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**SENSITIVITY ANALYSIS ABOUT **

**INFLUENCE OF OUT-OF-PLANE **

**DEFLECTIVE DEFORMATION UPON **

**COMPRESSIVE STRENGTH OF STEEL **

**PLATES **

**Akira Kasai **

Associate Profesor, Kumamoto University, 2-39-1 Kurokami, Chuo-Ku, Kumamoto, 860-8555, Japan

**Tatsuo Kakiuchi **

JR West Japan Consultants Company (PhD Candidate), 5-4-20 Nishinakajima, Yodogawa-Ku, Osaka, 532-0011, Japan

**Shohei Okabe**

GSST, Kumamoto University, 2-39-1 Kurokami, Chuo-Ku, Kumamoto, 860-8555, Japan

**ABSTRACT **

*In this study, it is aimed at verifying the relationship between amount of *
*the initial deflective deformation of simply supported steel plates and ultimate *
*compressive strength of them through elasto-plastic finite deformation *
*analysis. When initial deflection has been controlled smaller unitl now or *
*out-of plane deformation has become large after an earthquake, the current *
*compressive strength curve of steel plates in Japan cannot be applied. *
*Therefore, more accurate prediction method have been required in near *
*future, on behalf of rational design of steel structures. In other words, it is *
*needed to make clear the relationship between the initial imperfection and the *
*strength of simply supported steel plate. For this purpose, the parametric *
*study on compressive strength of steel plates taking the initial deflection and a *
*width-thickness ratio parameter into account was carried out. At first, a limit *
*width-thickness ratio parameter corresponding to maximum width-thickness *
*ratio parameter where compressive strength of a steel plate reaches yield *
*stress of steel material, was proposed through the parametric study. And, *
*predicted equation for calculating a limit width-thickness ratio parameter was *
*developed. Secondary, the relationship between the ultimate compressive *
*strength of a steel plate and amount of initial deflection was clarified. In *
*addition, estimated equations based on results of various numerical analysis *
*were developed. Finally, the prospect of this future study was discussed. *

**Key words**: Simply Supported Steel Plate, Compressive Strength, Initial
Deflective Deformation, Width-thickness ratio parameter, Allowable Stress

**Cite this Article:** Akira Kasai, Tatsuo Kakiuchi and Shohei Okabe.
Sensitivity Analysis about Influence of Out-of-Plane Deflective Deformation
upon Compressive Strength of Steel Plates. *International Journal of Civil *
*Engineering and Technology*, **6**(12), 2015, pp. 22-38.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=6&IType=12

**1. INTRODUCTION **

In this study, it is aimed at verifying the relationship between amount of the initial deflective deformation of simply supported steel plates and ultimate strength of them through elasto-plastic finite deformation analysis. It is a well-known fact that the initial deflection as initial imperfections has an influence on the ultimate compressive strength of simply supported steel plates. However, there are a few studies on the large initial deflection of steel plates relatively. On behalf of this fact, numerical analysis considering from small to large initial deflection are carried out in this study. Distinctive feature in this study is that it should be mentioned specially to consider not only the case of small initial deflection, but also large initial deflection. One of phenomena considered as a relatively small initial deflection is improvement of the production precision of steel plates. On the other hand, it is considered as one of the phenomena which consider large initial deflection relatively, to estimate the residual strength of steel plates after earthquake. Therefore, development of the strength evaluation technique according to amount of initial deflective deformation corresponds to make available to evaluate residual strength of steel plates uniformly, not only after starting a service of a infrastructure, but also after a certain special phenomenon occurring during a service of it.

There are many researches on ultimate compressive strength of steel plates. For example, Fukumoto et al. [1] compiled a data-base approach to the ultimate compressive strength of unstiffened steel plates in Europe, Japan and the United States. And, they described the strength formulas including the effect of initial imperfections such as residual stress and initial out-of-plane deflective deformation. In addition, Usami et al. [2, 3] summarized the experimental and analytical results on the strength of steel plates that both global and local buckling behaviors of welded box-sectioned compression members were considered. Furthermore, Nara et al. [4] developed ultimate strength evaluation formula of steel plates subjected to in-plane bending and compression through numerical analysis based on finite element method. There is a study on effective flange width of steel members by Usami et al. [5], and a study on strength prediction of thin-walled plate assemblies by Usami et al [6]. Komuro et al. [7] discussed about ultimate strength of steel plates from the standpoint of reliability design taking the material properties provided by Monte Carlo simulation in to account. Paik et al. [8] treated a study on ultimate compressive strength of dented steel plates. And then, Raviprakash [9] carried out numerical study on compressive strength of dented steel plate. Dent is one of general imperfections of thin-walled structures. Japanese Society of Steel Construction [10] summarized the ultimate strain of steel members used in the ultimate seismic design method based on the various study results performed so far, as well as these researches on the ultimate strength. On the other hands, only Komatsu et al. [11] and Ikeda et al. [12] mentioned study on the differences in out-of-plane deflections. Besides, there is no discussion

what amount of out-of-plane deflection has an influence on ultimate strength of steel plates. Specifically, the following 4 points are focused on, in the study.

1. Parametric study on the strength of steel plates about the amount of the initial deflection.

2. Consideration about relation between the strength of steel plates, the amount of the initial deflection and several structural parameters.

3. The proposal of the limit width-thickness ratio parameter.

4. The proposal of the formula estimating the strength of simply supported steel plate using amount of the initial deflection and structural parameters.

**2. FEM ANALYSIS OF A SIMPLY SUPPORTED STEEL PLATE **

**2.1. Analysis outline of numerical analysis method **

Figure 1 shows the FEM analytical model of a simply supported steel plate. Where, *a*
is depth of a plate, *b* shows width of a plate, and *t* indicates thickness of a plate. In
addition, it was decided to treat only 1 for the aspect ratio which was depth to width
ratio, in this paper. A simply supported steel plate considering in this paper was
models with 4-node quadrilateral finite-membrane-strain elements with reduced
integration (S4R). The number of elements was 100 meshes equally divided in the
loading direction and non-loading directions in consideration of the buckling
eigenvalue analysis. Numerical analysis using the ABAQUS [14] which is a
general-purpose analysis code was carried out.

**Figure 1 **Shape of Analytical model **Figure 2 **Distribution diagram of
residual stress

**Table 1** Parameters in the analytical model
Structural Properties
*R* 0.26-2.02
*= a* / *b* 1.0
*t *[mm] 32.0
Initial Imperfections
*wi,max* / *b* 1/50-1/5000
rt / y 1.0
rc /y 0.25
Material Properties
*E* [GPa] 200
y [MPa] 315
0.3

**2.2. Conditions of the analytical model **

Table 1 showed the structural and material properties of analytical model.
Width-thickness ratio parameter *R* is represented by the Equation (1).

*E*

*k*

*t*

*b*

*R*

###

*y*

###

###

_{}

###

###

### 12

### (

### 1

_{2}2

### )

(1)Where,* E* is Young's modulus, *y* is yield stress, is the Poisson's ratio, *k *(= 4) is

the buckling coefficient. *wi,max* is amount of initial deflective deformation, *rt* is

maximum residual stress in tension side, *rc* is maximum residual stress in

compression side.

The conditions of the analytical model were given in the following sections.

**2.2.1. Boundary conditions **

In order to estimate the compressive strength of steel plates, an numerical analysis was carried out under the most basic boundary condition in this study. Table 2 showed the boundary conditions of this analytical model. In this analysis, all edges of a plate were supported simply.

**Table 2** Boundary condition

Edge *u * *v * *w * *θx* *θy* *θz*
*x = *0 1 1 1 1 0 1
*x = a * 0 0 1 1 0 1
*y = *0 0 0 1 0 1 1
*y = b * 0 0 1 0 1 1
Free = 0 , Fix = 1

* u, v, w* : Displacements in *x, y, z* axis direction

**2.2.2. Residual stress **

In this numerical analysis, residual stress assumed at triangular distribution shown in
Figure 2. The main factor of such distribution depended on welding [2]. The
maximum value of the residual stress in tension side, which is distributed over
welding part, was assumed *rt*. On the other hand, the maximum residual stress of the

compression part was assumed 0.25*y* in the domain to remove around a weld.

**2.2.3. Initial deflective deformation **

It was assumed that the shape of initial deflective deformation shown in Figure 3 was
proportional to a half sine waveform which corresponded to buckling eigenmode [2].
The initial deflection was represented by the Equation (2). Further the origin of the
coordinate system was made point A in Figure 1. The directions of depth indicated *x*
axis, and the direction of width indicates *y* axis.

)
sin(
)
sin(
)
,
(
*b*
*y*
*a*
*x*
*w*
*y*
*x*
*w* * _{max}*

###

###

(2)**2.3. Materials & constitutive law **

The kind of steel materials to use in this study was SM490, named welding structural steel material. This steel material is generally used in Japan. And then, yield plateau and strain hardening region after yield plateau are existed as characteristic of this material as shown in Figure 4. In the strain hardening region, hardening coefficient follows below stress-strain relationship shown in Equation (3), according to Usami et al. [2]

###

###

##

##

###

*st*

###

*st*

*y*

*y*

*st*

*y*

*e*

*E*

*E*

_{}

_{}

_{}

_{}

_{}

_{}1 1 1 (3) Where, is the material-depended parameter,

*Est*is strain hardening coefficient,

*st* shows the strain arriving at strain hardening region. In the case of SM490, each

parameters are =0.06, *E / Est* =30, *st / **y* =7 [2].

**2.4. Loading pattern **

In order to carry out the numerical analysis in degradation area of load-displacement relation was also done in stable way, the displacement-based analysis was done. It was the analysis to give a uniform displacement of edge AD in loading direction as shown in Figure 1. The loading was monotonous increase only. After the total reaction force generated in edge AD has peaked, it controlled by displacement to the point where it was sufficiently reduced. Its maximum displacement was set to 30 times to be greater yield displacement.

**3. INFLUENCE OF INITIAL DEFLECTION ON STRENGTH **

**3.1. Definition of strength and ductility **

**Figure 3 **Shape of the initial deflection **Figure 4 **Stress-strain relationship

**Figure 5 **Average stress-average strain relationship

The strength and ductility of steel plate were defined for investigating the
analytical results specifically. At first, the relationship between load *P* and
displacement of steel plate was drown as shown in Figure 5. Here, the load to act on
edge BC in Figure 1 was defined as *P*. And then, the displacement of edge BC in *x*
-direction was defined as . Secondly, the average stress and the average strain
were used in Equations (4), to eliminate the influence of the plate width and the plate
thickness of the dimensional quantity.

= *P */ *A*, = / *a* (4a, 4b)

In addition, it should be noted that these average values do not express the local
stress or strain. And, the vertical axis in Figure 5 represented dimensionless average
stress which was the value that divided average stress by a yield stress, the horizontal
axis represented dimensionless average strain which was the value that divided
average strain by a yield strain. Here, maximum strength point was defined as *max*.

And then, the point corresponding to the peak strength was set to the ductility *max*.

**3.2. Strength toward sensitivity analysis of initial deflections **

In order to investigate influence of initial deflection on load-carrying capacity, parametric study was carried out. As main parameters, width-thickness ratio parameter and amount of the maximum initial deflection were dealt with. As results,

0 2 4 6
0
0.5
1
*/**y*
*/*
*y*

Maximum strength point

the relationship between compressive strength and width-thickness ratio parameter
was obtained as shown in Figure 6. Here, the vertical axis represented dimensionless
load-carrying capacity, and the horizontal axis represented width-thickness ratio
parameter *R* in Figure 6.

At first, the relationship between the load-carrying capacity in a certain *R* and
amount of initial deflection was described. It was found that the load-carrying
capacity tended to rise up, as an amount of maximum initial deflection *wi,max*

decreased in a region of* R* <1.17, as shown in Figure 6. On the other hand, the
strength tended to slightly drop, as an initial deflection decreased in the range of *R*
≥1.17. Therefore, it was summarized that the strength capacity of steel plates decrease
with an action of external force such as earthquakes in* R* <1.17.

**Figure 6 **Relationship between Strength and width-thickness ratio parameter

**Figure 7 **Increase ratio of the strength for each displacement

### 0

### 0.5

### 1

### 1.5

### 0

### 0.5

### 1

### 1.5

*R*

*u*

*/*

*y*Euler JSHB[13]

Results of Numerical Analysis

*wi,max/b*=1/50
*wi,max/b*=1/150
*wi,max/b*=1/300
*wi,max/b*=1/1000
*wi,max/b*=1/5000
0 0.5 1 1.5
-40
-20
0
20
40

Results of Numerical Analysis

*wi,max*=1/50
*w _{i,max}*=1/300

*w*=1/1000

_{i,max}*wi,max*=1/5000

*R*

*Inc*

*re*

*as*

*e*

*r*

*at*

*io*[%]

Secondary, the differences between the strength in the case of *wi,max / b *=1/150

and the others were discussed. As shown in Figure 7, the load-carrying capacity in the
case of *wi,max / b *=1/50, 1/300, 1/1000, 1/5000 were compared with that in the case of

1/150. The limit value of the maximum initial deflection is made 1/150 of plate width
as deviation from flatness of a plate by JSHB (JRA [13]). Therefore it was decided to
compare with numerical results in this limit value by this paper. The vertical axis
represented the increase rate for load-carrying capacity in the case of *wi,max / b* =1/150.

If the maximum initial deflection was large case *wi,max / b* =1/50, load-carrying

capacity was reduced in the range of *R* <1.17, load-carrying capacity could be
improved in the range of *R* ≥1.17 as shown in this Figure. If the maximum initial
deflection was small case *wi,max / b* =1/5000, load-carrying capacity was increased in

the range of *R* <1.17, load-carrying capacity could be worsened in the range of *R*
≥1.17 as shown in this Figure. Specifically, the strength of *wi,max / b* =1/5000

increased at most 37.4% at the *R* = 0.39, the strength of *wi,max / b* =1/50 decreased at

most 15.3% at the *R* = 0.33.

As results, it was found that load-carrying capacity tended to be increased as the
amount of maximum initial deflection would be small in the case of* R* <1.17.
Moreover, it is more effective to aim at amount of the initial deflection when
load-carrying capacity of steel plate is estimated, so that the strength of steel plate changes
larger according to amount of the initial deflection in *R* <0.5, which is equivalent to
the range of relatively thick plate in particular. Further, the strength in *R* ≥1.17 may be
considered hardly to change even if the initial deflection changes, because of below
two reasons, that the region of *R* ≥1.17 is more thin-walled area, and that quantity of
change is very small, although a reversal phenomenon has formed to a relation
between the load-carrying capacity and amount of initial deflection in *R* ≥1.17.

**3.3. Strength curve in Japanese Specifications for Highway Bridges **

As a result of above sensitivity analysis, if the maximum initial deflection amount
*wi,max / b* was small, such as 1/5000, *u */*y* crossed to 1 even if the width-thickness

ratio parameter *R* was greater than 0.7. These results indicate that the severe control of
amount of initial deflective deformation prompts the possibility using width-thickness
ratio parameter larger than current construction, or using more thin-walled structure
than now, when a construction design considering the behavior of buckling in steel
plates is carried out.

By the way, it was found that the load-carrying capacity evaluation of the present
analysis was less than the expression of JSHB (JRA [13]) the maximum initial
deflection *wi,max / b* was the range of *R* =0.5-0.7 even in the case of 1/150, when

Figure 6 was seen. Here, the design curve of JSHB (JRA [13]) shown in Figure 6 is
defined as 50% of the Euler buckling strength, in a relatively large area of the
width-thickness ratio (*R* >0.7), in consideration of the fact that the decrease in out-of-plane
deflection and stiffness is likely to occur at a low stress level [13]. Specifically, this
design curve is defined as follows.

)
7
.
0
(
5
.
0
)
7
.
0
(
0
.
1
2
*R*
*R*
*R*
*y*
*u*
*y*
*u*
(5)

However, a safety factor has been considered by the present buckling design in
Japan. Therefore, there are no cases that a numerical result falls below design curve.
And then, Figure 8 is prepared to indicate above fact. For example, the curve which
considered safety factor in standard design curve (serviceability curve) is drawn in
Figure 8, because of considering 1.14 (=1.7/1.5) as safety factor when an
earthquake-resistant design is carried out in Japan. It was found in Figure 8 that there was no case
that serviceability curve exceed numerical results, in the case that amount of initial
deflection was less than a width of steel plate divided by 150. On the other hand, there
were a few cases that numerical results were less than the serviceability curve in the
case that amount of initial deflection was larger than a width of steel plate divided by
150. For example, the numerical values in 0.5<*R*<0.772 fell below serviceability
curve, if amount of initial deflection was *b*/50. It was found that the steel plate which
exists in 0.5<*R*<0.772 needs attention about servicing after the earthquake, when
out-of-plane deflective deformation of a steel plate exceeded *b*/50 if out-of-plane
deflection of steel plate in a steel structure will be measured after severe earthquake.

**Figure 8 **Strength-*R* relationship at JSHB

**3.4. Definition of Limit width-thickness ratio parameter **

To perform allowable stress design of steel structures, a design at the region that the
allowable stress does not reduce as much as possible leads to utilize the performance
of the material used in this design. Therefore, it is very important to understand the
biggest width-thickness ratio parameter which is available for designing with yield
stress in the standard load-carrying capacity’s curve used with a design in steel
structures. A maximum width-thickness ratio parameter which corresponds to the
above-mentioned is 0.7 according to the Equation (5). This width-thickness ratio
parameter would be called a limit width-thickness ratio parameter, named *Rlim*, in this

study.

Now, elasto-plastic finite deformation analysis about the compressive strength of steel plates in this study is able to consider initial deflective deformation and residual stress as initial imperfections. It was possible to predict the compressive strength of steel plates in the state near the actual phenomenon, although it was analytic. Therefore, the width-thickness ration parameters when the ultimate compressive strength was parallel with the yield stress of constructed material, were calculated

0 0.5 1 1.5
0
0.5
1
1.5
*R*
*u*
*/*
*y* Euler
JSHB[13]

Results of Numerical Analysis

*w _{i,max}/b*=1/50

*wi,max/b*=1/150

*w _{i,max}/b*=1/300

*w _{i,max}/b*=1/1000

*wi,max/b*=1/5000

JSHB[13] considering of safety factor at seismic evaluation

through parametric analysis using this elasto-plastic finite deformation analysis
method. Generally speaking, the value of *Rlim* will be something different according to

amount of initial deflection. Table 3 summarized the relationship between amount of initial deflection and limit width-thickness ratio parameter. Moreover, this result was illustrated in Figure 9.

**Table 3** Limit width-thickness ratio parameter

*wi,max /b* *Rlim* *wi,max /b* *Rlim*

1/50 0.29 1/1000 0.59 1/100 0.36 1/2000 0.65 1/150 0.36 1/3000 0.67 1/200 0.39 1/5000 0.72 1/300 0.46 JSHB 0.70 1/500 0.52

**Figure 9 **Relationship between Limit width-thickness ratio parameter and maximum
initial deflection** **

Where, vertical axis indicated a limit width-thickness ratio parameter, and a transverse axis indicated the value defined as a width of a steel plate in divided by maximum initial deflection. And then, the estimated line calculated by a least squares method was also drawn in Figure 9. This estimated line was Equation (6).

###

ln 1.0###

1.0 0975 . 0_{}

*max*

*i,*

*lim*

*w*

*b*

*R*(50≤

*b*/

*wi,max*≤5000) (6)

It was found out that this equation is very highly precise through the fact that the
standard deviation was 0.0158, and watching Figure 9. As a result, this limit
width-thickness ratio parameter* Rlim* can be expected as an index for evaluating the strength

in response to the maximum initial deflection *wi,max*. Therefore, a load-carrying

capacity formula in accordance with the maximum initial deflection *wi,max* using the

limit width-thickness ratio parameter* Rlim* is proposed in Chapter 4.

101 102 103 104
0
0.2
0.4
0.6
0.8
1
*Rlim*
*b/wi,max*
* Rlim*
Predict

**4. PREDICTION METHOD OF COMPRESSIVE STRENGTH **

**CONSIDERING MAXIMUM INITIAL DEFLECTION **

In the previous chapter, it was found that amount of initial deflective deformation was
significant parameter for getting compressive strength of steel plates through
numerical study. In addition, limit width-thickness ration parameter, *Rlim*, shown in

Equation (6), was proposed in this study. In this section, by using Equation (6), the correlation equation between amount of the maximum initial deflection and the compressive strength is proposed in the present study.

**4.1. Pediction method for compressive strength of steel plates based on **
**Perry-Robertson formula **

It is well-known that the formula typed Perry-Robertson were used generally as the
prediction method for estimating compressive strength of steel compression members
like a column or a pier. Therefore, it is tried to make newer prediction method for
compressive strength of steel plates including effect of initial deflection in this study,
although the some past study [1, 4] have been proposed various calculation equation.
According to some studies by Usami et al. [5, 6], the equations predicting the
load-carrying capacity are complicated, although it is able to calculate the compressive
strength of steel plates. In addition, coverage of the initial deflection is 1/3233≤ *wi,max*

/ *b* ≤1/150. In order to improve these points and consider the case that out-of-plane
deflective deformation of steel plate is greater after severe earthquake, newer
predicting formula including more wide range of amount of initial deflection such as
1/5000≤ *wi,max* /* b* ≤1/50, was developed in this study, through a least squares method.

The following equations for calculating the compressive strength of steel plates were
developed.
0462
.
0
4
2
1 2
_{} _{}
*R*
*R*
*y*
(0.26≤ *R* ≤2.02, =0.0231) (7)

###

*R*

*R*

###

*R*

*a* 1 lim

###

_{ (0.26≤ }

_{R}_{ ≤2.02) }

_{(8) }

###

ln 1.0###

1.0 0975 . 0_{}

*max*

*i,*

*lim*

*w*

*b*

*R*(50≤

*b*/

*wi,max*≤5000) (9) Identical to Eq. (6) 229 . 0 0000324 . 0

_{}

*max*

*i,*

*w*

*b*

*a*(50≤

*b*/

*wi,max*≤5000) (10)

**4.2. Comparison of some strength equations **

The numerical results in this study and predicted equations in several previous studies
were shown in Figure 10, when amount of maximum initial deflection has been *b*/150.
Here, the vertical axis was non-dimensional the load-carrying capacity, the horizontal
axis was the width-thickness ratio parameter *R*. In addition, numerical results in this
study, the value calculated by the estimated equation in this study and the value
calculated by the equation developed by Usami et al. [5, 6] were shown in Table 4, to

compare several values concretely in case that amount of maximum initial deflection
is *b*/150.

In the range of *R* =0.36-0.65, the difference between the approximation equation
and numerical results were 0.04 to 0.07. And then, the difference between that
numerical results and the values calculated in the formula developed by Usami et al.
[5, 6] -0.02 to 0.04. In the range of *R* =0.65-2.02, the difference between the analysis
values and equations became almost the same. Figure 11 indicated the relationship
between the compressive strength of simply supported steel plates and
width-thickness ratio parameter, including effect of various amount of maximum initial
deflective deformation. Incidentally, the graphs separated in each amount of
maximum initial deflection were prepared in APPENDIX 2.

**Figure 10 **Strength curve (*wi,max/b* =1/150)

**Table 4** Comparison of strength curve (*wi,max/b* =1/150)

*R*
0.36 0.49 0.65 0.78 0.98 1.17 1.37 1.57 1.83 2.02
(1) *u* /y : Analysis 1.00 0.98 0.90 0.80 0.68 0.60 0.54 0.49 0.44 0.41
(2) *u* /y : Eq.(7-10) 0.96 0.91 0.83 0.76 0.65 0.56 0.49 0.43 0.37 0.33
(3) *u* /y : Usami [5, 6] 1.02 0.94 0.83 0.75 0.64 0.56 0.49 0.44 0.38 0.35
(1) - (2) 0.04 0.07 0.07 0.04 0.03 0.04 0.05 0.06 0.07 0.08
(1) - (3) -0.02 0.04 0.07 0.05 0.04 0.04 0.05 0.05 0.06 0.06

### 0

### 0.5

### 1

### 1.5

### 2

### 0

### 0.5

### 1

### 1.5

*R*

*u*

*/*

*y*Euler JSHB[13] Fukumoto[1] Nara[4](

*rc/*

*y*=-0.4)

Results of Numerical Analysis Eq.(7-10)

Usami[5,6]
*w _{i,max}/b*=1/150

*/*

*=0.25*

_{y}**Figure 11 **Strength curve of predicted equation

Future purposes of this study, it is possible to clarify the relationship between the maximum out-of-plane deflection and strength of the members, it is to establish the immediate seismic performance evaluation method measuring on site is a relatively easy out-of-plane deflection. The Equations (7-10) proposed in this study is available to decide how to use the steel structures after sever earthquake immediately.

**5. CONCLUSION **

In this study, it is aimed at verifying the relationship between amount of the initial
deflective deformation of simply supported steel plates and ultimate strength of them
through elasto-plastic finite deformation analysis. At first, the relationship between
the ultimate strength of steel plate and amount of initial deflection was clarified
through parametric studies on compressive monotonic analysis of simply supported
steel plates. Secondary, the limit width-thickness ratio parameter, *Rlim*, was defined. In

addition, estimated equations based on results of various numerical analysis were developed. The results obtained in this study are as follows.

The definition of the limit width-thickness ratio parameter, *Rlim*, was established. And

then, the correlation Equation (6) of limit width-thickness ratio parameter has been proposed.

The load-carrying capacity was significantly different due to the difference in the
maximum initial deflection *wi,max*.

It was found that the steel plate which exists in 0.5<*R*<0.772 needs attention about
servicing after the earthquake, when out-of-plane deflective deformation of a steel
plate exceeded *b*/50 if out-of-plane deflection of steel plate in a steel structure will be
measured after severe earthquake.

It was found that the ultimate strength of steel plates decreased sensitively, as the
initial deflection became large, in *R*<1.17. For example, the ultimate strength in the
case of *wi,max* /* b* =1/5000 was larger than the strength in case of *wi,max* /* b* =1/150 more

than 37%.

It was found that the limit width thickness ratio parameter may exceed 0.7, such as

*wi,max / b *= 1/5000, if the initial deflection is very small. Therefore, a limit

width-thickness ratio parameter can be raised up, if a limit level of initial deflections of steel plates has been managed strictly. And the good advantage for designing steel structures might have been hidden.

The equation for estimating the compressive strength of simply supported steel plate was proposed in this study. It was found that the proposed Equations (7-10) are

0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
Predict
w*i,max/b=1/50*
w*i,max/b=1/150*
w*i,max/b=1/300*
w*i,max/b=1/1000*
w*i,max/b=1/5000*
Result
w*i,max/b=1/50*
w*i,max/b=1/150*
w*i,max/b=1/300*
w*i,max/b=1/1000*
w*i,max/b=1/5000*
JSHB[13]
Euler

available to estimate the compressive strength in case of every initial out-of-plane deflection simply and accurately.

**REFERENCES **

[1] Y. Fukumoto and Y. Itoh, Basic Compressive Strength of Steel Plates From Test Data, Proc. of JSCE, Structural Eng./ Earthquake Eng., No. 344, pp. 129-139, 1984.

[2] T. Usami et al., Guidelines for Stability Design of Steel Structures, 2nd Edition, JSCE, Maruzen, 2005 (in Japanese).

[3] T. Usami, M. Suzuki, Iraj H. P. Mamaghani and H. Ge, A Proposal for Check of Ultimate Earthquake Resistance of Partially Concrete Filled Steel Bridge Piers, Journal of JSCE, No.525/I-33, 1995, pp.69-82 (in Japanese)．

[4] K. Nara, M. Tsuda and Y. Fukumoto, Evaluation of Ultimate Strength of Steel Plates with Initial Imperfections under In-plane Bending and Compression, Journal of JSCE, No.392/I-9, 1988, pp.259-264 (in Japanese)．

[5] T. Usami and H. Tenkumo, Ultimate Strength and Effective Width of Plates in Compression and Bending, Journal of JSCE, No.441/I-18, 1992, pp.78-85 (in Japanese).

[6] T. Usami and H. Ge, Strength Prediction of Thin-walled Plate Assemblies, Journal of Structural Engineering, JSCE, Vol.42A, 1996, pp.171-178 (in Japanese).

[7] M. Komuro, Y. Okui, K, Nozaka, T. Miyashita, K. Nogami and M. Nagai, Numerical Study on Ultimate Strength of Compressive Flange Considering Statistical Data for Distributions of Initial Displacement and Residual Stress, Journal of Structural Engineering, JSCE, Vol. 60A, 2014, pp.80-93 (in Japanese). [8] J. K. Paik et al., Ultimate strength of dented steel plates under axial compressive

loads, International Journal of Mechanical Sciences, 45, pp.433-448, 2003. [9] A. V. Raviprakash et al., Ultimate strength of a square plate with a

longitudinal/transverse dent under axial compression, Journal of Mechanical Science and Technology, 25(9), pp.2377-2384, 2011.

[10] T. Usami et al., Guidelines for Seismic and Damage Control Design of Steel Bridges, Gihodo Shuppan Co. Ltd., 2006.

[11] S. Komatsu, M. Ushio and T. Kitada, An Experimental Study on Residual Stresses and Initial Deformations of Stiffened Plates, Proceedings of the JSCE, Vol. 265, 1977, pp.25-32 (in Japanese).

[12] K. Ikeda, T. Kitada, Y. Yamakawa, M. Matsumura, A. Shibasaki and Y. Karino, Sensitivity Law and Probabilistic Variation of Elasto-Plastic Ultimate Strength of Imperfect Plates, JSCE Journal A, Vol. 64, No. 4, JSCE, 2008, pp.926-934 (in Japanese).

[13] Japan Road Association, Japanese Specifications for Highway Bridges, Part 1 Common, Part 2 Steel Bridges, 2012 (in Japanese).

[14] Dassault Systemes: ABAQUS Standard User’s Manual ver. 6.13-1, 2013.

[15] Mr. Shukla B.A. and Prof. Phafat N.G, Analysis of Co2 Welding Parameters on
The Depth of Penetration of AISI 1022 Steel Plates Using Response Surface
Methodology. *International Journal of Mechanical Engineering and Technology*,
**4**(6), 2013, pp. 31-36

[16] Mr. Shukla B.A. and Prof. Phafat N.G., Experimental Study of Co2 Arc Welding
Parameters on Weld Strength For AISI 1022 Steel Plates Using Response Surface
Methodology. *International Journal of Mechanical Engineering and Technology*,
**4**(6), 2013, pp. 37 - 42

**APPENDIX 1 **

The results in detail of the strength and ductility are given in Table A1. When initial
deflection was less than 1/300 in *R*=0.26, it was transcribed that the ultimate strength
is larger than the load at the point of 30 times of yield strain, and the ductility is
bigger than 30 times of yield strain, because load degradation did not occur after 30
times of yield strain which became limit value on the analysis condition in this study.

**Table A1** Compressive strength and ductility calculated by numerical analysis

*R*
*wi,max/b * 0.26 0.39 0.49 0.65 0.78 0.98 1.17 1.37 1.57 1.83 2.02
1/50 *u* /*y* 1.16 0.93 0.89 0.80 0.74 0.66 0.60 0.55 0.51 0.46 0.43
*u */*y* 20.88 1.81 1.98 1.68 1.68 1.68 1.68 1.68 1.68 1.98 1.98
1/150 *u* /*y* 1.36 1.00 0.98 0.89 0.80 0.68 0.60 0.54 0.49 0.44 0.41
*u */*y* 24.99 9.96 1.98 1.68 1.38 1.38 1.68 1.68 1.68 1.68 1.68
1/300 *u* /*y* >1.47 1.05 1.00 0.94 0.85 0.69 0.59 0.53 0.48 0.43 0.40
*u */*y* >30.0 10.59 2.19 1.55 1.43 1.38 1.68 1.68 1.68 1.68 1.98
1/1000 *u* /*y* >1.54 1.18 1.03 1.00 0.91 0.71 0.59 0.53 0.48 0.42 0.39
*u */*y* >30.0 13.98 8.58 1.89 1.46 1.03 1.38 1.68 1.68 1.68 1.98
1/5000 *u* /*y* >1.55 1.37 1.15 1.01 0.99 0.73 0.59 0.52 0.47 0.42 0.39
*u */*y* >30.0 21.48 12.18 2.58 1.73 0.90 1.41 1.73 1.83 1.98 1.98
**APPENDIX 2 **

Contents of Figures A1 indicate several cases of the relationship between the compressive strength of simply supported steel plates and width-thickness ratio parameter, including effect of each amount of maximum initial deflective deformation. Vertical axis indicates dimensionless ultimate compressive strength of steel plates, and horizontal axis indicates width-thickness ratio parameter, in every Figure. It's difficult to check the contents because all cases are included in Figure 11. So it is divided into a graph according to each initial deflection.

**(a) ****w****i,max****/b ****= 1/50** **(b) ****w****i,max****/b ****= 1/100**

0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*

**(c) ****w****i,max****/b ****= 1/150** **(d) ****w****i,max****/b ****= 1/200**

**(e) ****w****i,max****/b ****= 1/300**

**Figures A1 Relationship between ultimate strength and ****R**

**(f) ****w****i,max****/b ****= 1/500 ** **(g) ****w****i,max****/b ****= 1/1000 **
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
Euler
JSHB
Fukumoto
Nara(*rc/**y*=-0.4)

Results of Numerical Analysis

*wi,max/b*=1/50
*wi,max/b*=1/100
*wi,max/b*=1/150
*wi,max/b*=1/200
*wi,max/b*=1/300
*wi,max/b*=1/500
*wi,max/b*=1/1000
*wi,max/b*=1/2000
*wi,max/b*=1/3000
*wi,max/b*=1/5000
Predict
Eq.(7-10)
Usami[5,6],*/**y*=0.25
Euler
JSHB[13]
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*

**(h) ****w****i,max****/b ****= 1/2000 ** **(i) ****w****i,max****/b ****= 1/3000 **

**(j) ****w****i,max****/b ****= 1/5000 **

**Figures A1 Relationship between ultimate strength and ****R**** (Continued) **

0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
0 0.5 1 1.5 2
0
0.5
1
1.5
*R*
*u*
*/*
*y*
Euler
JSHB
Fukumoto
Nara(*rc/**y*=-0.4)

Results of Numerical Analysis

*wi,max/b*=1/50
*wi,max/b*=1/100
*wi,max/b*=1/150
*wi,max/b*=1/200
*wi,max/b*=1/300
*wi,max/b*=1/500
*wi,max/b*=1/1000
*wi,max/b*=1/2000
*wi,max/b*=1/3000
*wi,max/b*=1/5000
Predict
Eq.(7-10)
Usami[5,6],*/**y*=0.25
Euler
JSHB[13]