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 2 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 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 METHOD## 2.1. 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 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 Where: Is
the mass density Is the differentiation of displacement
with respect to time Is
the matrix format of the gradient vector operator at the deformed configuration
. 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 to
eqn. (3) and integrating over the domain V (which is the current configuration)
(Kim 2018; De Borst 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
having components (,,) is regarded as the unknown in the
assumed continuous displacement field ( Where: is
the shape or interpolation polynomial function expressed in natural
dimensionless coordinates . 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 ( L 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:
is the matrix vector of the displacement
degree of freedom at the nodal points of each element in the finite element
mesh. While
eqn. (9) maps the element displacement vector ( )
to the global displacement vector () using the incidence matrix ( )
(De Borst 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:
Is
the internal force vector Is the matrix that relates the strains
within the finite elements to the nodal displacements expressed as the
derivative of the shape function as given in eqn. (16) (The above definition
for holds for a linear expression, for a
non-linear expression, it relates the variations of strains within the finite
elements to the variations of the nodal displacement) (De Borst The
shape function has been expressed in dimensionless
isoparametric natural coordinates in order to facilitate the numerical
integration of the integrals where complex non-linear structural geometries as
well as material non-linearities are involved (Stein, 1993; De Borst Isoparametric
finite elements have been used for this formulation as the same shape function given in eqn. (17) has been used for both
the geometrical and displacement interpolations. Where:
equation 17 is a linear relationship in isoparametic coordinates are the shape functions gives all the finite element nodal points are the nodal point displacements 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 as in eqn. (11) is done incrementally,
that is, in load steps as opposed to the sudden application of the full
external load in one single load step (Stein, 1993). This
is because the set of algebraic equations which evolves as a result of finding
the solution to the finite elements is non-linear. Similarly, the application
of load incrementally enables the proper convergence of the solution (De Borst
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: Is when the stress component is known at
the time t
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 () where the integral in eqn. (20) applies
is not known at the time 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 to
time Hence
equation (28) can be rewritten as: orwith the external force vector additively
decomposed and expressed as in eqn. (29.1) 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: is the determinant of the Jacobian used
in the transformation between the natural isoparametric coordinates () and the corresponding Cartesian
coordinates () (see figure 1)?
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 evaluated using the stress from the first iteration (as given in
equation 33) is not in equilibrium with the external loads Is
the updated tangential stiffness matrix
Is the residual force 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.
## 3.
NUMERICAL EXAMPLE OF A
STEEL BEAM SUBJECTED TO LOW-SPEED IMPACT LOAD
A
numerical simulation of a 30m long steel beam (UB with a thickness of) subject to high-speed impact based on
the non-linear theoretical framework discussed above was carried out to
investigate the deformation and rotation capacity of the steel beam under the
impact. 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) loading where the unloading
cycle appears to be less than I/10000
## 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) and speed of loading as shown in
fig. 3, it shows that the beam is able to respond in a rather ductile manner to
the impact loading of a rigid impactor dropped from a height of However, from the deformed shape of the beam
element after the impact load, it is considered to have failed as its structural
integrity is lost due to excessive twisting of the beam. Similarly,
from the initial drop as shown in fig. 2.1, the beam rotated by and for the rebound displacement the rotation
was which is clearly over the maximum limit to
avoid any secondary hazard. Furthermore, the rotation at first mode was which is clearly above the maximum rotation
value of for steel structures to prevent collapse.
Implying that the beam has failed.
## 4.1. ANALYTICAL COMPARISMFrom the
above numerical simulations, the rigid impactor was only in contact with
the beam for a short period of time followed by a rebound of the beam due to
action of elastic interface restoring force (Aliyu 2019). This according to
Stronge (2018) can be classified as an elastic impact. Also form the Newtons
experimental law of impact, the coefficient of restitution for an elastic
impact is assumed to be 1 (Stronge, 2018). Hence, the kinetic energy for the
above elastic impact with a coefficient of restitution 1, according to Mughal
and Smith cited in Aliyu (2019), is evaluated using the following expression While
the maximum vertical downward displacement as
well as the available strain energy was evaluated using the following expressions 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. This closely matched the initial
vertical downward displacement from the Abaqus simulations which was found to
be (see fig. 4) ## 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 after initial impact. This was found to agree with the Abaqus vertical downward deflection of after initial impact. Although, the deflection appeared to be within the acceptable limits, the beam was considered to have failed as the rotation limits for safe design had been exceeded. It has turned out that this displacement-based method of non-linear finite element analysis as well as the energy momentum balance technique presented in (Aliyu, 2019) are viable for use practically as it is able to predict with accuracy the behaviour of structures subject to impact load (which is a non-linear dynamic problems) at a very reasonable cost. ## 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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