plane stress element Quad 4 node 182 and beam element 2 node 188 are used for analysis.
4. Analysis process
4.1 3D solid element analysis
(1) Define the job file name
. Click Utility Menu>File>Change Jobname in turn, enter: beam in the dialog box that appears, select “yes” in the “New Log and error files” field, and click “OK”.
(2) Define the unit type
- Click MainMenu>Preprocessor>Element type>Add/Edit/Delete in turn, a dialog box appears, click "Add", a "Library of Element Type" dialog box appears. Select "Solid" in the list column on the left of "Library of Element Type", select "Brick 8 node 185" in the list column on the right, and click "OK".
(3) Set the material properties
and click Main Menu>Preprocessor>Material Props>Material Models in turn. The "Define Material Model Behavior" dialog box appears. In the dialog box under "Material Model Available", double-click to open "Structural>Linear>Elastic >Isotropic", a dialog box appears, enter the elastic modulus EX = 2e11, Poisson's ratio PRXY = 0.28, click "OK", click "Material>Exit".
(4) Establish a geometric model
- Create a cuboid
Pick menu Main Menu>Preprocessor>Modeling>Create>Volumes>Block> By Dimensions. The dialog box shown in the figure below pops up, enter 0,1.6 in the "X1,X2" text box, enter 0,0.05 in the "Y1,Y2" text box, and enter 0,0.06 in the "Z1,Z2" text box And click OK.
(5) Division unit
- Pick menu Main Menu>Mesh>Mesh Tool. A dialog box pops up, select Volums in the Mesh option, Hex and Mapped in the Shape option, and then click the Mesh below to divide it.
(6) Applied load and boundary conditions
- MainMenu>Solution>DefineLoads>Apply>Structural>Displacement>On Areas, pick the left end face, and then click OK. And select All DOF in the pop-up dialog box, and click ok.
- Select MainMenu>Solution>DefineLoads>Apply>Structural>pressure>On Areas, pick the upper end face, and click OK. Enter 200000 in the pop-up dialog box.
(7) Solve the model
. Click Main Menu>Solution>Solve>Current LS to solve the problem. When the prompt Solution is Done, the solution is complete.
(8) To view the results,
click Main Menu>GeneralPostproc>PlotResults>Contour Plot>Nodal Solu, and a dialog box will pop up. Select "Stress" in the "Item to be Contoured" column, and in the column on the right, select "von Mises After "stress", click "OK", and the plane stress cloud diagram is displayed in the graphics window, as shown in the figure.
Click Main Menu>GeneralPostproc>Plot Results>Contour Plot>Nodal Solu in turn to pop up a dialog box, select "DOF Solution" in the "Item to be Contoured" column, and select "Displacement vector sum" in the column to the right , Click "OK", then a three-dimensional strain graph is displayed in the graphics window, as shown in the figure.
4.2 Plane stress element analysis
(1) Define the job file name
. Click Utility Menu>File>Change Jobname in turn, enter: beam in the dialog box that appears, select “yes” in the “New Log and error files” field, and click “OK”.
(2) Define the unit type
- Click MainMenu>Preprocessor>Element type>Add/Edit/Delete in turn, a dialog box appears, click "Add", a "Library of Element Type" dialog box appears. Select "Solid" in the list column on the left of "Library of Element Type", select "Quad 4 node 182" in the list column on the right, and click "OK".
(3) Set the material properties
and click Main Menu>Preprocessor>Material Props>Material Models in turn. The "Define Material Model Behavior" dialog box appears. In the dialog box under "Material Model Available", double-click to open "Structural>Linear>Elastic >Isotropic", a dialog box appears, enter the elastic modulus EX = 2e11, Poisson's ratio PRXY = 0.28, click "OK", click "Material>Exit".
(4) Establish a geometric model
- To create a rectangle,
select the menu MainMenu>Preprocessor>Modeling>Create>Areas>Rectangle>By 2 Corners. The dialog box shown in the figure below pops up, fill in 1.6 for Width and 0.06 for Height. Click OK.
(5) Division unit
- Pick menu Main Menu>Mesh>Mesh Tool. Select the set next to Lines in the Mesh option, and divide the two long sides into 15 parts and the two short sides into 3 parts; then click the bottom mesh, select the rectangle, and click OK.
(6) Applied load and boundary conditions
- MainMenu>Solution>DefineLoads>Apply>Structural>Displacement>On Lines, pick the left end line, and then click OK. And select All DOF in the pop-up dialog box, and click ok.
- Select MainMenu>Solution>DefineLoads>Apply>Structural>
pressure>On Lines, pick the upper end line, and click OK. Enter 200000 in the pop-up dialog box.
(7) Solve the model
. Click Main Menu>Solution>Solve>Current LS to solve the problem. When the prompt Solution is Done, the solution is complete.
(8) To view the results,
click Main Menu>GeneralPostproc>PlotResults>Contour Plot>Nodal Solu, and a dialog box will pop up. Select "Stress" in the "Item to be Contoured" column, and in the column on the right, select "von Mises After "stress", click "OK", and the plane stress cloud diagram is displayed in the graphics window, as shown in the figure.
Click Main Menu>GeneralPostproc>Plot Results>Contour Plot>Nodal Solu in turn to pop up a dialog box, select "DOF Solution" in the "Item to be Contoured" column, and select "Displacement vector sum" in the column to the right , Click "OK", then a three-dimensional strain graph is displayed in the graphics window, as shown in the figure.
4.3 Beam element analysis
(1) Define the job file name
. Click Utility Menu>File>Change Jobname in turn, enter: beam in the dialog box that appears, select “yes” in the “New Log and error files” field, and click “OK”.
(2) Define the unit type
- Click MainMenu>Preprocessor>Element type>Add/Edit/Delete in turn, a dialog box appears, click "Add", a "Library of Element Type" dialog box appears. Select "Beam" in the list column on the left of "Library of Element Type", select "2 node 188" in the list column on the right, and click "OK".
(3) Set the material properties
and click Main Menu>Preprocessor>Material Props>Material Models in turn. The "Define Material Model Behavior" dialog box appears. In the dialog box under "Material Model Available", double-click to open "Structural>Linear>Elastic >Isotropic", a dialog box appears, enter the elastic modulus EX = 2e11, Poisson's ratio PRXY = 0.28, click "OK", click "Material>Exit".
(4) Establish a geometric model
- Create a straight line
Create key point 1 (0, 0, 0) and key point 2 (1.6, 0, 0), then create key point 10 (0, 1, 0), which is the direction point of the section. Then connect points 1 and 2 to form a straight line.
- Define section properties
(5) Division unit
- Open the Mesh tool and select the line in the above set; select the key point 10 as the section direction point. After selecting it, click the set behind the middle line to divide it into 15 parts, then click the bottom mesh, select the line, and click OK.
(6) Applied load and boundary conditions
- Apply a fully restrained load to the left end point.
- Select MainMenu>Solution>DefineLoads>Apply>Structural>pressure>On Beams, and select the entire beam.
Since the load acts in the negative direction of the y-axis, enter 1 in the first air of the pop-up window. Since the load on the surface in the question needs to be converted into the load on the line, the second and third Enter 10000 in the air (200000*0.05).
(7) Solve the model
. Click Main Menu>Solution>Solve>Current LS to solve the problem. When the prompt Solution is Done, the solution is complete.
(8) To view the results,
click Main Menu>GeneralPostproc>PlotResults>Contour Plot>Nodal Solu, and a dialog box will pop up. Select "Stress" in the "Item to be Contoured" column, and in the column on the right, select "von Mises After "stress", click "OK", and the plane stress cloud diagram is displayed in the graphics window, as shown in the figure.
Click Main Menu>GeneralPostproc>Plot Results>Contour Plot>Nodal Solu in turn to pop up a dialog box, select "DOF Solution" in the "Item to be Contoured" column, and select "Displacement vector sum" in the column to the right , Click "OK", then a three-dimensional strain graph is displayed in the graphics window, as shown in the figure.
5. Results and discussion
The comparison between finite element calculation results and theoretical calculation values is shown in the following table.
It can be seen from the table that the results obtained by using three-dimensional solid elements and plane stress elements for analysis are quite different from the theoretical calculation results, and the results obtained by using beam element analysis are slightly different from the theoretical calculation results. The errors may be caused by model errors.
Therefore, it can be concluded that the difference between the analysis using beam elements and the theoretical calculation results is smaller.
Attachment: theoretical calculation process
