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Natural Frequency Extraction and Frequency Response Analysis of Reinforced Plate on Abaqus CAE

Executive Summary


In this project, natural frequency of reinforced plate has been found either on Abaqus
Software and with the stiffness-frequency formulas. Since there are two different
analysis methods, of course there are two different results. Abaqus software’s
value(37.8) is %30 higher than the simple calculations result (29.2). Since, software
has many inputs like; material, interactions, model dimensions etc.., in hand
calcualtions, only property used from model was the mass and stiffness. There may
also be one more reason for that which is; when stiffness was calculated, only elastic
deformation took place.


In the second step; it is obvious that the stress generated on the model due to applied
load exponentially increases while frequency approaches to natural frequency. It will
have huge impact if an earthquake engineer does not know about the frequency of
structures.

Problem Definition

Main objective of this project is to measure the natural frequency of the model created
in Assessment of the Reinforced Plate with Linear Elastic Analysis and Non-Linear Analysis. This model has 3 reinforcement bars attached to a plate and fixed on one
end.
Methodology of finding the natural frequency of the model is; defining a frequency in
Step-1 without applying any load. If any load is applied, there will not be a problem
since Abaqus software will override the input onto the assigned load. After this
process, analysis will give the natural frequencies of the model as 10 different modes
of outputs.


In the second step, we are asked to use the first step’s outputs as an input to find the
structural damping and mass damping values and apply a 150 Newtons of distributed
load on flat surface of the model.

Analysis Description

Natural frequency of the model has been calculated in the first step;

Table 1 Natural Frequencies of the model in 10 different frames

Ready to use excel sheet has been used to obtain the structural damping ‘β’ and mass
damping ‘α’ values. Damping ratio ‘ξ’ taken as %5 in this process.

Table 2 Obtained damping values due to frequencies on ready to use excel worksheet

Results and Discussion

There have been 10 different failure scenarios occurred in 10 different frames during
first step. Some of the cases are just mirrored versions of the other one’s but it is
crucial to have that knowledge in design phase.

Natural frequency could also be found via using 2 formulas below;

k = F/u

f = (1/2π)*(k/m)1/2

Where;
‘k’ is the stiffness,
‘f’ is the frequency
‘u’ is the displacement
‘F’ is the force
and ‘m’ is the mass.

Displacement of the model under 723N of total force from the previous analysis

Therefore;

k = F/u = 723N/5mm = 144.6 N/mm;

f = (1/2π)*(k/m)1/2 = (1/2π) * (144600/4.29)1/2 = 29.2 Hz

Frequency found via Abaqus software is 37.8 Hz where simple hand calculations
gives us the value of 29.2 Hz. In this case, software’s value is more reliable because at
the modeling and assembling phases; interactions between reinforcements to plate,
material properties, section properties have been assigned. Difference between 2
results occurs because of this. In hand calculations, there is not any materials or
interactions defined. Those calculations may be used in springs and then it will give
reliable and accurate results but, in this case, where there is a complex model, it is
hard to obtain a reliable result.

Table 3 Stress and Displacement values due to change in frequency

Table 3 shows that; there is a jump through frequency 27 to 36. Since natural
frequency is known as 37.8 of the model, with an engineering judgement, model gets
under huge stress while frequency catches the natural frequency. Extra of 150N’s
distributed load creates destructive effect within frequencies close to natural
frequency. Any load applied on the model will have direct incremental effect on the
stress generation and indirect effect on the deflection of the model.

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