This study presents the double Fourier cosine series method for solving the flexural problem of Kirchhoff plates resting on an elastic foundation of the Winkler type. The problem is a boundary value problem represented by a fourth order partial differential quantum. For the case of simply supported edges, the Dirichlet boundary conditions are identically satisfied by double Fourier cosine series if the plate centre becomes the origin of the Cartesian coordinates. A Fourier cosine series assumption for the unknown deflection function and the known load distribution results in an algebraic problem for the unknown Fourier parameters of the series; which is solved to obtain the deflection function. The paper presents general solutions for deflection and bending moments for arbitrary transverse load distribution and specific solutions for the deflections and bending moments for the specific cases of point load at arbitrary point, and at the centre, uniformly distributed load over the entire plate and sinusoidal load. It was found that the solutions obtained in this study were exact solutions and this was because the double Fourier cosine series used for the deflection shape functions were exact shape functions that satisfies all the Dirichlet boundary conditions. Furthermore, the trial solution was made to satisfy the boundary value problem at all points in the solution domain.
Keywords: Double Fourier cosine series, Kirchhoff plate, Winkler foundation, Boundary value problem
 Ike C.C.(2017). Flexural analysis of Kirchhoff plates on Winkler foundation using finite Fourier sine integral transform method. Mathematical Modelling of Engineering Problems (MMEP) International Information and Engineering Technology Association IIETA Vol 4 No 4 December 2017, pp 145 - 154 (eISSN 2369 - 0747).
 Ike C.C. (2018). Flexural analysis of rectangular Kirchhoff plate on Winkler foundation using Galerkin-Vlasov variational method. Mathematical Modelling of Engineering Problems (MMEP) International Information and Engineering Technology Association IIETA, Canada. Vol 5 No 2 June 2018 pp 83 - 92. https/doi.org/10.18280/mmep.050205.
 Ike C.C. (2017). Kantorovich-Euler Lagrange-Galerkin method for bending analysis of thin plates. Nigerian Journal of Technology NIJOTECH Vol 36 No 2 April 2017 pp 351-360. http//dx.doi.org/10.4314/nijtv36i2.5.
 Ike C.C. (2017). Variational formulation of Mindlin plate on Winkler foundation problem. Electronic Journal of Geotechnical Engineering 2017 (22.12) pp 4829 – 4846. Available at ejge.com.
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 Mama B.O., Ike C.C. Onah H.N., Nwoji C.U. (2017). Analysis of rectangular Kirchhoff plate on Winkler foundation using finite Fourier sine transform method. IOSR Journal of Mathematics (IOSR JM), Vol. 13, No 1, pp 58 - 66. DOI:10.9790/5728-1301065866.
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For most of the tunnels, the rock-support interface can vary between some full-slip to a non-slip condition, so both types should be investigated for static lining forces and deformations. In dynamic analysis, these conditions have a deterministic role on the interaction between the tunnel periphery rock mass and tunnel lining and evidently, on the load transfer intensity between them. In this paper, a dynamic analysis of tunnels under different conditions of the rock mass and lining interface has been carried out by analytical and FEM numerical methods. Also, the effect of the interface between tunnel lining and surrounding medium on earthquake intensity studied. In the following, the numerical results are verified based on an analytical method. The numerical and analytical results have a meaningful agreement with themselves. Results also showed that tunnel lining is more confronted with the induced stress due to the earthquake shake forces in non-slip condition, while in medium it is the other way around.
Keywords: Interface, Full-Slip, Non-Slip, Earthquake Loading, Tunnel Lining
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Case histories", Underground Space, 2, 106-114, 2017. DOI:
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Presented in this paper is an implicit finite difference solution for radial flow consolidation to prefabricated vertical drains (PVDs) under haversine cyclic loading with rest period assuming no smear and a constant coefficient of consolidation. Such long-term cyclic loads are induced by traffic on highways, airports and railway tracks. The effect of duration ratio of rest period to that of loading-unloading phase within a cycle, of 0 ,1 and 2, on the time variation of excess pore water pressure and effective stress was studied thoroughly. The effect of PVD spacing expressed by the ratio of radius of influence zone of drain well to that of PVD was studied for three different values of 5, 10 and 40. The findings revealed that for any rest period, the excess pore water pressure oscillates changing sign with time although the imposed loading is always positive. The oscillation comes to almost steady state at a time factor of unity. The effective stress increases with time but with some fluctuation without changing sign. The average curve for the region between the upper bound and lower bound of effective stress curve with time converges to the average of haversine cyclic loading with rest period (having amplitude q0), namely q0/2 ,q0/4 and q0/6 for the ratio of rest period to that of loading-unloading phase of 0, 1 and 2 respectively. This is independent of the time factor value at the end of first loading/unloading phase. This indicates that longer rest periods decrease the total primary consolidation settlement due to cyclic loading.
Keywords: Cyclic loading, Prefabricated vertical drains, Radial flow consolidation, Rest period, Wick drains
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The road construction industry is concerned with utilizing alternative and sustainable binder materials to aid in the production, placement and increase in the performance of flexible pavements. This research is aimed at evaluating the physical properties of bituminous mixtures modified with PET (Polyethylene Terephthalate), SBS (Styrene-Butadiene-Styrene) and EVA (Ethylene-vinyl Acetate) used as bitumen modifiers. The samples were prepared by heating the base bitumen and each polymers to their respective melting points to facilitate the miscibility of the materials; the base bitumen and each polymers were mechanically mixed together with a blending speed of 2000 ± 10 rpm for an hour. Penetration, ductility, softening point, flash and fire point and water in bitumen tests were performed to relevant standard specifications on bitumen and PMB (Polymer Modified Bitumen) with polymer composition varied from 10% to 50%. The penetration test result showed that polymers have hardening effect on bitumen; this was evidenced by the reduction in the penetration values of PET modified bitumen from 51 to 57 decimillimetre (dmm), 59 to 55 dmm for SBS modified bitumen and 68 to 59 dmm for EVA modified bitumen for all the percentage compositions. The PMB showed improved stiffening which was demonstrated by increase in the softening point of the PMB; the softening point of PET modified bitumen increased from 540C to 670C. Likewise, the softening point of SBS modified bitumen increased from 560C to 700C and the softening point of EVA modified bitumen also increased from 550C to 600C for all the percentage polymer compositions. The improvement in the water resistant capability of the PMB was confirmed by the reduction of moisture content of bitumen on addition of polymers from 3% for bitumen to 1.1% which is the least moisture content recorded at 50% composition of SBS modified bitumen.
Keywords: Physical properties, polymer additives, bitumen, bituminous mixture
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Monitoring and controlling soil behavior during the excavation of new tunnel nearby the existent tunnel is so crucial in providing safety for adjacent structures. The interaction between excavating tunnel (line 2) underneath the present metro station (Line 1) in their intersection was examined in this study. To simulate the conditions of stress and displacement caused by excavating over the current station, the process of tunnel excavation is modeled by Cut & Cover method. For the station's foundation, the pile-raft system is used to reduce settlement and the effects of interaction between the station and the new tunnel. The new tunnel under the station has been surrounded by soldier beams (soldier piles) in a distance of 20 cm from the tunnel wall. In the design phase for the tunnel Line 2 which will be excavated using a TBM-EPB (Tunnel Boring Machine-Earth Pressure Balance), the control of station’s safety near this line is a principle. Therefore, exploring the interaction between the excavating tunnel and the present station and also assessing the surface settlement in this area are of great importance. To investigate this mechanism, the limited threedimensional components analysis is applied by employing Abaqus 6-10.1 software. The Mohr-Coulomb Behavioral model is used for the soil. Results show that by the usage of pile-raft system, the surface settlement resulting from the excavation of new tunnel is reduced. Also, the results demonstrate a decline in the relative difference in terms of settlement over the raft.
Keywords: metro station, tunneling by TBM-EPB, settlement, Finite element, three-dimensional modeling
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