ACTA MECHANICA SINICA (English Series), Vo1.12, No.4, Nov. 1996 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567-7718

A N A N A L Y S I S OF T H E S H E A R F A I L U R E OF R I G I D - L I N E A R H A R D E N I N G B E A M S U N D E R I M P U L S I V E LOADING* Wang Lili (Li-lih Wang ~r

Norman Jones +

(Ningbo University, Ningbo 315211, China and also University of Science and Technology of China, Hefei 230026, China) +(Impact Research Centre, Department of Mechanical Engineering, University of Liverpool, P.O.Box 147, Liverpool L69 3BX, U.K.) A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which takes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with N,onaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams. ABSTRACT:

KEY WORDS: shear failure, rigid-linear hardening plastic beam, dynamic response, impulsive loading, adiabatic shear

1 INTRODUCTION The dynamic inelastic response of beams when subjected to large dynamic loads has been investigated by many authors [1]. It has been recognized that due to the additional influence of transverse inertia, transverse shear force effects play a more important role in the dynamic response than in the quasi-static behaviour of beams. Nonaka introduced the so-called plastic shearing 'hinge' (slide) for the dynamic shear response of beams, although it was limited to the simplest rigid-perfectly plastic beams [zl. Refs. [3~5] further developed the plastic shear hinge concept to take account of the influence of strain hardening for the dynamic plastic behaviour of ideal fibre-reinforced rigid-plastic beams. These studies focused largely on predicting the transverse displacements of beams. It should be noted that most of the above analyses are, explicitly or implicitly, based on the assumption that the beam material has a large enough ductility so that it only fails due to excessive displacement. However, an excessive displacement is only one of the possible modes of failure of a beam. In fact, as pointed out by Menkes and Opat [6] and in Ref. [7], for beams subjected to impulsive velacities, at least three failure modes may exist, i.e., large ductile deformations(Mode I), tensile-tearing (Mode II) and transverse shearing (Mode III). For the latter two modes, a beam fails due to excessive tensile (or bending) stress/strain (for Received 2 June 1995, revised 23 May 1996

Vo1.12, No.4

Wang & Jones: Shear Failure of Rigid-Linear Hardening Beams

339

Mode II) or an excessive transverse shear stress/strain (for Mode III), if a critical stress or strain criterion of failure is assumed. Thus, a satisfactory analysis for the dynamic failure of a beam requires predictions for the corresponding dynamic stresses and strains. The critical stress or strain criteria of failure mentioned above are generally material strain rate sensitive. Thus, in a complete analysis of a dynamic beam problem, it is necessary to predict the corresponding strain-rates too. All of these features were not studied in Refs. [2~5], nor examined by other researchers. Furthermore, with regard to the transverse shear failure mode, it has been recognized that two essentially different failure mechanisms exist especially at high strain rates. One is the conventional transverse shear failure, for which the fracture results mainly from the nucleation and growth of microcracks (for brittle failure) or micro-voids (for ductile failure). Another is an adiabatic shear failure which is characterized by a highly localized adiabatic shear band. Both experimental and theoretical investigations have shown that the process of adiabatic shearing is governed by a thermo-visco-plastic instability criterion with two control variables, which provide a critical relation between the strain and strain-rate for a given environmental temperature [s~11]. Thus, it is necessary to obtain the strains and strain-rates in order to predict the possible occurrence of an adiabatic shear failure in a dynamically loaded beam. In the present paper, the basic character of a travelling plastic shear hinge in a rigidlinear hardening plastic beam, is first studied. As an example, the dynamic shear responses, including shear strain, strain-rate as well as the transverse displacement, are analysed in detail for a simply supported beam loaded impulsively. Furthermore, the possible occurrence of different failure mechanisms, i.e., excessive displacement, excessive transverse shear strain and adiabatic shearing, and their relations are discussed. An analytical method for predicting the dominant failure mode is suggested. Finally, the special case for an infinitesimally small plastic zone is examined and compared with Nonaka's solution. It is found from the analytical solution obtained that the proposed travelling shear hinge approach, with the assumption that shear force will take precedence to yield than bending moment, is suitable for solving the posed impulsively loaded beam problem within the range

Q~ ( I

< 1

where L8 stands for the dimensionless length of the plastic zone, L8 = 1 - exp (

mCQVo ) Q0

in which, Q0 and M0 are yield shear force and fully plastic bending moment at yield respectively, L and m are half-span and mass per unit length of the beam respectively, while CQ and V0 are travelling velocity of the shear hinge and initial impulsive velocity respectively. The problem which the above range is exceeded may be solved by the travelling shear hinge approach too but with another assumption that joined yielding of shear force and bending moment must be considered. T R A V E L L I N G P L A S T I C S H E A R H I N G E (SLIDE) F O R A R I G I D - L I N E A R HARDENING PLASTIC BEAM The beam is assumed to be made from a rigid-linear hardening plastic material with

340

ACTA MECHANICA SINICA (English Series)

the relation ~=0

for Q < Q o

Q=Qo+Gp'~

for Q>_Qo

1996

qt and

~'_>0 Q0! (1) where Q is the transverse shear force, 7 is the | transverse shear strain, Q0 is the yield shear force and Gp the linear hardening modulus (Fig.l). Consider a shear plastic 'hinge' (slide) 0 travelling along the neutral axis x of a beam Y with a velocity ~ : d~/dt, where ~ is the loFig.1 An idealized rigid-linear hardening cation of the shear hinge on the x axis. Let plastic Q ,~ -/relation w denote the transverse displacement, w -Ow/at'the transverse velocity, p the density of the material, A0 the area of a beam crosssection and m = pAo the mass per unit length of a beam. Thus, momentum conservation and displacement continuity across a travelling shear hinge requireD,12] I I

1

[Q] =

(2)

=

(3)

respectively, where [P] denotes the difference between the values of quantity P on either side of a hinge or other singular interface. From Eqs.(2) and (3), we have

[Q] rob]

(4a)

which, in the present case of a rigid-linear hardening plastic beam becomes

= d=~/r-~p/m ----•

(4b)

where the plus or the minus sign corresponds to a transverse shear hinge travelling in the positive or the negative direction along the x axis, respectively. Eq.(4) shows that the travelling velocity of a shear hinge, ~, is determined mainly by the Q-'y characteristics of a beam. It should be emphasized that the values of Q0 and Gp in Eqs.(1) and (4b) are generally rate-dependent. It is evident that a shear hinge is stationary in a perfectly plastic material as observed by Nonaka and also in other theoretical solutions using a perfectly plastic material [1]. Note that in such a case the displacement continuity condition, Eq.(3), requires that the transverse particle velocity, dJ, has to be continuous across a stationary transverse shear hinge, although the shear strain 7 may be discontinuous. Thus, it transpires that a 'rigid-perfectly plastic beam' assumption leads to a stationary transverse shear hinge with non-unique values of the shear strain and shear strain rate. Neither shear strain nor strain-rate can be predicted in a failure analysis using a shear plastic hinge model when plastic hardening is neglected. The importance of retaining plastic hardening (dQ/d~, > 0) in a theoretical analysis provides a motivation for the following section.

Vol.12, No.4

Wang & Jones: Shear Failure of Rigid-Linear Hardening Beams

3 A SHEAR FAILURE ANALYSIS FOR A SIMPLY SUPPORTED LOADED IMPULSIVELY 3.1

341 BEAM

Transverse Displacement,

Shear Strain and Shear Strain-rate A simply supported beam subjected to a uniformly distributed impulsive velocity, V0, as shown in Fig.2, is analysed as an example of the rigid-linear hardening plastic shear hinge model discussed in the previous section.

~"

---X---

)

//

M

V0

ILllllllltl

t

L

Fig.2 Simply supported beam subjected to a uniformly distributed impulsive velocity, V0 Momentum conservation requires OM = Q Ox

v~ = m~ Ox

(5a, b)

where M is the bending moment acting on the cross-section of a beam. The rotatory inertia effects are disregarded in the derivation of (5b). The initial and boundary conditions for the problem in Fig.2 are w = 0

zb : V0

for

t = 0

(6a, b)

Q -- 0

for

x : 0

(6c)

M = 0

for

x = ~L

(6d)

Only one-hMf of the beam, 0 < x < L, is considered henceforth owing to the symmetry about x -- 0. If a plastic shear hinge travels in the negative direction of x with velocity - C Q from x -- L towards the beam mid-span, then the jumps in the transverse shear force, shear strain and transverse particle velocity across the plastic hinge are, respectively [Q] = ( - Q ~ ) - ( - Q o )

(7a)

[7] = (-~1) - 0

(7b)

[w] =

(7c)

0 -

w

where the subscript 1 denotes the quantity just behind the travelling hinge. Thus, Eq.(2) and Eq.(3) at the travelling hinge now, respectively, reduce to

ACTA MECHANICA SINICA (English Series)

342

q l - Q0 = m c q r

(8a)

(v = CQ7I

(8b)

Since both parts of the beam on either side of the travelling hinge are rigid, the transverse velocity distribution with respect to x for t > 0 is (Fig.3a) ~b -- 17d

for

0< x < ~

(9a)

= 0

for

~< x < L

(9b)

1996 L

oI

W

j~ (a)

(

~

~

~

b

)

Q~ Q,

where l~(t) denotes the transverse velocity at x = 0, ~ is the transient location of the transverse shear hinge and in the present case = L - CQt

(10)

The transverse shear force distribution can be obtained from Eq.5(a) Q=m~idx

for

0 0) is theoretically infinite when [~/] ~ 0. Therefore, in order to estimate the strain rate in a beam from an

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ACTA MECHANICA SINICA (English Series)

1996

engineering viewpoint, the absolute value of average shear strain, %v, over the localized plastic zone, Ls, is defined as

7av = I

fL-L,Z'7(x)dx[ = WI L8

(20a)

L8

when using Eqs.(17) and (18). The average strain rate, ~/av, can be determined approximately

as ~%v -- 7~v _

Ts

WI

(20b)

L,T,

Thus, not only the transverse displacement, but also the transverse shear strain and an estimate for the shear strain-rate can be predicted by means of the travelling rigid-linear hardening plastic shear hinge mode] suggested above. However, further analysis is required to obtain the bending m o m e n t distribution, in order to determine whether or not the foregoing theoretical analysis satisfiesthe associated

yield condition. From Eqs.(5b), (i0), (n), (12) and (19), together with the boundary condition for the bending m o m e n t at x = ( = (L - CQt), the bending m o m e n t distribution

M=QI(L-x):Q,L[I+

ln(1-

M=Q1CQt + _ ~ ( I _ CQt~[,I_ Qo [CQt ln(1 +EL-Z-

is

(L-CQt) 0. Therefore, the present inequality (35) is more reasonable, and, in fact, corresponds to the end of motion (t -- Ts). The case for v* > 1 will be analysed in a future paper.

4.3 Fibre-Reinforced Rigid-Plastic B e a m The analyses and discussions given above can be generalized to fibre-reinforced beams without significant difficulty, if the fibre-reinforced effect for beams is regarded as a strengthening effect which substantially leads to an increase of M0, and a consequent decrease of u and ~,*. In fact, the various analyses in Refs. [3] to [5] for the dynamic behaviour of ideal fibre-reinforced (strongly anisotropic) beams all correspond to the case of M0 --+ co or u*