Article Citation: O.H. Aliyu, and A.A
Salihu. (2021). NON-LINEAR FINITE ELEMENT ANALYSIS OF DYNAMIC PROBLEMS. International
Journal of Engineering Technologies and Management Research, 8(4), 79-93. https://doi.org/10.29121/ijetmr.v8.i4.2021.926 Published Date: 29 April 2021 Keywords: Nonlinear
Analysis Material
Non-Linearity Geometric
Non-Linearity Linearization Joint Rotation A formulation of the displacement based finite element method as well as the incremental analysis procedure which is considered suitable for analysis of non-linear dynamic problems is presented. The presented framework is used to investigate the influence of joint rotation on the failure of steel beam subject to high-speed impact load. The results from the non-linear numerical simulation are compared with those obtained from an analytical technique. Method: The non-linear Full Newton Raphson method was used for the simulation and results obtained were verified analytically using the energy momentum balance technique. Results: The beam suffered an initial vertical downward deflection of 27.7mm from the impactor load as well as a joint rotation of 20. Findings: From the results obtained the beam was considered to have failed due to excessive rotation. Similarly, from the comparism made between the analytical and non-linear numerical simulation results, it was concluded that the full Newton Raphson technique gave accurate results in simulating the dynamic problem which was achieved at an affordable cost.
1. INTRODUCTIONThe
analysis of damage in materials and structures subjected to dynamic loadings
such as blast, impacts, earthquake, fire and so on, is considered crucial. For
such dynamic loadings (generally involving plasticity and damage), the complete
displacement form of finite element analysis is mostly used especially where
material non-linearity is considered. This displacement version of finite
element analysis is generally easier to implement particularly for complicated
non-linear constitutive relations. It also offers the advantage of properly
modelling the element behaviour (Stein, 1993; De
Borst et al, 2012; Aliyu, 2019). In this
paper, the non-linear finite element method is applied to the material’s
non-linear problem involving a steel beam subjected to impact loads. The method
was used to study the deformation and rotation of the beam to the dynamic
impact loads. The accuracy of the non-linear finite element simulation results,
in assessing deformation was compared to those obtained from comparable hand
calculations and has proved to be quite satisfactory. 2. FORMULATION OF THE NON-LINEAR FINITE ELEMENT METHOD2.1. FORMULATION OF THE WEAK FORM OF THE EQUATION OF MOTIONIn
arriving at the non-linear finite element expression for solving dynamic
problems we begin by adopting the concept of equilibrium and virtual work in
obtaining an expression for the equation of motion where the idea of linear
momentum balance is assumed. First, we
consider the balance of momentum between the body V as well as its boundary S
with the stress vector t and gravity
acceleration put together in the vector g
resulting in the linear momentum balance expressed as: The
above expression momentum balance can be further adjusted to give the
expression as shown in eqn. (1.1) Now
applying the Gauss divergence theorem to the first expression in eqn. (1.1)
converts it from a surface integral to a volume integral expressed in eqn.
(1.2) after rearrangement. The
integrand in eqn. (1.2) must be equal to zero which gives the local form of the
balance of linear momentum also called the equilibrium equation or the equation
of motion in the strong form which is thus expressed
in eqn. (1.3) (Laursen,
2003; De Borst et al, 2012): Where:
By inserting eqn. (2) in eqn.(1) the equation of motion can
be rewritten in matrix form as In
adopting the principles of virtual work, equation (3) is further transformed
into a weak form as given in eqn. (5) after applying the divergent theory to
eqn. (4) which was arrived at by multiplying by a virtual displacement It
can therefore be seen from equation 5 that the weak form is a balance between
the internal virtual work and the external virtual work (i.e., the body forces
and surface traction) (Kim, 2018). It is important to stress that, in the
arriving at the expression in equation 5, no guess was made with respect to the
material behaviour or size of the spatial displacement gradients. Hence,
equation 5 can be used for both linear and non-linear behaviours. 2.2. DISCRETIZATION; FINITE ELEMENT FORMULATION
In
this section the displacement based finite element formulations are
discussed. This section starts with
finding the approximate solution to the above weak form of equation of motion.
For this method, the element nodal displacements Where:
In
order to evaluate the displacement field, for points within the elements which
are not key points (i.e., not nodal points) eqn. (7) is adopted. The
element strain, given in terms of the derivative of the displacement field (u) is obtained from eqn. (7.1) (Laursen, 2003; De Borst et al, 2012; Hartmann, 2005) It
should be noted that the measure of strain tensor to be used in equation 7.1
has not been specified as the equation is expected to be used in both linear
and non-linear regime as mentioned above in the formulation of the weak form of
the equation of motion.
Where:
While
eqn. (9) maps the element displacement vector ( Substituting
eqns. (7) and (9) into the weak form of equation (5) gives equation of motion
for the entire finite element mesh expressed as shown in eqn. (10): Solution
of equation (10) for every virtual displacement ( Where:
The
shape function Isoparametric
finite elements have been used for this formulation as the same shape function Where:
equation 17 is a linear relationship in isoparametic coordinates
2.2.1. LINEARIZATION Nonlinear
problems almost always result in nonlinear equations which have to be solved by
first linearizing the nonlinear equations and then solving the linearized
equations (i.e., getting the roots of the equation) iteratively using
appropriate techniques (such as: Newton Rahpson technique, Arc length or path
following method as well as the line search method) until the solution to the
problem is obtained. On the other hand, the linearized equations can be solved
directly using some other solution technique, which does not require iterative
solution method. The focus here is on
iterative solution technique using the Newton Rahpson method where correct
linearization of the nonlinear equation is key to getting an accurate solution
by ensuring quadratic convergence. 2.3. INCREMENTAL ANALYSIS
For
the non-linear analysis, the application of external load
Where
the inertial term is omitted. For
loading to be applied sequentially, the time parameter is used to order the
sequence of loading (that is the load steps or increments) (Laursen, 2003). In line with this preceding statement,
the unknown stress vector Where:
Using
eqn. (19) in eqn. (18) above with eqn. (15) gives the expression in eqn. (20) Where,
the second term in the right side of eqn. (20) above represents the internal
virtual work done. Since the volume of the element ( Equation
(20) above can equally be written as in eqn. (21) since we are concerned here
with linearizing the source of geometric non-linearity (i.e., the nodal
displacements) and removing the dependence on it. Equation (21.1) is the
unbalanced force at the nodes: Where:
Because
of the non-linear nature of equation (21) its solution necessitates the need
for an iterative approach. To achieve this, the non-linear equation must first
be linearised and then solved iteratively. The most frequently used iterative
method is the Newton-Raphson method which entails working out the residual as
well as the tangent stiffness matrix for each iteration from the weak form (Wang & Hsu, 2001; Ayoub & Filippou 1998; Hartmann,
2005; Tsavadaris, & Mello, 2012; Barth, & Wu, 2006; Kim 2018). As
mentioned before, the solution of non-linear equations, entails iterative
linearization of the governing equations.
For example, linearizing the dependence of the stress increment to the
displacement increment expressed as shown in eqn. (23) remembering of course
that no guess has been made
with respect to the material behaviour (i.e., the stress and strain measure) as
mentioned earlier on page 4 (Laursen, 2003; De
Brost et al.2012).
The increase in stress is expressed as in eqn. (22) which is then linearized as
expressed in eqn. (23) With
Where:
Equation (23) can therefore be rewritten as: (Laursen,
2003; De Brost et al.2012; Kim, 2018). With
the quasi-static loading, eqn. (18) written as: This
can be further expressed as in eqn. (27) Where:
Remembering
of course that no
assumptions have been made here, with respect to the nature of the stress or
strain measures. Considering
equations (7), (9), (16) and (25), in the above non-linear expression of eqn.
(21) leads to the linearized equation for finite load increment which is given
in eqn. (28)
With
the loading steps, ranging from time Hence
equation (28) can be rewritten as: or These
linearizations of the load steps as given in eqn. (29) as well as the
linearization of the governing stress-strain equation given in (25) tend to
move the solution away from the actual equilibrium solution (especially when
the initial estimate of the assumed displacement is too far from the actual
solution). This divergence can be reduced by incorporating equilibrium
iterations inside each loading step. With this incremental-iterative solution
technique, the initial displacement estimates are derived as follows: Where: The
subscript 0 signifies the beginning steps while subscript 1 refers to the first
iteration. Hence, the beginning internal force is expressed as: Where:
Figure 1: shows the concept of transformation
between isoparametric and Cartesian coordinates adapted from Stein 1993. The
stress As
initially mentioned, there is usually a divergence of the actual true solution
from the equilibrium solution which is why the original force vector
Following
the pattern of equations (34) and (35) higher iterations can be worked out
after the convergence of solution is attained for each lo
Where the last three steps in eqn. 36 are usually
performed for each integration point.
The
above figure shows the improvement in numerical results from the pure
incremental solution technique to incremental-iterative solution technique
where equilibrium iterations has been applied to each loading step. Fig. 2.1
shows a detailed graphical description of how the displacement corrections are
added for a single load step while table 1 gives the algorithm of the
incremental-iterative procedure. Table 1: Computational flow in a non-linear
finite element code adapted from De Brost et al.2012 3.
NUMERICAL EXAMPLE OF A
STEEL BEAM SUBJECTED TO LOW-SPEED IMPACT LOAD
A
numerical simulation of a 30m long steel beam (UB For
this finite element (F.E) simulation, the full Newton-Raphson method was used.
The material parameters are as follows: For
steel: · Young’s Modulus of elasticity; ·
Poisson
ratio 0.3 · Yield stress ·
Density 8E-9 For
rigid impactor: ·
Young’s
Modulus of elasticity; ·
Poisson
ratio 0.2 · Density The
displacement history time graph represents the true characteristic nature of
dynamic behaviour for a high frequency (33 Figure 3: Time history graph Figure 4: Deformation after initial drop Figure 4.1: Deformation after
initial drop Figure 4.2: Rebound displacement 4. NUMERICAL SIMULATION After the
initial displacement as shown in fig.4 and fig. 4.1 above and the rebound
displacement of the beam as shown in fig. 4.2, the beam continued to vibrate.
The amplitude of the vibration reduced to zero in less than a second as can be
seen from fig. 3. This suggests that the structure is overdamped as the
displacement had decayed rather exponentially. The vibration of the structure
after impact is as shown in figs. 5 to 7. From the frequency (33 Similarly,
from the initial drop as shown in fig. 2.1, the beam rotated by Figure 4.3: Mode shape 1 rotation Figure 6: Mode shape 2 Figure 7: Mode shape 3 4.1. ANALYTICAL COMPARISMFrom the
above numerical simulations, the While
the maximum vertical downward displacement The
maximum allowable vertical downward displacement was equally evaluated using
the following expression From
the above expressions, the maximum vertical downward deflection as well as the
allowable maximum vertical downward deflection were both found to be 5.
CONCLUSION
In
a pursuit to analyse non-linear dynamic problems the framework for the
displacement based finite element method was introduced where material and
geometric non-linearity had been considered. Similarly, the incremental
analysis procedure for non-linear problems have been discussed. The accuracy of
the results of deformation obtained from this non-linear finite element
analysis was compared to those obtained from the analytical technique using the
energy momentum balance technique as presented by Mughal and Smith cited in
Aliyu (2019) which gave a maximum vertical downward deflection of SOURCES OF FUNDINGThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. CONFLICT OF INTERESTThe author have declared that no competing interests exist. ACKNOWLEDGMENTNone. REFERENCES [1] Aliyu, OH. Design Recommendations for Steel Beams to
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