This example shows how to perform inertia relief in a static
analysis in
Abaqus/Standard.
The example involves stopping a pick-up truck, moving with an
initial velocity of 50.0 km/h (13.89 m/s), by applying braking loads. Inertia
relief is used here to supply inertia forces in a static analysis that oppose
the braking loads specified in the model. The solution provides the rigid body
deceleration and the static stresses in the pick-up truck. For comparison
purposes a dynamic analysis is performed with the same initial velocity and
braking loads.
A 1994 Chevrolet C1500 pick-up truck (see
Figure 1)
is modeled using approximately 55,000 elements. The model was obtained from the
Public Finite Element Model Archive of the National Crash Analysis Center at
George Washington University. The finite element model was converted into an
Abaqus/Standard
input file, and several missing constraints were added to carry out the
analyses. The model consists of various parts—such as cabin, truck bed, doors,
etc.—which are meshed with shell elements, three-dimensional beam elements, and
three-dimensional solid elements. The parts are attached with connector
elements, coupling elements, and multi-point constraints.
The materials used in the truck model are idealized as elastic or
elastic-plastic. Suitable adjustments are made to the material properties to
account for unmodeled features of various parts such as the internal details of
the engine, gearbox, etc. A summary of the material properties and the parts
for which they are used is given in
Table 1
and
Table 2.
Rigid body definitions are used for brakes and brake assemblies to take
advantage of the high stiffness of these parts relative to other parts.
Connector elements are used to model kinematic constraints governing relative
motions between various parts (see
Substructure analysis of a pick-up truck model
for details).
The finite element model of the truck is oriented such that the positive
1-direction goes from the rear to the front of the truck, the positive
2-direction goes from the passenger (right-hand) side to the driver
(left-hand) side, and the positive 3-direction is upward. In this system the
braking loads are applied at the respective wheel spindles as concentrated
forces in the negative 1-direction.
To simplify the analysis, normal contact between the tires and the road
surface is modeled through spring elements that have one node connected to the
wheel spindle and the other node fixed against displacement in the 3-direction
and kinematically constrained to the wheel spindle in the other directions.
Friction between the tires and the road surface is assumed to be nonexistent.
This allows the truck to translate freely in the 1- and 2-directions and rotate
freely about the 3-direction; the constraints on the spring nodes prevent
translation in the 3-direction and rotation about the 1- and 2-directions.
Loading
A separate static analysis is performed to obtain the correct initial
configuration and stress distribution under the applied gravity load. The
details of this analysis are explained in
Substructure analysis of a pick-up truck model.
This gives us the base state for the analysis of interest.
The total braking load for the truck moving at 13.89 m/s is computed by
assuming the truck to be a rigid body that comes to rest over a distance of 20
m after the brakes are applied. This gives a deceleration of 4.82
m/s2 in the 1-direction for the truck. The total mass of the truck
as computed from the finite element analysis is 1.72 × 103 kg, which
gives the total inertial force resisted by the brakes (or braking load) as 8.30
kN. Assuming that the front brakes provide 75% of the total resistance and the
rear brakes provide the remaining 25%, the braking load for each of the front
wheels is 3.11 kN and the braking load for each of the rear wheels is 1.04 kN.
The four braking loads applied to the truck are balanced in a static analysis
with an inertia relief load. The inertia relief load represents the dynamic
effects (not modeled otherwise in a static analysis) of a constant deceleration
from the truck's travel velocity to a complete stop.
Since the truck is free to translate in the 1- and 2-directions and rotate
about the 3-direction, inertia relief is performed in these three directions.
The other directions are constrained by boundary conditions as explained in the
previous section.
For comparison purposes a transient dynamic analysis is also performed
(after the initial static equilibrium under gravity load) in which the truck is
accelerated from zero velocity to the final uniform velocity of 13.89 m/s. This
dynamic analysis step is followed by another dynamic analysis step in which the
braking loads are applied to bring the truck to a complete stop. The braking
loads are ramped up smoothly from zero to the maximum value over 0.5 seconds
and then kept constant for 2.88 seconds—the time required to bring the truck to
rest from the initial velocity of 13.89 m/s with an average deceleration of
4.82 m/s2. To minimize the analysis time, substructures are used in
the dynamic analysis for all deformable parts except the chassis and suspension
components, which are modeled as fully deformable since they are the parts that
show significant stresses.
Results and discussion
The results for inertia relief in the pick-up truck model with braking loads show that the truck
decelerates at 4.83 m/s2 in the 1-direction. The truck has an angular
acceleration of 0.01 rad/s2 about the 3-direction at the center of mass due to
asymmetry in the distribution of mass. The vertical displacements at the wheel spindles (the
front wheel spindles dip about 0.7 mm, and the rear wheel spindles rise about 0.7 mm without
loss of contact between tires and the road surface) indicate that the truck pitches forward
due to the braking action. A plot of the von Mises stress shown in Figure 2 indicates that the largest stresses occur in the suspension components and the regions
where the suspension components are connected to the chassis. Plots of the active yield flag
and equivalent plastic strains (not shown) indicate that there is no plastic yielding in any
part of the truck.
The results for the transient dynamic analysis indicate that the average deceleration after the
full braking load has been applied is around 4.94 m/s2 in the 1-direction and the
average angular acceleration about the 3-direction is 0.03 rad/s2. The truck
pitches forward with the front wheel spindles dipping about 0.7 mm and the rear wheel
spindles rising about 0.7 mm in the braking load step. The von Mises stress for the dynamic
analysis, shown in Figure 3, shows a distribution similar to that obtained for inertia relief. There is no plastic
yielding in the chassis or suspension components.
Inertia relief relies on the assumption that the body undergoing loading is
free to translate and rotate as a rigid body. Therefore, no external or
internal constraints are allowed in the free directions (with the exception of
the case where statically determinant boundary conditions are applied and all
available directions are considered inertia relief directions). In a complex
model like the pick-up truck, with various kinematic constraints and large
geometry changes, it is necessary to ensure that the base state for the step
including inertia relief is converged to a tight residual tolerance. If it is
not converged to a tight tolerance, the out-of-balance forces and moments in
the base state will act as internal constraints on rigid body motions. Hence,
such unequilibrated forces and moments may prevent a geometrically linear or
nonlinear analysis from converging. In this example the solution controls
tighten the convergence tolerance in the gravity load step preceding the
inertia relief step.
The comparison of results for inertia relief and dynamic analysis of the
truck shows that inertia relief is an inexpensive alternative to dynamic
analysis for obtaining the steady-state response of a dynamic system for
certain loading situations. In this braking analysis, for example, the static
analysis with inertia relief runs about 10 times faster than the general
deformable transient dynamic simulation.
Static equilibrium analysis for gravity load and initial stresses followed
by dynamic acceleration to uniform velocity and deceleration with braking
loads.
Table 2. Properties for elastic materials used in the truck model.
Elastic Material Name
Properties
Part Name
E
(N/m2)
(kg/m3)
Steel
2.1 × 1011
0.3
7.89 × 103
A-arm mountings, fan, door lock beams, headrest
connector beams, radiator mounting beams, oil pan beams, rear axle, drive
shaft, steering, A-arm-rim connectors, A-arm-rail connectors, bed-rail
connector, dashboard support, steering column, rail connector, brakes, gearbox
CV joint, front rim
Steel
1.2 × 1011
0.3
3.89 × 103
engine gearbox
Steel
2.1 × 1010
0.3
1.82 × 103
engine front
Steel
2.1 × 1011
0.3
2.5 × 103
door lock
Rubber
2.461 × 109
0.323
8.0598 × 103
tires
Steel
2.1 × 1011
0.3
3.5765 × 103
rear suspension
Steel
2.1 × 1011
0.3
2.089 × 104
brake assembly
Steel
2.1 × 1011
0.3
6.911 × 103
brake assembly
Rubber-Metal Composite
2.1 × 1011
0.3
1.96 × 103
battery
Foam
2.0 × 109
0.3
2.527 × 102
seat bottom
Foam
2.0 × 109
0.3
7.55 × 102
seat top
Foam
2.0 × 109
0.3
1.69 × 102
seat headrest
Figures
Figure 1. Finite element model of 1994 Chevrolet
C1500 pick-up truck. Figure 2. Von Mises stress in front chassis and A-arm components for inertia relief. Figure 3. Von Mises stress in front chassis and A-arm components for dynamic analysis.
