Low Cycle Bending Fatigue of Steel Bars Under Random Excitation
Part II: Design Considerations
Wen-Chin Liu, Zhong Liang, George C. Leebv
Abstract
In this paper, an effort was made to quantify the low cycle bending fatigue phenomenon under
random loading histories. To obtain this goal, a series of tests were conducted. Based on the
experimental data, along with the considerations of the fatigue failure mechanisms, a more
appropriate damage accumulation index formula is established. This formula of this research can
serve as the foundation for further investigation on more complicated members with local buckling
effects, welding residual stress effects, and etc. Once these effects are clarified and considered, a
design principle is then developed.
Introduction
The current seismic design of steel structures includes the concept of probability, which is
resulted from two main considerations, safety and economy. Hence, it is possible that the structure
members exceed the elastic limit and get into the nonlinear stage under earthquake excitation.
Though the structures may not fail immediately, there is damage accumulated in the term of low
cycle fatigue inside the materials. After certain earthquakes, these structures may eventually
collapse.
Though some researchers, [1,2], have consider this possibility, their approaches are all mainly
based on the low cycle bending fatigue tests under constant loading amplitudes. Since it is proved
inappropriate to use test data under constant amplitude to predict low cycle bending fatigue strength
under random loading,[3], this traditional method should be modified to reveal the real material
behaviors under earthquake excitation.
Test data
In order to establish an appropriate formula to quantify the damage accumulation for steel
members, sufficient test data is needed. In this paper, the test data from reference [3] is used. This
data includes 2 parts; the first part provides information of low cycle bending fatigue under various
loading histories; the second part is used for quantifying the grain orientation effect.
These specimens are all A36 steel square bars and clamped instead welded or bolted to prevent
complicated boundary conditions. Thus, the data can be considered pure low cycle bending fatigue.
Formula development
The definition of failure is chosen as the same as that of reference [3]. That is, while the
moment fell below 80% of the maximum moment of the same strain, the specimen is considered
failed.
1
The formula development is described as the following sections:

Low cycle fatigue with constant amplitude
To develop a general formula for analyzing low cycle bending fatigue, the amplitude-life
relationship under constant bending amplitude reversals is established as a foundation. Here, the
modified Coffin-Manson, [4], formula is used:

  f  2 N f
2
where

c
（1）

is the total strain amplitude
2
 f is the fatigue ductility amplitude
2 N f is the total cycles to failure
c is the fatigue ductility exponent
The data of constant displacement amplitude tests from specimens part I is regressed in the
form of equation (1) as followed:

 0.2285  2 N f
2

0.3168
（2）
The square of the correlation coefficient r 2 of this regression is 0.9911.

Random displacement history low cycle fatigue
To quantify the damage accumulation of random bending history low cycle fatigue, the test
data of specimen SB from specimens part I is used. First of all, the Palmgren-Miner's rule, [5], is
applied as the basis for analysis. This method is assumed that the effect of each cycle is independent.
Accordingly, its damage index is a linear accumulation of each cycle, which is described as follow:
n
D i
（3）
i Ni
where D is the damage index
ni is number of cycles corresponding to the i th block of constant stress/strain amplitude
i
i
N i is the number of cycles to failure at
i
i
When D reaches 1, the specimen is considered failed.
The analysis of random displacement history tests using this method is shown in Table 1,
2
where the second row to the 5th rows and 7th row to 12th row are the analytical damage indexes
from different approaches. This method becomes unreliable when the complication of the
displacement history increases.
For example, the damage indexes of specimens SB-3 and SB-4 are over 1.57, which is far too
optimistic.
To improve this method, previous research results and the phenomena observed by the tests
are considered. The considerations include effect of sudden tension amplitude decrease, relative
strain undergone and partial cycle counting.
A. Effect of sudden amplitude decrease
According to the previous researches, the sudden amplitude decrease will affect the
characteristics of subsequent cycles. If the previous large amplitude is in tension, it will reduce the
estimated amount of damage accumulation in subsequent cycles, [6]. If the previous amplitude is in
compression, the damage accumulation in subsequent cycles will be underestimated, [7]. In these
two circumstances, the effect of the tension amplitude reduction is much more obvious, [8].
Furthermore, in low cycle bending fatigue behavior, the material grain orientation of the specimens
also plays an important role in this effect, [3].
To quantify this phenomenon, the amplitude-changing factor K] is introduced into equation (3).
To make the calculation simple, the compression amplitude reduction effect is neglected, and only
the tension amplitude reduction effect is considered. Equation (3) is rewritten as:
n m

D    i   K1 j 
i  Ni
j

 i j  i
1 
100




（4）
k 1 ka
bj
if the tension amplitude of previous j th cycle is large than
the current cycle`s
where K1 j 
 i j  i
1 
100




 ka
bj
if the compression amplitude of previous
j th cycle is large than the current cycle`s
1
otherwise
 i is the peak strain of i th cycle
k1 is the power coefficient of amplitude change
b j is the reduction coefficient of the sudden amplitude decrease effect
b j is increased as j increased, which means that the sudden amplitude decrease effect will be
3
gradually vanished as the cycles continued.
B.
Relative strain undergone
Second, consider the situation that the absolute value of maximum strain and minimum strain
is not the same, that is, if the value of the load ratio R minimum strain divided maximum strain,
is not -1. In this situation, the effective amplitude of these cycles should be modified. Some former
researches, [7], found that if the stress/strain level is large, the plastic stress relaxation would
progressively reduce the mean stress level to zero. That is, the effective strain amplitude would be
taken as:
e 
where  e
 max   min 
（5）
2
is the effective strain amplitude
 m a x is the maximum strain
 m i n is the minimum strain
The above method is developed from the constant amplitude cyclic loading tests. However, for
the random amplitude tests, because the amplitude keeps changing, there is not enough time for the
plastic stress relaxation to take place. Thus, using this modification of equation (5) along with
equation (3) can result in larger error, as seen in Table I. For example, specimens SB-3 and SB-4,
using this modification can cause almost 65% error. Besides, as the random amplitude history
become more complex, this modification is even harder to be practiced.
After certain detailed considerations in this study, it is concluded that the maximum relative
strain experienced between 2 consecutive cycles might be a reasonable measure. This concept
assumes that lager relative strain can introduce larger material damage. With this postulation, the
damage index is modified to the following equation to accommodate the condition of R being
unequal to-I or the random amplitude histories.
n
D   i  K1  K 2
（6）
i Ni
 
ri
where K 2  
 2  i 1   i





k2
k2 is the power coefficient of the relative strain undergone
 ri is the relative strain, (from peak to valley or valley to peak)
C.
Partial cycle counting and cycle counting method
When specimens are subjected to complex random strain histories, most cycles are
incomplete cycles. That is, at the end of a cycle, the strain does not return to where it was at the
beginning of that cycle. The damage effects of these partial cycles certainly are not the same as
those of full cycles. To quantify the effect of partial cycles, a third modification parameter K3 is
4
introduced:
D
i
where K3 
ni
 K1  K 2  K 3
Ni
（7）
 ti
2  i
 ti is the total strain of i th cycle undergone
In complex random amplitude low cycle bending fatigue, cycle counting is another
difficulty. As the complexity of the random strain history increased, this task becomes more difficult.
To overcome this difficulty, rainflow counting method is developed by former researchers, [9].
The major advantage of the rainflow method is its ability to eliminate half-cycle counting
and thus simplifying the process of examining the strain history. However, it is still a very tedious
method for complicated strain histories. In addition, it is also difficult to carry out the modifications
already established.
Following the general thought of the rainflow method, a new counting approach is developed
for random amplitude loading histories in this study. This counting procedure is illustrated in Fig. 1
with the following details:
i.
The counting unit used is 1/2 cycle. Although there will be stress saturation
during the loading reversals, the residual strain, (which causes grains to become elongated
or compressed), remains.
Thus, the unstressed point, e  0 , is important for describing the fatigue
phenomenon.
ii.
Plot the strain history. Find the peak points above 0 strain (pt.2, 6, 8, 10),
and the valley points below 0 strain (pt.4, 9, 11, 13). Each point is considered the extreme
point of each half cycle, and the amplitude is the absolute strain of the point.
The major difference between the conventional rainflow counting method and this proposed
cycle counting method is that the latter is simpler and the implementation of the modification
factors, K1 , K2 and K3 is straightforward. For instance, in half-cycle <567>, the amplitude is the
strain at point 6,  ri is the relative strain from peak/valley to valley/peak which is the total strain
from valley point 4 to peak point 6,  ti and is the sum of the length of 56 and 67 .
D.
Developed formulation
With the help of the new counting method and the K1 effect reduced to influence the
following 4 half-cycles, the modified damage index equation is then developed into the form as
followed:
n
1
D
 K1  K 2  K3
i 1 2 N i
5

 1
 
i 1  2 N i

n

i j  i
 4 
   1 

100
 j 

kj1

 j   
ri
  

 2  i 1   i
  

k2

  
  ti 
 2 i 


（8）
kj1 is the exponent of K1 to the previous j th half cycle and is as the form of:
 k1  ka 
the current cycle’s
kj1 
ka
if the tension amplitude of previous cycle is larger than
if the compression amplitude of previous cycle is larger
than the current cycle’s
0otherwise
If there is no previous cycles for reference, K1 , K2 and K3 should be taken as 1.
There is one thing in this equation that needs to be notified. Under random bending
loadings, the 2 sides of specimen separated by the neutral axis may experience different
tension/compression histories.
Thus, both sides should be checked and the larger index should dominate the failure judgment.
To verify the accuracy, the random test data of specimens SB from specimens part I is applied
into equation (8), along with equation (3) and (5) in Table 1. From this table, it can be seen that
when k1  11000 , and ka  5000 , the analytical error is dramatically reduced.

Quantification of grain orientation effect
The different grain orientations of the materials can cause the disparity of random low
cycle bending fatigue strength, [3]. This effect is quantified as introducing an angle coefficient " ka "
in K1 of equation (4). By introducing the test data from specimens part II, along with the change of
the ka values, the analysis damage indexes using equation (8) is provided in Table 2. In this table,
the first row is the identification of specimens; the second row are the orientation angles; the third
row is the damage accumulation using linear theory, equation (3); the 4th row is the damage index
using effective-strain method, equation (5), along with Miner's rule, equation (3); the 6th row is the
value of ka , the 7th row is the damage accumulation using the proposed formula, equation (8); the
8th row is the error reduced by comparing the third row and the 7th row; and the 9th through 16th
rows repeat the first portions of the table.
From Table 2, it can be seen that changing values of ka can increase the accuracy of
analysis. Besides, it also can be noticed that if the grain orientation of the specimens is near 45 ,
the specimens have the least low cycle bending fatigue strength and have the largest value of ka .
When the orientation angle is increased the fatigue life is increased and the ka value is decreased.
6
This trend matches the failure mechanism observed from reference [3]. To regulate and quantify the
relationship of ka and grain orientation angle, more tests should be conducted and more precise
method to measure the grain orientation should be developed.
Discussions on errors
The desired value of the indexes in Table 1 and Table 2 should be unity. The error is the
absolute value of the difference between the analytical indexes and 1. If the index value is greater
than 1, the results underestimate the truth. On the other hand, if the index value is less than one, it is
too conservative. Although equation (8) is more accurate compared with traditional approaches,
there still exist some errors.
These errors are caused by the following reasons:
I.
II.
The inherent error from the test setup and testing devices.
The modifications used in the proposed method only consider major
effects that have already been addressed and developed so far. There might be some other
factors that are not yet discovered.
III.
The simplifications of these methods, such as neglecting the sudden
compression amplitude reduction effect, condensing the sudden amplitude reduction effect
within the subsequent first cycle, neglecting the elastic strain effect, etc, are intended for easier
implementation in engineering practice.
Conclusions
Low cycle bending fatigue may be one of the major failure mode in steel structures
attacked by strong earthquakes. Conventional analysis is based on the cyclic loading tests, and it is
proven inaccurate for random loading tests. From the study of this paper, some conclusions are
listed as follow:
1.
Considering major effects and modifying the conventional method with 3
factors, K1 , K2 , K3 along with new cycle counting method, a more accurate and reliable
formula for predicting low cycle bending fatigue strength under the random amplitude
reversals is developed.
2.
The grain orientation of the steel cannot be easily acquired via traditional
material tests nor on the purchasing specification and the bending moment which the real
structure members are subjected under earthquake may not be in the desired direction. Thus,
for the conservative point of view, it is suggested that k1  11000 , ka  5000 and k 2  1 .
3.
Despite ka, because the modifications of the formula are based on the
failure mechanisms that are valid under either bending or axial loading formula developed is
still useful if specimens are under axial cyclic (random) loadings reversals. To practice it, the
power coefficients of K1 and K2 should be adjusted for this purpose.
7
Suggestions on further investigations
This paper is focused on the A36 steel square bars clamped, subjected with one direction
bending force without axial load induced. However, in real world, structural members are more
complicated whether in shapes, in connection constrain or on the forces they subjected to. To make
the results more feasible and provide design principles, further investigations are needed. The future
investigation may include:
1.
To find out the contribution of local buckling effects, welding (material
properties changed) effects and different connection configuration effects to the low cycle
bending fatigue.
2.
3.
To find out the effects of the combination of axial and bending loads.
To find out the behavior of low cycle fatigue under multiple directional
bending loads.
4.
And to find out the effects of the strain rates.
The first three are to reveal the conditions of real structure members, and the last one is
to concern the dynamic loading effect of earthquakes. Considering these supplements along with
our proposed formula, a method for structure design is than established to prevent low cycle
bending fatigue caused by earthquakes.
Reference
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4.
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