Homework Project 5
Reduced Order Models: Beam Plates and Shells
Advanced Topics in Finite Elements
Summer 2015
John Connor
System Description
Reduced order models are used to reduce the run time of finite element models. Reduced order models can be very advantageous when evaluating large analyses which would normally require more computational power. Reduced order models reduce the amount of degrees of freedom from them model and creates an approximation of a more detailed model.
In this homework I will explore 2 types of reduced order models the first will use beam elements the second will use plate elements. Beam elements are used to simulate pipes, structural beams
(ex: “I” beams), and other simple shapes which can be extruded along a profile. Beam elements are 1 dimensional and used to represent a 3 dimensional object. Plate elements are used to model shapes that can be represented as 2 dimensional objects which can then have a thickness applied to represent a 3 dimensional object. The project will model a canileaver beam with uniform distribution along its length using a beam model and a simply supported square plate subject to a uniformly distributed pressure.
Governing Equations, Boundary Conditions and Input Data
Beam and plate equations are defined in
Roark’s Formulas for Stress and Strain
, the equations presented in Roark’s provide stress and displacement equations for several types of boundary conditions and loading scenarios. Figure 1 shows an excerpt from Roak’s used to determine the stress and deflection in the cantilever beam, Figure 2 shows the same thing but for the plate problem.
Figure 1 – Roark’s Beam Bending Equations
Figure 2 – Roark’s Plate bending Equations

Input data from the homework assignment sheet, the calculated inputs, and the assumed values for the problem are shown below in Table 1.
Table 1- Input Data
Name Value [Unit]
Beam Length 1 [m]
Source
Assignment Sheet
Beam Height
Beam Width
0.1 [m]
0.1 [m]
Assignment Sheet
Assignment Sheet
Plate Length/Width
Plate Thickness
Elastic Modulus
Poisson’s Ratio
Density
1 [m]
0.05 [m]
200 [GPa]
0.3
7850 [kg/m 3 ]
Assignment Sheet
Assignment Sheet
Assignment Sheet
Assignment Sheet
Assignment Sheet
Second Moment of
Inertia
Line Load
Uniform Pressure
8.33[µm]
1000 [N/m]
1000 [Pa]
Calculated Input
Assumed Value
Assumed Value
The Galerkin finite element variation formulation used for the finite element analysis is:
∫ 𝑢̇𝑣𝑑𝑥 + ∫ 𝑢 ′ 𝑣 ′ 𝑑𝑥 = 0
Description of the Finite Element Model
The first model that will be reviewed is the beam model which was created using Abaqus CAE.
The beam model was made by drawing a line in the part creator; the line then had a profile had a profile with the specified height and width. Then an orientation to the profile on the beam was then applied and the beam was then meshed and material properties were applied. A line load was applied such that the beam was loaded. One end of the beam is then fixed in all degrees of freedom. Figure 1 shows the beam model prior to the analysis being run.
Figure 1 – Beam Model Overview

The second model is the plate model which was created in Abaqus CAE using the plate tool. The plate model was created and then a section was applied which the thickness of the plate has been specified. The plate then has a uniform pressure applied to the top of the plate and a boundary condition which locks the displacement at the edges of the plate but allows the edges to rotate.
The plate model is shown in Figure 2.
Figure 2 – Plate Model Overview
Results, Discussion and Conclusion
The stress and displacement results of the beam and plate models are shown in Figures 3-6.
Figure 3 – Stress Plot for Beam Model

Figure 4 – Displacement Plot of Beam Model
Figure 5 – Stress Plot of Plate Model

Figure 6 – Displacement Plot for Plate Model
The results from the finite element analyses were compared to the classical beam and plate equations to validate the models the results are shown in Table 2.
Table 2 – Results Validation
FEA Results
Roark’s Equation Results
Stress [MPa] Displacement
[µm]
Stress [MPa] Displacement
[µm]
Cantilever
Beam
Simply
Supported Plate
2.99
0.15
75.76
1.87
3.00
0.11
75.00
1.78
The results show that the beam model predicts nearly the same value as the Roark’s equations, the flat plate model results are also within the expected range given that general flat plate equations are not as exact as circular flat plates.
References
1.
Low Carbon Steel; MatWeb;
<http://www.matweb.com/search/DataSheet.aspx?MatGUID=034970339dd14349a8297d
2c83134649&ckck=1>
2.
Young, Warren; Bundynas, Richard; Roark’s Formulas for Stress and Strain Ed. 7;
McGraw-Hill; Dated 2002
3.
Zienkiewicz, O. (2013). Finite Element Method: Its Basis and Fundamentals.
Butterworth-Heinemann Ltd.