The issue of prescribing the support requirements for stratified roofs of no major discontinuities otherthanhorizontalbeddingplanesisherebyapproached ideallyaswellaspragmatically. Firstly, the unreinforced case is analytically defined; the solution acquired by elementary beam theory for a fixed beam under distributed load is compared to an Airy stress function solution for a fixed beam under its own weight based on Timoshenko beam theory. Finally, a finite difference numerical solution is performed and verified. The model is then used to investigate the behavior of a two-member stratified roof with contact plane governed by the angle of friction and tightened inordertomobilizetheshearingreactionforceatthediscontinuity. Parametric analyses to investigatethepossibleeffectsofelasticparameterssuchasthe modulusofelasticityandthe Poisson's ratio and also the interbedding friction angle and its effect on the response of the model conclude thissection. The last part involves the numerical implementation of a bolting support system providingthe previously determinedforce andthe prescription ofits characteristics,i.e. length,spacing,diameterandpretensionofbolts. The impact of applying concentrated compressive forces instead of the theoretical distributed support is also outlined.


The response of layered and competent rock in the roof of underground openings has often been approximated by that ofanelasticfixed-endbeamunderdistributedload equal to the beam self-weight. The stability of such a roof iscloselyrelatedtothethicknessofthebeamsandthe bedding plane strength characteristics. Confining our research to formations containing as the only major discontinuity a cohesionless bedding plane, an investigation ofasupportsystem providingsufficient compression ofthestratatogenerateshearstrengthby frictional resistance is carried out.

2.1 Elementary beam theory

The solution for a two-dimensional beam fixed at two opposing lateraledgessaggingunderitsown weight,is provided undertheapproximationofreplacingtheself weight withadistributedloadqaccordingtoequation (1):

  • (Equation in full paper)

The maximum stress components σmax (x=0, y=t/2) and τmax (x=t/2, y=0) are only affected by small L/t ratios. However, the vertical stress component σyy is generally overlooked in the elementary theory. Should it not be ignored, it would yield a different stress distribution on the yz plane of Figure 1 than the one provided by the stress function solution because of the replacement of self-weight with a distributed load on the upper surface of the beam (Timoshenko and Goodier, 1970).

2.3 Numerical solution - Verification

A commercial finite difference code was used to provide the numerical solution for the fixed-end beam.

The modeled plane strain beam was of L/t=12.5, zero Poisson's ratio, 1 m length, density 2400 Mg/m3 and an elasticity modulus of 28 GPa. Only the left half beam was modeled in order to take advantage of th

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