Given the lack of positive evidence for the cause of failure, it was then essential to consider the conditions under which the ladder was being used and the mechanics involved. A ladder leaning against a wall is an application of simple static analysis to determine the critical variables that can affect stability (Box 15). The key variables are the angle of repose of the ladder, and the coefficient of friction between the feet and the ground.
Box 15 Static mechanics of ladders
The ladder problem is a classic in applied mechanics where it is important to know the safe angle at which to lean a ladder against a wall (Figure 50). This angle – the angle of repose, labelled θ – is made by the ladder with the ground. The ladder is at equilibrium, so the forces in the system are balanced. However, if the system of forces is changed by someone climbing to the top, the safe angle is changed. A change in frictional forces would also be critical.
The forces in Figure 50 include the reaction of the ladder (length x) against the ground and wall (N, R) and the weight of the ladder (Mg). When the equilibrium equation is simplified, N and R can be eliminated, giving
where μ0 is the coefficient of friction of the ladder feet against the ground, and μ1 is the coefficient of friction of the ladder tips against the wall. The static coefficient of friction being considered here is the value when all surfaces are completely still – the coefficient becomes smaller when the surfaces start to move.
If a rather low value of 0.25 is assumed for the coefficients of feet and tips, the angle of repose of the ladder at equilibrium is 62°. This means the ladder will slip down at lower angles, but is stable at angles above this critical value. What happens if the user climbs to near the top? The equilibrium equation tends to
The critical angle rises to 76°, so the ladder is less stable. Any angle below 76° results in instability, when the ladder can slip away. Note the exact position of the user on the ladder is not specified in the equation owing to simplification.
What about the value of the friction? In fact, values of the static coefficient of friction are usually higher than 0.25, a typical value for a wooden foot against a wooden floor being about 0.5. This gives a lower critical angle of repose, and makes the ladder less likely to slip. A wet floor has low friction, so must be avoided. It is recommended that the repose angle be about 75°, and for extra security someone stands on the lowest rung, or the ladder is tied to the wall at the top or the base.
4.4.1 Site visit
A visit to the site of the accident was essential. The witness statement could be checked, and any further evidence that could clarify the circumstances could be examined directly. The visit showed the visible evidence of the intermittent contact between the tips of the ladder and the wall below the window (Figure 43(a)) and lower down the traces of an impact with the small sill above the patio doors (Figure 51). The trace contact marks corroborated the witness statement, showing how the tips and hence the ladder structure itself, had oscillated as it slipped, giving intermittent contact with the wall. No traces of marks from the feet on the concrete slabs of the patio itself could be found. The key information about the stability of the ladder, the angle of repose and coefficient of friction of the feet, remained unknown, however.
The injured user attempted to guess the information requested – especially the angle of repose and degree of extension of the ladder – but it is always best to determine such information independently. After traumatic injury, someone is more likely to forget details, even if in normal circumstances they could recall them.
The ladder had not been photographed in situ just after the accident. The lack of trace marks from the feet meant the extended length of the ladder could not be determined. However, there was a key bit of information that seemed indisputable: the tips of the ladder were resting on the window sill, which was 3.69 metres vertically from the ground. This was measured during the site visit.
4.4.2 Objectives of the reconstruction
The events were reconstructed using the ladder fitted with new tips. As a starting point, the ladder was fully closed rather than being extended. Another ladder of different design was placed alongside for the purposes of comparison and safety during the stability tests. The objectives of the test were as follows.
To estimate the static coefficient of friction of the feet.
To establish the angle of repose.
To determine the footprint of the feet.
To load the ladder statically at the estimated height of use.
To examine dynamics by simulation of movement.
4.4.3 Coefficient of friction
The first task was to estimate the static coefficient of friction of the polypropylene feet. A simple way of doing this for very light loads is to place the feet on a surface similar to that being used for the real ladder and tilt it until the feet just slip (Figure 52). In this case, the surface is a concrete slab. Theory (similar to that in Box 15) shows the critical condition for slip on the inclined plane is
where θ is the angle of inclination of the slab to the level ground. Two experiments established that the coefficient of friction for the feet – mass about 100 gm – was approximately 1.0.
Such a value is typical for the elastomeric polymer concerned acting on smooth concrete. But did the feet fitted to a ladder give the same value in the reconstruction?
Direct determination from the unladen accident ladder showed that it slipped at an angle of about 40°; so, neglecting friction at the tips, the coefficient of friction is about 0.6, considerably lower than the value estimated from an inclined plane experiment. There are several reasons for this situation, as Box 16 relates.
Box 16 Coefficient of friction of polymers
A classical law taught to generations of engineers states that the static coefficient of friction – friction force divided by normal force – is independent of load and contact area, and it is roughly true for most rigid metals and ceramics. Polymers, rubbers, fibres, thermoplastics and thermosets, composites and all natural materials, such as wood, bone, and skin are quite different and deviate substantially from the classical law. The effect has been well studied, as Figure 53 shows for an important speciality polymer, PTFE.
The effect has interesting consequences. For example, it means fibrous polymers hold together well in textiles where a high coefficient of friction acts between the fibres. PTFE fibres in GoreTex waterproof matting are stable and won't pull apart easily. On the other hand, PTFE sheet, with a low coefficient of friction of about 0.04 under load, will ease the movement of heavy machinery where the sheet is used as a bearing pad.
Elastomers generally show high frictional coefficients (greater than 1) because most imposed loads will strain all of the material, and not just that at the contact zone. This is a unique and desirable property, and is used in products like tyres, which must grip the road under variable road conditions.
In the first place, the coefficient of friction of polymers is known to be dependent on load, decreasing as the load increases. Secondly, the feet of the accident ladder had been designed for a repose angle of about 75° (Figure 54), any other angle reducing the contact surface against the ground. This was demonstrated by recording the footprint of the feet at several angles of repose (Figure 55). The edge in contact with the ground thus becomes even more heavily loaded over a much smaller area of contact.
4.4.4 Repose angle
The next issue to be addressed was what angle the ladder had been leaning at. The claimant stated that it had been leant against the sill of the upper window. So the angle of repose could be calculated for two situations: an unextended ladder and the ladder extended by one rung. Greater degrees of extension would create progressively lower angles of repose. As a working hypothesis, it was assumed the ladder had been used either unextended, or with one rung extended. Leaning the ladder against the sill would produce a repose angle given by
The length of the unextended ladder was 4.46 m, and extended by one rung it was 4.71 m. With the sill being 3.69 m from the level ground, then θ = 56° unextended, or 52° for an extension of one rung. The situation for the unextended ladder is shown in Figures 56(a) and 56(b), with the plastic tips leaning against the window sill. In this position, a slight movement of the ladder would allow the ladder to jump down onto the adjacent wall, where either the aluminium tips of the lower section of the ladder, or the plastic tips on the upper section would make contact with the brick wall.
The tips from the accident showed abrasion, most visibly obvious at their upper corners (Figure 57). In addition, matching the ladder to the wall with known lengths and heights showed the unextended ladder had most likely been used. The angle of repose of 56° was well below the recommended angle of repose of 75° for this design of ladder. But what could explain the curious set of marks below the window and the comment from the user about ‘walking down the wall’?
4.4.5 Stability experiments
The final phase of experiments involved simulating the weight of the user when working near the top of the ladder. Both the weight and height of the user were known, and he thought he was standing on the fourth, fifth, or sixth rung from the top when the ladder slipped. The user could thus be simulated by simply suspending a fixed mass of 72.6 kg (force of about 726 newtons) to represent his weight on a rope from the upper rungs. The mass of the ladder was about 20 kg (force of 200 N). It was leant at an angle of 56° against the wall, representing the unextended ladder (Figure 58).
The exact position of the suspended mass was important, because when ascending or descending a ladder, the user would have shifted his weight from foot to foot. The user could not remember which rung he was standing on at the time of the accident, but it was likely that the fourth, fifth and sixth rungs from the top were in use at the time. As the user was cleaning the right-hand pane of glass, his left hand was probably holding the left-hand part of the uppermost rung. By leaning over, his weight would have shifted to the right-hand part of the rung he was standing on, so the right-hand part of the fifth rung was used to hang the mass.
One feasible trigger for the accident could have been momentary loss of contact of the left-hand tip with the sill against which it was resting, by reaching over to the right, for example. A spring balance was used to determine how much force was needed to pull the left-hand stile at right angles from the sill – it was about 100 N. The torque at the top of the ladder simultaneously pulled the left-hand foot out of contact with the ground, and the loss of contact caused the right-hand tip to slip down to a slightly lower position against the wall. Repeating the effect led to progressively lower positions against the wall until the whole ladder became dangerously unstable and the experiment was halted.
What might be the expected effect of trying a stability experiment using a suspended weight from the sixth and fourth rungs? What would you expect to happen for each experiment? Using experimental values for the static coefficient of friction for various loading conditions, estimate the effect of increasing mass on the critical angle of repose for top-loading conditions.
It is easy to see the stability of the ladder decreases as the user ascends the ladder, so suspending the weight from the sixth step will make the ladder more stable than the fifth step. Conversely, suspension from the fourth step will make it less stable.
The stability equation for the situation of a top-loaded ladder is given approximately by
Adding a suspended weight to represent the mass of the user however, increases the total mass of the system acting at the ground, and hence the normal reaction. This will, in turn, increase the load on the polymer feet. The first experiment to measure the static coefficient of friction with an inclined plane gave a value of about 1.0 for a mass of 100 gm. But it was reduced to about 0.6 for the ladder fitted with the feet (20 kg). The total mass of the ladder plus suspended mass was 20 + 72.6 = 92.6 kg. For top loading (user near the top of the ladder), the critical angle θc will be described by the above equation, with the following estimates using the stability equation above.
The effect of the increased mass on the feet would be to lower the coefficient further, but by how much? Figure 53 suggests the biggest drop occurs at lower loads, the curve tending to flatten out as the load is increased further. So there is a smaller decrease as the load is increased further. Assuming that polypropylene behaves like PTFE, then a decrease of the coefficient of friction to about 0.5 might be expected. The critical angle of repose would be expected to rise even further to about 63°. Therefore, a ladder inclined to the ground at 56° would be unstable when top-loaded, and would slip. Moving the weight higher would decrease stability, while moving it lower would increase stability.
It was felt important to determine the limits of the stick-slip motion seen in the hung-mass experiment. There was no downward motion observed at all with the mass on the sixth step. However, the results for the fourth step were more dramatic.
Application of a force of about 120 N to the left-hand tip of the accident ladder led rapidly to uncontrollable stick-slip motion down the wall, and the ladder fell away (Figure 59). The marks left on a painted board visibly demonstrated the stick-slip motion, with a series of impacts showing the intermittent contact of the ladder tips down the wall (Figure 60).
What did the reconstruction show? Firstly, it suggested the ladder had been leant against the wall at a maximum angle of about 56°, well below the recommended angle of repose. Secondly, it confirmed that at relatively low angles of repose of this particular design of ladder, stick-slip motion could occur after momentary instability – even if the angle of repose was above the critical angle of repose. The instability was produced by a torque load that moved one tip of the ladder away from the wall. Provided the user was near the top of the ladder, catastrophic and uncontrollable loss of the ladder was inevitable.
The reconstruction of the accident showed the user initially leant the ladder at too low an angle for a reasonable safety margin. Although it was above the critical angle of repose, it was susceptible to stick-slip instability when the user was near the top of the ladder. The visible contact evidence from the wall (Figure 43(a) and (b)) confirmed stick-slip motion of the ladder tips.
But the question of the fractured tip remained unanswered. It could still have caused the accident if the tip had broken at the sill, and initiated stick-slip motion. However, both the broken and intact tips showed abrasion against brickwork, so it is more likely that it survived for some way down the wall. The final piece of evidence was an impact mark on the small wooden sill above the patio doors (Figure 51). This was probably caused when the tip hit the sill, and the tip broke from its weakest point. The fracture was a result of the accident, not the cause.
Construct flow diagrams for the separate sequences of events of the accident based on:
the initial evidence of the fractured tip;
all the available evidence.
Indicate what effect each sequence would produce on the tips of the ladder. Examine the actual condition of the tips to show which was the most likely sequence of events.
The sequence of events from initiation of the accident based on the initial evidence, assuming the tip broke to cause the accident, is as follows.
(a) Ladder tip fractures.
(b) Ladder slips from sill.
(c) Ladder starts to slip down wall.
(d) User thrown from ladder.
(e) Ladder hits ground.
The broken tip would not show any abrasion, because it would have been thrown clear after the first break, while the intact tip would be deeply abraded. Also, the sharp edge of the broken tip still on the stile might also be abraded by contact with the wall.
The sequence of events using all the available evidence is:
(a) Ladder tips slip from sill.
(b) Ladder tips impact wall.
(c) Repose angle drops.
(d) Ladder walks down wall with increasing velocity.
(e) Repose angle decreases rapidly.
(f) Sideways rocking motion of ladder throws user from ladder.
(g) Ladder tip hits sill above patio door and fractures.
(h) Ladder hits ground.
In this case, both the tips will show equal abrasion due to motion against the brick wall, as was found by inspection (Figure 57). The evidence points towards the tip breaking as a result of the accident, and not itself breaking to cause the accident.