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The conditions of end supports of straight ladders are often the cause of major injuries. The firm and secure ladder ends against instability in general and sliding of the top and bottom ends in particular are among the check list of most ladder safety training books and manuals. However, the restraint to the free expansion of a ladder can cause a catastrophic failure due to buckling even at intermediate loads and should be presented in the latter as a serious potential hazard. This paper deals with an investigation of an extension. An analytical structural model that simulates the buckling behavior of an axially restrained ladder subjected to static and dynamic loading is developed. It compares two different ladder end conditions and shows that instability due to buckling can occur during ascension or descent in the case of an axially restrained ladder. The analytical results are supported and validated by a finite element model simulation conducted in parallel. This study may explain the root cause of similar incidents involving falls from portable ladders worldwide.

Reports on ladder safety have shown some startling statistics related to ladder accidents worldwide. Every year millions of people are injured and thousands are killed. In the US the Consumer Product Safety Commission reported more than 90,000 people receive emergency room treatment from ladder-related injuries every year and over 300 people die from ladder falls annually. According to the Canada’s occupational safety and health magazine, statistics on lost-time injuries list approximately 1500 accidents in Canada involving ladders with over 40 deaths per year. Based on a study conducted on the 24,882 ladder falls of 1987, OSHA estimates that 53% of the ladders are involved in the accidents broke during use. The main cause of ladder failure is the sliding of the bottom and the top ladder ends [

The investigation of the cause of a fall is often followed by a thorough inspection of the incident seen and an examination of the used ladder but seldom simulation whether analytical, numerical and experimental are undertaken. This is because the assumptions, simplification and conditions for the simulations to reproduce the incident are difficult to make. Kenner et al. [

When ladders fail, it is important to gather information about operating conditions and to obtain a professional engineering assessment of the cause. This is one of the recommendations that were given in the report on the incident of a New York City fire fighter rescue attempt that caused a collapse of an aerial ladder [

This study deals with axially restrained columns. A model based on beam-column is used to treat an aluminum extension portable ladder and to explain its buckling failure. The study shows that the axially restrained side beams can create instability at even intermediate loads. The restraint to the free expansion of ladders and their end support conditions are not treated by standards such as CSA and ANSI [

Ladders are design against buckling due to compression. The normal buckling load limit is determined on the basis of the weight of the worker, the carried load and his tools. The ladder load capacity limit is obviously greater to allow for a margin of safety. However portable ladders are not designed to carry certain loads such as pure lateral loads or load generated during operation by supports and attachments other than those mentioned previously. If the free expansion of a ladder is limited as shown in

First, Beam shortening is a reduction of the length of a structural beam when subjected to loading. It can be produced by an axial compressive force or thermal contraction but also by bending. With reference to _{c} due to compression in the two parts of the ladder P_{b} for the length b and P_{h} for the length (L-b) is given by:

when a beam is subjected to bending, its length decreases, that is to say, its neutral axis becomes shorter due to curvature [_{f} as a result of bending is given in [

Beam shortening effect in portable ladder

FBD of the portable ladder with (a) roof support (b) wall support

The relationship between the moment M and the deflection v is well established (Timoshenko et al. 1961). Singularity functions are used to determine the moment:

After substitution, integration and application of the boundary conditions, the beam shortening of a simply supported inclined ladder is given by

The total beam shortening is therefore given by:

During unloading, if the beam ends are constrained this will generate an important axial compression. In this case, the ladder end is blocked against the rigid inclined roof and depending on the stiffness coefficient of the combined end pads and the roof contact system ka, the magnitude of the generated compressive force Equation (6) can cause buckling of the ladder. The stiffness coefficient can have any value between 0 and 1; 0 means that the compressive load is completely absorbed and 1 the complete beam shortening is transformed to a maximum compressive load. The compressive force is given by:

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The ladder is treated as a beam-column because it is subjected to the simultaneous action of axial compression and flexion in this particular case. Lateral loads creates an instability because the moment is amplified considerably even in cases of small deflection. The two loads cannot be treated separately and the theory of the buckling of beam-columns must be used [

The allowable axial compressive and bending load limits of a beam-column is given by a linear combination of the maximum bending and compressive loads expressed by the following relationship adopted by most standards:

C_{r}, M_{rz} et M_{ry} are the maximum compressive load and bending moments when taken individually with all other loads set to zero. M_{z} and M_{y} are the amplified moments in the two directions obtained by Equation (7). In the ladder case, only the bending in the plane normal to the steps is considered and therefore M_{y} = 0. A similar equation is suggested by CSA 157-05 standard [

The analytical beam shortening study is supported and validated by a numerical finite element buckling study that was run in parallel. The FE model shown in

Partial view of the portable ladder FE model

depending on ascension or descent. The amplification factors are taken as 1.5 and 2 in a first case and 1.75 and 2.5 in a second case respectively in order to cover a possible range. The weight of the ladder is also considered in the numerical analysis.

To simulate the additional compressive load generated by the difference between the dynamic and static loading due to the ladder expansion constraint, a second run is conducted imposing the amount of an axial displacement obtained from the non-linear analysis of a first run with the ladder simply supported. This generates the resulting compressive force which obviously depends on the worker position on the ladder. A third run combining axial compression and bending with the buckling option is finally conducted to evaluate the capacity of the ladder to resist buckling. The analysis was run under Ansys software [

According to the information gathered by the CSST “Commission de la Santé et Sécurité du Travail” which is the Quebec occupational health and safety organization, a worker of about 100 kg died from a fall from a ladder deployed at a height of rung 19 or 5.8 m with a 10˚ relative to the vertical. The lower supports with durable shoes rested on a terrazzo floor while the upper end caps were in contact with a rigid ceiling inclined at 29˚ with respect to the horizontal, as shown in

The first intervention was to eliminate the possibility of ladder bottom end sliding failure. This was achieved by conducting a friction test experiment at the accident seen.

Portable ladder buckling

Load application location on the ladder shoe for friction tests

shows only the maximum and minimum friction coefficient values indicating that the tests are repeatable. Depending on the shoe and the load position, the friction coefficient varies between 0.3 and 0.6. The higher values are indicative of the rubber being the predominant material contact surface

Friction coefficient test results

Friction coefficient requirement to avoid sliding

dynamic factors due to the ascension/descent and the load absorption coefficient due to the end pads and the roof flexibility. In general it could be said that the results of the analytical model are in a good agreement with those of FEM and in particular the axial compressive displacement or beam shortening and the axial compressive force shown.

The effect of the dynamic factors is illustrated for ratios of 2.5 to 1.75 and 2 to 1.5 are used in the simulation [

Dynamic factors effect on (a) axial displacement and (b) axial force

Dynamic factors effect on (a) bending and (b) buckling

bending moment and lateral displacements to amplify causing instability. It is to be noted the Euler buckling load for a column is given only for a reference in the axial force vs. position on the ladder graph of

Stiffness coefficient effect (a) axial displacement and (b) axial force

Stiffness coefficient effect on (a) bending and (b) buckling

The calculations are done with the dynamic factors of 1.75/2.5. The higher the stiffness coefficient is the less compression load is absorbed and therefore the higher the four parameters are. Once again, as the worker climbs the ladder towards rung 13 the loads increase and the risk of instability is higher and in particular with the 0.5 value. The bending moment and its amplified values due to the effect of compression are shown in

The ladder collapse due to buckling is confirmed as the probable cause of failure by both the developed analytical model and the CSA 157-05 which is more conservative. The reduction of the dynamic factors from 1.75 and 2.5 to 1.5 and 2 leads to a reduction in the compressive load but are not enough to prevent the amplification of the deflection and bending. The instability occurred when the worker made his descent from the 13th to the 12th foot step to prepare for reception of the spare fluorescent tubes. Indeed, as soon as the worker began the descent to land on the 12th rung from the 13th rung, the collapse of the ladder was inevitable due to dynamic loading. The flexibility of the parts in contact with the legs (shoes, caps, floor and ceiling) can partially absorb the load created by the beam shortening but cannot prevent the collapse because even with a stiffness coefficient of 0.3 (70% of load absorption) buckling is reached.

Beams and columns axially restrained to free expansion such as the ladder under investigation experience high axial loads that make them buckle. Bending creates beam shortening that can produce buckling failure if axial displacement is restrained and the load is released. Six recommendations have been made: 1) do not use this type ladder of ladder for ceiling jobs; 2) ladders should not be axially restrained to free expansion; 3) use ends with rollers if the top end rests against an inclined wall; 4) caution against the use of a ladder with end restraint; 5) caution against this danger in the safety instructions, operating manuals and maintenance of ladders; and 6) analyze similar incidents involving temperature expansion restriction (i.e., collapse of firefighter areal ladders).

The author acknowledges the financial support of the Quebec provincial health and occupational safety “Commission de la Santé et de la Sécurité du Travail CSST”. He also would like to thank this organization for authorizing the publication of this investigation.