You are allowed to use a calculator or a computer calculator (including spreadsheet tools) in this assessment. If you use a spreadsheet to calculate any results, record in your submitted answer the algebraic expressions you have used to derive your answers. Clearly label the rows and columns and units in any tabular data you include in your answers.
You are allowed to use your own dictionary in this exam and/or the Spell Checker facility on your computer.
·There are 4 pages to this online assessment.
·You will have 48 hours to complete the assessment.
·This assessment is worth 80% of the overall module mark.
·Questions 1 and 2 are worth 40 marks each. Question 3 is worth 20 marks. The mark distribution is given alongside the questions.
·Answer all 3 questions.
·It is anticipated that this task should take you approximately 1 hour per question.
·The deadline for submission of your assessment is between: 09:00 Thursday
·Please submit your assessment to the ‘Exam Submission’ area in the module’s Minerva page.
·Please include your Student Identification Number (SID) in the title of your submission.
·If there is anything that needs clarification or you have any problems, please email [email protected] and we will respond to you as quickly as possible within normal working hours UK time (9:00-17:00 hours, Monday-Friday). Remember to include your student name, ID and the module details.
·You may not communicate with academic staff or other students during this online assessment period.
1.A frame is designed with the following nodal coordinates for nodes 1 to 4 respectively: coordinate system is positive rightwards (x), downwards (y) and clockwise (θ). There are four elements: 1-2, 2-3, 2-4 and 3-4.
(NB: In each of the following questions a – e explain your answers as suggested and illustrate with selected sections of your spreadsheets; do not just cut and paste the spreadsheets).
(a)Write down the values of sin α and cos α for each of the four elements that should be used in the transformation matrix. Use exact quantities, not decimals.
(b)If nodes 1 and 4 are fixed, what will be the size of the reduced structure stiffness matrix Kr? Explain your answer with reference to the full structure stiffness matrix K and include a table mapping the local (element) degrees of freedom onto the global (structure) degrees of freedom.
(c)Assuming at this stage that E (Young’s modulus), I (second moment of area of the section) and A (the area of the section) are all equal to 1, assemble Kr. Explain the steps in your calculations, making it clear at what stage you have performed the reduction and why, referring to the table produced in 1(b) above. Please report Kr to 3 decimal places, and build your spreadsheet such that E, I and A are variables.
(d)The frame is to be manufactured from a square hollow section (SHS 80/3.2] with the following properties: Weight = 7.63 kg/m; Area = 972 mm2; Second moment of area I = 0.9495 x 106 mm4. Calculate the structure deflection vector D and hence describe the movement of nodes 2 and 3 if a vertical load of 2 kN is applied at node 3. Explain the steps in your calculation, reporting Kr-1. Ensure that the correct units are reported alongside each coefficient in your vector (to 3 decimal places); marks will be deducted for incorrect units. Comment on the suitability of this section size for this application.
(e)Calculate the support reactions on the frame at nodes. Check that your answers are consistent with the structure being in equilibrium and include a free body diagram.
(a)Determine the maximum elastic moment resistance MY, the maximum plastic moment resistance Mp and hence the shape factor k with respect to the axis indicated by the chain-dotted line for the section shown in (a square L-section of width/height H and thickness t). You may assume that t << H. Please state any other assumptions you make (NB do not use a method which involves a tabulated value for the second moment of area of the section).
(b)For the two-bay frame in , determine the factor of safety against plastic collapse, and draw the bending moment diagram at collapse for the frame. Include critical sections and redundancies, sketches of individual collapse modes, and a sketch of the final collapse mode. Assume the vertical loads are in the centre of the spans.
A solid beam with rectangular cross-section (length = 8 m, depth = 2 m, breadth = 0.5 m) is modelled using three triangular finite elements . What downwards deflection will this model predict at the mid-span in response to a 100 kN load applied as shown? Compare your answer to the classical solution, and comment on any difference. Please include copies of your A, B and k matrices for element 1, and your reduced structure stiffness matrix Kr; the stiffness matrices should report three decimal places in the “1.000e10” style. You may assume that the elastic modulus of the material from which the beam is made is 20 GPa.