Question 1. Figure 1 shows a steel portal frame loaded with a snow load, p (kNIm), where p is 1.8. The frame's internal Joints B, C and D are all rigidly-connected.

The fully-plastic moment of resistance, Mp (kNM), varies between members as shown in the Figure below. The frame and its members are fully-restrained against buckling.

Figure 1. A steel portal frame

a) Choose an appropriate "Instantaneous Centre Point" (IPC), and use the Plastic Theory to determine the required value of Mp that Will cause the frame to collapse under the snow load. Take the Collapse Loads, P, to be the equivalent concentrated loads applied on Spans BC and CD.

b) If the Collapse Load, P, is to be taken as a single equivalent concentrated load applied at the Apex Point C, what would be the corresponding Internal and External Virtual Works?

Question 2. With reference to the same portal frame shown above in Figure 1, the rafter BC, which is a Universal Beam 254 x 146 x 43, is to be analyzed to withstand the snow load, p (kN/m). Again, p is 1.8.

Figure 2 depicts the isolated rafter.

Figure 2. The rafter BC

Use the Principle of Superimposition, and the Mohr Area-Moment Theorems 2 to determine the maximum deflection, vfmax (measured perpendicular to the rafter's axis) at point F, which is at mid-span of BC. Take E = 210 GPa. For this analysis, use the equivalent concentrated load applied to Spans BC.

Detailed analyses and diaqrams of the rafter in question are required.

Question 3. The rectangular reinforced concrete slab shown in Figure 3 is supported in such way that its two longer edges are fixed and its two shorter edges are pinned. The slab is orthotropic with a positive (sagging) ultimate moment in the direction of the smaller span being 35% qreater than the ultimate sagging moment (M) in the direction of the longer span. The value of the negative (hogging) ultimate moment is 40% greater than the value of the ultimate sagging moment (M) in the direction of the longer span.

The slab is required to support the following loads at the Ultimate Limit State (ULS):
i) A uniformly-distributed load of 33.5 kN/m2, and
ii) A point load of F located at the centre of the slab.
Unique slab geometry and the load input data should be calculated as follows:
i) Lx = 5.38 (m),
ii) Ly = 4.1 m,
iii) F = 18 (kN).

Figure 3. A rectangular reinforced-concrete slab

Propose the yield lines pattern for the given slab and use the Yield Line Analysis to determine the value of M, the ultimate moment per unit length.