Reaction Forces on a Fixed Ladder in Static Equilibrium: Analysis and Definitive Experimental Test of the Ladder Problem

Abstract

The development of a theoretical model to predict the four equilibrium forces of reaction on a simple ladder of non-adjustable length leaning against a wall has long remained an unresolved matter. The difficulty is that the problem is statically indeterminate and therefore requires complementary information to obtain a unique solution. This paper reports 1) a comprehensive theoretical analysis of the three fundamental models based on treating the ladder as a single Euler-Bernoulli beam, and 2) a detailed experimental investigation of the forces of reaction as a function of applied load and location of load. In contrast to previous untested proposals that the solution to the ladder problem lay in the axial constraint on compression or the transverse constraint on flexure, the experimental outcome of the present work showed unambiguously that 1) the ladder could be modeled the best by a pinned support at the base (on the ground) and a roller support at the top (at the wall), and 2) the only complementary relation needed to resolve the static indeterminacy is the force of friction at the wall. Measurements were also made on the impact loading of a ladder by rapid ascent and descent of a climber. The results obtained were consistent with a simple dynamical model of the ladder as a linear elastic medium subject to a pulse perturbation. The solution to the ladder problem herein presented provides a basis for theoretical extension to other types of ladders. Of particular importance, given that accidents involving ladders in the workplace comprise a significant fraction of all industrial accidents, the theoretical relations reported here can help determine whether a collapsed structure, against which a ladder was applied, met regulatory safety limits or not.

The Role of Friction in the Static Equilibrium of a Fixed Ladder: Theoretical Analysis and Experimental Test

“…In the paper [1] designated Part I, the author derived theoretically and confirmed experimentally a solution to the long-standing "ladder problem", i.e. the problem to predict the 4 forces of reaction on a fixed ladder resting on horizontal ground and inclined against a vertical wall.…”

“…• The maximum force of friction is directly proportional to the applied load. (1) • The force of friction is independent of the apparent area of contact.…”

“…Application of Equation (1) to the contact at the wall related the vertical reaction 4 R to the horizontal reaction 3 R by a linear expression…”

“…In a recent publication [1] to be referred to as Part I, the author provided a DOI: 10.4236/wjm.2018.812032 446 World Journal of Mechanics theoretical solution and supporting experimental test of the long unresolved problem to determine the four equilibrium forces of reaction on an inextensible ladder [2] inclined against a rough vertical wall, as schematically illustrated in Figure 1. The ladder was modeled as an Euler-Bernoulli [E-B] beam [3] [4] [5] in which the force of friction between the ladder and the wall constituted a critical part of the solution.…”

“…Moreover, the author is unaware of any previously published experimental measurement of these reactions apart from [1].…”