CIVE50003 Computational Methods IICoursework – Influence lines and bridge structuresThis project is to be carried out individually using the Matlab programmingenvironment. Please make an electronic submission on Blackboard of a report (nomore than 12 pages in pdf format) and your complete Matlab code (all m files). Yourreport should include a detailed critique of your findings in the context of finiteelement and structural theory, including computational considerations whereappropriate. Please write carefully and professionally and use appropriate formattingin your document. You may collaborate and use any code released to you, but youmay 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 amoving train locomotive with the aid of influence lines. An influence line tracks thechange of a force or moment at a single location depending on how a load patternmoves 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 aresupported by ‘anchor arms’. Assume that all members, or ‘chords’, are pin-ended barscarrying only an axial force. Such bridges were commonly built as part of railwaynetworks in the 19th century all over the world (the Forth Railway Bridge inEdinburgh is a famous example), and were often statically determinate to enable asimple analysis of the statics. The action of the locomotive may be represented as a 2MN 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 tothe right. Assume a modulus ofelasticity of 200 GPa and, whereapplicable, a yield stress of 355 MPa.2Fig. 1 – Determinate truss steel railway bridge and its various idealisations. Thegeometry is also given, with each ‘bay’ having a 7.6 m span.3NOTE: In what follows you may find it beneficial and intuitive to ‘animate’ the bridgeas the locomotive passes over it. The following code snippet can be usefully adaptedfor this:figure(‘unit’, normalized’,‘outerposition’,[0 0 1 1]);% insert intermediate code herefor (each locomotive position)% insert intermediate code hereclf;% execute FE plotting code here and a ‘hold on’drawnow; pause(0.5);endQ1. Considering only global static equilibrium and without using finite elementanalysis yet, draw the influence lines for the vertical reactions at A and B as thelocomotive moves from A to F. Draw the influence line for the vertical force thatmust be resisted by the DEF portion of the bridge when the locomotive is on theABCD 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 afinite element model of the entire structure ABCDEF is not necessary.Q3. Write a general Matlab Class BRIDGE to represent a finite element model ofonly the ABCD portion of the bridge, treating the effect of the DEF portion as ahorizontal roller support at D (why?). Research Cholesky decomposition (the cholcommand) and use it in the most efficient way possible in your FE solver, justifyingwhy it can be very beneficial to performance. Show that you are able to obtain thesame influence lines for the vertical reactions at A, B and D as in Q1. Comment onhow you could, in fact, reduce the size of the finite element model even further.NOTE: the very general Matlab Class BRIDGE should abstractly represent anypossible bridge model you will encounter here. The generic modules for matrixassembly, FE solution, plotting (perhaps with force calculation) should be coded asseparate methods of such a Class, in addition to the ‘create instance’ method. Sinceevery truss model will have the same nodes, dofs and element connectivity and willdiffer only through the locations of the point load representing the movinglocomotive, you should have this ‘base arrangement’ come as standard each time youcreate an instance of the Class and this information should not be present in your4main 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 ofany 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 nowobtain with those from Q1 and Q3. Comment on the possible reasons for anydifferences.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 theinfluence lines for the bottom chords marked B1 to B10 in Fig. 1 and plot these all onthe same properly-annotated figure. Which of these chords exhibits the maximumpossible axial force and where is the locomotive located when it does so? Do the sameanew 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 axialforce 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 onwhether the top and bottom chords of the anchor arm can safely resist the passinglocomotive. Structurally, where is the worst place to put the locomotive?5NOTE: The critical Euler column buckling load for pin-ended member of length L isPcr = EI(π/L)2, where I is the 2nd moment of area in minor-axis bending, while thesquash load is Ps = Aσy. The truss chord members are so slender that buckling willprobably always control under compression.Q8 Repeat Q6 and Q7 with the additional horizontal chord at O1O2. How do theinfluence lines and factor of safety change?Q9* (OPTIONAL BONUS). Consider Fig. 2 below, and then watch Buster Keatonover the Easter holidays.Fig. 2 – A scene from Buster Keaton’s 1926 film The General. Yes, they actuallydestroyed a train and a bridge in order to film this. It was the most expensive filmscene shot during the silent film era.ReferencesSnyder L.K. (1932) “A study of influence line analysis of stresses in a cantilever stylehighway bridge” MSc Thesis, School of Mines and Metallurgy, The University ofMissouri, USA.