3 Applications for One-Dimensional Problems
3.2 Static Problems
3.2.2 Calculation of Static Deflection Through FEM
Table 1 Cantilever beam: Condition number
κ(K)= ||K|| · ||K−1||
of the matrix of coeffi- cients,K
e0a/L Gauss points
Amod k1=0.01 k1=0.1 k1=0.2 k1=0.5
0.01 500 4875.431 100.9415 10.00192 4.984897 1.990991
1000 2335.448 101.0362 10.00464 4.985585 1.991034 2000 2240.871 101.1099 10.00701 4.986253 1.991106 3000 1663.168 101.1350 10.00786 4.986504 1.991137
0.1 [54] 500 93.13096 97.12981 9.651288 4.82809 1.951197
1000 92.71524 97.19989 9.653703 4.82881 1.951289
2000 92.54316 97.25277 9.655588 4.829385 1.951368
3000 92.51211 97.27045 9.65623 4.829583 1.951395
The linear system corresponding to the TPNI stress model is stable. The condition number of the coefficient matrixKtends to 1 whenk1-parameter increases. Also, the condition number is slightly affected by the nonlocal parameter (Table1).
What is more, the condition number of the coefficient matrixKcorresponding to the model of the modified kernel is acceptable for the case ofe0a/L=0.01. When the nonlocal parameter takes such a small value, the model of the modified kernel approaches the first kind Fredholm integral equation. On the other hand, the condition number gives excellent results whene0a/L =0.1, and the linear system deriving is stable (Table1).
The numerical transverse deflections regarding the nonlocal integral models are deduced by the following FEMs [48]:
N n=1
⎡
⎣4
j=1
wnj xn
xn−1
d2φni
d x2 E Iξ(x,e0a)d2φnj
d x2 d x
+ xn
xn−1
d2φni d x2 E I
M m=1
4 j=1
wmj sm
sm−1
A(|x−s|,e0a)d2φmj ds2 dsd x
⎤
⎦
= N n=1
xn xn−1
φni(x)q(x)d x−φiN(L)Q¯N L(L)+φ1i(0)Q¯N L(0) +
φiN
(L)M¯N L(L)− φ1i
(0)M¯N L(0) (53)
N n=1
⎡
⎣k1
4 j=1
wnj xn
xn−1
d2φni
d x2 E Id2φnj d x2 d x
+k2 xn
xn−1
d2φni
d x2 E I M m=1
4 j=1
wmj sm
sm−1
A(|x−s|,e0a)d2φmj
ds2 dsd x
⎤
⎦
= N n=1
xn xn−1
φni(x)q(x)d x−φNi (L)Q˜N L(L)+φ1i(0)Q˜N L(0) +
φiN
(L)M˜N L(L)− φ1i
(0)M˜N L(0), (54)
where the arbitrary finite elements of the beam are denoted byVn =(xn−1,xn)and Vm =(sm−1,sm)and the numbers of finite elements byNandM, respectively. Also, Hermite shape functions are denoted byφni(x)(i=1, . . . ,4), the approximate trans- verse deflection byw(x)=4
j=1wnjφnj(x)as well as the generalized displacements bywj, respectively. Gauss quadrature rule is used for the integrals’ calculation of Eq.
(53) and Eq. (54) as well.N =M =100 is also selected for all the below investigated problems.
(a) Cantilever beam with a concentrated load at the middle of the beam The essential and the natural BCs of a cantilever beam subjected to a concentrated load at the middle of the beam are given by:
w(0)= dw d x
x=0=0 (55)
M¯(L)= ¯Q(L)=0, M(L)˜ = ˜Q(L)=0 (56)
Fig. 5 aNormalized transverse deflections of a cantilever beam, with a concentrated load applied to the beam’s middle, fore0a/L=0.03.bDetail of an area at the free end of the beam
The maximum classic-local transverse deflection takes the following value at the free end of the beam:wclmax=5P0L3/48E I.
In Figs.5and6, the responses of the nonlocal integral models present to be flexible in comparison with that of the classic-local model.
What is more, the transverse deflection of the nonlocal differential model is iden- tical to that of the classic-local model at the interval(0,L/2), yet it is smaller than that of the classic-local model at the interval(L/2,L). In other words, the beam’s stiffness increases in the interval ranging from the load point to the free end of the beam. It is critical to mention that the nonlocal differential and the nonlocal integral models present totally different results.
Furthermore, the deflection of the TPNI stress model exceeds that of the model of the modified kernel in Figs. 5and6. This behavior is expected by taking into consideration Fig.3. The difference between their values increases as the nonlocal parameter augments (Fig.6).
Based on Fig.6a, the deflections of the model of the modified kernel linearly increase when 0.01<e0a/L<0.03, but their increase rate reduces when 0.03<
e0a/L <0.04. The attenuation function widens as the nonlocal parameter increases.
Its area close to the boundary and ergo the information’s quantity decreases, yet the intervals in which the area of the attenuation function does not approach 1 increase (Fig.1b). Hence, the created gap is occupied by the locality, and this is obvious in Fig.6a.
(b) Cantilever beam with a concentrated load at the free end of the beam The essential BCs of a cantilever beam with a concentrated load applied to the free end of the beam are given by Eq. (55) and the natural BCs by:
M¯(L)= ˜M(L)=0, Q(L)¯ = ˜Q(L)=P0 (57) The maximum classic-local transverse deflection takes the following value at the free end of the beam:wclmax= P0L3/3E I.
Fig. 6 Normalized transverse deflections of a cantilever beam, with a concentrated load applied to the beam’s middle, for different values ofe0a/Lat pointsa0.25L,b0.5L,c0.75L,dL
Based on Figs. 7 and8, the responses of the nonlocal integral models appear to be flexible in comparison with that of the classic-local model. The transverse deflection of the TPNI stress model presents a monotonically increasing behavior as the nonlocal parameter augments.
What is more, the transverse deflection of the nonlocal differential model is iden- tical to that of the classic-local model (Figs.7and8). In other words, the nonlocal differential model is insensitive to the locality effect. It is worth mentioning that the nonlocal integral models appear to have a qualitatively different behavior from that of the nonlocal differential model.
Finally, the response of the model of the modified kernel of a cantilever beam with a concentrated load at the free end of the beam presents a similar behavior with that of a cantilever beam subjected to a concentrated load at the beam’s middle close to the fixed point (Fig.8a). This behavior is more intense than that presented in Fig.6a.
(c) Cantilever beam with a uniformly distributed load
The essential and the natural BCs of a cantilever beam subjected to a uniformly distributed load are given by Eqs. (55) and (56).
Fig. 7 aNormalized transverse deflections of a cantilever beam, with a concentrated load applied to the free end of the beam, fore0a/L=0.03.bDetail of an area at the free end of the beam
Fig. 8 Normalized transverse deflections of a cantilever beam, with a concentrated load applied to the free end of the beam, for different values ofe0a/Lat pointsa0.25L,b0.5L,c0.75L,dL
Fig. 9 aNormalized transverse deflections of a cantilever beam, with a uniformly distributed load, fore0a/L=0.03.bDetail of an area at the free end of the beam
The maximum classic-local transverse deflection takes the following value at the free end of the beam:wclmax=q0L4/8E I.
The responses of the nonlocal integral models present to be flexible in comparison with that of the classic-local model in Figs.9and10. In Fig.10a, the locality effect is evident for the model of the modified kernel. On the other hand, the transverse deflection of the TPNI stress model presents a monotonically increasing behavior as the nonlocal parameter augments.
The responses of the model of the modified kernel appear to have an extremum when e0a/L =0.03, since the locality’s contribution increases as the nonlocal parameter increases (Fig. 10a). The deflection of the model of the modified ker- nel takes larger values than that of the classic-local model irrespective of the above behavior.
Moreover, the deflection of the classic-local model exceeds the deflection of the nonlocal differential model. In other words, the nonlocal differential model stimulates the stiffness of the beam. It is noteworthy that the nonlocal integral models present a qualitatively different behavior from that of the nonlocal differential model.
(d) Clamped-clamped beam with a concentrated load at the middle of the beam
The essential BCs of a clamped-clamped beam with a concentrated load applied to the middle of the beam are given by:
w(0)= dw d x
x=0=0, w(L)= dw
d x
x=L
=0 (58)
The maximum classic-local transverse deflection takes the following value at the middle of the beam:wclmax= P0L3/192E I.
Fig. 10 Normalized transverse deflections of a cantilever beam, with a uniformly distributed load, for different values ofe0a/Lat pointsa0.25L,b0.5L,cL
On the basis of Fig.11, the responses of the nonlocal integral models appear to be flexible in comparison with those of the classic-local and the nonlocal differential models.
What is more, the transverse deflection of the model of the modified kernel, near to the fixed boundaries of the beam, takes smaller values than that of the TPNI stress model with bothk1=0.01 andk1 =0.1 (Fig.11b). At the same time at the middle of the beam, the response of the model of the modified kernel appears to have a flexible behavior compared to the responses of the other explored models (Fig.11c). Also, the model of the modified kernel is intensely affected by the locality added to the constitutive equations as the nonlocal parameter increases.
As regards the TPNI stress model, their deflections monotonically evolve with respect tok1-parameter and when the nonlocal parameterer increases (Fig.11b, c).
It is of importance that the nonlocal integral models do not appear to have a qualitatively different behavior from that of the nonlocal differential model.
(e) Clamped-clamped beam with a uniformly distributed load
The essential BCs of a clamped-clamped beam subjected to a uniformly distributed load are given by Eq. (58).
Fig. 11 aNormalized transverse deflections of a clamped-clamped beam, with a concentrated load applied to the beam’s middle, fore0a/L=0.03. Normalized transverse deflections for different values ofe0a/Lat pointsb0.25L,c0.5L
The maximum classic-local transverse deflection takes the following value at the middle of the beam:wclmax=q0L4/384E I.
In Fig.12, the responses of the nonlocal integral models appear to have a flex- ible behavior in comparison with that of the classic-local model. Furthermore, the transverse deflection of the nonlocal differential model is identical to that of the classic-local model. The deflections concerning the TPNI stress model monotonically evolve with respect tok1-parameter and when the nonlocal parameterer increases (Fig.12c, d).
Moreover, the deflections of the TPNI stress model with both k1=0.01 and k1=0.1, close to the fixed boundaries of the beam, take larger values than that of the model of the modified kernel (Fig.12c). At the same time at the middle of the beam, the response of the model of the modified kernel and the response of the TPNI stress model with k1=0.1 compete with each other (Fig.12d). By all the investigated models, the deflection of the TPNI stress model withk1=0.01 takes the largest values (Fig.12c, d).
What is more, the model of the modified kernel is intensely affected by the locality, which is added to the constitutive equations as the nonlocal parameter increases.
Fig. 12 aNormalized transverse deflections of a clamped-clamped beam, with a uniformly dis- tributed load, fore0a/L=0.03.bDetail of an area at the middle of the beam. Normalized transverse deflections for different values ofe0a/Lat pointsc0.25L,d0.5L
It is critical to mention that the nonlocal integral models present a qualitatively different behavior from that of the nonlocal differential model.
(f) Clamped-pinned beam with a concentrated load at the middle of the beam The essential and the natural BCs of a clamped-pinned beam with a concentrated load applied to the middle of the beam are given by:
w(0)= dw d x
x=0 =0, w(L)=0 (59)
M¯(L)= ˜M(L)=0 (60)
The maximum classic-local transverse deflection takes the following value at the point of the beamx=
1−√ 5/5
L:wclmax=√
5P0L3/240E I.
Based on Fig. 13, the responses of the nonlocal integral models present to be flexible in comparison with those of the classic-local and the nonlocal differential models. What is more, the transverse deflection of the model of the modified kernel,
Fig. 13 aNormalized transverse deflections of a clamped-pinned beam, with a concentrated load applied to the beam’s middle, fore0a/L=0.03. Normalized transverse deflections for different values ofe0a/Lat pointsb0.25L,c0.5L
close to the fixed boundary of the beam, takes smaller values than that of the TPNI stress model withk1=0.01 andk1=0.1 too (Fig.13b). At the same time at the middle of the beam, the response of the model of the modified kernel appears to have a flexible behavior compared to the responses of the other investigated models (Fig.13c). Besides, the model of the modified kernel is intensely affected by the locality added to the constitutive equations as the nonlocal parameter increases. The deflections regarding the TPNI stress model monotonically evolve when the nonlocal parameter augments (Fig.13b, c).
It is of significance that the behavior demonstrated by the nonlocal integral models is not qualitatively different from that of the nonlocal differential model.
(g) Clamped-pinned beam with a uniformly distributed load
The essential and the natural BCs of a clamped-pinned beam subjected to a uniformly distributed load are given by Eqs. (59) and (60).
The maximum classic-local transverse deflection takes the following value at the point of the beamx=(15−√
33)L/16:wclmax=(q0L4/E I)
39+55√ 33 65536
.
Fig. 14 aNormalized transverse deflections of a clamped-pinned beam, with a uniformly dis- tributed load, fore0a/L=0.03. Normalized transverse deflections for different values ofe0a/Lat pointsb0.25L,c0.5L
In Fig.14, the responses of the nonlocal integral models appear to have a flex- ible behavior in comparison with that of the classic-local model. Furthermore, the transverse deflection of the nonlocal differential model is identical to that of the classic-local model. As regards the TPNI stress model, its deflections monotonically evolve when the nonlocal parameter increases (Fig.14b, c).
Moreover, the deflections of the TPNI stress model with both k1=0.01 and k1=0.1, close to the fixed boundary of the beam, take larger values than that of the model of the modified kernel (Fig.14b). At the same time at the middle of the beam, the response of the model of the modified kernel and the response of the TPNI stress model with k1=0.1 compete with each other (Fig.14c). By all the investigated models, the deflection of the TPNI stress model with k1=0.01 takes the largest values (Fig.14b, c). The model of the modified kernel is also affected by the locality, which is added to the constitutive equations as the nonlocal parameter increases.
It is critical to mention that the nonlocal integral models present a qualitatively different behavior from that of the nonlocal differential model.
(h) Simply supported beam with a concentrated load at the middle of the beam The essential and the natural BCs of a simply supported beam with a concentrated load applied to the middle of the beam are given by:
w(0)=w(L)=0 (61)
M¯(0)= ˜M(0)=0, M¯(L)= ˜M(L)=0. (62) The maximum classic-local transverse deflection takes the following value at the middle of the beam:wclmax= P0L3/48E I.
In Fig.15, the responses of the nonlocal integral models present to be flexible in comparison with that of the classic-local model.
Moreover, the transverse deflection of the model of the modified kernel takes larger values than those of the other models that studied. A extremum point is demonstrated in Fig.15c as well. The extremum point is caused by the fact that the modified kernel adds locality to the gap created by the nonlocality on the boundary. The effect of the
Fig. 15 aNormalized transverse deflections of a simply supported beam, with a concentrated load applied to the beam’s middle, fore0a/L=0.03.bDetail of an area at the middle of the beam.
Normalized transverse deflections for different values ofe0a/Lat pointsc0.25L,d0.5L
modified kernel is therefore greater on points close to the boundary. Similar results are deduced for the case of a cantilever beam.
Besides, the deflections of the TPNI stress model monotonically evolve with respect tok1-parameter and when the nonlocal parameterer increases in Fig.15c, d.
The nonlocal deflection additionally tends to that of the classic-local model because of the augmentation ofk1-parameter.
The transverse deflection of the nonlocal differential model also appears to have a flexible response in comparison with that of the classic-local model. It is of signif- icance that the nonlocal integral models present a qualitatively similar behavior to that of the nonlocal differential model.
(i) Simply supported beam with a uniformly distributed load
The essential and the natural BCs of a simply supported beam subjected to a uniformly distributed load are given by Eqs. (61) and (62).
The maximum classic-local transverse deflection takes the following value at the middle of the beam:wclmax=5q0L4/384E I.
Based on Fig.16, similar conclusions are drawn for the nonlocal models as in case of a concentrated load.
Fig. 16 aNormalized transverse deflections of a simply supported beam, with a uniformly dis- tributed load, fore0a/L=0.03.bDetail of an area at the middle of the beam. Normalized transverse deflections for different values ofe0a/Lat pointsc0.25L,d0.5L
The model of the modified kernel presents a flexible behavior in comparison with the other investigated models. When e0a/L >0.03, however, the aforementioned model is affected by the locality more than the corresponding model in case of a concentrated load.
In addition, the deflections of the TPNI stress model monotonically evolve with respect tok1-parameter and when the nonlocal parameterer increases in Fig.16c, d.
The nonlocal deflection also tends to that of the classic-local model because of the augmentation ofk1-parameter.