ON THE HOLED PLATE UNDER TENSION LOAD.

No. 28 Vol.1 Thn. XIV November 2007 ISSN: 0854-8471

TeknikA

21

ON THE HOLED PLATE UNDER TENSION LOAD

Sabril Haris HG

Laboratorium Material dan Struktur, Jurusan Teknik Sipil, Universitas Andalas

ABSTRACT

Two types of holed plate subjected tension load are examined by using Abaqus numerical software for non linear calculation. The absence of the element around the hole becomes the interesting issues such as the effective area and the stress concentration. In this paper, variations on diameter of hole and stress-strain curve based on Voce’s law are investigated regarding the plate response in the elastic and plastic region. The maximum load from the numerical work is compared by the analytical result and a good agreement is obtained.

1. INTRODUCTION

The holed plate subjected tension load will have experience the phenomena called by stress concentration due to the absence of the elements around the hole. This increasing stress makes those element will attain yield stress first meanwhile the others are still in the elastic region. Since the yielding process occurred, it will affect the response of the whole plate because yielding elements now have the less stiffness regarding in the plastic state.

The subject of this paper is to observe the stress concentration on the holed plate by using the nonlinear software Abaqus. Two types of holed plate are subjected static monotonic load until the ultimate condition reached. A detailed analysis of this simulation is presented in view of the yielding in the elements around the hole and the propagation process to the others.

Several aspects are discussed, such as response in the elastic region that comprises determination of first yield-load and significant yield-load, and response in the plastic region that covers ultimate conditions of plate. Furthermore, the effect of diameter of hole and parameter of stress-strain curve material model are also analyzed.

2. DESCRIPTION OF MODEL

2.1 Dimension of Plate

There are two types of plate that will be used in this simulation. The first is plate with one hole in the center of plate and the other is plate with two-half hole in the out-fiber side.

Dimension of plates are fixed for all of the models. They are the length of plate L0 = 300 mm,

the width of plate h = 100 mm, and plate thickness t = 10 mm. Then, the full cross section area A0 can be

calculated as 1000 mm2.

Diameter of hole (d) are varied for d = 0, 10, 20, and 30 mm. So, for d = 0 mm, plate 1 and type-2 are identical.

For both types, in the cross section 1-1, area of plate is defined as the net area (Anet) which is

calculated as Anet = t (h - d). The models are shown

in Fig. 1.

a. Plate Type - 1

b. Plate Type - 2

Figure 1 Types of Plate

2.2 Material

Material used in this paper is the mild steel grade 41, which is widely used in many steel structures with yield stress σy = 250 MPa. In order to define

the input to Abaqus, Voce’s law [1] is used to construct stress-strain curve for the material characteristic. This formula is shown in Eq. (1).

(

-C.ε

)

0

σ = σ + Q 1 - e (1)

Base on some experimental data of uniaxial tensile test [2,3], basic parameter in Eq. (1) determined as follows:

σ0 = 250 MPa

Q = 250 C = 15 h = 100 mm

L0= 300 mm

d 1

10 varied constan maximu and then Piec stress-s

Elas used fo follows von Mi

2.3 Mo

By solid w handle hole, fin

2.4 Loa

The is subje the othe unexpec effect, t are rele Stat for some valu nt. Plastic strai

um value 0.18 n stress values cewise-linear r train are obtain

Figure 2

stic modulus E or Poisson’s rat

s isotropic hard ses criterion fo

deling

using Abaqus which is regular

stress concen ner meshing el

Figure 3

ading Conditio

e tension loadin ected the unifo er side becom cted additiona the vertical and eased unless the tic monotonic

0.1

Plastic Strain strain-stress cu ue, i.e. 75 and n data are set 8 with uniform s are generated relationships fo ned as well as

Stress-strain C

E is 200000 M tio (μ). For the dening for wor or the effective

software, pla rly meshed as s ntration at the lements are cho

Element Mes

on

ng conducted i orm load in th mes the fixed su

al stress rega d lateral fixatio e nodes in the m

loading are su

0.15 0.2 m increment 0

by Voce’s law for three types

shown in Fig. 2

Curve

MPa while 0.3 e yield surface rk hardening a e stress.

ate is modeled shown below.

area around t osen.

shing

in this simulati he one side wh upport. To avo arding Poisson ons in the supp

middle one. ubjected until t

be

RESPONSE REGION

For two types o holes (d) and p ss relationship ysis is observa ond one is in th

Effective Area

Loading the p tic response ween load and d Elastic respon mula:

σ = E . ε

For uni-axial te divided by the ned as the e ded by the leng er to use initi

for the conditio Replacing σ by

F = k ΔL

Eq. (3) shows

ΔL by the stiff Differences of to the hole, bo he hole, theref effective area that fulfills Eq Aeff / L0

From the load h model, the

arization in th nning curve a ation of load-d eved by using lts shown in Ta

Tab

Difference valu s not influence

in the elastic re It is interestin

Plate Type d

No Hole

Type 1

Type 2

OF PLATE IN

of plate, variat parameter Q i ps are obser ation in the el he plastic region

a

plate for small denoted by displacement. nse of the str

ension σ can be cross section a elongation or gth of the plat ial dimension on in this elast y F/A0 and ε by

; where k =

the linear rela fness k = E A0

cross section oth for two typ fore it is neces

(Aeff) instead

q.(3). The stiffn

d-displacement stiffness k is he elastic part

are taken to c displacement.

the formula. A able 1.

ble 1 Effective

ue of Q in the e the result sinc

egion

ng then to fin d (mm) k

0 666.6 10 661.5 20 645.7 30 618.6 10 661.2 20 645.5 30 621.0

N THE ELAS

tion on the dia in the stress s rved below. astic region an n.

l deformation linear relatio

ructures meet

e defined as th area. Meanwhi displacement te. Further then

than instanta tic region. y ΔL/L0, we ge

ationship betw / L0.

area will take pes of plate. Be ssary then to d d of the initia fness then beco

t curve plotte easy to obta . Some data a construct the

Then, Aeff w

Aeff = k L0 / E

Area

e material prop ce the respons

nd the relatio Aeff (mm

667 1000.00 578 992.37 708 968.56 663 927.99 260 991.89 594 968.39 038 931.56

STIC

ameter

strain-First nd the

gives onship

ts the aneous

et:

(3)

ween F

place ecause define l area omes k

ed for ain by at the linear will be

E. The

perties es are

No. 28

the fact plate th

3.2 Firs

It is be take observi Abaqus stress d If w uniform can be c

The their nu

No H

Typ

Typ Plate

8 Vol.1 Thn

ional purpose, ized value Aef

ross sectional a

gure 4 Norma

m the curve in n normalized A

eff tor for the red hat is limited fo

st Yield

s widely know en place on the

ng the stress s simulation, w due to that phen we assumed t m, the load that calculated as th

Fyield = Aeff .

e yield-load va umerical results

Table 2

Hole 0

area for no hol

alized Effective

n Gig.4, we ge duction area du or this study.

wn that stress c e element arou values whic we can see the

nomenon. the stresses on

t generates firs he formula in E

σyield

alues calculated s are shown in

2 First Yield Lo

formula n 250.000

ember 2007

re replaced by

respectively. A e plate.

e Area vs d/h

et the relationsh ter of hole as:

(

(d/h)2 in Eq.4 ue to hole in t

oncentration w und the hole. ch get from t

increasing of t

n the plates a st yielding (Fyi

Eq. (5). 153.125 1. 125.075 1. 106.525 2. 155.750 1. 128.400 1. 115.125 2. kN) Table 2 gives centrations occ need the small greater diame ss concentratio ds for plate type h the same diam The stress conc wn in Fig. 5.

Figure

Significant Yie ve

The load that lted from num e after that poi rly linear. The e

lement around nge the stiffnes Significant yi tional load. T ificant-yield ched. First line ore the significa esents some da instance, for p meter and Q = curs due to hol lest load to ach eter we have, on we get. It c e 2 are greater meter of hole.

centration for b

a.Plate Typ

b.Plate type

5 Stress Conc

eld on The Lo

t generates fi merical result int the respons effect of local d the hole, is ss of the structu ield point c To calculate

load, two e represents the

ant point as we ata just after th plate type 1 w

250, the sign the first yield

re 6 Significan

SSN: 0854-8

ormation that le on the plate hieve the first the higher ra can be also see

than for plate t

both types of p

e-1

e-2

entration

oad-Displacem

irst yielding ( is not mean se of the plate i

yielding at the not significan ure.

comes after the value of straight lines e data on elasti ell as the secon he significant with 20 mm in

nificant-yield lo is 125.075 kN

nt Yield

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stress e since

Significant-yield loads for different value of Q are almost same because the effect of hardening which occur in some of element around the hole is still in the beginning of hardening (in the small strain). As shown in Fig. (2), in the beginning of material hardening, the difference between varied Q is small. It can be seen in the Fig. (7.c) and (7.d).

4. RESPONSE OF PLATE IN THE PLASTIC

REGION

4.1 Yield Propagation

To observe the response of plate in the plastic region, it is beneficial to spot the load-displacement curve on the part after significant yield point. The load displacement curves for two types of plate

those vary on of diameter of hole as well as variation on Q respectively are shown in Fig. (7).

Plate type 1 with one hole in the center of plate, in general, has less significant-yield load than plate type 2 with two-half hole in the out fiber side. It is agree with the load at the first yielding (Table 2).

As shown in Fig. (7.a) and (7.b), for both type, the greater diameter of hole will make the stiffness of the load-displacement curve decrease faster. From the curves, the smooth rounding path is created after significant-yield point until reach the stage with nearly linear as the new stiffness of the plate. The greatest diameter is always positioned on the lowest curve.

a. Plate Type 1, variation on d, Q = 250 b. Plate Type 2, variation on d, Q = 250

c. Plate Type 1, variation on Q, d = 20 mm d. Plate Type 2, variation on Q, d = 20 mm

Figure 7 Load-displacement Curve for Two Types of Plate

Yield propagation can be analyzed from the shape of the plate. For plate type 1, on the holed-cross section, area of plate is divided into two parts, the upper and the lower. The distance from the corner to the out-fiber is (h-d)/2 for both parts. For plate type 2, on the holed-cross section, there is only one area of plate from upper-hole side to lower-hole side and the distance is (h-d). So, propagation model for plate type 2 produce the greater value of load

than plate type 1. Yield propagation for both type of plate follows the grey-black contour as shown in Fig. (8).

a. Plate Type 1

100 150 200 250 300 350

0 2 4 6 8

displacement (mm)

L

o

ad

(

k

N

)

d = 0 mm (no hole)

d = 10 mm

d = 20 mm

d = 30 mm

100 150 200 250 300 350

0 2 4 6 8

displacement (mm)

L

o

ad

(

k

N

)

d = 0 mm (no hole)

d = 10 mm

d = 20 mm

d = 30 mm

10 0 150 2 0 0 2 50 3 0 0

0 0 .5 1 1.5 2

displacement (mm)

L

o

a

d

(

k

N

)

Q = 250

Q = 160

Q = 75

100 150 200 250 300

0 0.5 1 1.5 2

displacement (mm)

L

o

ad

(

k

N

)

Q = 250

Q = 160

No. 28

Figure

As effect t the sign after th Q.

4.2 Ma

The = 0. It c the cros

Diffuse

Applyin

Tensile defined

Aga hole, as The numeric simulat are defi the hole as (σu).

mentioned be to the load-dis nificant-yield r at point that is

ximum Load

e maximum loa can be attained ss section with

e necking for th

dσ

s mentioned pr ese analytical r

cal ones whic tion. Maximum fined when the

e, reaches the

Table 3 it is er of hole d = 2

n. XIV Nove

Plate Type 1

agation for Tw

efore, variation splacement cur

reached. The d s proportional w

ad in the plate d when diffuse hole (section

hese cases lead

and maximum

area in the cr eviously. results will be ch are obtaine m load from n e critical point

true uniform s

s shown that 20 mm or Anet =

ember 2007

wo Types of Pla

n on Q has l rve of plate af differences occ with the value

defined when necking occur 1-1 in Fig. 1.).

ross section w

compared to t ed from Abaq numerical resu , element arou strain (εu) as w

the results = 800 mm2.

There are som lts and the nu 1 and Q = 2 y close each oth anwhile, for the h differences le

ONCLUSION

Numerical wor es which are s in the beginni ne the stiffnes responses in th effective area ameter Q in th

ple relationship the effective ur around the h fness of the str d attained. Max determined fr ws a good agre

FERENCES

T.Beylytschko finite elemen Wiley, 2000. R.Tornqvist, Structures, Ph Denmark, 200 Civil Engin University of 2007.

IS

le 3 Maximum

me variation umerical ones. 250 - plate typ her with differ e others, the re

ss than 10 %.

N

rks have been subjected the ng of the elast ss of the struct

he elastic regio in the cross se he Voce’s law p between the

one is propos hole is not signi ructures until t ximum load for rom the num ement with the

o, W.K.Liu an nts for contin

Design of hD Thesis, Tec 03.

neering Mate f Mexico, Ten

SSN: 0854-8

m Load

between anal For Q = 75 -pe 2, the resul ences less than esults are quite

done for the tension load. ic region are u tures. It is obt on are determin ection. The eff w is immateri e full cross s sed. The first ificant to chan the significant r the holed pla merical work w

e analytical res

d Moran, Non nua and struc

Crashworthy hnical Univers

erials Labor nsile Test of

8471

lytical - plate tained ned by fect of ial. A ection

yield ge the t-yield ate can which sult.

nlinear ctures,

Ship sity of