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As a basis for specifications for the testing of spring loaded cammingdevices used as rock climbing anchors, two successive models of thesedevices are developed. The rigid body model assumes an infinitelystrong, undeformable cam; it determines the cam's basic shape and therelationship between this shape and the cam's frictional holdingability. Based on the rigid body model, the second, elastic modeltakes deformation and material strength into account; this elasticmodel determines the relationship between cam shape and the shearstrength of the cam. The models predict two failure modes for cammingdevices placed in vertical, parallel cracks: inadequate friction andshear yield failure. A measure of the limitations imposed by thesefailure modes would provide climbers with information necessary forintelligent anchor placement. Introduction
Spring loaded camming devices (SLCDs) are a type of anchor that rockclimbers use to connect themselves to cracks or other irregularies ina rock cliff. These devices work through a combination of the wedgeprinciple and friction: once the cams are placed in a crack, any forcepulling the cams down causes the cams to press tightly against thecrack walls, generating a frictional force that retards the downwardpull. Figure 1 (80K) shows a typical SLCD placed in a test fixture crack.Springs built into the device (not visible) keep the cams snug againstthe walls. Any downward load applied at the carabiner rotates thecams and forces them against the walls, effectively wedging the devicemore tightly in place. Over the past two decades, SLCDs have provento be an effective anchor, and now SLCDs are manufactured in a varietyof sizes, shapes, configurations, and constructions.
The current methods of testing the spectrum of available SLCDs arebased either on anecdotal evidence that lacks an experimental controlor on pull-to-failure tests that determine the maximum forceobtainable before the device collapses. Such tests provide neither afull nor accurate picture of a device's holding ability. As a first steptoward developing better methods for the testing of SLCDs, thisreport develops a mathematical model for how SLCDs work, suggeststheoretical limitations of SLCD performance determined by the model,shows preliminary verification of this model, and suggestsimplications for both cam testing and the use of SLCDs.
Modeling SLCDs in Parallel Cracks
A succession of two models is used to develop an understanding of SLCDperformance in parallel cracks. The rigid model assumes that theactual materials that compose the cams are perfect: perfectly rigidand strong. This rigid model provides a basic theory of why cammingdevices work, defines the cam's shape, and determines the relationshipbetween the shape and the friction coefficient. The elastic modelbuilds on the rigid model by additionally taking elasticity and shearyield into account. The elastic model provides estimates of camdeformation, the resulting shear loading, and the maximum appliedforce to failure.These models are developed for the condition shown in Figure 1, an SLCD subject to a downwardapplied force while placed in a vertical, parallel crack. Theparallel crack constraint is chosen for two reasons. From the pointof view of the climber, any evaluation of these devices must addressperformance in parallel cracks because SLCDs are designed explicitlyfor use in such a situation. The parallel crack criterion is alsoimportant because it bounds narrowing and flaring cracks; parallelcrack results define a performance limit for each regime.
The Rigid Model -- Explaining How Camming Devices Work
As a first approximation, a camming device can be modeled as a rodwedged or ``cammed' in a parallel crack as shown in Figure 2a below. If the rod does not slip out, adownward tug or force applied to the upper end wedges the rod harderinto the crack, causing the rod to push against the walls with anormal force. Whether or not the rod slips out is determined by athird force, that of friction. Figure 2b showsthe forces on the rod; the friction force on the upper left end of therod is intentionally omitted. For camming devices that rely onfriction, this upper left end corresponds to an internal axle (see Figure 1 or Figure 4) andthus experiences negligible friction.
If the rod shown in Figure 2 does not move, both the forces andtorques must each sum to zero. These equalities and the definition ofthe friction force yield four equations that define the relationbetween the applied, normal and friction forces. R is the rodlength, is the friction coefficient, and is theangle the rod makes with the horizontal.
Two important results are derived from algebraic manipulation of theseequations:
Equation 1 shows the wedging effect that multiplies the applied forceby a factor of
To produce a camming device that works over a wide range of cracksizes, multiple rods are combined into a two dimensional surface inthe shape of a cam. This shape satisfies the criterion shown in Figure 3a below: the angle between the surface andthe line perpendicular to the radius must remain a constant, thecamming angle, . In polar coordinates, such a shape is defined by:
Rearranging and integrating yields:
Equation 3 defines the logarithmic spiral plotted inFigure 3b. A section cut from this spiral produces thecam (Figure 3c). Typically three or four such cams aremounted together on an axle to produce a complete SLCD.
Figure 4 shows the forces on a complete SLCD in a parallel crack. Thefree body diagram is very similar to the rod in Figure2, though now the contact forces are spread out evenly over allfour cams. The camming angle is set by the tightness of the spiralrather than by the rod length and crack width. The relationshipbetween the camming angle and the friction coefficient is stilldetermined by Equation 2: any given cam can only hold if tan < . If the friction coefficientbetween the cam and the rock surface drops below tan ,the device fails due to a lack of frictional holding ability.
The Elastic Model -- Materials Failure Modes
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Because the rigid model presumes that the cams are perfectly strongand do not bend or deform when weighted, it does not take into accountthe cam material properties: Young's Modulus, Poisson's Ratio, andshear yield strength. The elastic model assumes the cam shapedetermined by the rigid model but deforms this shape elastically underan applied force. This deformation produces a contact area as shown inFigure 5. The size of this contact area determines how large theapplied force can become before the cam's shear yield strength isexceeded and the device fails.
An approximate size of this contact area can be obtained by applyingHertz's theory2 of the contact betweentwo elastic bodies.
- The logarithmic curvature of the cam can be approximated as thecurvature of a circle. 4
- The cam is arbitrarily wide relative to the radius W >>R; in typical cams, W and R are of the same orderof magnitude.
- The cam has no holes or teeth; manufactured devices are often``drilled out' to reduce weight and have teeth to provide ``bite.'
- The effects of plastic deformation are not accounted for.
Given the contact area from Equation 4, the shear stress at thecam/crack boundary is approximately
| Figure 6: Estimated maximum applied forcesustainable for two types of SLCDs. |
These estimates do not necessarily reflect actual performance; rather,the force estimates are plotted to show the wide range of maximumforces that devices might be expected to sustain. These forceestimates span the range of forces that climbers generate. Thehighest force that a climber can expect is about 12,000N and a climberhanging from an anchor applies a force of about 1500N. According tothe elastic model, larger SLCDs can sustain larger forces and thesmaller devices are expected to fail under conditions commonlyexperienced by climbers. Even though several estimations andsimplifications are incorporated into the elastic model, the range ofperformance expectations remains large. The lower end of the appliedforce range is of particular importance because the smaller devices mayfail under nominal conditions.
Implications of the Model
The perfect SLCD is light weight and durable, fits in a wide range ofcrack sizes, anchors reliably to the most slippery rock, has a narrowprofile (width) that allows placement in shallow cracks, and cansustain the largest applied forces a falling climber can exert. Theseoptimal characteristics can only be obtained through variation of thecam's geometry and material properties: the camming angle, radius,width, Young's Modulus, Poisson's Ratio, and the shear yield. Everycam is a compromise; efforts to lighten or narrow the cam reduce bothdurability and maximum load, and greater range is traded for lessfrictional holding ability. These compromises are implicit inEquation 2, which constrains the camming angle tobe less than the arctan of the friction coefficient, andEquation 5, which defines the maximum load that can beapplied before the onset of shear failure. Even though theseequations provide only estimates of cam performance, the modelpredicts both the wide variation of performance among different SLCDsand the likelihood that some SLCDs, especially smaller ones, fail atforces commonly produced by falling climbers.
Published measurements of force to shear yield and frictional holdingability would provide information that allows climbers to makeintelligent decisions concerning SLCDs. Most importantly, climberswould be aware when devices have dangerously low frictional holdingability or maximum load to shear failure. Of similar importance, anexplicit awareness of cam limitations allows climbers to place thedevices more securely and with greater confidence. Finally,performance information would provide climbers with a way of comparingSLCDs made by different manufacturers. To date, this information iseither not explicitly stated or simply not available to the climbingcommunity. Making performance information available would alertclimbers to the existence of SLCD limitations and provide climberswith the information required to successfully work around theselimitations.
Footnotes
Power Of Two Cracks
2from Ball and Roller Bearings, Eschmann, Hasbargen, Weigand, JWiley and Sons, pg 113.
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3The literature suggests that the effect of the non-compressive forceis small (Contact Mechanics, KL Johnson, Cambridge UniversityPress, 1985).
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4Finite element analysis work by Luke Sosnowski suggests a negligibledifference between logarithmic and circular curvature.
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