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Dynamic plastic behaviour of a notched free–free beam subjectedto step-loading at one end

Y. Zhang, J.L. Yang*

The Solid Mechanics Research Centre, Beijing University of Aeronautics and Astronautics, Beijing 100083, People’s Republic of China

Received 29 January 2002; revised 8 April 2002; accepted 30 April 2002

Abstract

A rigid perfectly plastic model is developed to study the initial, small deflection response of a free–free beam with an initial notch along its

span under a concentrated step-loading suddenly applied at one end of the beam. Complete solutions are obtained for various combinations of

the magnitude of the load, the location of the notch and its defect-severity. The partitioning of the initial energy dissipation rates is discussed

for some typical situations. It is concluded that: (i) the different initial deformation mechanisms and the initial energy dissipation rate of the

beam depend not only on the magnitude of the load but also on the defect-severity and location of the notch; (ii) because of the influence of

the notch, the structural response of the beam is far more complicated than that of the un-notched free–free beam; and (iii) for some cases the

maximum rate of energy dissipation in plastic hinges will be more than 1/3 of the total input energy rate, while for an un-notched free–free

beam, it has been demonstrated that the plastic dissipated energy is always less than 1/3 of the input energy [Int. J. Impact Engng 21 (1998)

165].

q 2002 Published by Elsevier Science Ltd.

Keywords: Notched free–free beam; Rigid perfectly plastic model; Concentrated step-loading; Energy dissipation rate

1. Introduction

One of the catastrophic accidents in a nuclear power

plant is a rupture in the piping system. In many cases, the

accident is caused by stress corrosion cracks that initiate at

the inner surface of the pipe. These cracks tend to grow

rapidly in the circumferential and radial directions, when the

pipe is subjected to unforeseen loading, for example, an

intense dynamic load, such as an explosion, impact by a

projectile or by other breaking pipes (called pipe-whip). Inorder to obviate brittle fracture and its potentially

catastrophic consequences in applications for nuclear

reactor piping, it is useful for designers to understand the

effects of cracks or notches on the structural response and

integrity of ductile piping under impact condition.

In recent years, several papers [2–6] have been

published to study the dynamic plastic behaviour of

structures with an initial crack or notch using a rigid,

perfectly plastic (r-p-p) model. The advantage of the model

is that the r-p-p material idealization significantly simplifies

the deformation mechanism of the structure without losing

the key features of its dynamic response. Another important

reason for adopting the r-p-p model is based on the

observation of impact experiments for those structures

containing cracks or notches. Woodward and Baxter [7]

reported an experimental study on impact bending of

continuous and notched free– free steel beams. Their

experimental results showed that the effect of notches is to

change the strain profile in the beam, localize the plastic

deformation, which is more like a stationary plastic hinge

and provide a site for fracture initiation. They commentedthat the rigid-plastic approach might be expected to be

applicable in some cases. Yang et al. [8] reported an

experimental study on dynamic behaviour of clamped

cracked steel beams subjected to impact at mid-span. This

study also verified that the crack has the effect of

concentrating the deformation locally, making it more like

a stationary plastic hinge at the crack site, where a sharp

increase in rotation is observed. The influence of the crack

or notch on the localization of deformation, however,

depends on its position and the severity of the defect (crack

or notch). In some situations, the localization of deformation

is insensitive to the defect and in other situations its

influence is rather significant. For this reason, some

researchers have studied a so-called ‘defect sensitive

0308-0161/02/$ - see front matter q 2002 Published by Elsevier Science Ltd.

PII: S 0 3 0 8 - 0 1 6 1 ( 0 2 ) 0 0 0 4 4 - 3

International Journal of Pressure Vessels and Piping 79 (2002) 741–752www.elsevier.com/locate/ijpvp

* Corresponding author.

E-mail address: [email protected] (J.L. Yang).

http://www.elsevier.com/locate/ijpvp

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region’ for which Chen and Yu [9] examined various typical

one-dimensional structural members, such as a cantilever

beam, a circular ring and a free–free beam, each containing

a crack or notch.

Symond and co-workers [10,11] showed that the r-p-p

material idealization is applicable for the structural impact

problem if the input energy is much larger than the

maximum elastic energy that can be stored in the structure

and the applied pulse is shorter than the fundamental period

of the elastic vibration of the structure. It should be noted

that for most cases of structures subjected to intensedynamic loading, the energy ratio R of the kinetic energy

input to the maximum possible elastic energy is at least

greater than 3 and the loading time is short enough.

Therefore, the r-p-p model can serve as a reasonable first-

order theory.

The remaining problem which needs to be solved is how

to select a deformation mechanism. Theoretically, a perfect

r-p-p model should be established on so-called ‘complete

solutions’ which satisfy the equation of motion, the force

boundary condition and do not violate the yield criterion at

any point in the structure. These kinds of complete solutions

provide a good estimation of the deformation mechanisms

of the structure during the response process, including the

permanent deformation and the distribution of the dissipated

energy in the structure, compared with that predicted by a

mode solution.

In the present paper, a free–free beam with an initialnotch along its span (see Fig. 1) under a concentrated step-

loading suddenly applied and maintained force at one end of

the beam is studied. Because of the effect of the notch on the

dynamic response of the beam, the admissible deformation

mechanisms are much more complicated. The main purpose

is to obtain a complete solution based on the r-p-p model

through multi-hinge deformation mechanisms, which

was first suggested by Hua et al. [12] for analysis of

either a stepped or sharply bent beam and later was

adopted to study the dynamic behaviour of a simply

supported beam with an initial notch under step and

impulsive loading [4].

The basic assumptions are the same as in Ref. [6]; theyare:

(1) The material of the beam is rigid, perfectly plastic and

rate independent, so that the dynamic fully plastic

bending moment M p is constant.

(2) The beam has a uniform section and density and its

Nomenclature

C stationary hinge at the notch siteE c energy dissipation rate at the notch

E h energy dissipation rate in the hinge system

E in total input energy rate

H 1 positive-yielding hinge located on the rightside of the notch

H 2 positive-yielding hinge located on the left side

of the notch

H r reverse-yielding hinge

M bending moment

M p fully plastic bending moment

P transverse step load

Q shear force

a transverse accelerationb distance of notch away from loaded point

f non-dimensional load PL = M p f c non-dimensional critical value of load

L length of beam

m mass per unit length of beam

t time

w transverse displacement

x distance of hinge away from loaded point

g reduction factor

a angular acceleration

u rotation anglel non-dimensional bending moment M = M p

t non-dimensional time t ð M p=mL 3Þ1=2

b non-dimensional distance of the notch away

from loaded point b= L

j non-dimensional variable x= L

h non-dimensional variable xhr= L

I region corresponding to mode rigid-body

motion

II region corresponding to mode C

III region corresponding to mode H 1IV region corresponding to mode H 2V region corresponding to mode C – H 1VI region corresponding to mode H 2 – C

VII region corresponding to mode H r – C

VIII region corresponding to mode H r – C – H 1

Fig. 1. (a) Rigid, perfectly plastic free–free beam. (b) Internal forces acting

on an element.

Y. Zhang, J.L. Yang / International Journal of Pressure Vessels and Piping 79 (2002) 741–752742



cross-section possesses an axis of symmetry parallel to

the direction of the load.

(3) The deflection of the beam is small, so that only the

initial phase of the response is considered.

(4) The effect of the shear force is neglected in the yield

condition.

(5) The transverse dynamic load, P, is a concentrated step

force, i.e. a suddenly applied maintained, constant

force.

2. Analysis

2.1. Rigid-body motion

Consider a free– free beam with a notch at any site alongthe beam as shown in Fig. 1. The length of the beam is L and

mass mL . At initial time t ¼ 0; the beam is subjected to a

transverse step load P at one end as shown in Fig. 1. The

dynamic fully plastic bending moment of any cross-section,

apart from the notched section, is M p; and that of the

notched section is g M p with 0 , g # 1; where g depends on

the specific notch geometry [5]. Then g ¼ 1 corresponds to

no notch. When the deflection w is small, the translational

and rotational equations of motion for an element of the

beam are

›Q

› xþ m

›2w

›

t

2¼ 0 ð1Þ

› M

› x¼ Q ð2Þ

where M ð x; t Þ and Qð x; t Þ denote the bending moment and

transverse shear force, respectively. For the case of small

loads, the bending moment does not reach the fully plasticbending moment M p (or g M p). Consequently, the beam

moves as a rigid body and the motion can be divided into

two parts, i.e. translational motion of the mass centre with

acceleration, ac; and rotation with angular acceleration, ac:These are given by

ac ¼ P=mL ð3Þ

€ac ¼6P

mL 2ð4Þ

Thus, the acceleration of an arbitrary element located at x in

the beam is given by

›2w

›t 2¼

P

mL 2

L

22 x

!6P

mL 2ð5Þ

By substituting Eq. (5) into Eqs. (1) and (2), the bending

moment and shear force distribution are obtained as

Q ¼ x2P

L 2

3Px

L 2

!ð6Þ

M ¼ x2 P

L 2

Px L 2

!ð7Þ

for 0 # x # L : Let Q ¼ 0; Eq. (6) gives

x1 ¼ 0 or x2 ¼2

3L ð8Þ

According to the distribution of the bending moment along

the beam, the maximum absolute value of the bending

moment could occur at

x2 ¼2

3L ð9Þ

where the bending moment is

M 1 ¼4

27PL ð10Þ

On the other hand, the bending moment at the notch site is

M 2 ¼ ð L 2

bÞ

2

L 2Pb ð11Þ

Using non-dimensional quantities b ¼ b= L ; l ¼ M = M p; f ¼ PL = M p; then Eqs. (10) and (11) can be recast as

l1 ¼4

27f ð12Þ

l2 ¼ f b ð12 b Þ2ð13Þ

Setting l1 ¼ 1 and l2 ¼ g ; respectively, in the above

equations, yields

f 1 ¼27

4ð14Þ

f 2 ¼ g b ð12 b Þ2

ð15Þ

To make a comparison between f 1 and f 2; let f 1 ¼ f 2; then we

have

4g 2 27b ð12 b Þ2¼ 0 ð16Þ

This cubic equation has two positive real roots: 0 , b 1 #

b 2 , 1 for given g . With increasing magnitude of the step-

loading, if b 1 , b , b 2; the fully plastic moment is first

attained at the notched section. In contrast, for b 2 , b , 1

or 0 , b , b 1; the fully plastic moment is first attained at

an un-notched section. Therefore, according to the analysis,

a critical magnitude of f c ¼ f c1 is obtained. When f # f c1;the notched beam only moves as a rigid body. The

magnitude of f c1 can be expressed by

f c1 ¼

27

40 # b # b 1; b 2 # b # 1

g

b ð12 b Þ2b 1 , b , b 2

8>><>>: ð17Þ

In Sections 2.2–2.4, the parameter b is discussed for three

particular cases, i.e. 0 , b # b 1; b 1 , b , b 2 and b 2 #

b , 1:

2.2. Case 1 0 , b # b 1

The deformation mechanism of structures under dynamic

Y. Zhang, J.L. Yang / International Journal of Pressure Vessels and Piping 79 (2002) 741–752 743



loading is far more complicated than that of the structure

under static load. It not only depends upon the structural

parameter but also depends upon the intensity of the load.

For a particular magnitude of step loading, the initial

deformation mechanism does not change with time. There-

fore, the so-called change of the load in the following

analysis means the different values of the magnitude of the

step loading.

Analysis shows that there are eight deformation mech-

anisms of the structural response for this case.

2.2.1. Single-hinge mechanism ð H 2Þ

When f . f c1; a plastic hinge forms at H 2; which is a

distance of x from the loaded point B as shown in Fig. 2.

Here, €u 1 and €u 2 denote the angular accelerations of

segments AH 2 and H 2 B relative to AH 2; respectively. The

equation of motion for the single-hinge mechanism can be

expressed as

mð L 2 xÞ3 €u 1

12¼ M p ð18aÞ

mxL

2€u 1 þ

x

2€u 2

!¼ P ð18bÞ

mx2 3 L 2 x

12€u 1 þ

x

6€u 2

!¼ M p ð18cÞ

For convenience, the equation are rewritten in terms of

dimensionless variables defined as,

j ¼ x L

; t ¼ t ffiffiffiffiffiffiffi

M p

mL 3

s ; ð

_Þ ¼ d

dt ð Þ; ð

€Þ ¼ d2

dt 2ð Þ

Then Eqs. (18(a)–(c)) becomes

€u 1 ¼12

ð12 j Þ3

ð19aÞ

j 1

2€u 1 þ

j

2€u 2

!¼ f ð19bÞ

j 232 j

12€u 1 þ

j

6€u 2

!¼ 1 ð19cÞ

The solution for Eqs. (19(a)–(c)) can be expressed in terms

of j by

€u 1 ¼12

ð12 j Þ3ð20aÞ

€u 2 ¼6ð12 3j Þ

j 3

ð12

j Þ3

ð20bÞ

f ¼3ð12 2j Þ

j ð12 j Þ2ð20cÞ

It should be noted that the single-hinge mode may be

degenerated into the critical case of pure rigid-body motion

by setting €u 2 ¼ 0: Therefore, Eq. (20b) gives j ¼ 1=3 and

substituting this value into Eq. (20c) to obtain the critical

value for single mode, f c1 ¼ 27=4 which is coincident with

Eq. (17) for the case of 0 , b # b 1: However, the single-

hinge mode is invalid when f exceeds f c2 which is another

critical value (see Section 2.2.2) at which the single-hinge

mode changes to a double-hinge mode of response, because

the yield moment in the beam is reached at the notched siteother than that at the first hinge.

2.2.2. Double-hinge mechanism ð H 2 – C Þ

After examining the yield criterion for the previous

single-hinge mechanisms, when f c2 , f # f c3; a double-

hinge mode ð H 2 – C Þ is found as shown in Fig. 3 in which €u 1;€u 2 and €u 3 denote the angular accelerations of segments AH 2;

Fig. 2. Free body diagram for the mode ð H 2Þ:

Fig. 3. Free body diagram for the mode ð H 2 – C Þ:




H 2C relative to AH 2 and CB relative to H 2C ; respectively,

and x is the distance of the hinge H 2 from the loaded point B.

The dimensionless governing equations are

€u 1 ¼12

ð12 j Þ3ð21aÞ

j

2€u 1 þ

j 2

2€u 2 þ

b 2

2€u 3 ¼ f ð21bÞ

ðj 2 b Þ2 32 j 2 2b

12€u 1 þ

j 2 b

6€u 2

!¼ 12 g ð21cÞ

b 23 þ 3j 2 2b

12€u 1 þ

3j 2 b

6€u 2 þ

b

3€u 3

!¼ f b 2 g ð21dÞ

The equations can be solved by expressing€u 1;

€u 2;

€u 3 and f in

terms of j . They are given by

€u 1 ¼12

ð12 j Þ3ð22aÞ

€u 2 ¼ 26ðg 2 1Þ

ðj 2 b Þ3þ

6ð32 2b 2 j Þ

ðj 2 b Þðj 2 1Þ3ð22bÞ

f ¼

3ð2b 3j þ gj 2ðj 2 1Þ2 þ 2jb ð2j 2 1Þ þ b 2ð12 2j 2 3j 2ÞÞ

b ðb 2 j Þ2ðj 2 1Þ2

ð22dÞ

The double-hinge mode ð H 2 – C Þ may degenerate into a

single mode ð H 2Þ when €u 3 ¼ 0; and this leads to the

transition conditions from a single-hinge mode to a double-

hinge mode. From Eq. (22d), we have

f c2 ¼3 2b 3j c þ gj 2c ðj c 2 1Þ2 þ 2j cb ð2j c 2 1Þ þ b 2 12 2j c 2 3j 2c

À ÁÀ Áb ðb 2 j cÞ2ðj c 2 1Þ2

ð23Þ

where j c is obtained by the equation

3bj 2c ð12 2j cÞ2 gj 3c ðj c 2 1Þ2

þ b 3 12 2j c 2 2j 2c

þ 3j cb 221 þ 2j c þ j 2c

¼ 0 ð24Þ

Further analysis shows that with increase of f ð f . f c2Þ; the

relative angular acceleration, €u 2; between AH 2 and H 2C

decreases and when €u 2 ¼ 0; the double-hinge mode ð H 2 – C Þ

degenerates into a single mode (C ), i.e. only one hinge

forms at the notched site and the transition conditions

should satisfy

f c3

¼3 2b 3j c þ gj 2c ðj c 2 1Þ2 þ 2j cb ð2j c 2 1Þ þ b 2 12 2j c 2 3j 2cÀ ÁÀ Á

b ðb 2 j cÞ2ðj c 2 1Þ2

ð25Þ

where j c is given by

6ðg 2 1Þ

ðj c 2 b Þ32

6ð32 2b 2 j cÞ

ðj c 2 b Þðj c 2 1Þ3¼

0ð26

Þ

2.2.3. Single-hinge mechanism (C )

For the magnitude of step-loading further increasing to

the range of f c3 , f # f c4; the response of the deformation

mechanisms is in a single-hinge mode as shown in Fig. 4,

where the hinge (C ) forms at the notched site, in which a1;€u 1 and a2; €u 2 are accelerations of the mass centre and the

angular accelerations of sections AC and CB, respectively.

The dimensionless governing equation can be obtained from

Fig. 4, thus

ð12 b Þ €j 1 þ b €j 2 ¼ f ð27aÞ

ð12 b Þ2

2€j 1 þ g ¼

ð12 b Þ3

12€u 1 ð27bÞ

b 2

2€j 2 þ

b 3

12€u 2 ¼ f b 2 g ð27cÞ

€j 1 þ

12 b

2 €u 1 ¼ €j 2 2

b

2 €u 2 ð27dÞ

where €j 1 ¼ a1mL 2= M p; €j 2 ¼ a2mL 2= M p: The solutions are

given by

€j 1 ¼2 f b ð12 b Þ2 þ ð32 6b Þg

2b ðb 2 1Þ2ð28aÞ

€u 3 ¼6ð3bj 2ð12 2j Þ2 gj 3ðj 2 1Þ2 þ b 3ð12 2j 2 2j 2Þ þ 3jb 2ð21 þ 2j þ j 2ÞÞ

b 3ðb 2 j Þ3ðj 2 1Þ2ð22cÞ

Fig. 4. Free body diagram for the mode (C ).




€u 1 ¼3ð f b ð12 b Þ2 þ ð2b 2 3Þg Þ

b ðb 2 1Þ3ð28bÞ

€j 2 ¼2 f b ðb 2 3Þðb 2 1Þ þ ð32 6b Þg

2b 2ðb 2 1Þð28cÞ

€u 2 ¼3ð f b ðb 2 2 1Þ þ ð1 þ 2b Þg Þ

b 3ðb 2 1Þð28dÞ

2.2.4. Double-hinge mechanism (H r – C and C – H 1)

Through carefully examining the yielding condition, it is

found that for a given g , with increase of the magnitude of

the load, either a reverse yielding on the segment AC or a

positive yielding on the segment CB may take place. Which

one is first attained depends on the notch position, i.e. thevalue of b . Here, these two situations are discussed.

2.2.4.1. Reverse yielding of segment AC first attained .

Corresponding to the situation, for the magnitude of the load

in the range f c4 , f # f c5 the deformation mechanism of a

double-hinge mode ð H r – C Þ is shown in Fig. 5, where €u 1; €u 2and €u 3 are the angular accelerations of segments AH r; H rC

and CB, respectively, and j ¼ x= L represents the dimen-

sionless distance of the hinge H r from the loaded point B.

The non-dimensional governing equations are given by

€

u 1 ¼

12

ð12 j Þ3 ð29aÞ

2j ð12 j Þ

2€u 1 þ

b 2 2 j 2

2€u 2 þ

b 2

2€u 3 ¼ f ð29bÞ

ðj 2 b Þ2 12 j

4€u 1 þ

j 2 b

6€u 2

!¼ 1 þ g ð29cÞ

b 2 212 j

4€u 1 2

j 2 b

2€u 2 þ

b

3€u 3

!¼ f b 2 g ð29dÞ

whose solution can be obtained in terms of j by

€u 1 ¼12

ð12

j Þ3

ð30aÞ

€u 2 ¼18

ðb 2 j Þð12 j Þ22

6ð1 þ g Þ

ðb 2 j Þðj 2 b Þ2ð30bÞ

€u 3 ¼ 23ð12 j Þðb 2 2j Þ

2b 2€u 1 þ

3j ðj 2 b Þ

b 2€u 2 þ

6g

b 3ð30cÞ

f ¼ 26j

ðj 2 1Þ2þ

3gj 2 þ 6bj 2 3b 2

b ðb 2 j Þ2ð30dÞ

When €u 2 2 €u 1 ¼ 0; the above double-hinge mode ð H r – C Þ

degenerates into the single-hinge mode (C ), and this leads

to the transition condition from mode (C ) to mode ð H r – C Þ:The transition load obtained from Eq. (30d) is

f c4 ¼ 2 6j cðj c 2 1Þ2

þ 3gj 2c þ 6bj c 2 3b 2

b ðb 2 j cÞ2ð31Þ

where j c satisfies the following equation

22b 3 2 1 þ g ðj c 2 1Þ3

þ 3j c 2 6bj c þ 3b 2ðj c þ 1Þ ¼ 0

ð32Þ

2.2.4.2. Positive yielding of segment CB first attained .

When the magnitude of the load f is larger than f c4; a new

positive hinge H 1 will appears in the segment CB, and

therefore, a double-hinge mode ðC – H 1Þ becomes valid (see

Fig. 6). For this situation, the higher the magnitude of the

load, the closer the points between H 1 and B will be. In Fig.6, a1; €u 1; €u 2 and €u 3 are accelerations of the mass centre of

segment AC and the angular accelerations of segments AC ,

CH 1 relative to AC and H 1 B relative to CH 1; respectively,

and j ¼ x= L represents the dimensionless distance of the

hinge H 1 from the free end of the beam. The dimensionless

governing equations can be expressed as

ð12 b Þ2

2€u1 þ g ¼

ð12 b Þ3

12€u 1 ð33aÞ

Fig. 5. Free body diagram for the mode ð H r – C Þ: Fig. 6. Free body diagram for the mode ðC – H 1Þ:




ð12 j Þ€u1 þ ðb 2 j Þ12 j

2€u 1 þ

b 2 j

2€u 2

!¼ 0 ð33bÞ

ðb 2 j Þ2 1

2€u1 þ

3 þ b 2 4j

12€u 1 þ

b 2 j

3€u 2

!¼ 12 g

ð33cÞ

j €u1 þ1 þ b 2 j

2€u 1 þ

2b 2 j

2€u 2 þ

j

2€u 3

!¼ f ð33dÞ

j 21

2

€u1 þ3 þ 3b 2 4j

12

€u 1 þ3b 2 2j

6

€u 2 þj

6

€u 3 ! ¼ 1

ð33eÞ

where €u1 ¼ a1mL 2= M p: The solution is given by

€u1 ¼3ð1 þ b 2 2 2b ð1 þ gj 2 g Þ þ g ðj 2 2 1ÞÞ

2ðb 2 1Þ2ðj 2 1Þðb 2 j Þð34aÞ

€u 1 ¼3

ðb 2 1Þ3

3ðb 2 1Þðg 2 1Þ

b 2 j þ

3ð12 b Þ

j 2 12 g

ð34bÞ

€u 2 ¼ 23ð1 þ 2b 3 þ g ðj 2 1Þ3

2 3j þ 6bj 2 3b 2ðj þ 1ÞÞ

ðb 2 1Þ3ðb 2 j Þ3

ð34cÞ

€u 3 ¼ 6j 32 3

j €u1 2 3 þ 3b

2

4j 2j

€u 1 2 3b 2

2j j

€u 2 ð34dÞ

By setting €u 3 ¼ 0; the transition load from mode ðC – H 1Þ to

mode (C ) is obtained, i.e.

where j c is determined by

2b 4ðj c 2 1Þ2 6b 2j c þ bj 2c ð3 þ g ðj c 2 3Þðj c 2 1ÞÞ

þ 2gj 3c ðj c 2 1Þ þ b 3 2 þ 4j c 2 3j 2c

¼ 0 ð36Þ

2.2.5. Single-hinge mechanism ð H 1Þ

For some large values of g (for example, g ¼ 0:95),

when the external load is sufficiently high, the double-

hinge mode ðC – H 1Þ may develop into a single-hinge

mode ð H 1Þ: Single-modes H 1 and H 2 are the same

except that the positions of the hinges are different, i.e.hinge H 1 locates at the right segment of the notch but

hinge H 2 locates at the left segment of the notch. The

transition condition from a double-hinge mode to a

single-hinge mode can be determined by means of a

procedure similar to that in the previous sections. The

analysis for the single-hinge mode ð H 1Þ is similar to that

for the single-hinge mode ð H 2Þ: Here, the transition load

from the double-hinge mode ðC – H 1Þ to a single-hinge

mode ð H 1Þ is given in the same form as that of Eq.

(35), i.e.

where j c should satisfy

1 þ 2b

3

þ g ðj c2

1Þ

32

3j c þ 6bj c2

3b

2

ðj c þ 1Þ ¼ 0ð38Þ

2.2.6. Three-hinge mechanism ð H r – C – H 1Þ

The double-hinge mode ð H r – C Þ may develop into a

three-hinge mode when the external load is sufficiently high.

The transition condition from a double-hinge mode to a

three-hinge mode can be determined by means of a

procedure similar to that in the previous sections. Now

successive study is made on the double-hinge mode ð H r – C Þ

(containing a reverse yielding hinge) in Section 2.2.4.1 to

develop into the three-hinge mode ð H r – C – H 1Þ which isshown in Fig. 7 where €u 1; €u 2; €u 3 and €u 4 denote the angular

accelerations of segments AH r; H rC ; CH 1 and H 1 B;respectively. The notations, j ¼ xh1= L and h ¼ xhr= L are

the dimensionless distances of the hinges H 1 and H r from

the loaded point B, respectively. The dimensionless

f c5 ¼3 2b 3ðj c 2 1Þ þ 2bj cðj c 2 2Þ þ j 2c þ gj 2c ðj c 2 1Þ2 þ b 2 2 þ 2j c 2 3j 2c

À ÁÀ Á2j cðb 2 1Þðj c 2 1Þðb 2 j cÞ2

ð37Þ

f ¼3ð2b 3ðj 2 1Þ þ 2bj ðj 2 2Þ þ j 2 þ gj 2ðj 2 1Þ2 þ b 2ð2 þ 2j 2 3j 2ÞÞ

2j ðb 2 1Þðj 2 1Þðb 2 j Þ2ð34eÞ

f c4 ¼3 2b 3ðj c 2 1Þ þ 2bj cðj c 2 2Þ þ j 2c þ gj 2c ðj c 2 1Þ2 þ b 2 2 þ 2j c 2 3j 2c

À ÁÀ Á2j cðb 2 1Þðj c 2 1Þðb 2 j cÞ2

ð35Þ

Fig. 7. Free body diagram for the mode ð H r – C – H 1Þ:




governing equations are

€u 1 ¼ 12ð12 h Þ3 ð39aÞ

j 2 212 h

4€u 1 2

h 2 b

2€u 2 þ

b 2 j

2€u 3 þ

j

6€u 4

!¼ 1 ð39bÞ

j 212 h

2€u 1 2 ðh 2 b Þ €u 2 þ ðb 2 j Þ €u 3 þ

j

2€u 4

!¼ f ð39cÞ

ðh 2 j Þð12 h Þ

2€u 1 þ

ðh 2 b Þðb þ h 2 2j Þ

2€u 2

2ðb 2 j Þ2

2€u 3 ¼ 0 ð39dÞ

2ð12 h Þððh 2 b Þ2

2 ðb 2 j Þ2Þ

4

€u 1

þðb 2 j Þ2ðh 2 b Þ

22

ðh 2 b Þ3

6

" #€u 2 2

ðb 2 j Þ3

3€u 3 ¼ 22

ð39eÞ

and the solution is given by

€u 1 ¼12

ð12 h Þ3ð40aÞ

€u 2 ¼ 26ð22 þ h 2 þ 2bj þ j 2 2 2h ð22 þ b þ 2j ÞÞ

ðh 2 1Þ2ðh 2 b Þðh 2 j Þ2ð40bÞ

€u 3 ¼ 2 12ðh 2

j Þð12 h Þ2ðb 2 j Þ2 þ ðh

2b Þðb þ h

22j Þ

ðb 2 j Þ2€u 2 ð40cÞ

€u 4 ¼6

j 3þ

18

j ð12 h Þ2þ

3ðh 2 b Þ

j €u 2 2

3ðb 2 j Þ

j €u 3 ð40dÞ

The three-hinge mode ð H r – C – H 1Þ degenerates into a

double-hinge mode ð H r – C Þ when €u 4 2 €u 3 ¼ 0; and there-

fore, from Eq. (40e), the transition load is found

where j c and h c are determined by the following equations

bh cðh c 2 1Þ2

ðbh c 2 2h cj c 2 2bj cÞ2 j 2c ðb 2 h cÞ

Â h cðh c 2 1Þ2

þ b 21 þ 2h c þ 2h 2c

þ 2j 3c b þ h c þ 2h cb 22 2h cb 2 2h 2c

2 j 4c 1 þ b 2 þ h cb 2 2h c 2 h 2c

¼ 0 ð42aÞ

12 2b 2 þ g ðh c 2 1Þ2 þ 4bh c 2 2h c 2 h 2c

ðh c 2 1Þ22

2ðb 2 h cÞ2

ðh c 2 j cÞ2

þ2ðh c 2 b Þ3

ðh c 2 j cÞðh c 2 1Þ2¼ 0 ð42bÞ

On the other hand, for some values of b , the three-hinge

mode ð H r – C – H 1Þ would degenerate into a double-hinge

mode ðC – H 1Þ when setting €u 2 2 €u 1 ¼ 0; and therefore,

from Eq. (40e), the transition load is found in the same form

as that of Eqs. (41), i.e.

and j c and h c should satisfy

222 3h 2c þ h 3c 2 2bj cðj c 2 1Þ þ j 2c

þ h c 6 þ 2b ðj c 2 1Þ2 4j c þ j 2c

¼ 0 ð44aÞ

12 2b 2 þ g ðh c 2 1Þ2 þ 4bh c 2 2h c 2 h 2c

ðh c 2 1Þ22

2ðb 2 h cÞ2

ðh c 2 j cÞ2

þ2ðh c 2 b Þ3

ðh c 2 j cÞðh c 2 1Þ2¼ 0 ð44bÞ

2.3. Case 2 b 1 , b , b 2

For this case, a plastic hinge forms at the notched section

in the beam indicating that a single-hinge mode (C ) is validwhen f . f c1: With increase of the magnitude of the load,

the deformation mechanism mode development is similar to

that of the analysis in Case 1, in which both reverse and

positive yielding situations need to be discussed, respect-

ively, and therefore, the details are omitted here. For this

case, there are six dynamic deformation mechanisms, which

are summarized in Appendix A.

f ¼23b ðh 2 j 2 1Þðh 2 j Þðh þ j 2 1Þ þ 3j ðh 2ðh 2 2Þ þ h ðj 2 1Þ2 þ j Þ

j ðh 2 1Þ2ðh 2 j Þðj 2 b Þð40eÞ

f c5 ¼23b ðh

c

2 j c

2 1Þðh c

2 j c

Þðh c

þ j c

2 1Þ þ 3j c

h 2

c

ðh c

2 2Þ þ h c

ðj c

2 1Þ2 þ j c

À Áj cðh c 2 1Þ2ðh c 2 j cÞðj c 2 b Þ ð41Þ

f c6 ¼23b ðh c 2 j c 2 1Þðh c 2 j cÞðh c þ j c 2 1Þ þ 3j c h 2c ðh c 2 2Þ þ h cðj c 2 1Þ2 þ j c

À Áj cðh c 2 1Þ2ðh c 2 j cÞðj c 2 b Þ

ð43Þ




2.4. Case 3 b 2b # 1

For this case, only the single-hinge mode ð H 1Þ is

available when f . f c1: The analysis is similar to that

described in Case 1.

3. Partitioning of energy dissipation rate

For f , f c1; the input energy to the beam will be totally

transferred into the kinetic energy of rigid-body motion

including both the translational and rotational motions.

When f . f c1; however, a part of the input energy will be

dissipated in the form of plastic work at hinges. For the early

time response of the beam, a convenient way of quantifying

this effect is to calculate the ratio of the rate of energydissipation in the hinge system E h; to the rate of energy

input to the beam by the applied force, E in: As the

deformation modes of the beam depend upon both the

magnitude of the load and the position of the notch,

the partitioning of the input energy rate varies for different

cases, which will be discussed in Section 4. Since the initial

value of _u i is zero, while €u i remains constant, we have

_u i= _u j ¼ €u i= €u j ð45Þ

According to the analyses in the previous sections, thefollowing equations are easily obtained:

E h=E in ¼g ð €u 2 2 €u 1Þ

f ½ €j 2 þ 0:5b €u 2

Mode ðC Þ ð46Þ

E h=E in ¼€u 2

f ½0:5ð1 þ j Þ €u 1 þ j €u 2Mode ð H 1 and H 2Þ

ð47Þ

E h=E in ¼€u 2 þ g €u 3

f ½0:5ð1 þ j Þ €u 1 þ j €u 2 þ b €u 3Mode ð H 2 – C Þ ð48Þ

E h=E in ¼g €u 2 þ €u 3

f ½€u1 þ 0:5ð1 þ b Þ €u 1 þ b €u 2 þ j €u 3

Mode ðC – H 1Þ

ð49Þ

E h=E in ¼

2 €u 1 þ ð1 þ g Þ €u 2 þ g €u 3

f ½20:5ð12 j Þ €u 1 þ ðb 2 j Þ €u 2 þ b €u 3

Mode ð H r – C Þ

ð50Þ

E h=E in ¼2 €u 1 þ ð1 þ g Þ €u 2 þ ðg 2 1Þ €u 3 þ €u 4

f ½20:5ð12 h Þ €u 1 þ ðb 2 h Þ €u 2 þ ðb 2 j Þ €u 3 þ j €u 4

Mode ð H r – C – H 1Þ ð51Þ

4. Numerical calculation and discussion

According to the above analyses, for a given defect-severity of the notch the complete solution of the

deformation mechanisms of a rigid, perfectly plastic free–

free beam subjected to step-loading at one end with a notch

at any cross-section along the span can be obtained. The

defect-severity and notch position can be expressed by the

parameters g and b , respectively. The initial deformation

mechanisms for the notched beam depend not only on the

magnitude of the load, f , but also on the defect-severity and

the position of the notch, i.e. g and b . The numerical

solutions are shown in Figs. 8– 10 for the deformation

mechanisms corresponding to g ¼ 0:1; 0.5 and 0.8,

respectively. On the other hand, from the above

discussion, it is known that the partitioning of energy

dissipation rate also depends upon the values of the

parameters f , g and b .

(1) For a given g , the different initial deformation

mechanisms for the beam can be represented by a map in the f 2 b plane (see Figs. 8–10). The maps consist of eight

regions corresponding to different deformation mechan-

isms. They are pure rigid-body motion (region I); single-

hinge modes H 1 or H 2 (regions III and IV); single-hinge

mode C (region II); double-hinge mode C – H 1 (region V);

double-hinge mode H r – C (region VII); double-hinge mode

H 2 – C (region VI); and three-hinge mode H r – C – H 1 (region

VIII), respectively. It is shown that the smaller is the value

of g , the wider is the range in which the first hinge will form

at the notched position when f . f c1 and it is necessary to

consider the reverse yielding situation. The relation between

g and b is expressed by Eq. (16).(2) Figs. 8–10 show that when b satisfies the condition

b 2 , b # 1 (region III), no plastic hinge forms at the

notched position no matter what magnitude of the step-

loading is applied at the free end of the beam. This implies

that the deformation mechanisms of the notched beam are

the same as those of an un-notched beam. For this case, if

f # 27=4; there is only a rigid-body motion, and a single-

hinge mode ð H 1Þ is always valid if f . 27=4 even as f tends

to be infinitely large. The distance between the loading point

Fig. 8. Response modes of a notched free–free beam subjected to step-

loading (map in f 2 b plane for g ¼ 0:1).




and the hinge H 1 can be expressed from Eq. (20c) by

f ¼3ð12 2j Þ

j ð12 j Þ2ð52Þ

with j , 1=3: The equation indicates that j ! 0 as f !1:(3) For b ¼ 0:06 (i.e. 12 b ¼ 0:94) and g ¼ 0:8 (see

Fig. 10) which is a typical and most complicated case for

which the beam will undergo all of the response modes

represented by the eight regions, it is found that 0 , f #

f c1 ¼ 6:75 corresponds to pure rigid-body motion; f c1 ,

f # f c2 < 21 corresponds to a single-hinge mode ð H 2Þ;

f c2 , f # f c3 < 35 corresponds to a double-hinge modeð H 2 – C Þ; f c3 , f # f c4 < 64 corresponds to a single-hinge

mode (C ); f c4 , f # f c5 < 66 corresponds to a double-

hinge mode ð H r – C Þ consisting of a reverse yielding hinge;

f c5 , f # f c6 < 73 corresponds to a three-hinge mode

ð H r – C – H 1Þ and finally, f c6 , f corresponds to a double-

hinge mode ðC – H 1Þ:(4) The partitioning of the input energy rate under given

situations is shown in Fig. 11. It is observed that the

maximum rate of energy dissipation in plastic hinges is

always less than 1/3 of the total input energy rate if b 2 ,

b # 1 indicating that the notch has no influence on the

deformation mode of the free–free beam. In other words,

when f . f c1 a single-hinge mode ð H 1Þ is always valid. With

increase of load, the energy dissipation rate will be close to 1/3

of the input energy. This behaviour is the same as that of the

un-notched free–free beam for which it has been demon-

strated that the plastic dissipated energy is always less than

1/3 of the input energy [1]. However, for some other values

of b (0 , b # b 1 or b 1 , b # b 2) the maximum rate of

energy dissipation will greater than 1/3 of the total input

energy rate. Moreover, the smaller is the value of b , the

greater is the maximum rate of energy dissipation.

(5) Figs. 12 and 13 show the variation of the ratio of the

energy rate dissipation in the hinge at the notch section, E c;to E in with notch position b for f ¼ 15 and 50, respectively.

It is observed from Fig. 12 that the maximum values of theratio are about 75, 50, 38 and 35% for g ¼ 0:1; 0.5, 0.9 and

0.95, respectively. In particular, there are four regions

corresponding to different defect-severity, in which the

plastic hinge at the notch section dissipates more input

energy. For example, the regions for g ¼ 0:1 and 0.5 are

close to b ¼ 0:015 and 0.09, respectively. This indicates

that if a notch falls into the region, fracture may easily take

place. It is clear that the curve describing the region

decreases sharply for small g , such as g ¼ 0:1; 0.5 and

decreases gently for large g , such as g ¼ 0:9; 0.95. Fig. 13

shows that for sufficiently high load, the region, where the

notch dissipates more input energy is gradually close to the

impact point. This indicates that the notch section mayeasily be broken when it is close to the impact point for a

high load situation. However, for a notch which is away

from the region, the fracture cannot happen at the notch

section even for small value of g .

5. Conclusions

The complete solution based on a r-p-p model for the

initial, small deflection response of a free–free beam with





Fig. 11. Dissipated energy rates of the hinges under different positions of

the notch.




an initial notch along its span consists of eight deformationmechanisms, characterized by different combinations of

multi-hinge modes. In the analysis, particular attention has

been paid to the partitioning of energy dissipation rate of the

hinges, as well as that of only the hinge at the notch site for

some typical situations. This will be helpful for the failure

analysis of the notched (or cracked) beam. It is concluded

that: (i) the different initial deformation mechanisms and the

initial energy dissipation rate of the beam depend not only

on the magnitude of the load but also on the defect-severity

and location of the notch; (ii) because of the influence of the

notch, the structural response of the beam is far more

complicated than that of the un-notched free–free beam;(iii) for some cases, the maximum rate of energy dissipation

in plastic hinges will be more than 1/3 of the total input

energy rate, while for an un-notched free – free beam, it has

been demonstrated that the plastic dissipated energy is

always less than 1/3 of the input energy [1]; and (iv) the

analysis in this paper verified that the effect sensitive region,suggested by Chen and Yu [9], is highly dependent on the

severity of the defect and that a more severe defect leads to a

large sensitive region as shown in Figs. 8–10.

Acknowledgments

The work described in this article was supported

financially by grant number 19972006 under the Chinese

Nature Foundation. The authors would like to acknowledge

the support.

Appendix A. The conclusion of the deformation modes

For 0 , b # b 1

Fig. 12. Dissipated energy rates of the hinge at the notch site for f ¼ 15: Fig. 13. Dissipated energy rates of the hinge at the notch site for f ¼ 50:




For b 2 # b # 1

For b 1 , b , b 2

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