Embedded Files
Amsterdam bench
Suggested viewing: St. Pancras Bench
DESCRIPTION AND LOCATION
These stylish burgundy benches can be found scattered around the Museumplein in Amsterdam, Netherlands. (More famously known for the ‘I Amsterdam’ sign, near the Van Gough museum).
These benches style an economic and contemporary look that fits in nicely with the surrounding environment.
APPLIED LOADS
We start off by looking at the loads that are applied to the bench; we have the self-weight of the bench itself although it seems to be designed to have minimal weight (thin steel sheets).
We have the vertical load from people sitting, luggage etc. Less likely loads are people trying to raise the bench causing uplift.
Horizontal loads from wind or people leaning on the bench should also be considered although the holes in the structure reduce the effect of wind loading.
CONCEPTUAL/QUALITATIVE BEHAVIOUR
Supports:
As we analyse the bench it is apparent there are only two supports; the two legs of the bench embedded in the ground as nominally pinned supports (the giveaway would be the materials thickness). The legs are very thin when we look at a front view of the bench; in fact the whole bench is made of the same material seemingly bent into its shape. So very little fabrication is involved, which gives an economic solution for manufacturing the benches.
We also notice that the legs are wide; this helps in stability preventing the bench from toppling forward or backwards.
Furthermore, the legs are inclined in one direction (as opposed to being vertical) which helps to resist lateral loads.
At first one would suspect that the legs are subject to buckling, but at closer inspection we see that the bench is made of a sheet that’s been folded into this shape which gives the bench ‘form resistance’.
Dead Loads (Permanent Actions):
In this example, it is possible to consider this bench as a simple supported beam spanning between the two legs; however, in this case we will consider it as a portal frame. A portal frame is exactly what the name suggests, a frame that looks like a portal, or a goal-post (as some people call it). You can also see for simplification we are ignoring the incline of the legs; the analysis is very similar though.
The dead load of the bench is the weight of the bench itself, the legs and the back rest. The cross section of the bench is more or less uniform, so we can assume that the dead load is a UDL. Now if we consider how the bench would deform under its own load what would it look like? (Notice in the figure below that the beams rotate at the pinned connections).
Now we know that the horizontal beam would have a maximum bending moment at the middle and that the curve of the BMD will be a parabola., but what happens at the ends (top of the columns)?
We could divide this into steps to construct the actual BMD. First step draw the BMD of the upper beam regardless of its supports (i.e. assuming it is simply supported). We know the beam will have a parabolic BMD with the maximum at the middle and zero at the ends.
Now we adjust the BMD according to the case; starting with the corners. Corners (or joints) are often very rigid structures that (for analysis purposes) don’t tend to roatate, so they are continuous around the corner, that rotates affecting the members attached to it. Notice how in the figure below the corners remain at a constant angle under bending; this is where the assumed rigidity comes from.
As we can see corners can transfer moment to other members. So, for the sake of this tutorial let's assume that the metal sheet is sufficiently stiff to transfer moments at the corners.
The corners can transfer (resist) moment and because the structure is symmetrical then we’ll have a constant moment resisting the downward bending; so we’ll have a moment that would cause upwards bending if it was isolated.
This reaction moment will act to resist the downward bending (sagging) of the beam, as shown below.
The best way to go about this case is using the principle of superposition. If we superpose the simply supported beam BM above the resisting BM from the corners we arrive at the actual BMD, as shown in the figures below.
As noted in the St. Pancras Bench beginner tutorial; the total BM of the fixed end beam is equal to that of a simply supported beam (i.e. WL2/8), in other words we just push the BM diagram upwards when we add fixity to the supports. Now as we stated earlier the moment that is applied to a corner must be transferred to the other members, we can obviously see that the legs will bend outwards and that the moment will have no other place to go so it will decrease throughout the legs finally becoming zero when it reaches the pinned connection. (Remember pin connections allow rotation, therefore they cannot resist moment).
Now an easy way to check is to cross-reference this BMD with the original deflection we sketched for the portal frame.
We see that where we have zero BM on the BMD, we observe a point of contraflexure (switch between hogging and sagging) in the deflected shape.
Moreover, we draw the BMD on the side of the member that experiences tension (convention).
Imposed Loads (Variable Actions):
An imposed load would be a person sitting on the bench or some luggage etc.
this can be modelled as a point load and has relatively simple analysis using the principle of superposition. We add the BMD of a point load which is a triangle to the existing BMD we have.
Page updated
Report abuse