Term Project:
Pure Elastic 2D Beam Bending - Using Numerical Methods
Scott M. Steffan
December 9, 1999
Numerical Analysis for Engineering MEAE 4960H02
Rensselaer Polytechnic Institute – Hartford
TABLE OF CONTENTS
Section
Page Number(s)
1.0
Abstract
1
2.0
Introduction
1
3.0
Theory
1-3
4.0
4.1
4.2
4.2
4.4
Numerical Methods
Finite Element Method
Linear “Shooting” Method
Finite Difference Method
Raleigh-Ritz Method
3-6
3-4
4-5
5
5-6
5.0
Results and Error Analysis
5.1
Uniform Load
5.1.1
Finite Element Method
5.1.2
Linear “Shooting” Method
5.1.3
Finite Difference Method
5.1.4
Raleigh-Ritz Method
5.2
Linearly Varying Load
6-14
6-12
6-9
9
10
11-12
12-14
6.0
Conclusion
15
7.0
References
15
8.0
Appendices
Appendix A
Appendix B
Appendix C
Appendix D
Appendix E
Appendix F
Appendix G
16-33
Glossary of Notation

The following is a list of symbols used throughout this report.
Symbol

Label
Stress
Units
lbs./in2

Strain
in./in.
P
Load
lbs.
A
Cross-Sectional Area
in2
M
Bending Moment
lbs./in.
c
Radial Distance from Centerline of Beam
in.
I
Second Moment of Inertia
in4
E
Young's Modulus of Elasticity
lbs./in2
h
Interval Spacing
in.
w
Distributed Beam Load
lbs./in
L
Beam Length
in.
b
Beam Width
in.
H
Beam Height
in.
1
1.0
ABSTRACT
This term project studied four different numerical methods that could be used to approximate pure
elastic beam bending deflections. The first study involved a linear 2-noded beam element, finite
element method. A parametric mesh density study was performed with seven different element
breakups. Results showed the higher element meshes generated better approximations to the
actual beam deflections. The coarser meshes were too stiff and underestimated the actual
deflected shape. Then a detailed look into using the linear “shooting” method, the finite difference
method, and the Raleigh-Ritz method to approximate beam deflections was completed. Analysis
showed the finite difference method to be the least accurate of the group, however, each
technique produced respectable results. Higher accuracy could be obtained from any of the
methods by simply decreasing the spacing.
2.0
INTRODUCTION
The objective of this project involves solving for deflections along a prismatic, linear elastic bar
subject to pure bending loads. The report will begin with a brief discussion of the Strength of
Materials and Differential Equations theory involved in generating closed form solutions for a
specific problem. Then, a detailed look into four numerical methods that could be used to
generate approximate solutions will begin, followed by numerical deflection approximations for
two different loading situations. Finally, using the information provided a discussion on the
advantages and disadvantages of the various numerical techniques will be completed.
3.0
THEORY
A prismatic bar is a beam having constant cross-sectional area in which the length is typically
much greater than any of its cross-sectional dimensions. Figure 1 is an example showing a
prismatic bar bent by equal and opposite couples (Shigley & Mischke, 1989):
Figure 1: Beam Bending – Equal and Opposite Couples
There are many different aspects to the study of beam bending, however, this report is done
assuming the following:
(1)
(2)
(3)
(4)
The beam is subject to pure bending
The material is isotropic and homogeneous
The deflections are small
The bending is pure elastic
Assumption (1) allows axial loading to be neglected. Rather than having two loading types
involved in the stress solution, there is only one as shown in equation 1:

2
Mc P

I
A
Equation 1: Elastic Stress Equation
Where:
 = the stress at a specified location
M = the bending moment
c = the height above or below the beam center line
I = the beam's second moment of inertia
Assumptions (2) - (4) allow nonlinear effects to be neglected, therefore Hooke's Law is obeyed:
  
Equation 2: Hooke’s Law for Stress and Strain
Where:
 = the stress at a specified location
 = the strain at a specified location
E = Young's Modulus of the Material (uniform in all directions)
The problem used for this research involves a simply supported beam subject to a distributed,
uniform load. The question is: How much will a beam deflect under a specified load? The
answer involves the manipulation of the following fourth order differential equation:
d2
dx 2

d2y
 E I 2   w
dx 

Equation 3: 4th Order Differential Equation - Distributed Load
Where:
E = Young's Modulus of the Material
I = The beam's second moment of inertia
w = The load per unit length over the beam (including the weight of the beam)
This is done using the diagram and coordinate system shown in figure 2 below:
w
y
y
x
H
z
L
b
Figure 2: Beam Geometry, Boundary Conditions, and Coordinate System
The fourth order differential equation can be solved fairly painlessly when both the load and
cross-section are uniform. First, boundary conditions for the specified end conditions (simply
supported) are determined. They are shown in equation 4 below:
y(0) = y(L) = 0 = y’’(0) = y’’(L)
Equation 4: Boundary Conditions
Successive simple integration aids in obtaining the exact solution:
d3y
 w x  C1
dx 3
d2y w
E I 2  x 2  C1 x  C 2
2
dx
dy w 3 C1 2
EI
 x  x  C 2 x  C3
dx 6
2
w 4 C1 3 C 2 2
EI y 
x 
x 
x  C3 x  C 4
24
6
2
EI
3
Equations 5-8: Differential Equation Integration
Where C1, C2, C3, C4 are arbitrary constants. Using y’’(0)=0 implies C2=0, and allows for
C1=-wL/2 to be determined. Also, y(0)=0 implies C4=0, and with y(L)=0, it follows that C3=wL3/24.
Using this information, an equation for deflection as a function of distance along the beam, y(x) is
generated:
y ( x) 
w

x 4  2 Lx 3  L3 x 
24 E I
Equation 9: Deflection as a Function of Beam Location
Symmetry also reveals that at x=L/2, the maximum deflection occurs:
y max 
5wL4
384 EI
Equation 10: Max Deflection
This derivation provides a baseline for which numerical methods can be compared with.
Equation 9 above is commonly referred to as a singularity function. A table of singularity
functions for various different loading and supporting conditions can be found in reference (1), pg.
735-742, Table A-9. Differential Equation boundary conditions for the other types of end
conditions are supplied in Appendix A.
Once these deflections are determined, elastic stresses and strains can be solved for. As
mentioned earlier, when the cross-sectional dimensions of the beam are constant, as well as the
load over the surface, the closed form solution can be solved through integration. This is the
situation that will be studied. Further, a more complex situation will be discussed to show the true
benefits of numerical approximation methods.
4.0
NUMERICAL METHODS
The 4th order beam differential equation 3 discussed in the previous section is an example of a
boundary value problem. Rather than taking the mathematical problem and generating analytical
solutions, a numerical method takes the mathematical problem, makes approximations, converts
the problem into a numerical problem, and generates a numerical solution. The problem being
discussed involves constant second moment of inertia (I(x)=I0) and load (w(x)=w0), making it easy
to solve analytically and provide solid ground for generating numerical approximations and
comparing them to the actual values. There are many numerical techniques that could be used to
generate approximate deflection results. The four that will be investigated in this research are the
Finite Element Method, the Linear “Shooting” Method, the Finite Difference Method, and the
Raleigh-Ritz Method.
4.1
Finite Element Method
One of the most widely used techniques today is the Finite Element Method (FEM). FEM is a
numerical approach commonly used to solve engineering mechanics problems. Many general
finite element code packages have been written over the years with user friendly windows and
menus (GUI) which allow for easy geometry setup, boundary condition manipulation and
evaluation/post-processing of common structural problems. Some of the most popular
commercial codes in the industry are ANSYS, MSC Nastran, and MARC. ANSYS will be the
code used for the work in this research.
The FEM subdivides the model, in this case a beam, into smaller sections (elements) and
determines values at each end (nodes) of these smaller beams. A physical, finite element mesh
is converted into a mathematical model. The differential equations involved in the model are
minimized and solved through algebraic matrix manipulation of a linear system of equations. One
advantage this method has over some of the others to be studied is the way the boundary
4
conditions are handled. The boundary conditions are treated as integrals in a functional, which is
to be minimized. By doing this, the model construction is independent of the actual boundary
conditions. This allows the user to easily modify the loading, constraints, and actual geometric
shape. It is very beneficial when there are partial differential equations, derivatives, and irregular
shapes involved.
Various element types could be used for this particular problem. One is known as a 2D-beam
element. This type of element can be used for beam structures that have either a constant crosssectional area, or can be divided into elements, varying the cross-sectional dimensions
appropriately. A two-noded linear beam element as described above will be used for this work.
Figure 3 represents a thin beam subdivided into 8, two-noded beam elements:
Elements
Nodes
Figure 3: 2-Noded Linear Beam Element Mesh – 8 Elements
This element type allows the user to input each element’s cross-sectional properties, material
properties, and loading/boundary conditions.
4.2
Linear Shooting Method
The Linear Shooting Method uses the following 2nd order differential equation boundary-value
problem:
y ' '  p( x) y '  q( x) y r ( x)
Equation 11:
2nd
a  x  b, y (a)   , y (b)  
Order Boundary Value Problem
The boundary value problem is replaced by two initial-value problems shown below:
y ' '  p( x) y '  q( x) y r ( x)
y ' '  p ( x) y '  q ( x) y
a  x  b, y (a)   , y ' (a)  0
a  x  b, y (a)  0, y ' (a) 1
Equations 12-13: 2nd Order Initial Value Problems
Then, using the 4th order Runge-Kutta Method, approximate initial value solutions for y1(x) and
y2(x) are generated from equations 12 and 13 respectively with the input of boundary conditions
at the endpoints. Once y1(x) and y2(x) are known, the following can be solved:
y( x)  y1 ( x) 
  y1 (b)
y 2 (b)
y 2 ( x)
Equation 14: Solution for y(x)
Now, the deflection at any given location along the length of the beam can be approximated,
provided the appropriate number of subintervals is specified. The method is shown graphically in
figure 4 below (Burden & Faires, 1997):
5
Figure 4: Graphical Representation of “Shooting” Method
4.3
Finite Difference Method
The third method used in this research is the Finite Difference Method. FDM uses the linear
second-order boundary-value problem shown in equation 11 of the previous section. Unlike the
Finite Element Method, each of the derivatives in the differential equation is replaced by
difference-quotient approximations. This method effectively replaces a continuous differentiation
with a finite difference over a discrete region. Equation 11 would require difference-quotient
approximations for y’’ and y’, however the beam equation 6 only has the y’’ term. First, the
spacing, h is set by equation 15 where a and b are the beginning and end of the beam
respectively, and N is the number of intervals (stations):
h
(b  a )
( N  1)
Equation 15: Station Spacing for FDM
Then, using Taylor Polynomial Expansion, the Intermediate Value Theorem, and the centereddifference formula, y’’(xn) is simplified and replaced for each interior station along the beam with
equation 16:
y' ' ( xn ) 
y n1  2 y n  y n1
 r ( xn )
h2
Equation 16: Central Difference Approximation for 2nd Order DE
Using equation 16, together with the end conditions, helps to create a linear system of equations,
which can be expressed in matrix form as shown in equation 17:
Ay  b 

Equation 17: Matrix form of FDM Linear System
Where:
y represents the vector of unknowns (deflections)
A represents the matrix of coefficients
b represents the vector of known quantities.
The tridiagonal, positive definite, and simply diagonally dominant matrix can be reduced and
solved using an iterative linear system solution technique.
4.4
Raleigh-Ritz Method
The Raleigh-Ritz Method uses an approach very similar to the Finite Element Method, but
different from the linear “shooting” or finite difference methods. The method begins with the
differential equation shown in equation 18:

d 
dy 
 p( x)   q( x) y  f ( x)
dx 
dx 
where 0  x  1
6
Equation 18: Boundary Value Problem DE for Raleigh-Ritz Method
For the simply supported beam differential equation, the q(x) term is non-existent, p(x)=1, and the
boundary conditions are set to zero at each end. Then, rather than making approximations for
the boundary value problem shown in equation 18, the method looks for a y(x) which will
minimize the integral shown in equation 19:
1
I [ y]   { p( x)[u ' ( x)] 2  q( x) [u ( x)] 2  2 f ( x)u ( x)}dx
0
Equation 19: Integral to be minimized
The method is used to approximate the solution of y over a small set of functions produced by a
combination of linearly independent basis functions, i(x) where I=1,2,…,n. The first
approximation to the solution of y(x) is made by equation 20:
n
y ( x) i   ci i ( x )
1
Equation 20: First Approximation to Solution
The coefficients, ci are chosen so that equation 19 is minimized. The entire technique can then
be put into matrix form, and i(x), the basis function is determined based on the spacing, h, and
the set of piecewise linear polynomials shown in equation 21:
0  x  xi 1
 0,
 x  xi 1
, xi 1  x  xi

 hi 1
 i ( x)  
x x
 i 1
, xi  x  xi 1
 hi
 0,
xi 1  x 1








Equation 21: Piecewise Linear Basis Function
The tridiagonal matrix can then be solved numerically, for the unknown vectors, ci.
5.0
RESULTS AND ERROR ANALYSIS
5.1
Uniform Load – Simply Supported Beam
In order to do some comparisons between the actual solution and approximations determined by
the various numerical methods some beam specifications are given:
E = 1e+11 lbs/in2
Width, b = 1 in.
Height, H = 1 in.
Second Moment of Inertia, I(x)=I0=bh3/12 = 1/12 in4
Length = 10 in.
Uniform Load, w(x)=w0 = -1e+5 lbs/in2
 Results recorded every 0.25in along the beam.
5.1.1 Finite Element Method
Using linear two-noded 2D-beam elements in ANSYS, a parametric mesh density study was
performed. This allowed us to see how close the true beam deflection was approximated.
Meshes of 2, 4, 5, 8, 10, 20 and 40 elements were used. The linear results were extracted from
ANSYS and plotted in Excel. The lower element meshes were capable of determining deflections
7
exact to within the computer epsilon at the ends of each element (node locations), however, the
result between nodes is very inaccurate. This is because the code is simply linearly interpolating
between nodes to generate results. Figure 6 below shows 2, 4, 5, and 8 element mesh densities
and their results relative to the strength of material solution:
Finite Element Method: 2D Simply Supported Beam
Deflection vs. Axial Beam Location
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
-0.0002
-0.0004
Deflection (in.)
-0.0006
-0.0008
-0.0010
-0.0012
Y disp - 2E
Y disp - 4E
Y disp - 5E
Y disp - 8E
Closed Form
-0.0014
-0.0016
Axial Beam Location (in.)
Figure 6: 2, 4, 5, and 8 Beam Element Meshes – Deflection vs. Axial Beam Location
The finite element method first solves for displacements, and then stresses and strains. For this
reason, it is critical to model the true deflected shape. Using too few elements, results in a
geometry model that is too stiff to accurately predict the peak stresses in the beam. This means
a small error in displacement results in higher order error in the stress solution. Figure 7 shows
the 10, 20, and 40 element mesh densities and their results relative to the strength of material
solution:
Finite Element Method: 2D Simply Supported Beam
Deflection vs. Axial Beam Location
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
-0.0002
-0.0004
Deflection (in.)
-0.0006
-0.0008
-0.0010
-0.0012
Y disp - 10E
Y disp - 20E
Y disp - 40E
Closed Form
-0.0014
-0.0016
Axial Beam Location (in.)
8
Figure 7: 10, 20, and 40 Beam Element Meshes – Deflection vs. Axial Beam Location
It is visibly clear that the higher number of elements, the more accurate the deflected shape.
Querying the deflections every 0.25” along the beam allowed for some error calculations. The
plot in figure 8 shows the percent error of the various mesh densities vs. axial beam location:
Finite Element Method Mesh Density Study
% Error in Deflection vs. Beam Location
40.0
Error-2E
Error-4E
Error-5E
Error-8E
Error-10E
Error-20E
Error-40E
35.0
% Deflection Error
30.0
25.0
20.0
15.0
10.0
5.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Beam Location (in.)
Figure 8: % Error vs. Axial Beam Location for Different Mesh Densities
Due to the very small deflections, all error analysis was generated in percent error from the
closed form solution. Using the data presented above, the l-infinity norm for each different mesh
density was determined. This was done by taking the percent error at every location along the
beam for each method and sorting out the highest. Then, the highest error of the four methods
was separated. Table 1 shows the results:
Table 1: FEA Mesh Study: % Error l-infinity norm
The analysis data obviously supports a finer mesh density, with the 40-element mesh giving
approximations at every node to within the computer epsilon. It must be mentioned, however,
that increased accuracy comes with additional solution time per iteration (possibly more iterations
necessary as well). For this simpler problem, the difference was not very noticeable (a matter of
seconds), but when these engineering mechanics problems, become larger, and more complex
(3D, more involved boundary conditions, etc.), engineering judgement to the accuracy of the
9
approximation comes into play. It is impractical and sometimes impossible to check
approximations with the exact solutions. A parametric mesh density study to show a converged
stress solution is being reported is also a good check. These studies typically require at least
three mesh densities with the same geometry and boundary conditions. One other method to
generate higher accuracy is to use a higher order element. A quadratic element, rather than a
linear element can usually achieve the required accuracy with a coarser mesh. Their additional
terms allow a better displacement approximation.
Note: All the data in this section can be found in tabular form in appendix B. The finite element
macros used to build the models and run the solutions can be found in appendix C.
5.1.2 Linear “Shooting” Method
The second method used to approximate deflections for the problem was the Linear “Shooting”
Method. Algorithm 11.1 from the textbook was used in Maple V (Burden & Faires, 1997).
The spacing was set to output results every 0.25” along the beam, the same breakup as the 40
element FEM, which would allow for good error comparison. The percent error relative to the
closed form solution is plotted in figure 9 below:
Linear "Shooting" Method: % Error in Deflection vs. Beam Location
0.0040
0.0035
% Deflection Error
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Axial Beam Location (in.)
Figure 9: “Shooting” Method: % Error vs. Axial Beam Location
The solution produces good results with very quick solution time. They are not as accurate as the
40-element finite element method approximation, but the trade for solution time may make the
“shooting” method a valid approach for this particular problem. The computational error is highest
at the edges, near the boundary conditions. Algorithm 11.1 required a 2 nd order differential
equation. This involved solving two steps of integration of the 4th order DE as discussed in the
theory section of this report, prior to using the code. In cases that vary in cross-sectional
dimensions or load over the length of the beam, the integration becomes more complex, as will
be shown in the section 5.2.
Note: All the data in this section can be found in tabular form in appendix B. A printout of the
algorithm as well as the actual solution file can be located in appendix D.
10
5.1.3 Finite Difference Method
Algorithm 11.3,also provided by our textbook (Burden & Faires, 1997) was used in Maple V to
generate deflection approximations along the beam using the linear finite difference method. The
method’s accuracy relative to the closed form solutions using N=39 or h=0.25 are shown in the
figure 10 below:
Finite Difference Method: % Error in Deflection vs. Beam Location
0.0700
0.0600
% Deflection Error
0.0500
0.0400
0.0300
0.0200
0.0100
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Axial Beam Location (in.)
Figure 10: Finite Difference Method: % Error vs. Axial Beam Location
Again, the results are reasonable, however this method recorded higher overall errors than FEM
or linear “shooting”.
Note: All the data in this section can be found in tabular form in appendix B. A printout of the
algorithm as well as the actual solution file can be located in appendix E.
5.1.4 Raleigh-Ritz Method
The final method chosen to approximate beam deflections is the Raleigh-Ritz Method. Again, the
course textbook provided an algorithm (11.5) written in Maple V which would allow the user to
input the 2nd order DE and interval spacing on a scale linearized from 0 to 1 (Burden & Faires,
1997). The chosen spacing was 0.25” to remain consistent with the other analysis. The
uniformly loaded beam deflections were again approximated very well. The percent error relative
to the closed form solution is plotted in figure 11 below:
11
Rayleigh-Ritz Method: % Error in Deflection vs. Beam Location
0.0040
0.0035
% Deflection Error
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Axial Beam Location (in.)
Figure 11: Raleigh-Ritz Method: % Error vs. Axial Beam Location
The plot looks very similar to the percent error plot for the linear “shooting” method, figure 8. In
fact, the errors for the two methods turned out to be exactly the same at each measured axial
beam location.
Note: All the data for the Raleigh-Ritz method analysis can be found in tabular form in appendix
B. A printout of the algorithm as well as the actual solution file can be located in appendix F.
Once all the data was gathered and plots were generated, the error given by each numerical
approximation method was analyzed. The l-infinity norm for the four methods, each with the
same spacing of 0.25” was determined. This was done by taking the highest % error for each
location along the beam for each method. Then, the highest error of the four was separated.
Table 2 below shows what was discovered:
Table 2: % Error l-infinity norm: Simply Supported Beam w/ Uniform Load
The finite difference method turned out to be the least accurate of the group, while the linear
“shooting” and Raleigh-Ritz methods produce the same error. The reason the linear “shooting”
method performs better that the finite difference method is due to the use of 4th Order RungeKutta for approximating solutions to the initial value problems. This technique has a truncation
error of order O(h4) vs. O(h2) for the FDM. There are two ways to get a better FDM solution. The
first, which is applicable to all techniques presented, is to reduce the spacing size, h. The second
would be to use a higher order Taylor series expansion for approximating y’’(x). Although, both of
these changes will increase accuracy, the increase in intervals and terms will add to solution time.
The best method for this loading/boundary condition situation turned out to be the 40-element
(0.25” spacing) finite element model.
12
5.2
Varying Load – Simply Supported Beam
The purpose of this section is to show the power of using a numerical analysis technique to
approximate beam-bending deflections. As mentioned earlier, not all beams have a constant
load and cross-section making the actual 4th order differential equation much more difficult to
evaluate. This is when numerical methods and engineering judgement as to the accuracy of the
solution become very valuable. To illustrate, solutions for the following beam properties will be
generated using the three methods:
E = 1e+11 lbs/in2
Width, b = 1 in.
Height, H = 1 in.
Second Moment of Inertia, I = bh3/12 = 1/12 in4
Length = 10 in.
Distributed Load, w*x= -100000x lbs/in2
 Results recorded every 0.5 in length along the beam.
For this problem, it is slightly more mathematically intensive to generate a closed form solution,
because there are more terms to be integrated due to the varying load. The following set of
equations reflect the actual integration solution:
EI
EI
EI
EI
EI
d4y
 wx
dx 4
d3y
w
 x 2  C1
3
dx
2
2
d y
w
 x 3  C1 x  C2
2
dx
6
dy
w 4 C1 2

x 
x  C 2 x  C3
dx 24
2
w 5 C1 3 C2 2
y 
x 
x 
x  C3 x  C 4
120
6
2
Equations 22-26: Differential Equation Integration
Using the boundary conditions given in equation 4, the coefficients, C1, C2, C3, C4 were
determined and are shown in table 3:
w 2
L
6
C2  0
C1 
7 ( w) 4
L
360
C4  0
C3 
Table 3: Coefficients for Deflection Equation 18
Also, due to the fact that the beam-bending problem begins with a 4th order differential equation,
one would need to reduce this into a 2nd order boundary value problem to make the finite
difference method, or linear “shooting” algorithms supplied by the textbook generate
13
approximations. The simple geometry and boundary condition set-up involved in using a
commercial finite element code are displayed in figure 11:
Figure 11: ANSYS Finite Element Method Physical Model
The results of the four methods and their percent errors are shown in figures 12 and 13 below:
Simply Supported Beam w/ Linearly Increasing Load
Deflection vs. Beam Location
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
-0.0010
Deflection (in.)
-0.0020
-0.0030
-0.0040
-0.0050
-0.0060
Closed Form
20E-Finite Element
Linear "Shooting"
Finite Difference
Raleigh-Ritz
-0.0070
-0.0080
Axial Beam Location (in.)
Figure 12: Deflection vs. Axial Beam Location
10.0
14
Simply Supported Beam w/ Linearly Increasing Load
% Error vs. Beam Location
0.3000
% Error FEM
% Error "Shooting"
0.2500
% Error FDM
% Error Raleigh-Ritz
% Error
0.2000
0.1500
0.1000
0.0500
0.0000
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Axial Beam Location (in.)
Figure 13: % Error vs. Axial Beam Location
Figure 12 shows all methods do a good job of approximating the actual beam deflection. Figure
13, the percent error plot shows the Finite-Difference Method generates the least accurate results
using the same refinement as the other three numerical methods. Their error is so small relative
to the FDM that they do not visually appear on the chosen % error scale. The Linear “Shooting”
and Raleigh-Ritz produce deflections closest to the actual, however the Finite Element Method
ranks a close second. The l-infinity norm for the four methods of equal spacing (h=0.5”) was
again determined and shown in table 4 below:
Table 4: % Error l-infinity norm: Simply Supported Beam w/ Linearly Varying Load
All errors were within 0.3% of the closed form, Strength of Materials calculation, with the finite
difference again producing the least accurate results. The linear “shooting” and Raleigh-Ritz
methods showed the same % error at each location, and under this loading condition, were
slightly better than the finite element method.
This problem supports the use of fast, accurate numerical approximation techniques for beambending deflection analysis. They become more and more valuable as the problem’s degree of
difficulty increases.
Note: All the data used to make the plots in this section can be found in tabular form in appendix
B. Finite Element Method, Linear "Shooting" Method, Finite Difference Method, and Raleigh-Ritz
Method analysis files can be located in appendices C, D, E, and F respectively.
15
6.0
CONCLUSION
Although there are many different types of numerical methods that could be used to generate
approximations for beam bending boundary value problems, the four chosen showed good
results. All methods require a good mesh spacing to obtain a worthy representation of the true
beam deflection. The Finite Element Method is the most universal, but also the most computation
intensive of the three. When the geometry and/or boundary conditions become more complex,
this method still allows for good results if modeled correctly. The Linear “Shooting” Method
produced good approximations under both studied loading conditions, however it can often have
instability problems. For this reason, the Finite Difference Method, although it requires more work
to obtain the specified accuracy, was also investigated and performed well. The Raleigh-Ritz
Method, with its similarities to the finite element method also showed very good deflection results
for these simple beam-loading conditions. Table 5 below was created to rank the three methods
studied as to their accuracy and usefulness:
Criteria
Ease Of Use
Solution Time
Accuracy of Approx.
Versatility
Availability
Totals
Finite Element
1
3
1
1
1
7
Linear "Shooting"
2
1
1
4
1
9
Finite Difference
2
2
4
2
1
11
Raleigh-Ritz
2
1
1
3
1
8
Table 5: Method Selection Chart
The most objective criteria are the accuracy of the approximation and the measurement of
iteration/solution time. The others are somewhat subjective to the codes used for this particular
research.
Although mathematical computation time is the longest for the Finite Element Method, the
versatility and accuracy when used correctly, with a good, refined mesh in areas of concern
makes it the first choice for solving linear elastic beam-bending type problems. The similarities of
the Raleigh-Ritz method standout in its ranking close behind the FEM. Linear “Shooting” and
Finite Difference Methods rank 3rd and 4th respectively. Note, this may not be consistent with all
boundary value problems and is based solely on the research and analysis studies performed
specifically for this project. Some other types of numerical approximation methods that could be
studied for a beam deflection problem are Collocation, Subdomain, and the Least Squared
Method.
7.0
REFERENCES
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Inc., New York, NY, 1989
(2) Burden, R., and Faires, J.D., Numerical Analysis 6th Edition, Brooks/Cole Publishing
Company, Pacific Grove, CA, 1997
(3) Timoshenko, S.P., Goodier, J.N., Theory of Elasticity 3rd Edition, McGraw-Hill, Inc., New
York, NY, 1970
(4) Edwards, C.H. Jr., and Penney, D.E., Elementary Differential Equations With Boundary Value
Problems 3rd Edition, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1993
(5) Bathe, K.J., and Wilson, E.L., Numerical Methods In Finite Element Analysis, Prentice-Hall,
Inc., Englewood Cliffs, New Jersey, 1976
(6) Desai, C.S., Elementary Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, New
Jersey, 1979
(7) Craig, R.R. Jr., Mechanics of Materials, John Wiley & Sons, Inc., New York, NY, 1996
(8) Conte, S.D., and de Boor, C., Elementary Numerical Analysis, An Algorithmic Aproach,
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