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Figure 19. Stability plots for 6; =0.2, g=90.0I, 7 =0.1, and ye = 0.2.

Figure 19 Stability plots for 6; =0.2, g=90.0I, 7 =0.1, and ye = 0.2.

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Figure |. Schematic diagram for exponential and linear taper configurations. The literature assessment divulges that, selected study has been conducted on stability of non-uniform beams for various types of foundations. According to Irie et al.!!, it obtained that the consequence of rise of the flexural rigidity is more than that of mass, that’s why authors are interested for study of stability of an exponentially tapered beam. Nevertheless, no research has been conducted before to analyze an exponentially tapered sandwich beam having asymmetric configuration placing over a Pasternak foundation of variable stiffness with temperature gradient. Therefore, the present article deals with the above said configuration. Here the importance has been given to the exponentially tapered beam rather than linearly tapered beam because for this type of study, for same maximum value and same range of second moment of area, the exponentially tapered beam will be economical in comparison to the linearly tapered beam which is clear from the Figure 1. The shaded regions in Figure 1, represents the extra materials which will be required if the linear tapered beam will be considered instead of exponentially tapered beam. With the above constraint the
Figure |. Schematic diagram for exponential and linear taper configurations. The literature assessment divulges that, selected study has been conducted on stability of non-uniform beams for various types of foundations. According to Irie et al.!!, it obtained that the consequence of rise of the flexural rigidity is more than that of mass, that’s why authors are interested for study of stability of an exponentially tapered beam. Nevertheless, no research has been conducted before to analyze an exponentially tapered sandwich beam having asymmetric configuration placing over a Pasternak foundation of variable stiffness with temperature gradient. Therefore, the present article deals with the above said configuration. Here the importance has been given to the exponentially tapered beam rather than linearly tapered beam because for this type of study, for same maximum value and same range of second moment of area, the exponentially tapered beam will be economical in comparison to the linearly tapered beam which is clear from the Figure 1. The shaded regions in Figure 1, represents the extra materials which will be required if the linear tapered beam will be considered instead of exponentially tapered beam. With the above constraint the
Figure 2. Configuration for the system. An exponentially tapered sandwich beam having asymmet- ric configuration of length / placing over a Pasternak foun- dation having variable spring stiffness is shown in Figure 2. A combination of static and dynamic external excitation P(t) =PRt+ Pcos(at) is applied along the X-axis through the centroid of the cross-section of the beam. In the expres- sion for external excitation, P, represents the static load and F represents the dynamic load. The top and bottom elastic layers have thickness 2h, , 2h, at one end while 2h, and 2h, at the other. It has a viscoelastic core with complex shear modulus G3 = G,(1+ jn) and thickness of 2h, and 2h, at the ends.
Figure 2. Configuration for the system. An exponentially tapered sandwich beam having asymmet- ric configuration of length / placing over a Pasternak foun- dation having variable spring stiffness is shown in Figure 2. A combination of static and dynamic external excitation P(t) =PRt+ Pcos(at) is applied along the X-axis through the centroid of the cross-section of the beam. In the expres- sion for external excitation, P, represents the static load and F represents the dynamic load. The top and bottom elastic layers have thickness 2h, , 2h, at one end while 2h, and 2h, at the other. It has a viscoelastic core with complex shear modulus G3 = G,(1+ jn) and thickness of 2h, and 2h, at the ends.
The expression for Strain Energy (V), Kinetic Energy (7) , and Non-conservative work done W,, will be
The expression for Strain Energy (V), Kinetic Energy (7) , and Non-conservative work done W,, will be
The different section dependent parameters are,
The different section dependent parameters are,
Determination of critical buckling loads for static stability and instability regions The non-dimensional critical buckling loads for static con- dition and the regions of instability are figured out respec- tively from the following equations
Determination of critical buckling loads for static stability and instability regions The non-dimensional critical buckling loads for static con- dition and the regions of instability are figured out respec- tively from the following equations
Figure 5 presents the influence of 6, on critical buck- ling loads for static condition. The critical loads decrease with increase in 6,. Also 6, has the similar effect on the system stability as shown in the Figure 6.
Figure 5 presents the influence of 6, on critical buck- ling loads for static condition. The critical loads decrease with increase in 6,. Also 6, has the similar effect on the system stability as shown in the Figure 6.
the ratio of rigidity modulus of the shear layer to modulus of elasticity, elastic foundation variation parameter and ratio of thickness of shear layer to the beam length. With increase in G,/E, , the increase in Gs increases the shearing resist- ance and decrease in £, will cause the reduction in stability. For the values of G,/£, considered in this case, the effect of G, dominate the effect of E,. So, overall stability of the system increases. With increase in the value of /,, stiff- ness increases. Hence, it stabilizes the system. While there is increase in d//, the increased value of d_ increases the shearing resistance, so improves the stability of the system. Also for smaller length the buckling resistance increases, hence stability of the system increases.
the ratio of rigidity modulus of the shear layer to modulus of elasticity, elastic foundation variation parameter and ratio of thickness of shear layer to the beam length. With increase in G,/E, , the increase in Gs increases the shearing resist- ance and decrease in £, will cause the reduction in stability. For the values of G,/£, considered in this case, the effect of G, dominate the effect of E,. So, overall stability of the system increases. With increase in the value of /,, stiff- ness increases. Hence, it stabilizes the system. While there is increase in d//, the increased value of d_ increases the shearing resistance, so improves the stability of the system. Also for smaller length the buckling resistance increases, hence stability of the system increases.
Figure 12. Stability plots for B,=0.5, g=0.01, py, = 0.05, M2 =0.5, and y, =0.2.
Figure 12. Stability plots for B,=0.5, g=0.01, py, = 0.05, M2 =0.5, and y, =0.2.
Figure 14. Stability plots for Pi =|, 6) = 62 =0, g=0.01 » Mi =0.05, uw =0.5, and y. =0.2. Figure 13. Stability plots for B, =I, g =0.01, 11, = 0.05, M2 =0.5, and y. =0.2.
Figure 14. Stability plots for Pi =|, 6) = 62 =0, g=0.01 » Mi =0.05, uw =0.5, and y. =0.2. Figure 13. Stability plots for B, =I, g =0.01, 11, = 0.05, M2 =0.5, and y. =0.2.
Figure 15. Stability plots for Bi =|, 6, = 0.4, 62 =0, g =0.01, MW =0.05, pw =0.5, and y. = 0.2.
Figure 15. Stability plots for Bi =|, 6, = 0.4, 62 =0, g =0.01, MW =0.05, pw =0.5, and y. = 0.2.
Figure 16. Stability plots for Bi =|, dz =0.4, dz, =0, g=0.01, u, =0.05, wo =0.5, and y, = 0.2.
Figure 16. Stability plots for Bi =|, dz =0.4, dz, =0, g=0.01, u, =0.05, wo =0.5, and y, = 0.2.
Figure 18. Stability plots for Bi =|, dz, = 0, dz, = 0.4, g=0.01, u, =0.05, pw. =0.5, and y, = 0.2. Figure 17. Stability plots for Bi = 0.5, 6 = 0, 62 = 0.4, g=0.01, = 0.05, p12 =0.5,and ye = 0.2.
Figure 18. Stability plots for Bi =|, dz, = 0, dz, = 0.4, g=0.01, u, =0.05, pw. =0.5, and y, = 0.2. Figure 17. Stability plots for Bi = 0.5, 6 = 0, 62 = 0.4, g=0.01, = 0.05, p12 =0.5,and ye = 0.2.
Figure 22. Stability plots for 6s =0.4, g=0.0l, 7 =0.1, and ye =0.2. Figure 21. Stability plots for 6 =0.2, g=0.0l, 7 =0.1, and ye =0.2.
Figure 22. Stability plots for 6s =0.4, g=0.0l, 7 =0.1, and ye =0.2. Figure 21. Stability plots for 6 =0.2, g=0.0l, 7 =0.1, and ye =0.2.
Figure 20. Stability plots for 6,=0.4, g=0.01, 7 =0.1, and ye =0.2.
Figure 20. Stability plots for 6,=0.4, g=0.01, 7 =0.1, and ye =0.2.
Figure 23. Stability plots for 52 = 0.2, g = 0.01, 7 =0.1, and ye =0.2.
Figure 23. Stability plots for 52 = 0.2, g = 0.01, 7 =0.1, and ye =0.2.
Figure 25. Stability plots for dz, =0.4, g=0.01, 7 =0.1, and ye =0.2. Figure 24. Stability plots for 52 = 0.4, g = 0.01, 7 =0.1, and ye =0.2..
Figure 25. Stability plots for dz, =0.4, g=0.01, 7 =0.1, and ye =0.2. Figure 24. Stability plots for 52 = 0.4, g = 0.01, 7 =0.1, and ye =0.2..
From Figures oT to 29 it is seen ‘that for Ye <1 (Figure 27), the unstable regions shifted toward lesser excitation frequencies than Figure 28 but wheny, >1
From Figures oT to 29 it is seen ‘that for Ye <1 (Figure 27), the unstable regions shifted toward lesser excitation frequencies than Figure 28 but wheny, >1
Figure 26. Stability plots for ky =|, ko =5, g=0.01, 7 =0.1, and y. =0.2 The results from the above study reveal that, stiffening of the system due to Pasternak foundation enhanced the static as well as the dynamic stability of the system by moving the instability zones from origin and narrowing the zones areas in some cases. G, is the modulus of rigidity the sup- porting shear layer, so it improves the stiffness of the sys- tem. The value G;/E, should be more for more stability of the system. While there is rise in d// the increased value of d improves the shearing resistance, which improves the stability. The system stability improves with the spring stiffness and elastic foundation parameter, this is because these parameters improve the strength of the founda- tion which help in improving the system stability. With increase in the values of temperature gradients (uniform and variable) for upper and lower elastic layers reduce the value of EJ and EA for which there is reduction in stiff- ness and ultimately reduction in the stability of the sys- tem. With increase in taper parameters the stability of the system worsen as the slenderness of the system increases
Figure 26. Stability plots for ky =|, ko =5, g=0.01, 7 =0.1, and y. =0.2 The results from the above study reveal that, stiffening of the system due to Pasternak foundation enhanced the static as well as the dynamic stability of the system by moving the instability zones from origin and narrowing the zones areas in some cases. G, is the modulus of rigidity the sup- porting shear layer, so it improves the stiffness of the sys- tem. The value G;/E, should be more for more stability of the system. While there is rise in d// the increased value of d improves the shearing resistance, which improves the stability. The system stability improves with the spring stiffness and elastic foundation parameter, this is because these parameters improve the strength of the founda- tion which help in improving the system stability. With increase in the values of temperature gradients (uniform and variable) for upper and lower elastic layers reduce the value of EJ and EA for which there is reduction in stiff- ness and ultimately reduction in the stability of the sys- tem. With increase in taper parameters the stability of the system worsen as the slenderness of the system increases
Figure 27. Stability plots for ye = 0.1, 4 = 0.05, uw. = 0.5, and g= 0.01.
Figure 27. Stability plots for ye = 0.1, 4 = 0.05, uw. = 0.5, and g= 0.01.
Figure 28. Stability plots for y. =|, 4, = 0.05, uw. =0.5, and g=00l. Madhusmita Pradhan @
Figure 28. Stability plots for y. =|, 4, = 0.05, uw. =0.5, and g=00l. Madhusmita Pradhan @
Figure 29. Stability plots for ye =5, uw; = 0.05, pi. =0.5, and g=0.01.
Figure 29. Stability plots for ye =5, uw; = 0.05, pi. =0.5, and g=0.01.
Figure 30. Stability plots for Gs/E: = 0.001, 7 = 0.1, g = 0.01, and y. = 0.2.
Figure 30. Stability plots for Gs/E: = 0.001, 7 = 0.1, g = 0.01, and y. = 0.2.
Figure 31. Stability plots for Gs/E, = 0.005, 7 =0.1, g = 0.01, and y. = 0.2.
Figure 31. Stability plots for Gs/E, = 0.005, 7 =0.1, g = 0.01, and y. = 0.2.
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