CIVE50003 Computational Methods II Coursework – Dynamic analysis of a cable-stayed bridge 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 10 am on Friday 25th March 2022. You are asked to investigate the dynamic response of a cable-stayed bridge shown in Fig. 1. The structure shown is an idealisation of the main portion of the Theodor Heuss Bridge in Düsseldorf. It is famous for being the first cable-stayed bridge in Germany, built between 1953 and 1957. Assume that L = 108 m and H = 40 m. It is proposed to model the bridge deck ABMCD with 2D general beam elements d1 to d14, for which you can assume A = 1.88 m2 and I = 1.8036 m4. The pylons BE and CF can be modelled with 2D general beam elements p1 to p6, for which you can assume A = 0.49 m2 and I = 0.4904 m4. The pin-ended cables can be modelled with 2D bar elements c1 to c6 for which you can assume A = 0.0531 m2 (why do you not need I ). There should be 32 elements in total. These properties assume only one plane through the bridge. Assume a modulus of elasticity of steel of 200 GPa and a density of ρ = 7850 kg/m3. Assume that the bridge is already prestressed to support its own self-weight and any other ‘permanent’ loads, which you do not need to consider. However, in Q1 & Q3 the bridge deck is acted on by a ‘live’ vertical point load P = 100 kN (i.e. a truck) at the midpoint M. You will learn more about bridges and dynamics in years 3 & 4, the aim here is to write structural analysis software and to develop an intuition for dynamic behaviour. 2 Fig. 1 – Idealisation of the Theodor Heuss Bridge in Düsseldorf, Germany 3 The Matlab code that you submit must make use of object-oriented programming through the creation of a single Class Bridge with appropriate properties and methods (constructor, assembler, various FE solvers and post-processors / plotters). This Class should then be instantiated inside other calling scripts, one per Q below. A premium is placed on minimising code repetition and good code structure. You may find it beneficial to simply get an FE model running correctly before turning it into a Class. Static linear stress analysis This analysis involves the solution of the static equilibrium equation below in the same way as we have been doing through the module so far (with matrix partitioning). [ ]{ } { }k d f= Q1. Perform a static linear stress analysis of the structure under the action of the midspan point load P. Verify that the reaction forces satisfy global equilibrium. Report the vertical sag at M and the horizontal sway at E. Plot the deformed shape of the structure and comment. Note: the vertical sag at M should be 6.8018 mm and horizontal sway at E should be 1.1586 mm. Make sure you can compute these values exactly (for validation only, almost nothing in civil engineering is ever known to within 10-4 mm or 0.1 μm!). Linear frequency analysis This analysis involves the solution of the eigenvalue problem [ ]{ } [ ]{ }2FF F FF Fk mφ ω φ= where ω is the angular frequency of vibration in radians / second, which you should then transform into a cyclic frequency f = ω/2π in cycles / second (or Hertz) and a period T = 1/f in seconds. [mFF] is a partitioned sub-matrix of [m], the global mass matrix, and { F} is a vector of free global dofs representing the vibration mode associated with the ω vibration frequency. A vibration mode is a deformation state where the internal elastic forces (caused by stiffness-resisting deformation) are in balance with the inertia forces (caused by mass-resisting acceleration). Q2. Perform a linear frequency analysis to compute all of the eigenvalues
(how many are there ) and plot the eigenmodes corresponding to the lowest four frequencies and comment. Compute also the total weight of the structure. 4 Note: the lowest four vibration frequencies should be 0.47347, 0.63481, 0.97582 and 1.2728 Hz. The highest frequency should be 205.7568 Hz. The structure should weigh 7716581.2295 kg. Make sure you can compute these values exactly. Explicit response-history analysis This analysis involves the incremental solution of the dynamic equilibrium equation [ ]{ } [ ]{ } [ ]{ } { }t t t tm d c d k d f′′ ′+ + = where {d}t, {d′}t and {d″}t are the nodal displacement, velocity and acceleration vectors respectively at time t, [k], [c] and [m] are the stiffness, damping and mass matrices respectively and {f}t is the vector of external forces at time t. Specifically, solution requires integrating the above to track the evolution of {d}t with time. An explicit integration scheme uses, say, central differences to approximate the velocity and acceleration vectors as { } { } { }( )12t t t t td d dt + ′ ≈ and { } { } { } { }( )2 1 2 t t t t t t d d d d t + ′′ ≈ + respectively to express {d}t+Δt (displacement vector at a time step Δt in the future) explicitly in terms of its present {d}t and previous {d}t–Δt values. Convince yourself that this gives: { } [ ] [ ] { } [ ] [ ] { } [ ] [ ] { } 1 2 2 2 1 1 2 1 1 2 2t t t t t t d m c f k m d m c d t t t t t + = + In what follows, you may assume that: The damping matrix [c] is given by [ ] [ ] [ ]c m kα β= + where α = 0.0583 and β = 3.6×10-5 which are the Rayleigh damping parameters to get 1% damping for relevant vibration modes (more about this in your 3rd year dynamics module). The time increment Δt does not exceed 1 2 cr t t = where max 2 crt ω = where ωmax is the highest cyclic vibration frequency (in radians/second) from Q2. If you exceed Δtcr (a ‘critical’ time step), your solution for {d}t may diverge because explicit integration schemes are only conditionally stable. For t ≤ 0, initialise the displacement vector to zero: { } { } { }0 0t t td d= = = = 5 Q3. In Matlab, create a vector of times ts from 0 to 50 seconds in steps of Δt. Create a second vector fs that computes P·(1 + 0.1·sin(Ωt)) at each time point (i.e. a time- history record for a ‘cyclic truck’ point load that is vibrating vertically at an angular frequency of Ω in radians / second). This record is fed into the correct dof of {f}t, all others zero. You must use [kFF], [cFF], [mFF], {dF} and {fF} during integration. Perform an explicit response-history analysis of the bridge under a midspan cyclic point load assuming Ω = 1, 2, 3, 4, 5 and 6 radians / second (separate analysis for each). Compare the time-history of the structural response (the horizontal sway at E and the vertical sway at M) under each vibrating frequency on a plot (also show the static value from Q1) and comment. Hint: you are investigating ‘resonance’. Q4. Download the ElCentro1940.m and LomaPrieta1989.m files and explore them. Given an input Δt, they will output horizontal (x-direction) and vertical (y-direction) ground acceleration records ax(t) and ay(t) in m/s2 for two strong earthquakes that hit California in the 20th century. Add these into the external loading vector as { } [ ] { } ( ) { } ( )( )x x y ytf m a t a tι ι= + where {ι}x is a vector of 1’s in the locations of only x-direction dofs and 0’s elsewhere, while {ι}y is a vector of 1’s in the locations of only y-direction dofs and 0’s elsewhere (these are known as influence vectors). No other loading should be present (ignore the truck), and you must use the full unpartitioned [k], [c], [m], {d} and {f} during integration. Perform an explicit response-history analysis of the bridge for both earthquake records (separately). Compare the time-history of the structural response (the horizontal sway at E and the vertical sway at M) under both records on a plot and comment. Which earthquake is worse for the structure and why do you think that is END AJS Jan’ 22 with very special thanks to Alfredo Camara 6