Test No. NVL04 Find reactions at the fixed ends and
maximum displacement of a bar axially loaded beyond plasticity.
Definition
Figure 1.
Bar dimensions are 10 x 10 x 200 mm. Distance between loaded point and left end A=50
mm. Strain-stress curve of the bar material is defined by the power
law:
(1)
σ
=
K
ε
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
Gaeq4WdmNaeyypa0Jaam4saiabew7aL9aadaahaaWcbeqaa8qacaWG
Ubaaaaaa@3C8B@
Where,
K
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
Strength coefficient
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
Must be in the range [0,1]
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
=0
Material is perfectly plastic.
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
=1
Material is elastic.
The material properties are:
Properties
Value
K
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
530 MPa
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
0.26
Poisson's Ratio
0
Figure 2. Corresponded strain-stress curve
The study was performed for the following load F values: 30000 N, 47000 N, 55000 N,
and 60000 N. These loads cover the full range of elastic-plastic response of the
bar.
Reference Solution
One-dimensional analytical reference solution is described here.
The length of the bar does not change under the load.
(2)
∫
0
A
ε
1
d
x
−
∫
0
L
−
A
ε
2
d
x
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
WaaybCaeqal8aabaWdbiaaicdaa8aabaWdbiaadgeaa0WdaeaapeGa
ey4kIipaaOGaeqyTduMaaGymaiaadsgacaWG4bGaaiiOaiabgkHiTi
aacckadaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaamitaiabgkHi
Tiaadgeaa0WdaeaapeGaey4kIipaaOGaeqyTduMaaGOmaiaadsgaca
WG4bGaaiiOaiabg2da9iaacckacaaIWaaaaa@5048@
or,
(3)
∫
0
A
N
/
(
K
*
A
)
n
d
x
−
∫
0
L
−
A
(
F
−
N
)
/
(
K
*
A
)
n
d
x
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
WaaybCaeqal8aabaWdbiaaicdaa8aabaWdbiaadgeaa0WdaeaapeGa
ey4kIipaaOWaaOqaa8aabaWdbiaad6eacaGGVaWaaeWaa8aabaWdbi
aadUeacaGGQaGaamyqaaGaayjkaiaawMcaaaWcpaqaa8qacaWGUbaa
aOGaamizaiaadIhacaGGGcGaeyOeI0IaaiiOamaawahabeWcpaqaa8
qacaaIWaaapaqaa8qacaWGmbGaeyOeI0Iaamyqaaqdpaqaa8qacqGH
RiI8aaGcdaGcbaWdaeaapeWaaeWaa8aabaWdbiaadAeacqGHsislca
WGobaacaGLOaGaayzkaaGaai4lamaabmaapaqaa8qacaWGlbGaaiOk
aiaadgeaaiaawIcacaGLPaaaaSWdaeaapeGaamOBaaaakiaadsgaca
WG4bGaaiiOaiabg2da9iaacckacaaIWaaaaa@5C73@
Where,
ε
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
GaeqyTduMaaGymaaaa@386E@
Tensile strain at the left span of the bar.
ε
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
GaeqyTduMaaGymaaaa@386E@
Compressive strain at the right span of the bar,
N
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
Reaction force at left end of the bar.
R
=
F
−
N
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGsbGaeyypa0JaamOraiabgkHiTiaad6eaaaa@3D1F@
Reaction force at the right end of the bar.
A
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGlbaaaa@3987@
Bar cross-section area
From this equation you can find the reaction at the left end of the
bar.
(4)
N = F / (
1 +
(
a / b
)
n
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
GaamOtaiaacckacqGH9aqpcaGGGcGaamOraiaac+cadaqadaWdaeaa
peGaaGymaiabgUcaRmaabmaapaqaa8qacaWGHbGaai4laiaadkgaai
aawIcacaGLPaaapaWaaWbaaSqabeaapeGaamOBaaaaaOGaayjkaiaa
wMcaaaaa@4461@
and
R
=
F
−
N
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape
GaamOuaiaacckacqGH9aqpcaGGGcGaamOraiabgkHiTiaad6eaaaa@3CBC@
at the right end.
Results
Bar was modeled as a 3D solid with immovable ends. Axial force F could not be applied
precisely at the solid bar axis, so four line spots were created at the bar sides
and total load F was uniformly distributed over the spots (
Figure 3 ).
Figure 3.
The following table summarizes the reaction force results.
Force F [N]
SOL Reference, Reaction
[N]
SimSolid , Reaction [N]
% Difference
30000
17128
18151
5.97%
47000
26834
27146
1.16%
55000
31401
31788
1.23%
60000
34256
34591
0.98%
Typical von Mises stress distributions are shown in
Figure 4 and
Figure 5 . The distribution has high gradients at load
application lines; yet the reactions values correlate to the 1D solution because the
reactions are applied far from the active force.
Figure 4. von Mises stress distribution at load F=30000
N
Figure 5. von Mises stress distribution at load F=60000
N