CIVE50003 计算方法

CIVE50003 Computational Methods II

Coursework – Influence lines and bridge structures

This project is to be carried out individually using the Matlab programming
environment. Please make an electronic submission on Blackboard of a report (no
more than 12 pages in pdf format) and your complete Matlab code (all m files). Your
report should include a detailed critique of your findings in the context of finite
element and structural theory, including computational considerations where
appropriate. Please write carefully and professionally and use appropriate formatting
in your document. You may collaborate and use any code released to you, but you
may not share any files or results. The deadline is 5 pm on Wednesday 5th April 2023.

You have been asked to investigate the statics of a truss bridge under the action of a
moving train locomotive with the aid of influence lines. An influence line tracks the
change of a force or moment at a single location depending on how a load pattern
moves across the structure. It is an important concept in the study of bridges.

A symmetric truss bridge is illustrated in Fig. 1. It consists of a simply-supported
‘central span’ resting on the tips of two opposing ‘cantilever arms’, which in turn are
supported by ‘anchor arms’. Assume that all members, or ‘chords’, are pin-ended bars
carrying only an axial force. Such bridges were commonly built as part of railway
networks in the 19th century all over the world (the Forth Railway Bridge in
Edinburgh is a famous example), and were often statically determinate to enable a
simple analysis of the statics. The action of the locomotive may be represented as a 2
MN point load moving along the bottom line of horizontal chords from A to F.

Assume that each chord is a built-up I-
section with dimensions as shown to
the right. Assume a modulus of
elasticity of 200 GPa and, where
applicable, a yield stress of 355 MPa.

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Fig. 1 – Determinate truss steel railway bridge and its various idealisations. The
geometry is also given, with each ‘bay’ having a 7.6 m span.
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NOTE: In what follows you may find it beneficial and intuitive to ‘animate’ the bridge
as the locomotive passes over it. The following code snippet can be usefully adapted
for this:
figure(‘unit’, normalized’,‘outerposition’,[0 0 1 1]);
% insert intermediate code here
for (each locomotive position)
% insert intermediate code here
clf;
% execute FE plotting code here and a ‘hold on’
drawnow; pause(0.5);
end

Q1. Considering only global static equilibrium and without using finite element
analysis yet, draw the influence lines for the vertical reactions at A and B as the
locomotive moves from A to F. Draw the influence line for the vertical force that
must be resisted by the DEF portion of the bridge when the locomotive is on the
ABCD portion of the bridge. Treat this henceforth as a vertical ‘reaction’ at D.

Q2. With reference to your hand calculations in Q1 and also Fig. 1, explain why a
finite element model of the entire structure ABCDEF is not necessary.

Q3. Write a general Matlab Class BRIDGE to represent a finite element model of
only the ABCD portion of the bridge, treating the effect of the DEF portion as a
horizontal roller support at D (why?). Research Cholesky decomposition (the chol
command) and use it in the most efficient way possible in your FE solver, justifying
why it can be very beneficial to performance. Show that you are able to obtain the
same influence lines for the vertical reactions at A, B and D as in Q1. Comment on
how you could, in fact, reduce the size of the finite element model even further.
NOTE: the very general Matlab Class BRIDGE should abstractly represent any
possible bridge model you will encounter here. The generic modules for matrix
assembly, FE solution, plotting (perhaps with force calculation) should be coded as
separate methods of such a Class, in addition to the ‘create instance’ method. Since
every truss model will have the same nodes, dofs and element connectivity and will
differ only through the locations of the point load representing the moving
locomotive, you should have this ‘base arrangement’ come as standard each time you
create an instance of the Class and this information should not be present in your
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main calling program. Neat and elegant code is required, with proper commentary,
annotation and indentation (consider using CTR+I to automate this).

Q4. Using both structural and computational arguments, demonstrate why the loss of
any diagonal chord is a Bad Thing.

Q5. Add a single horizontal chord at the location identified by O1O2 in Fig. 1.
Compare the influence lines for the vertical reactions at A, B and D that you now
obtain with those from Q1 and Q3. Comment on the possible reasons for any
differences.
NOTE: whether or not to include this chord should be a method of your Class.

Q6. Remove the additional horizontal chord that you added in Q5. Compute the
influence lines for the bottom chords marked B1 to B10 in Fig. 1 and plot these all on
the same properly-annotated figure. Which of these chords exhibits the maximum
possible axial force and where is the locomotive located when it does so? Do the same
anew for the top chords marked T2 to T10 (on their own common figure).
NOTE: use the dofs that you compute by FE for each model to calculate the axial
force in each chord of interest via the transformation matrix for a 2D bar element.
For the bottom chords, your influence lines should resemble something like this:

Q7. What is the factor of safety of the bridge against a) Euler column buckling and b)
plastic collapse and what is the critical chord in AB for both conditions? Comment on
whether the top and bottom chords of the anchor arm can safely resist the passing
locomotive. Structurally, where is the worst place to put the locomotive?
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NOTE: The critical Euler column buckling load for pin-ended member of length L is
Pcr = EI(π/L)2, where I is the 2nd moment of area in minor-axis bending, while the
squash load is Ps = Aσy. The truss chord members are so slender that buckling will
probably always control under compression.

Q8 Repeat Q6 and Q7 with the additional horizontal chord at O1O2. How do the
influence lines and factor of safety change?

Q9* (OPTIONAL BONUS). Consider Fig. 2 below, and then watch Buster Keaton
over the Easter holidays.

Fig. 2 – A scene from Buster Keaton’s 1926 film The General. Yes, they actually
destroyed a train and a bridge in order to film this. It was the most expensive film
scene shot during the silent film era.

References
Snyder L.K. (1932) “A study of influence line analysis of stresses in a cantilever style
highway bridge” MSc Thesis, School of Mines and Metallurgy, The University of
Missouri, USA.

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