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The boundary conditions given in Eq. (53) are replaced into Eq. (46), provides four equations and a system with four unknown constants, c;,i ¢ N. The set of equations is then solved to determine those unknown coefficients for clamped beam displayed in Eq. (54). Fig. 4. Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Simply Supported nanobeam

Figure 4 The boundary conditions given in Eq. (53) are replaced into Eq. (46), provides four equations and a system with four unknown constants, c;,i ¢ N. The set of equations is then solved to determine those unknown coefficients for clamped beam displayed in Eq. (54). Fig. 4. Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Simply Supported nanobeam

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The strain energy of the beam is mathematically expressed in Eq. (5) as the integral of the stress and strain components over the volume. The expression is further reformed by substituting Eqs. (3) and (4) into Eq. (5). Fig. 1. Material distribution in the beam domain.
The strain energy of the beam is mathematically expressed in Eq. (5) as the integral of the stress and strain components over the volume. The expression is further reformed by substituting Eqs. (3) and (4) into Eq. (5). Fig. 1. Material distribution in the beam domain.
Fig. 2. Distribution of the kernel function when (a) c, = 1, (b) c; = — 1 So now the constitutive equation can be written by separating the variable functions based on Lemma 1 as
Fig. 2. Distribution of the kernel function when (a) c, = 1, (b) c; = — 1 So now the constitutive equation can be written by separating the variable functions based on Lemma 1 as
The nondimensional form of the other parameters are defined as Now Eq. (24) can be nondimensionalized with cross sectional area integration over the stress component to evaluate it more decently.
The nondimensional form of the other parameters are defined as Now Eq. (24) can be nondimensionalized with cross sectional area integration over the stress component to evaluate it more decently.
To better represent the solution this study, introduce new parameter defined as follows Substituting the constants stated in Eq. (49) into Eq. (46) provides the explicit displacement profile for nonlocal stress driven simply supported beam under a uniformly distributed load across the beam domain presented in (51)
To better represent the solution this study, introduce new parameter defined as follows Substituting the constants stated in Eq. (49) into Eq. (46) provides the explicit displacement profile for nonlocal stress driven simply supported beam under a uniformly distributed load across the beam domain presented in (51)
Fig. 3. Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Simply Supported beam The maximum displacement of a simply supported nanobeam typically occurs at the midspan of the beam domain. However, this position can vary depending on the # values. Higher / values shift the maximum displacement location to the left clamped support. According to the nonlocal stress driven beam theory when f approaches zero, the maximum displacement is described by eq. (52)
Fig. 3. Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Simply Supported beam The maximum displacement of a simply supported nanobeam typically occurs at the midspan of the beam domain. However, this position can vary depending on the # values. Higher / values shift the maximum displacement location to the left clamped support. According to the nonlocal stress driven beam theory when f approaches zero, the maximum displacement is described by eq. (52)
Comparison of maximum displacements for different support condition x 10-3. Table 1
Comparison of maximum displacements for different support condition x 10-3. Table 1
Effect of axial inhomogeneity on maximum displacements of Simply Supported beam x 10-°.
Effect of axial inhomogeneity on maximum displacements of Simply Supported beam x 10-°.
Effect of transverse gradation on maximum displacements of Simply Supported beam x 10-°. Table 3
Effect of transverse gradation on maximum displacements of Simply Supported beam x 10-°. Table 3
And also, those 4; terms are Afterward, the constants listed in Eq. (54) are substituted into Eq. (46), yielding the explicit expression for the displacement of clamped nanobeams under a uniformly distributed load according to the stress-driven nonlocal theory. The maximum displacement of this boundary condition is observed at the midspan similar to the simply supported beam though the position of the maximum displacement may vary due to the effect of axial FGM parameter / as shown in Fig. 5(b). The maximum displacement of a clamped nanobeam under these loading conditions, as predicted by the stress-driven nonlocal theory, is specified in Eq. (57) when a limit with P= 0 is applied.
And also, those 4; terms are Afterward, the constants listed in Eq. (54) are substituted into Eq. (46), yielding the explicit expression for the displacement of clamped nanobeams under a uniformly distributed load according to the stress-driven nonlocal theory. The maximum displacement of this boundary condition is observed at the midspan similar to the simply supported beam though the position of the maximum displacement may vary due to the effect of axial FGM parameter / as shown in Fig. 5(b). The maximum displacement of a clamped nanobeam under these loading conditions, as predicted by the stress-driven nonlocal theory, is specified in Eq. (57) when a limit with P= 0 is applied.
Fig. 5. Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Clamped beam
Fig. 5. Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Clamped beam
Effect of axial inhomogeneity on maximum displacements of Clamped beam x 107°. Table 4 Substituting the four constants from Eq. (59) into Eq. (46) yields the explicit expression for the nanocantilever displacement under a
Effect of axial inhomogeneity on maximum displacements of Clamped beam x 107°. Table 4 Substituting the four constants from Eq. (59) into Eq. (46) yields the explicit expression for the nanocantilever displacement under a
Fig. 6. Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Clamped nanobeam. 3.3. Nanocantilevers The boundary conditions for nanocantilever fixed at the near end and free at the far end are specified at Eq. (58
Fig. 6. Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Clamped nanobeam. 3.3. Nanocantilevers The boundary conditions for nanocantilever fixed at the near end and free at the far end are specified at Eq. (58
Effect of transverse gradation on maximum displacements of Clamped beam x 107°. Table 5 uniformly distributed load based on the stress-driven nonlocal theory, shown in Eq. (62)
Effect of transverse gradation on maximum displacements of Clamped beam x 107°. Table 5 uniformly distributed load based on the stress-driven nonlocal theory, shown in Eq. (62)
Effect of axial inhomogeneity on maximum displacements of Cantilever beam x 107°. Table 6
Effect of axial inhomogeneity on maximum displacements of Cantilever beam x 107°. Table 6
Effect of transverse gradation on maximum displacements of Cantilever beam x 1073. Table 7
Effect of transverse gradation on maximum displacements of Cantilever beam x 1073. Table 7
Fig. 7. Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Cantilevered beam.
Fig. 7. Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Cantilevered beam.
Fig. 8. Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Cantilevered nanobeam.
Fig. 8. Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Cantilevered nanobeam.
Related topics:
NanotechnologyConstitutive ModelingFunctionally graded materialsLaplace TransformNonlocal ElasticityMode ShapesStructural Dynamics (Random Vibrations)
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