University of Mississippi Department of Civil Engineering
CE407: Civil Engineering Laboratory II
Experiment S2a: Stress Analysis of a Circular Arch-Numerical Experiments
Dr. Chris L. Mullen
Last Updated: 10-24-01
1.0 INTRODUCTION
1.1 The SABLE or SAP200 computer programs installed in the Civil Engineering
Computer Graphics Laboratory, Room 213 Carrier Hall, will be used to find
the internal forces in the circular arch portion of the Three-Hinged Arch
studied in Experiment S1. From these forces, stress and strain may then
be computed using the elementary beam bending theory of the mechanics of
materials.
1.2 The objective is to use the finite element method to estimate the
horizontal thrust reactions for comparison with those measured in Experiment
S1, and to further esticmate the internal forces, stresses, and strains
for comparison with the strains to be measured in Experiment S2b) .
2.0 THEORY
2.1 Definitions.
Finite Element Method- numerical method used in this application
to solve the equations of static equilibrium. A special case of this method
is the Displacement FEM which computes the displacements at discrete points
in a body called the nodes or joints. The body is assumed to be comprised
of finite elements which are connected at these joints and provide stiffness
to loads applied to the body.
Beam Element Internal Forces-in the case of a beam or beam element
used to model a planar arch, there are three forces which are necessary
to maintain static equilibrium at that section, namely, the axial force,
N,
the shear force, V, and the bending moment, M.
Stress- the intensity of force acting per unit of area at a
point. Only the strain can be measured directly. The stress is obtained
from a constitutive law (e.g. Hooke's law) which relates stress to strain.
2.2 Analysis
Compute the internal forces using the method of sections taken at an
arbitrary location along the arch. The system is statically determinate
when the arch acts as a single unit (without the deck and spandrels).
Compute the normal stress as a function of distance from the neutral
axis using N and M and elementary beam bending theory where plane sections
remain plane during deformation. Use Hooke's law and the assumption of
plane stress to obtain the normal strain.
3.0 SOFTWARE
The SABLE or SAP2000 finite element programs are to be used to construct
a model of the arch including the pinned supports and the hinge at the
crown. The beam elements provided in the program are to be used to define
the stiffness of the arch with respect to vertical loads applied to the
deck.
Prepare an Excel spreadsheet that computes the coordinates of
points on a circular arch in a Cartesian system with origin where the horizontal
line passing throught the points A and B intersects a vertical line at
midspan.
4.0 NUMERICAL PROCEDURE
For the symmetrical arch configuration in Experiments S1, consider
the three load cases. Use the finite element software to define a planar
model of the arch without the deck or spandrel elements.
Key modeling features are:
In order to obtain printed results for all three internal forces at
the gage position, it is necessary to place a joint at that location. Otherwise,
it will be necessary to plot the individual diagrams and graphically determine
the values.
The loads in the vertical direction are most conveniently applied using
joint loads instead of beam loads which are defined in a local beam coordinate
system (inclined in this case).
The pinned support joints should be restrained in all three DOF (x-translation,
y-translation, and z-rotation).
The hinge joint should be restrained only in the third DOF (z-rotation).
The beam elements that attach to the support and hinge joints should
have a pin member end condition at that end so that no bending moment will
be permitted.
All other member end conditions should be fixed so that a bending moment
will be developed as required.
Perform a mesh sensitivity analysis to determine the best accuracy for
the least number of elements. Use elements of equal arclength and
a criteria based on the effect on changes in computed bending moment.
Summarize your results in an Excel spreadsheet.
5.0 RESULTS AND CALCULATIONS
5.1 Computed Results Table
Summarize the numerical results in a simple table of the style developed
for the recorded data. Calculate theoretical predictions from the elementary
theoretical analysis, and compute percent differences (error) between the
numerical and theoretical stresses for the case of a central load
of 50 N, taking the theoretical stresses to be the "correct" ones.
6.0 DISCUSSION
Comment on sources of error in the computed data.
7.0 CONCLUSIONS
State what you accomplished through the experiment. Were your numerical
experiments reasonable? Was the software well suited for this experiment?
Do you have any suggestions to improve the procedure?