Strain estimation for external event loads

Nuclear Engineering and Design 96 (1986) 437-440 North-Holland, Amsterdam

STRAIN K.A.

ESTIMATION

PETERS

FOR EXTERNAL

437

EVENT

LOADS

and K.A. BUSCH

lnteratom GmbH, Friedrich Ebert Strasse, 5060 Bergisch-Gladbach, Fed. Rep. Germany Received December 1985 It is common knowledge that structural design on the basis of stress limitation leads to an overestimation of the damage potential of low cycle dynamic loads (e.g. seismic events). Until now the high effort needed for step-by-step numerical evaluation of nonlinear dynamic structural response is a serious obstacle for a more realistic design based on strain limitation. The concept of energy dissipation and structural hardening as functions of strain range leads to the definition of a set of l-DOF-substitute structures gained by a limited number of quasi-static non linear computations or by a certain knowledge of the yield behaviour of structural members. Step-by-step numerical computations are restricted to the substitutes and thus the effort for a strain limitation design is reduced to a bearable level.

1. I n t r o d u c t i o n

estimation m e t h o d relying on certain non-linear structural parameters. For a more stringent treatment see ref.

F o r low cycle d y n a m i c load cases (e.g. aircraft crash or earthquake) a design on the basis of strain limitation is desirable for economical reasons as well as with respect to safety aspects. C o m p a r e d to stress limitation design the following p r o b l e m s have to be resolved: - load characterization; the so-called response spectrum is not sufficient. - d e t e r m i n a t i o n of material data; allowable strain ranges of 2% would lead to significant impacts on the design. - d e v e l o p m e n t of manageable strain evaluation methods. All three problems are treated at the moment. Especially the third p r o b l e m has already found some attention in the past without being solved satisfactorily. Two m a i n approaches can be distinguished: the m e t h o d of reduced response spectra based on the concept of displacement ductility (e.g. ref. [1]) a n d iterative procedures to gain the socalled equivalent stiffness and d a m p i n g matrices (e.g. ref. [2]). The first a p p r o a c h actually characterizes the excitation while the concept of equivalence of the second a p p r o a c h uses detailed knowledge a b o u t the structure but is essentially based on steady state response (see ref. [3]). Further approaches have been formulated by H a b i p and Widuch [4], Keintzel [5] and W o e r n e r [6]. The present p a p e r delivers an outline of a strain

[7]. 2. L o g i c The main idea is to describe a structure by a sufficient set of characteristic parameters. If these parameters are known, simple " s u b s t i t u t e s " representing the given p a r a m e t e r set are examined. Therefore, a description of the actual structural motion is not available so that the characteristic parameters must be functions of the failure relevant strain quantity. It is assumed that the m a x i m u m equivalent strain range delivers the failure criterion. As a matter of fact this assumption influences also the investigations to be carried out for material and excitation.

3. I d e n t i f i c a t i o n o f c h a r a c t e r i s t i c p a r a m e t e r s

F o r the sake of simplicity a n d without too great a loss of generality elastic-plastic material defined by Y o u n g ' s modulus E a n d yield stress % is assumed (see fig. 1).

3.1. A special class of 1-DOF-structures To develop ideas with respect to what quantities control the nonlinear structural behaviour, a class of

0029-5493/86/$03.50 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)

438

K.A. Peters, K.A. Busch / Strain estimations

6y ~

orc~g

Fig. 1. Elastic-plastic stress-strain relation. 1-DOF-structures consisting of a special configuration of massless truss elements with areas A i and lengths l, carrying a rigid mass m is investigated (see fig. 2). As excitation a fixed point motion Z ( t ) is assumed. X ( t ) is the relative desplacement response, l 3 is chosen big enough so that yielding of truss 3 is excluded. Obviously, truss 2 will not yield due to A 1 < A 2. It can easily be seen that with respect to strain behaviour these structures can be described by four parameters. The linear behaviour is fixed by co = circular frequency, p - u = linearily c o m p u t e d stress due to unity acceleration. The nonlinear b a h v i o u r is to be described as a function of strain range c. Let F and X be the (static) force acting on m and the corresponding displacement respectively. Let the hardening coefficient s, the ductility coefficient d and the volume coefficient v be defined according to figs. 3 5. In this simple case s, d and v do not depend on c. The meaning of s and d is obvious. It can easily be shown that

S(E)=fgtgo O ~ Fig. 3. Hardening coefficient. energy dissipated within the "yielding volume" A 1/1 to the m a x i m u m possible energy dissipation a n d therefore measures strain concentration. Theorem:

d=s+v.

With respect to strain response the four parameters

x(~.]X(~y)

I

t

I

I

~y

E

-_

t9 S

d ( E ) = - -~g ~ Fig. 4. Ductility coefficient.

~ -'Y F(,y)X((y)

(y

is the m a x i m u m possible energy dissipation for given c, ~, P a n d m. So the volume coefficient v compares the

F(~)F(Ey)-

A2

,

F-Z]

~

l

I

I

I

I

I

I

I

[--,--z(t)

L,..x(t), z(t)

I

X (£y)

z(t)~

Fig. 2. A special class of 1-DOF-structures.

v(E)=

A1 I~ ~y ( E - Ey) E-~y F(Ey)'X(£y) Ey

Fig. 5. Volume coefficient.

X(£)

439

K.A. Peters, K.A. Busch / Strain estimations

w, p, s and v describe the I-DOF-structure in question completely which therefore are referred to as the four parametric class R(o, p, S, u). 3.2. Generalization Let now a multi DOF-structure be given: = natural circular frequencies, a, to 1 with respect to ‘PC = mode shapes normalized mass matrix M, factors (with respect to exY, = modal participation citation in question), = modal force distributions, F, = modal stress distributions (linear). 0, Fundamental for the generalization of s and u is the concept of the quasistatic process. A quasi-static process in cp,-direction 0 + aF, is a force time-history f( t)F, so that 0 t y let 0 + (Y,(6) F, be the process inducing the equivalent strain z within the structure. Let X,(C) be the corresponding displacement distribution (in general not proportional to cp,) and H,(C) the energy dissipated by the process.

That means that u,(t) is the ratio of the energy to be stored in the yielding volume T:(E) and the total modal energy. Taking advantage of the additivity of the yielding volume it is possible to estimate u and s using nonlinear properties of singular structural members and so avoiding the explicit computation of quasi-static processes (see ref. [7]).

4. Strain estimation In the case of structural response essentially given by the ith mode it would suggest itself to use the I-DOFstructure R( w, p, s,(c), u,(c)) as a substitute structure if an allowable strain range is given. Naturally this assumption is not correct in general. The idea is to use modal interaction to weaken the substitutes and leave the selection of the “worst case” to the excitation. Definition by =

2

v,(e)

=

@t

min ~ is called the “modal P Y,O,(P> ficient” ( p structural locations).

=

is

called

H,(e)

,

=

=

b 2

is called “structural

load coefficient”.

OY

is called a “representative of the to E and Z”, if for the maximum strain range of R(w,, p, si(c), u,(c)) undergoing the excitation Z(t) E, < c is valid. Obviously, by is the acceleration which can be taken by the structure within elastic limits if RSS-super-imposition of modal stresses is assumed. We are now ready to formulate the strain estimation (hypothesis of the worst mode): let an excitation Z(t) be given and an allowable strain range E. If for all i the with reR(w,, 6, S,(E), u,(e)) are modal representatives spect to Z and C, then t is an upper limit for the strain range of the structure undergoing Z(t). It should be remarked that modes with p, > fi must not be taken into account with respect to the worst mode hypothesis. R(w,>

the

“modal

yielding

is

I

called

the

“modal

volume coefficient”,

s,(e)

b

j3,

s,(c),

U,(E))

i th mode with respect

H,(e)

=

load coef-

ey(e-ey) volume”. o,(e)

called “structural

design levkl”.

Definitions PI

eY(m~~($e,~p))‘l~‘is

--%(C> %(ey) c --1

1 is called

the “modal

hardening


coefficient”. d,(e)

= s,(c)+u,(c) is called the “modal ductility coefficient”. It is an easy task to show that w,, p,, s, and o, are generalizations of the corresponding values of section 3.1. s, and v, in general depend on c. The volume coefficient can be interpreted as follows:

V,(E)=

+V,(c)cV%

$xf(E,)Wf

5. Examples The strain estimation method has been tested by numerous examples. It always gave conservative results. The overestimation of the strain range is about 1.5 to 2.0 for a structure of some complexity (and naturally lower for the more simple ones). Two examples shall be cited explicetely.

440

K.A. Peters, K.A. Busch / Strain estimations

2A F I

A

H

Table 1 Strain range Test

Measured

Estimated

V78 V57b

7.4×10 3 1.2×10 -2

1.3x 10-2 1.8x10 -2

z(t) Fig. 6. A special 2-DOF-structure.

5.1. A simpe 2-DOF-structure The 2-DOF-structure is defined as a c o m b i n a t i o n of two truss elements of length l a n d areas 2A a n d A respectively carrying two rigid masses m (see fig. 6).

a n d V57b, b o t h leading to significant strain ranges have been recalculated using the described estimation m e t h o d (see table 1). It is worthwhile to remark that the effort needed is a b o u t 1 / 1 4 0 of the effort needed for a full nonlinear c o m p u t a t i o n (computer costs).

References Pl.2 = A Z ( v ~ -Y-1), A ,7P= mV~ ,

P2 >>/~,

s,(e)=s2(,)=O c,(,)=02(,)=0.5

(forall,), (forall,).

Using an artificial earthquake excitation of a duration of 8 s four examples have been computed. Frequencies (~max = 2 ~min) a n d design level (bmax = 2 bmi n ) have been varied. The best result delivered an overestimation of strain range of 1.13, the worst of 1.47.

5.2. SAE-experiments (KWU) These experiments have been carried out to show up the safety margins of a seismic code design and as a calibration i n s t r u m e n t for nonlinear c o m p u t a t i o n methods. The tests are described in ref. [8]. Two tests, V78

[1] N.M. Newmark and W.J. Hall, Procedures and criteria for earthquake resistant design, Building Science Series 46 (1973). [2] V. Tansirikongkol and D.A. Pecknold, Approximate Modal Analysis of Bilinear MDF Systems, J. Engrg. Mech. Div. ASCE (1980). [3] T.K. Canghey, Sinusoidal excitation of a system with bilinear hysteresis, Trans. ASME (1960). [4] L.M. Habip and L. Widuck, Nichtlineare Antwort mechanischer Systeme unter einfachen und mehrfachen transienten Belastungen, VDI-Berichte 496 (1983). [5] E. Keintzel, Zahigkeitskriterien fiar Stahlbetonhochbauten in deutschen Erdbebengebieten, Dissertation TH-Karlsruhe (1981). [6] J.D. Woerner, Absch~.tzung des Verhaltens von Strukturen mit begrenzten Nichtlinearit~,ten mit Hilfe eines Traglastverfahrens, l l t h MPA Seminar, Stuttgart, 1985. [7] K.A. Peters and K.A. Busch, Strain estimation for low cycle dynamic loads, Res. Mechanica (1985), to appear. [8] D. Mikatsch, B. Charalambus and E. Haas, Comparison of dynamic test data with results of various analytic methods, 1 l th MPA Seminar, Stuttgart, 1985.