Theoretical approach with analytical and numerical procedure for determination initial displacement of a reinforced and prestressed concrete members, simple and cantilever beams, loaded by axial forces and bending mom...Theoretical approach with analytical and numerical procedure for determination initial displacement of a reinforced and prestressed concrete members, simple and cantilever beams, loaded by axial forces and bending moments is <span style="font-family:Verdana;">proposed. It is based on the principle of minimum potential energy with</span><span style="font-family:Verdana;"> equality of internal and external forces. The equations for strain internal energy have been derived, including compressed and tensile concrete and reinforce</span><span style="font-family:Verdana;">ment. The energy equations of the external forces with axial flexural dis</span><span style="font-family:Verdana;">placement effects have been derived from the assumed sinusoidal curve. The trapezoid rule is applied to integrate the segment strain energy. The proposed method uses a non</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">-</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">linear stress-strain curve for the concrete and bilinear elastic-plastic relationship for reinforcement;equilibrium conditions at a sectional level to generate the strain energies along the beam. At the end of this article are shown three specific numerical examples with comparative, experimental (two tests)</span></span></span><span><span><span style="font-family:;" "=""> </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">results with the excellent agreement and one calculation result with a great disagreement, by obtaining results of virtual principle method.</span></span></span><span><span><span style="font-family:;" "=""> </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">With this method is avoiding the adoption of an unsure (EJ), as in the case of underestimating or overestimate initial flexural rigidity.</span></span></span>展开更多
A new method for analysis of counter beams is presented in the paper. The analysis has taken into account their stiffness EI, Winkler’s space with modulus of subgrade reaction k and equality deformities of the founda...A new method for analysis of counter beams is presented in the paper. The analysis has taken into account their stiffness EI, Winkler’s space with modulus of subgrade reaction k and equality deformities of the foundation beam with the ground. The solution is found by using the numerical analysis of the Winkler’s model, with variation of different moduli of the subgrade reaction k2 outside the force zone r, while under the force P exists the modulus of the subgrade reaction k, up to the definition of minimum bending moments. The exponential function k2(r), as the geometric position of the minimum moments is approximately assumed. From the potential energy conditions of the reciprocity of displacement and reaction, the width of the zone r and the modulus of the subgrade reaction k2 are explicitly determined, introducing in the calculation initial and calculation soil displacement wsi successively. At the end of the paper, it presented numerical example in which the influence of k and k2 values on bending moments of the counter beam is analyzed. The essential idea of this paper is to decrease the quantity of the reinforcement in the foundations, beams, i.e. to obtain a cost-efficient foundation construction.展开更多
文摘Theoretical approach with analytical and numerical procedure for determination initial displacement of a reinforced and prestressed concrete members, simple and cantilever beams, loaded by axial forces and bending moments is <span style="font-family:Verdana;">proposed. It is based on the principle of minimum potential energy with</span><span style="font-family:Verdana;"> equality of internal and external forces. The equations for strain internal energy have been derived, including compressed and tensile concrete and reinforce</span><span style="font-family:Verdana;">ment. The energy equations of the external forces with axial flexural dis</span><span style="font-family:Verdana;">placement effects have been derived from the assumed sinusoidal curve. The trapezoid rule is applied to integrate the segment strain energy. The proposed method uses a non</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">-</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">linear stress-strain curve for the concrete and bilinear elastic-plastic relationship for reinforcement;equilibrium conditions at a sectional level to generate the strain energies along the beam. At the end of this article are shown three specific numerical examples with comparative, experimental (two tests)</span></span></span><span><span><span style="font-family:;" "=""> </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">results with the excellent agreement and one calculation result with a great disagreement, by obtaining results of virtual principle method.</span></span></span><span><span><span style="font-family:;" "=""> </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">With this method is avoiding the adoption of an unsure (EJ), as in the case of underestimating or overestimate initial flexural rigidity.</span></span></span>
文摘A new method for analysis of counter beams is presented in the paper. The analysis has taken into account their stiffness EI, Winkler’s space with modulus of subgrade reaction k and equality deformities of the foundation beam with the ground. The solution is found by using the numerical analysis of the Winkler’s model, with variation of different moduli of the subgrade reaction k2 outside the force zone r, while under the force P exists the modulus of the subgrade reaction k, up to the definition of minimum bending moments. The exponential function k2(r), as the geometric position of the minimum moments is approximately assumed. From the potential energy conditions of the reciprocity of displacement and reaction, the width of the zone r and the modulus of the subgrade reaction k2 are explicitly determined, introducing in the calculation initial and calculation soil displacement wsi successively. At the end of the paper, it presented numerical example in which the influence of k and k2 values on bending moments of the counter beam is analyzed. The essential idea of this paper is to decrease the quantity of the reinforcement in the foundations, beams, i.e. to obtain a cost-efficient foundation construction.
