Amaechi Anyaegbunam | University of Nigeria, Nsukka (original) (raw)
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Papers by Amaechi Anyaegbunam
The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material... more The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the halfspace. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic halfspace deduced from Michell's expressions. A surprising conclusion is arrived at, namely: there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic halfspace. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of theory of elasticity.
Using Simpson's integration rule a highly convergent series expression for Euler's consta... more Using Simpson's integration rule a highly convergent series expression for Euler's constant ,γ, has been developed and presented herein. It is shown that this new definition of γ converges to the true value far much more rapidly than the original definition. With the summation of only 509 terms of the harmonic series the new expression calculates γ to 12-decimal places of accuracy. On the other hand it needs the summation of more than 10 12 (one trillion) terms of the harmonic series to achieve this same degree of accuracy via the original definition. Consequently it becomes obvious that this new expression for γ is extremely useful for the rapid and accurate evaluation of γ.
This note utilizes the analog models of forced vibration to express the mass of foundation block ... more This note utilizes the analog models of forced vibration to express the mass of foundation block in terms of damping ratio and thereafter obtains the expression for the minimum foundation mass required to limit vertical machine vibration amplitude to a prescribed limit. The resulting formula, which accounts for internal damping, is constrained by the limitations of the original analog solutions. Moreover, formulas are derived for determining the damping ratio for conditions different from optimum that enable the evaluation of the non-optimum foundation mass required to limit vibration. Since the ideal optimum solution may not be practical, expression has been derived for the best non-optimum foundation mass. The method presented in this paper is believed to be easier to implement and to yield smaller foundation blocks than the existing traditional method, and thus will enable cheaper machine foundations to be constructed. It is believed that the proposed optimal solution may lead to the near elimination of machine vibration nuisance produced by high-frequency machines during operation.
This paper which is presented from the perspective of Nigerian bankers, explores the perception o... more This paper which is presented from the perspective of Nigerian bankers, explores the perception of risk in Public-Private Partnership project finance and delivery. It discusses the evolution of the Public-Private Partnership Projects (PPP) concept building on broad inter-organizational relationships. A review of literature indicates that normally these relationships are fraught with challenges as well as opportunities which both parties in a Public-Private partnership can either take advantage of or be confounded by. Data was gathered from professionals working in three major banks in Nigeria using a five point Likert scale questionnaire. The paper concludes that Nigerian banks are generally wary of bearing risk and would instead seek to transfer risk to other parties
A simple and direct approach is presented for the formulation of the dynamic stiffness matrix of ... more A simple and direct approach is presented for the formulation of the dynamic stiffness matrix of a beam-column element. The traditional approach for analysis of a beam-column considers the mass as being lumped and then considers the system as having a single degree of freedom (SDOF). In this work the author considers the model as a system with distributed mass thereby treating the system as having an infinite number of degrees of freedom. The differential equation of motion of this system, in which the axial compressive force is accounted for, is derived by applying Newton's second law of motion. By imposing the appropriate boundary conditions the dynamic stiffness matrix which includes the effect of axial compressive force is synthesized. Using this matrix an existing result was reproduced, namely that for a simply supported beam-column w1N = w1*sqr(1-lamda) where w1N = the fundamental vibrational frequency of a beam-column, w1=the fundamental vibrational frequency of vibration...
Using Simpson's integration rule a highly convergent series expression for Euler's constant ,γ, h... more Using Simpson's integration rule a highly convergent series expression for Euler's constant ,γ, has been developed and presented herein. It is shown that this new definition of γ converges to the true value far much more rapidly than the original definition. With the summation of only 509 terms of the harmonic series the new expression calculates γ to 12-decimal places of accuracy. On the other hand it needs the summation of more than 10 12 (one trillion ) terms of the harmonic series to achieve this same degree of accuracy via the original definition. Consequently it becomes obvious that this new expression for γ is extremely useful for the rapid and accurate evaluation of γ.
Journal of Geotechnical and Geoenvironmental Engineering
This closure derives the viscous damping coefficient DI equivalent to the hysteretic loss factor ... more This closure derives the viscous damping coefficient DI equivalent to the hysteretic loss factor β, and gives the limiting value of radiation damping coefficient in terms of the loss factor. It is shown that no single value of DI can make both the frequency and amplitude of response of viscously damped and hysteristically damped systems to agree. It is emphasized that the response of a rigid disk on a half space is accurately predicted by the Lysmer analog model over a frequency range of ao of 0 to 2
Proceedings of the ICE - Engineering and Computational Mechanics, 2011
This paper presents the explicit stiffness matrix of the very useful 15-noded cubic strain triang... more This paper presents the explicit stiffness matrix of the very useful 15-noded cubic strain triangle, for cross anisotropic medium, that was evolved through the use of area coordinates. This stiffness matrix would be found to be very useful in finite element analysis of such geomechanics problems as collapse, vibration and consolidation because it will drastically reduce the computer run-time through a faster assembly of the global stiffness matrix.
Journal of Geotechnical and Geoenvironmental Engineering, 2011
This note utilizes the analog models of forced vibration to express the mass of foundation block ... more This note utilizes the analog models of forced vibration to express the mass of foundation block in terms of damping ratio and thereafter obtains the expression for the minimum foundation mass required to limit vertical machine vibration amplitude to a prescribed limit. The resulting formula, which accounts for internal damping, is constrained by the limitations of the original analog solutions. Moreover, formulas are derived for determining the damping ratio for conditions different from optimum that enable the evaluation of the non-optimum foundation mass required to limit vibration. Since the ideal optimum solution may not be practical, expression has been derived for the best non-optimum foundation mass. The method presented in this paper is believed to be easier to implement and to yield smaller foundation blocks than the existing traditional method, and thus will enable cheaper machine foundations to be constructed. It is believed that the proposed optimal solution may lead to the near elimination of machine vibration nuisance produced by high-frequency machines during operation.
Journal of Geotechnical and Geoenvironmental Engineering, 2011
The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material... more The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the halfspace. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic halfspace deduced from Michell's expressions. A surprising conclusion is arrived at, namely: there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic halfspace. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of theory of elasticity.
International Journal of Geomechanics, 2013
Nonlinear power-type failure envelopes of the form τ = (a + bσ ) n have been examined in this pap... more Nonlinear power-type failure envelopes of the form τ = (a + bσ ) n have been examined in this paper. It is shown that equations for which 0 < n < 1/2 are legitimate failure envelopes provided that 'a' is greater than some function of 'b' contrary to earlier assertions. The principal stress σ 1 -σ 3 relations corresponding to these laws have been derived explicitly for the quadratic law (n = 1/ 2) and implicitly for n =1/3, 2/3, 3/4. For other 'n' values a numerical algorithm for deducing the principal stress relations has been given. The procedure for evaluating the parameters 'a' and 'b' from triaxial test data for a specified 'n' value is presented in detail and it parallels Baker's earlier effort. Almost all previous studies on non-linearity have concentrated on its effect on the factors of safety of slopes. This study provides a numerical method for evaluating the earth pressures on smooth retaining walls, under plane strain conditions, for the case n 1/2. When n = 1/2 closed-form equations, that are non-existent in the literature, have been derived for both the earth pressures and the slip surfaces in two-dimensional plane strain active and passive stress states. A new explicit formula is presented for the depth of tension cracks in plastic soils for n = 1/2 while new implicit formulas are developed for n =1/3, 2/3, 3/4. The assumed value of this depth has a profound influence on the calculated factor of safety of a slope. Existing Rankine, Bell and Coulomb formulas over-estimate the passive resistance of geomaterial and this study shows that the use of a non-linear law predicts more realistic reduced values of passive resistance. Therefore, the factor of safety of two or more hitherto applied to passive resistance in the design of embedded walls can now be reduced to a lower value. A computer program in QBASIC 4.5 has been included for automatically determining the best 'n' value that matches the triaxial test data together with the associated 'a' and 'b' and also for doing the rest of the calculations rapidly. As a consequence, a best-fit non-linear power-type envelope can now be fitted effortlessly to the Hoek-Brown criterion. ________________________________________________________________________________________ has been the nonlinear equation of choice for analyzing the stability of rock masses and has been used by numerous investigators for analyzing stability of rock slopes . Considerable difficulty is encountered when Hoek-Brown equation is used directly in strength-reduction finite element type of slope stability analysis ) hence it has been common to obtain its M-C equivalent. This approximation can have dire consequences on the calculated factors of safety . shows that the power-type failure envelope τ = (a + bσ ) n provides an excellent fit to the transformation of the Hoek-Brown criterion to the σ-τ plane and describes the procedure for determining the coefficients 'a' and 'b' . The routine for doing the latter was coded using Mathematica software that is not easily accessible.
International Journal of Geomechanics, 2014
The elastic solution of a loaded cross-anisotropic half-space is dependent on the type of anisotr... more The elastic solution of a loaded cross-anisotropic half-space is dependent on the type of anisotropy that is governed by whether the characteristic equation has real and distinct or equal or complex roots. Most previous solutions have been for the case of real and distinct roots and most have also been incomplete. This work presents complete expressions for all stresses and displacements in a homogeneous, linearly elastic cross-anisotropic half-space due to a surface vertical point load for all the three types of anisotropy that are believed to be more compact and simpler than those given by the only currently existing complete solution. In the present work only 38 intermediate parameters need to be calculated in order to define all stresses and displacements as compared to the 62 parameters required in the other existing complete solution. Also, an important discovery is made namely: that the surface settlement of a half-space is given by the same formula irrespective of the type of anisotropy and this parallels the previous discovery that the contact stress of a rigid punch on a half-space is independent of anisotropy. This overrides the current notion that different formulas for the settlement apply when the characteristic equation has real roots and when it has complex roots. Hence, it is concluded that all existing formulae for the surface settlement of a cross-anisotropic half-space due to distributed surface loads-for the case of rsal and distinct roots-are valid for all types of anisotropy. It is discovered that a much publicized solution for the problem of a surface vertically loaded cross-anisotropic half-space is in error. Parametric studies carried out show that all the elastic constants strongly influence the horizontal normal stresses and radial displacement. It is believed that the compact formulas presented herein will be appealing to engineers in all parts of the world.
The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material... more The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the halfspace. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic halfspace deduced from Michell's expressions. A surprising conclusion is arrived at, namely: there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic halfspace. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of theory of elasticity.
Using Simpson's integration rule a highly convergent series expression for Euler's consta... more Using Simpson's integration rule a highly convergent series expression for Euler's constant ,γ, has been developed and presented herein. It is shown that this new definition of γ converges to the true value far much more rapidly than the original definition. With the summation of only 509 terms of the harmonic series the new expression calculates γ to 12-decimal places of accuracy. On the other hand it needs the summation of more than 10 12 (one trillion) terms of the harmonic series to achieve this same degree of accuracy via the original definition. Consequently it becomes obvious that this new expression for γ is extremely useful for the rapid and accurate evaluation of γ.
This note utilizes the analog models of forced vibration to express the mass of foundation block ... more This note utilizes the analog models of forced vibration to express the mass of foundation block in terms of damping ratio and thereafter obtains the expression for the minimum foundation mass required to limit vertical machine vibration amplitude to a prescribed limit. The resulting formula, which accounts for internal damping, is constrained by the limitations of the original analog solutions. Moreover, formulas are derived for determining the damping ratio for conditions different from optimum that enable the evaluation of the non-optimum foundation mass required to limit vibration. Since the ideal optimum solution may not be practical, expression has been derived for the best non-optimum foundation mass. The method presented in this paper is believed to be easier to implement and to yield smaller foundation blocks than the existing traditional method, and thus will enable cheaper machine foundations to be constructed. It is believed that the proposed optimal solution may lead to the near elimination of machine vibration nuisance produced by high-frequency machines during operation.
This paper which is presented from the perspective of Nigerian bankers, explores the perception o... more This paper which is presented from the perspective of Nigerian bankers, explores the perception of risk in Public-Private Partnership project finance and delivery. It discusses the evolution of the Public-Private Partnership Projects (PPP) concept building on broad inter-organizational relationships. A review of literature indicates that normally these relationships are fraught with challenges as well as opportunities which both parties in a Public-Private partnership can either take advantage of or be confounded by. Data was gathered from professionals working in three major banks in Nigeria using a five point Likert scale questionnaire. The paper concludes that Nigerian banks are generally wary of bearing risk and would instead seek to transfer risk to other parties
A simple and direct approach is presented for the formulation of the dynamic stiffness matrix of ... more A simple and direct approach is presented for the formulation of the dynamic stiffness matrix of a beam-column element. The traditional approach for analysis of a beam-column considers the mass as being lumped and then considers the system as having a single degree of freedom (SDOF). In this work the author considers the model as a system with distributed mass thereby treating the system as having an infinite number of degrees of freedom. The differential equation of motion of this system, in which the axial compressive force is accounted for, is derived by applying Newton's second law of motion. By imposing the appropriate boundary conditions the dynamic stiffness matrix which includes the effect of axial compressive force is synthesized. Using this matrix an existing result was reproduced, namely that for a simply supported beam-column w1N = w1*sqr(1-lamda) where w1N = the fundamental vibrational frequency of a beam-column, w1=the fundamental vibrational frequency of vibration...
Using Simpson's integration rule a highly convergent series expression for Euler's constant ,γ, h... more Using Simpson's integration rule a highly convergent series expression for Euler's constant ,γ, has been developed and presented herein. It is shown that this new definition of γ converges to the true value far much more rapidly than the original definition. With the summation of only 509 terms of the harmonic series the new expression calculates γ to 12-decimal places of accuracy. On the other hand it needs the summation of more than 10 12 (one trillion ) terms of the harmonic series to achieve this same degree of accuracy via the original definition. Consequently it becomes obvious that this new expression for γ is extremely useful for the rapid and accurate evaluation of γ.
Journal of Geotechnical and Geoenvironmental Engineering
This closure derives the viscous damping coefficient DI equivalent to the hysteretic loss factor ... more This closure derives the viscous damping coefficient DI equivalent to the hysteretic loss factor β, and gives the limiting value of radiation damping coefficient in terms of the loss factor. It is shown that no single value of DI can make both the frequency and amplitude of response of viscously damped and hysteristically damped systems to agree. It is emphasized that the response of a rigid disk on a half space is accurately predicted by the Lysmer analog model over a frequency range of ao of 0 to 2
Proceedings of the ICE - Engineering and Computational Mechanics, 2011
This paper presents the explicit stiffness matrix of the very useful 15-noded cubic strain triang... more This paper presents the explicit stiffness matrix of the very useful 15-noded cubic strain triangle, for cross anisotropic medium, that was evolved through the use of area coordinates. This stiffness matrix would be found to be very useful in finite element analysis of such geomechanics problems as collapse, vibration and consolidation because it will drastically reduce the computer run-time through a faster assembly of the global stiffness matrix.
Journal of Geotechnical and Geoenvironmental Engineering, 2011
This note utilizes the analog models of forced vibration to express the mass of foundation block ... more This note utilizes the analog models of forced vibration to express the mass of foundation block in terms of damping ratio and thereafter obtains the expression for the minimum foundation mass required to limit vertical machine vibration amplitude to a prescribed limit. The resulting formula, which accounts for internal damping, is constrained by the limitations of the original analog solutions. Moreover, formulas are derived for determining the damping ratio for conditions different from optimum that enable the evaluation of the non-optimum foundation mass required to limit vibration. Since the ideal optimum solution may not be practical, expression has been derived for the best non-optimum foundation mass. The method presented in this paper is believed to be easier to implement and to yield smaller foundation blocks than the existing traditional method, and thus will enable cheaper machine foundations to be constructed. It is believed that the proposed optimal solution may lead to the near elimination of machine vibration nuisance produced by high-frequency machines during operation.
Journal of Geotechnical and Geoenvironmental Engineering, 2011
The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material... more The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the halfspace. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic halfspace deduced from Michell's expressions. A surprising conclusion is arrived at, namely: there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic halfspace. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of theory of elasticity.
International Journal of Geomechanics, 2013
Nonlinear power-type failure envelopes of the form τ = (a + bσ ) n have been examined in this pap... more Nonlinear power-type failure envelopes of the form τ = (a + bσ ) n have been examined in this paper. It is shown that equations for which 0 < n < 1/2 are legitimate failure envelopes provided that 'a' is greater than some function of 'b' contrary to earlier assertions. The principal stress σ 1 -σ 3 relations corresponding to these laws have been derived explicitly for the quadratic law (n = 1/ 2) and implicitly for n =1/3, 2/3, 3/4. For other 'n' values a numerical algorithm for deducing the principal stress relations has been given. The procedure for evaluating the parameters 'a' and 'b' from triaxial test data for a specified 'n' value is presented in detail and it parallels Baker's earlier effort. Almost all previous studies on non-linearity have concentrated on its effect on the factors of safety of slopes. This study provides a numerical method for evaluating the earth pressures on smooth retaining walls, under plane strain conditions, for the case n 1/2. When n = 1/2 closed-form equations, that are non-existent in the literature, have been derived for both the earth pressures and the slip surfaces in two-dimensional plane strain active and passive stress states. A new explicit formula is presented for the depth of tension cracks in plastic soils for n = 1/2 while new implicit formulas are developed for n =1/3, 2/3, 3/4. The assumed value of this depth has a profound influence on the calculated factor of safety of a slope. Existing Rankine, Bell and Coulomb formulas over-estimate the passive resistance of geomaterial and this study shows that the use of a non-linear law predicts more realistic reduced values of passive resistance. Therefore, the factor of safety of two or more hitherto applied to passive resistance in the design of embedded walls can now be reduced to a lower value. A computer program in QBASIC 4.5 has been included for automatically determining the best 'n' value that matches the triaxial test data together with the associated 'a' and 'b' and also for doing the rest of the calculations rapidly. As a consequence, a best-fit non-linear power-type envelope can now be fitted effortlessly to the Hoek-Brown criterion. ________________________________________________________________________________________ has been the nonlinear equation of choice for analyzing the stability of rock masses and has been used by numerous investigators for analyzing stability of rock slopes . Considerable difficulty is encountered when Hoek-Brown equation is used directly in strength-reduction finite element type of slope stability analysis ) hence it has been common to obtain its M-C equivalent. This approximation can have dire consequences on the calculated factors of safety . shows that the power-type failure envelope τ = (a + bσ ) n provides an excellent fit to the transformation of the Hoek-Brown criterion to the σ-τ plane and describes the procedure for determining the coefficients 'a' and 'b' . The routine for doing the latter was coded using Mathematica software that is not easily accessible.
International Journal of Geomechanics, 2014
The elastic solution of a loaded cross-anisotropic half-space is dependent on the type of anisotr... more The elastic solution of a loaded cross-anisotropic half-space is dependent on the type of anisotropy that is governed by whether the characteristic equation has real and distinct or equal or complex roots. Most previous solutions have been for the case of real and distinct roots and most have also been incomplete. This work presents complete expressions for all stresses and displacements in a homogeneous, linearly elastic cross-anisotropic half-space due to a surface vertical point load for all the three types of anisotropy that are believed to be more compact and simpler than those given by the only currently existing complete solution. In the present work only 38 intermediate parameters need to be calculated in order to define all stresses and displacements as compared to the 62 parameters required in the other existing complete solution. Also, an important discovery is made namely: that the surface settlement of a half-space is given by the same formula irrespective of the type of anisotropy and this parallels the previous discovery that the contact stress of a rigid punch on a half-space is independent of anisotropy. This overrides the current notion that different formulas for the settlement apply when the characteristic equation has real roots and when it has complex roots. Hence, it is concluded that all existing formulae for the surface settlement of a cross-anisotropic half-space due to distributed surface loads-for the case of rsal and distinct roots-are valid for all types of anisotropy. It is discovered that a much publicized solution for the problem of a surface vertically loaded cross-anisotropic half-space is in error. Parametric studies carried out show that all the elastic constants strongly influence the horizontal normal stresses and radial displacement. It is believed that the compact formulas presented herein will be appealing to engineers in all parts of the world.