MoM - I
Assignment
Complex Engineering Problem (CEP)
Class: ME-SIV-B
Group Members:
Name
Roll Number
Muhammad Haris
ME-1813
Ghazi Zain ul Abidin
ME-1800
S/Lt Adnan
P.No 3084
Contents
1. Problem Diagram .................................................................................................... 2
2. (A-1) Analytical Methods for Solving Statically Indeterminate Beams .............. 2
3. (A-2) Solving for reaction forces for specified loading ....................................... 3
4. (A-3) Finite Element Analysis (FEA) for analytical results verification .............. 4
5. Conclusion .............................................................................................................. 8
Page 1 of 10
1.
Problem Diagram
2.
(A-1) Analytical Methods for Solving Statically Indeterminate Beams
Method of Superposition
i.
1.
2.
3.
4.
5.
6.
The following steps are used in the method of moment area for statically indeterminate
beams:
Determine the support reactions using the equations of static equilibrium.
Identify the degree of indeterminacy of the beam, and draw the shear and bending
moment diagrams.
Select a section of the beam where the bending moment is unknown, and compute the
first moment of area, Ix, and the second moment of area, Ix2, of the section about the
neutral axis.
Use the compatibility equation method to determine the bending moment at the section.
This involves analyzing two determinate structures: the original beam with the section
removed, and a fictitious beam with the section removed and replaced by a spring. The
spring stiffness is determined by the compatibility of displacements at the section
between the two structures.
Repeat steps 3 and 4 for additional sections along the beam until the entire bending
moment diagram is determined.
Finally, use the shear force and bending moment diagrams to calculate the deflection of
the beam using standard methods such as the double integration method.
Page 2 of 10
ii.
Method of Moment Area
1.
2.
3.
4.
3.
The basic steps involved in using the method of superposition for statically indeterminate
beams are as follows:
Assume that the beam is composed of several simpler beams, each of which is statically
determinate and can be solved using basic principles of statics.
Apply the loadings to each of the simpler beams individually, and solve for the reactions,
shear forces, and bending moments in each case.
Combine the individual solutions using the principle of superposition. This involves
adding together the reactions, shear forces, and bending moments from each of the
individual solutions to obtain the overall solution for the original beam.
Check the overall solution to ensure that it satisfies the boundary conditions and any
other requirements of the problem.
(A-2) Solving for reaction forces for specified loading
i.
Method of Superposition
The following results were obtained from an analysis through the above method (See
appendix 1 for detailed working):
Reaction force 1 (fixed support)
Reaction force 2 (roller support)
Bending Moment
Deflection (Mild Steel)
Deflection (Aluminium)
ii.
3290.625 lb
309.325 lb
-141750 lb in
1.50 x10-5 in
4.25 x10-5 in
Method of Moment Area
The following results were obtained from an analysis through the above method (See
appendix 2 for detailed working):
Reaction force 1 (fixed support)
Reaction force 2 (roller support)
Bending Moment
3206.25 lb
393.75 lb
-121500 lb in
Page 3 of 10
4.
i.
(A-3) Finite Element Analysis (FEA) for analytical results verification
Simplified FEA through SkyCiv Online Beam Calculator
An initial simplified analysis was made through a rudimentary online beam calculator,
which provided the following results:
Figure 1: Problem Diagram
Figure 2: Tabular Results
Page 4 of 10
Page 5 of 10
ii.
Detailed FEA through Ansys Static Structural Analysis
An in-depth static structural analysis was made Ansys Mechanical Software, which
provided the following results:
Figure 3: Problem Diagram
Figure 4: Reaction Force 1 (Fixed Support)
Figure 5: Reaction Force 2 (Roller Support)
Page 6 of 10
Figure 6: Bending Moment
Figure 7: Total Deformation (Mild Steel)
Figure 8: Shear Stress (Mild Steel)
Page 7 of 10
Figure 9: Total Deformation (Aluminium)
Figure 10: Shear Stress (Aluminium)
5.
Conclusion
Based on the results, the following conclusions can be made:
The computer-based FEA results closely corroborate the results gained through both analytical
method, with a maximum true error of 9.43% (<10%), which can be explained due to the
assumptions and simplifications either analytical methods use in order to calculate a result.
While a 3-D analysis would be further required to make a definitive statement upon the safety of
the beam under the given loading conditions, one can reach a conclusion based on the above
results that the beam would be safe under the above loading provided that it has at least a
square cross-sectional area. Furthermore, a beam of mild steel would be a safer choice than a
beam of aluminium as it resists bending more and has lower deflection values and total
deformation.
Page 8 of 10