1. The truss structure ABCDEFG shown in Figure 1 is subjected to an Ultimate Limit State (ULS) factored vertical load of 1500 kN at C. Self-weight of the members themselves may be neglected, and the connections between members are pinned.

(a) Determine whether or not the truss structure is statically determinate, giving your reasoning.

(b) Identify which members in the truss are in tension and which are in compression.

(c) Calculate the axial forces in members BD and CD, stating clearly whether they are in tension or compression under the applied loading.

(d) Using the Blue Book online at http://tsbluebook.steel-sci.org/, or a similar resource, select Hot-finished Celsius Circular Hollow Sections (CHS) that are of sufficient strength to be used for members BD and CD, assuming Grade S355 steel.

(e) Sketch a possible arrangement for the connection between the members that meet at the point D, with an explanation of the key features of the sketch.

2. A steel-framed building has eight storeys, including the ground floor. The columns are laid out on a regular grid at 12m centres. The structural arrangement of a typical floor in the building follows the concept shown in Lecture 3 Slide 72, and the spacing of the secondary beams is 3m.

The building is used entirely for classroom space with loading specified by BS 6399 Part 1. Simply supported construction may be assumed, with the floor slab spanning one-way between secondary beams. The floor slab is made of concrete, with nominal thickness 200mm and self-weight of 24 kN/m3

Note that the ‘Concentrated load' component of the design minimum imposed floor load from BS 6399 may be neglected.

The structural members are to be designed to the Eurocodes. Figure 2 shows a free body diagram of one of the secondary beams in a typical floor, subjected to the design Ultimate Limit State (ULS) factored load w (kN/m) that the beam experiences. The supports each end are from the primary beams on which the secondary beam sits.

(a) Show that the value of the design ULS factored load w is 32.9 kN per metre length of beam.

(b) Sketch the bending moment diagram (BMD) and shear force diagram (SFD) for the beam, indicating maximum values on your sketches. You may neglect the contribution from the self-weight of the beam.

(c) Select from the Blue Book or similar resource an Advance UK Beam of sufficient bending strength to resist the applied bending moment, assuming Grade S275 steel.

(d) Confirm that the Advance UK Beam section selected in (c) is adequate to resist the maximum shear force acting on the beam.

(e) Sketch the distribution of horizontal stress on a vertical cross-section through the beam at midspan in service under dead load only (i.e. with zero imposed floor loading), indicating peak values of stress on your sketch.

(f) Show that the deflection of the beam is within the guidelines for a beam supporting a floor slab.

(g) Describe the phenomenon of lateral-torsional buckling of steel beams. Discuss whether lateral-torsional buckling is likely to occur in the beam that is the subject of this question.

3. Figure 3 shows a single-storey reinforced concrete (RC) frame structure ABCD, in which support A is fixed in position and pinned, and support D is a horizontal roller which is also pinned. The structure is to be designed according to the Eurocodes, and the dead load of the
frame itself may be neglected.

The frame is subjected to two variable actions applied simultaneously - a nominal horizontal load due to wind of 10 kN at B, together with a nominal vertical imposed roof load of 30 kN acting at midspan of the beam BC.

(a) Identify all the reactions acting on the frame. Is the structure statically determinate (state your reasoning)?

(b) Calculate the design reactions at the supports A and D under each of the two possible load combinations: (i) Imposed load as the leading variable action; (ii) Wind as the leading variable action.

(c) By drawing a free body diagram of column AB, confirm that the magnitude of the worst case design bending moment at the top of the column at B is 75 kNm. What is the bending moment at C?

(d) Draw a free body diagram of beam BC, and show that the worst case design bending moment at midspan of BC is 108.8 kNm. Sketch the bending moment diagram for this case for the beam BC, indicating key values on your sketch.

(e) Beam BC is a rectangular section RC beam, with width b = 150 mm and effective depth to the reinforcement d = 400 mm, made of Grade 40 concrete (fck = 40 N/mm2). Calculate the reinforcement required in the cross-section at midspan to resist the applied bending moment from (d). Assume that high tensile reinforcement is used, with fyk = 500 N/mm2.

(f) Sketch the worst case design shear force diagram for beam BC, indicating key values, and determine the reinforcement required for the beam to resist these shear forces

4. Figure 4 shows a pin-jointed structure ABC consisting of two members pinned together and simply supported at the supports. The product of the cross-sectional area and Young's modulus for both members is AE = 20000 kN.

(a) Obtain the member stiffness matrix k in global co-ordinates for each of the members AB and BC, in terms of AE.

(b) Assemble the overall structure stiffness matrix [K], in terms of AE.

(c) A vertical downwards 10 kN load is applied at the joint B. Calculate the horizontal and vertical displacement of the joint B. Self-weight of the members themselves may be neglected.