Design Exercise

A major upgrade of a mineral processing plant is underway, involving the design and construction of several new areas, including a materials handling depot. At one part of this depot, loaded trucks cross over a down-ramp to a basement loading bay, necessitating a bridge. The bridge will comprise large parallel steel Welded Beams (W.B.), which support a reinforced concrete slab.

The loading analysis for this structure has already been undertaken and the design loading and span for each beam has been assessed to be as shown in the diagram below:

The concentrated loads are a simplified representation of the front and rear wheel loadings from the truck that will use this bridge and the uniformly distributed loading is due to the concrete slab.

You are required to select a suitable W.B. size for this application, based on bending strength. This means selecting a beam which is as small as possible, but which does not have bending stresses anywhere in it which exceed specified allowable values when the maximum design bending moment occurs.

The design sequence should be as follows:

Step 1

Firstly, you need to determine the design bending moment for the beam. The complication here is that you don't know exactly where the pair of point loads should be positioned in order to generate the maximum bending moment (B.M.) in the beam. There are two ways to determine this; in this Step 1) you will simply analyse a series of different positions for the truck and, from these, determine the maximum B.M. 'Move' the truck across the bridge in 2.0m increments, with the first position having the right-hand load at 2.0m from the right-hand support, then 4.0m from it, 6.0m, 8.0m and so on, until it is clear that the maximum B.M. produced by each case is getting less. For each truck location, derive and draw the BMD. At this stage, ignore the UDL on the beam. For each, note the maximum B.M. occurring anywhere along the beam and then produce a graph of maximum B.M. versus distance of the right-hand load from the right-hand support. Note that good quality graphs, with fully labelled axes, etc, are expected. From this, you should be able to determine what is known as the absolute maximum bending moment (AMBM). This is the maximum bending moment occurring anywhere along the span of the beam as a set of loads at fixed spacings (known as a 'wheel train') moves across it.

Step 2

This is the second, more direct, method of determining the AMBM. There is a well established procedure for determining the position of the wheel train which will give the AMBM. It is stated here without proof:

i. Determine the location of the resultant of the wheel train loads

ii. The AMBM generally always occurs under the wheel loading which is closest to the resultant. Call this load, say, P_i.

iii. Position the wheel trains along the span such that the resultant and the load designated P_i are located equidistant either side of midspan.

iv. Now analyse the beam with the set of wheel loads in this position. The AMBM occurring in the beam will be the B.M. at the location of P_i.

Determine the AMBM using this approach and compare it to that determined in Step 1)

Step 3

You now need to add to the AMBM the additional B.M. due to the uniformly distributed load. Note that the loading given in the diagram only allows for the weight of the concrete slab sitting on the beams. An additional allowance needs to be made for the self weight of the steel beam. Since the steel beam size has not been selected yet, the actual self weight is unknown. However, by inspecting the W.B data sheet and assuming a self weight UDL (kg/m, converted to kN/m) which is towards the heavier end of the sizes available, this problem is overcome. (Later, when the beam size is selected, you should confirm that the loading assumed is not less than that of the beam adopted.) Thus, determine the B.M. due to this UDL, at the position of the AMBM. Add this to the AMBM arrived at in Step 2), to arrive at the design B.M. for the beam.

Step 4

Now design the beam. As already mentioned, this is done by selecting the smallest beam size which can be used without the bending stresses in it exceeding specified allowable values when the design bending moment occurs. Further data is as follows:

i. a steel welded beam (W.B.) section is to be used. A copy of the data sheet for W.B. sections is attached.

ii. steel yield stress f_y = 450 MPa.

iii. maximum allowable bending stresses is 0.66fy in both tension and compression. The maximum bending stress in the beam can be computed using the flexure formula:

σ_max = -M_maxy/I_x

where y is the perpendicular distance from the neutral axis of the cross section, M_max is the maximum bending moment determined from Step 3 in each beam and I_x is the second moment of area of the cross section (see supplied W.B sheet data).

Attachment:- Data.rar