Westergaards 1926 pavement formulas have been traditionally used to validate the accuracy of predictions made by a fem model. The formulae that are most widely used are the boussinesq and westergaard formulas murthy. Application of bousinesqs and westergaards formulae in analysing foundation stress distribution for a failed telecommunication mast. The doweled joint model has been used to evaluate the joint load transfer efficiency, the nondoweled model results have been compared to the values calculated using the westergaards theory. Equation 1 b employs westergaards special theory in which radius b replaces the true radius, a, of the loaded area. Stress distribution in soils gate 2019 civil boussinesq. However, the actual pavement conditions are often different from westergaards ideal assumptions of infinite or semiinfinite slab size and full contact between the slabsub grade interfaces. They are all similar and differ only in the assumptions made to represent the elastic conditions of the soil mass. He considered rigid pavement slab as a thin elastic plate resting on soil subgrade, which is assumed to be a dense liquid. Westergaards theory is considered good to design the rigid pavements. Westergaards longheld reverence for testing propositions with an appeal to evidence permeates the book, in contrast with a tendency among some marxists to let theory run away with them. The average velocity is the same for all the 3 parts.
Basic assumptions, moment of inertia, settlements, modification of moments, maximum moments and shear, beams curved in plananalysis for torsion, redistribution of moments for single and multispan beams, design examples. Westergaard formalized the compound material with the following assumptions. Vehiclepavement interaction, viscoelastic theory and assumptions, layered systems concepts, stress solutions for one, two and three layered systems, fundamental design concepts. The key assumptions of his solution are the following. The westergaards developed closedform stress solutions for calculating wheel load stress and warping stress for jointed concrete pavement. They are all similar and differ only in the assumptions made to represent. In this set of notes the assumption is made that the soil can be treated as an elastic body without excessive. Determine the allowable compressive force manual table 414 assumptions. Others have discussed the problems involved in ignoring the issue of underlying assumptions and fo cusing only on the benefits of combining both approaches donmoyer, 1985. Equation 1 b employs westergaards special theory in which radius b. The wiki article is quite incomplete unfortunately, but it does give the following as a general reference, which may be useful. Many formulas based on the theory of elasticity have been used to compute stresses in soils. From an educators perspective, we have made assumptions about the needs and readiness of learners, and assumed that there are best teaching and learning strategies for particular learners under particular learning context.
Westergaards paper 1 dating back to 1933 is considered as the milestone for the evaluation of hydrodynamic loads on water retaining structures, given its simple yet elegant solution and physical interpretation of the phenomenon. Financial management theory and practice brigham 11e. Validation of thermal strain gradients the product td. Vertical stress in a soil mass university of waterloo. Hydro pole, light stand, column, etc lines loads rack or rail loading, strip foundation rectangular area raft or rectangular footing circular area tank earth embankment road, railway, fill, ice, etc. For undrained condition, the westergaard equation is shown below.
Pdf application of bousinesqs and westergaards formulae in. Design of rigid pavements is based on westergaards analysis, where modulus of subgrade reaction, radius of relative stiffness, radius of wheel load distribution are used. So, here the upward reaction is assumed to be proportional to the deflection, i. Denseliquid foundationthe primary im plications of using a dense liquid or winkler zimmerman s assumption are shown in figure 2. Assumptions of westergaard s theory of stress distribution. When a point load q acting on the surface of a semi infinite solid, a vertical stress. Stresses in dowel bars and tie bars pavement design factors. If for nothing else, this book should be read for the judiciousness and calm reason of westergaards case.
List the assumption of boussinesq analysis for the. Bull department of civil engineering, the university, newcastle upon tyne, ne1 7ru, uk abstract the result of violating westergaards assumption that a rigid road pavement raft unit is, for all load conditions, in full. In this theory, we are making assumptions about learning from different perspectives. Westergaards assumptions mainly reduce the stresses obtained directly below the center of the load as compared to the boussinesq solution.
According to westergaard the vertical stress at a point p at a depth z below the concentrated load q. For example, if you are doing a study on the middle school music curriculum, there is an underlying assumption that music will continue to be important in the middle school program. However, at a certain distance away from the center, stresses obtained from westergaard begin to exceed those calculated from boussinesq. Threedimensional finite element model for analysis of. For the latter, on the other hand, there exists no reliable theory to be based on except the cbr method which has been found out by experiences. Soil mechanics material by sky promod mobile version.
It is a common observation that the theory of elasticity is useful beyond the range of elasticity. The first systematic work on the history of the theory of structures. Westergaards theory supplemented by empirical modifications 1, all these methods consider a slab on grade as a homogeneous, isotropic, and elastic infinite slab resting on a winkler subgrade that exerts, at all points, a vertical reactive pressure proportional to the slab deflection. Assumptions underlying quantitative and qualitative. Conceptual diagram for com paring and contrasting plate and layered theory. Financial management theory and practice brigham 11e38. We will limit ourselves to mode i, and the summary of the procedure given.
This validation method verifies that the model predicts responses in accordance with the assumptions used in westergaards analysis to develop his equations. It is a common observation that the theory of elasticity. Use of linearelastic layered theory for the design of. Methodology results and discussion collection of samples. Westergaards model of subgrade reaction elastic layered theory cant be applied for jointed rigid pavements due to the fact that one of the assumptions of the layered theory was that layers are infinitely long in horizontal direction no effect of discontinuity at joints.
This was introduced to account for the effect of shear stresses in the vicinity of the load, which is neglected in the ordinary theory of mediumthick plates. The formulas that are most widely used are the boussinesq and westergaard formulas. Islab2000 graphical results for the comparison with westergaards solution appendix d. Detailed solution 11022018 afternoon session ce f126, katwaria sarai, new delhi 110 016 ph. In this theory, the elastic soil is intermixed with the infinite thin rigid layer that. For soil, the soil mass is elastic, isotropic, homogeneous and semiinfinite.
Westergaard the boussinesq method is one of the most commonly used methods for obtaining stress distributions in elastic media. A slab resting on a winkler foundation was run and compared with westergaards interior and edge loading equations. This method of analysis is usually applied in the calculation of the immediate and consolidation settlements of foundations because the factor of safety of such foundations against a bearing capacity failure is usually greater than three. The settlement problem stresses in a soil mass due to footing pressure the boussinesq method for evaluating soil pressure westergaards method for evaluating soil pressures 55 immediate eiastic settlement computationtheory 56 immediate settlements application 57 alternati ve methods of computing elastic settlements 52. Pdf foundation analysis and design free download pdf. In contrast, those falling within the situationalist perspective focus on the. Bina nusantara westergaard s theory of stress distribution assumptions. At point p of above figure due to a point load q, vertical stress. Equation lb employs westergaards special theory, first proposed in 1926, in which radius b replaces lhe true radius, a, of the loaded area. Boussinesq 1985 evolved equations that can be used to determine stresses at any point p at a depth z as a result of a surface point load. Include the shear and bending moment diagrams for the beam in your pdf. Computers and geotechnics 12 1991 3147 technical note the violation of westergaards full contact assumption and its effect on raft unit pavements john w.
In 1926 he investigated the mechanical behaviour of concrete pavements for roads westergaard, 1926, which would soon become the basis of the relevant regulations. Stress solution at crack tip westergaard 1934 general stress solution the procedure for computing the stress state near a crack tip was established by westergaard 1934. Boussinesqs equation geotechnical engineering civil. The violation of westergaards full contact assumption and. For those who are unfamiliar with it, wikipedia does have an article about peter westergaards tonal theory. Soil mechanics material by sky iii the selfweight of the soil is ignored. Application of bousinesqs and westergaards formulae in analysing foundation stress distribution for a failed telecommunication mast article pdf available july 2012 with 2,678 reads. The formulae that are most widely used are the boussinesq and westergaard formulas murthy, 1992. Hi, westergaard developed a solution to determine distribution of stress due to point load in soils composed of thin layer of granular material. Different types of joints are required like expansion and contraction. Comparison between islab2000 results and westergaards solution appendix c. For critical design, a combination of load stress, frictional stress and warping stress is considered. Application of bousinesqs and westergaards formulae in. Westergaards theory and assumptions, stresses due to curling, stresses and deflections due to loading, frictional stresses.
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