non linear project
TRANSCRIPT
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Nonlinear Structural Analysis Project
1. The two degree of freedom perfect structure shown below consists of three rigid links and
two elastic rotational springs, and is to be considered under the depicted load case. The
spring deformations and the vertical nodal displacements can be expressed to a second-
order in terms of the two system parameters as:
2 2
1 1 2
1 2
3U 127 U L Uv ; v
4 128L 2= + = +" "
2 2 2 2
1 1 2 1 1 2I II 22 2
7 U 163U U 3U 127 U Ud ; d U
4L 2 4 L 2128L 128L= + + = + + +" "
0P0P
0P75.0
0.75L L
L
k k
4
0
k 4 10 N.m / r ad
L 4 m
P k / L
=
=
=
1U
2U
(I) (II)
[1]
[2]
[3]
1v 2v
a. Show that the structure exhibits a trivial fundamental equilibrium path.
b. Show that the geometric stiffness matrix along the trivial path is identically obtained
from the virtual work method and from the rotational spring analogy. What is therelative benefit of the rotational spring analogy?
c. Determine the lowest buckling load factor, and sketch the corresponding buckling
mode.
d. Employ ADAPTIC to validate your buckling load and buckling mode predictions,
making use of the supplied data file frame_sample.dat, and considering an
imperfection of L/1000 in the location of the roller support.
e. Compare using ADAPTIC the post-buckling responses associated with positive and
negative imperfections, and comment on any differences. Classify the buckling
behaviour with regard to symmetry and stability.
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2. Consider the elastic buckling of a perfect column, as depicted below, which is to be
modelled using nonlinear finite elements.
L/2 L/2
2
EIP
L=
1U
2U 3U
a. Given that along the fundamental path the element end moments are zero and the
element shortening may be ignored, assemble the terms of the overallE
[K ] andG
[K ]
related to1 2 3
U U U using two cubic elements accounting for local geometric
nonlinearity.
b. Demonstrate that the tangent stiffness matrix is positive definite for the theoretical
lowest buckling load factor (c
20.19 = ). What conclusions can you draw from this?
c. Estimate the lowest buckling load factor from the finite element model using thetheoretical buckling mode expressed in terms ofT
1 2 3U U U :
T
t
1.8 5.6{ } 1.3
L L =
Confirm that the buckling load factor obtained from the assumed mode overestimates
the actual finite element solution. What conclusions can you draw regarding the
buckling load factor predicted by the finite element model?
d. Employ ADAPTIC to verify the predictions of the finite element model, making use of
the supplied data file column_sample.dat, and considering an imperfection of
L/1000 represented by an axial load eccentricity at the roller support.
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3. An elasto-plastic cantilever with a steel I-section is modelled with ADAPTIC using the
cubic beam-column element, which is based on two Gauss stations and cross-sectional
discretisation. The cantilever is initially modelled with one cubic element using a bilinear
material model ignoring the effects of strain-hardening, where the applied load is assumed
to increase to 10 times the plastic bending capacity. A data file cant_sample.dat is
provided for this initial model.
L
pMP
L= 5
p
6
p
L 4 m
M 2.923 10 N.m
F 2.745 10 N
=
=
=
a. Considering the results from ADAPTIC for the last load step, extract the local element
forces and displacements from the output file cant_sample.num. Show that the
extracted local element displacements are obtained correctly as the transformation of
the global nodal displacements.
b. Determine the local element forces for the local displacements of (a) considering a
cubic element with two Gauss stations based on explicit interaction between the
generalised stresses, as given by:
g g
g g
p p
F M(F ,M ) 1 0
F M = + =
and assuming zero initial plastic strains.
Compare the determined local forces against those of the ADAPTIC element based on
cross-sectional discretisation as obtained in (a).
c. Use ADAPTIC to establish the minimum number of equal length elements required to
predict the plastic bending capacity and the final tip displacements to within 5%.
Comment on the sensitivity of the solution to the extent of cross-sectional discretisation
considering at least 6 monitoring areas.
d. Use the last element mesh in (c) to obtain the cantilever response for steel with a strain-
hardening parameter of ( 0.02 = ). Comment on the difference between the perfectly-
plastic and strain-hardening structural responses.
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4. An imperfect four-storey steel frame is to be analysed
under earthquake loading, taking into account the effects
of material plasticity. For this purpose, the adaptive
element of ADAPTIC (qdp2) is to be employed, which is
an elastic quartic element subdividing into a mesh pattern
of elasto-plastic cubic elements (cbp2)1 only when and
where necessary. This enables the structure to beaccurately modelled with one qdp2 element per member.
a. A data file 4storey_stat_sample.dat is provided
for analysing the frame under static vertical loading
allowing for material nonlinearity. Modify this file as
appropriate and use ADAPTIC to i) estimate the
elastic buckling load using Hornes formula by using
( r 0.001= ) with a nominal load ( 5P 10 N= ), ii)
compare this to the elastic buckling prediction from
ADAPTIC (by setting the yield strength to very high
value), and iii) establish the elasto-plastic buckling
load.
b. Use the data file 4storey_freq_sample.dat to
obtain the natural frequencies for the first 5 modes2
of
vibration, considering lumped masses at the points of vertical loading. Specify in the
data file masses corresponding to half of the elasto-plastic buckling load determined in
(a), allowing for the acceleration of gravity ( 2g 9.81m / sec= ).
c. Consider the elasto-plastic dynamic analysis of the frame under an earthquake
acceleration record specified in file earthquake1, scaled by the acceleration of
gravity (g). Select a suitable time step in relation to the 3rd
mode of vibration, and use
the supplied data file 4storey_earth_a_sample.dat, modifying the masses similar
to (b) as well as the corresponding initial vertical load. Plot the relative drift3
and
vertical deflection of the top floor against time, and discuss the obtained results.Investigate the effect of halving the time step on this prediction, commenting on the
significance of the outcome in relation to the achieved accuracy. Plot the final deflected
shapes, showing element types qdp2 and cbp2 in different colour, and highlighting the
significance of the different corresponding element regions.
1Elements of different type can be shown in different colours with adaptic s by clicking Crtrl+H.
2The mode shapes can be plotted with adaptic s, by setting Labels on after Ctrl+G, and specifying a large
displacement scale after Ctrl+D.3
Relative drift in this case can be obtained as the difference between the X displacement at the support and that
of the 4th
floor, which may be determined using Arithmetic Expressions after Ctrl+N with adaptic g.
P
P
P
P/2
P
P
P
P/2
6 m
4 m
3 m
3 m
3 m