A model for predicting the failure of structural steel elements

J. Construct. Steel Research 14 (1989)41-64

A Model for Predicting the Failure of Structural Steel Elements

Luis Calado & Jofio A z e v e d o

CMEST, Department of Civil Engineering, Instituto Superior T6cnico, Lisboa, Portugal

(Received 6 October 1988; revised version received 24 November 1988; accepted 20 February 1989)

A B S T R A C T

A numerical model for predicting the failure of structural steel elements subjected to cyclic loading is presented. The cyclic response is considered to be inelastic to seismic loading. The model for damage accumulation is expressed as a function of the inelastic strain and the dissipated hysteretic energy. It is shown that damage accumulation does not depend on the maximum strain but does depend on the inelastic deformation. The numerical model is used for the evaluation of the behavior coefficient of steel structures.

1 INTRODUCTION

The observation of past seismic events such as the San Fernando 1971 earthquake or the Mexico 1985 earthquake, has shown that a certain level of structural damage has always to be expected. In the case of steel structures damage is due to several causes such as: local buckling of slender plates and webs, fracture of the welded zones, low cycle fatigue, buckling of bracing elements and plate shear buckling.

Some research has been performed on these phenomena in order to develop detailing criteria to confine structural damage during earthquakes to acceptable levels. Nevertheless, it has been difficult to predict the behavior of structural steel elements subjected to seismic loading based on static cyclic testing. This problem arises from the fact that the failure of steel structural elements depends on the loading histories, t The European

41 J. Construct. Steel Research 0143-974X/89/$03-50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

42 L. Calado, J. Azevedo

Convention for Constructional Steelwork (ECCS) through its technical group TWG 1.3 - Seismic Design 2 suggests a loading history to be used in quasi-static cyclic loading. The suggestion of this loading history intends to make uniform the loading histories to be used in experimental testing, in order to make possible the comparison of results obtained in different research centers. Simultaneously it intends to make possible the assessment of the seismic behavior of steel elements based on cyclic quasi static results using a specified loading history. A joint research project for the analytical and experimental behavior of steel structural elements using the referred loading history is currently under way, involving the Centre for Structural Mechanics of the Technical University of Lisbon (CMEST) and the Polytechnic of Milan.

In this paper it is shown that it is possible to predict the seismic failure of structural steel elements based on static cyclic results and that these are independent of the loading history. The use of the numerical model for the assessment of behavior coefficients for steel structures is described.

2 SIMULATION OF THE STRUCTURAL B E H A V I O R OF THE CROSS-SECTION

Experimental cyclic testing in full scale structural elements shows the physical phenomena related to damage accumulation. Some tests performed in beams under cyclic bending 3 and bracing systems subjected to horizontal cyclic loading 4 have shown that the deterioration of the strength capacity of such elements is closely related to phenomena of local buckling of the webs and flange, fracture of the welded zones and the webs-flange connection and also to low cycle fatigue as shown in Figs 1 to 3. This implies that a correct simulation of the cross-section behavior must take all these phenomena into account.

The numerical model assumes that the cross-section is divided into a finite number of strips, and that for each strip the observed phenomena are simulated based on an adequate formulation.

2.1 Local buckling

Some of the variables that account for inelastic buckling under cyclic loading may be assumed to be more important than others. Using the elastic plate buckling theory as a guideline, the topology measure of greatest importance appears to be the ratio of the panel dimension to the thickness of the plate b/t. For a plate uniformly loaded and without

A model for predicting the failure of structural steel elements 43

Fig. I. Failure mode of an H cantilever beam. Fig. 2. Failure mode of a box-shaped cantilever beam.

transversal stiffeners, the elastic plate buckling theory gives the following equation for the critical stress: 5'6

k(~2 E) ° 'or = 1 2 ( 1 - v2)(b/t) 2 ( 1 )

being k a coefficient dependent on the constraints along the non-loaded edges on the plate, E the Young's Modulus, v the Poisson coefficient, b the width of the plate and t the plate thickness. If the cross-section is divided into a finite number of strips, it is possible to associate with each strip the


Fig. 3. Failure mode of a bracing system.

critical strain for local buckling. Dividing eqn (1) by the limit stress try, it can be written in a non-dimensional form:

ec....2 r = k( Tt 2 E/o'y)

ey 12(1 - ~2)(x/t)2 (2)

being ecr the critical strain for local buckling and x the distance between the centroid of the strip and the plate connection. The codes do not provide a precise definition of the conventional limit stress try; however, it should be noted, that they are based on results obtained in the elastic field. Thus, in order to prevent local buckling, the elastoplastic behavior of the structure and the interaction between overall and local stability must be considered. On account of this, in this investigation, the value of the material yield


stress determined by a uniaxial tension test was assumed as the limit stress ,O'y.

The value of k can be evaluated based on the ideal constraint conditions of the non-loaded edges of the plate, the cross-section type or the plate's slenderness.

In accordance with the German Code 7 and considering H shaped cross-sections or box sections, k can be made equal to 1, assuming that the constraining part buckles simultaneously with the constrained part and, therefore, does not provide an efficient rotational constraint. In this case, the value of k corresponds to the ideal constraints.

2.2 Fracture

In the course of the tests, it became evident that the cracks were mainly located in the area surrounding the web-flange connection. This was observed both for cold formed sections and composed sections. Based on this experimental evidence, this phenomenon was simulated using the theory of fracture mechanics.

In order to increase the stress on each strip of the cross-section, a value dependent on the stress concentration was used. This value is a function of two coefficients. The first is based on the initial stress concentration (cold process, welding, contour irregularities, etc.) and governs the beginning of crack formation. The second is a function of the crack length and governs its propagation. The first coefficient is assumed to be constant and equal to 1 for all the strips except for those in the vicinity of the web-flange connection. In this zone, which has a length two to three times the thickness of the web or the flange, according to the fracture direction, a linear variation of the coefficient was assumed, with the maximum value at the web-flange connection. The second coefficient has a linear variation and amplifies the stresses in the strips within the zone of crack propagation with a maximum value in the crack's limit.

The crack may propagate in the cross-section, in both directions, in the web or in the flange. For the failure criteria, the method of the strain energy density criterion proposed by Sih and Madenci, s was used. It admits that the failure of a given strip occurs when a certain amount of critical energy per unit volume is accumulated during the loading process. This critical value is equivalent to the value absorbed per unit volume of a specimen subjected to a standard tension test.

2.3 Fatigue

In order to simulate the stiffness deterioration under cyclic loading, the classical fatigue model for steel elements subjected to a low number of


cycles was utilized. Using Miner's rule 9 and thus assuming a linear accumulation of structural damage, this phenomenon can be expressed using the following equation:

AD = c(AB) d (3)

where c and d are parameters that depend on the element's properties, AD is the deterioration in terms of strength, stiffness or dissipated hysteretic energy and AB is the rate of decrease of each cycle for the variable under study.

The scarce information available in the literature is not enough to make possible a statistical evaluation of parameters c and d. Consequently, it seems acceptable to use a linear equation with all the uncertainties lumped on c. Equation (3) can thus be written as:

Ai = Aoi(1 - cN d) (4)

and be applied to each strip of the cross-section. In this equation A i

represents the area of the strip after N cycles of deterioration, Aoi the initial area of the strip, c a coefficient that depends on the number of cycles necessary to cause the failure by fatigue of the strip (-<1), N the order number of the deterioration cycle and d a coefficient that, as previously mentioned, is considered to be equal to one. This way the phenomenon is simulated through reductions of the cross-section area, as a function of the number of previous cycles.

3 A C C U R A C Y OF THE CROSS-SECTION MODEL

To verify the validity of the numerical model previously referred, the model was included in physical theory models representing the cyclic behavior of steel beams 3 and diagonal bracing systems 4 already available. The last ones assume that the structural elements can be simulated by rigid bars connected by elasto-plastic cells, where all the geometric and mecha- nical properties of the elements are concentrated. They are also based on physical considerations that make possible the evaluation of the cyclic nonlinear behavior of the structural elements. The only necessary information concerns the material properties and the geometric properties of the elements. In Figs 4(a) and 5(a) the experimental results for a steel cantilever beam under cycling bending and a bracing system are shown. The numerical results for the same elements, subjected to the same loading history, are illustrated in Figs 4(b) and 5(b).


F (KN) / F L 4~ ,7s+

I' 1. I,,o %0' c , = ~ ~. 25 o . o . rTIt g l l l l l - ~ l I

~. = 25 = sec t i on

. . ~

-,e o v,¢., ,oo l ! ,o.o

(a) (b)

Fig. 4. Cyclic behavior of an H canti lever beam.

v (cm)

F (KN) F 9OO

IN' ~.= 57 1[.8o

P (KN) ]

INI -~ 1

;,: N i l l • , , f j ' v " ! ! !J

-5".0 r//// ~j ~ 5.0 v (cm) ; . . . . . . . 5.0 v (©m)

(a) (b)

Fig. 5. Cyclic behavior of a bracing system.


The studies that were carried out allow the conclusion that the numerical model for the behavior of the cross-section gives a good representation of the several aspects of the experimental behavior up to failure. Local buckling, fracture and fatigue that were observed during the experimental tests are adequately simulated. The small differences that can be observed between the numerical and experimental results are basically due to anticipation or delay of the numerical model in what concerns the arising of local buckling. Nevertheless, the force-displacement diagrams and the dissipated hysteretic energy are quite analogous to the ones observed during the experimental tests.

These results are encouraging and suggest that the model can be used in the study of the cyclic behavior of steel structural elements up to failure.

4 SIMULATION OF DAMAGE ACCUMULATION

The objective of a damage accumulation model is to determine a simple formulation that can predict the failure of steel structural elements. The failure in this context is defined as an unacceptable limit state associated with a reduction in the hysteretic energy dissipation capacity.

For this damage accumulation model, the fatigue model for steel elements under a small number of cycles is used. 1° According to this model, the number of cycles of constant amplitude that lead to failure, Nf, can be related to the plastic deformation of each cycle by the equation:

Ny = c - l ( a s p ) -c (5)

where C and c are parameters related to the structural behavior. The amplitude of the plastic deformation is described in Fig. 6.

The amount of deformation to be used in this equation can either be a strain, a distortion angle, a rotation, a displacement or other deformation quantity, according to the type of failure to be analysed.

Using the hypothesis of linear damage accumulation (Miner's rule), the damage for each cycle of plastic deformation A~p is 1/Np, and the accumulated damage after N different amplitude A6p/is given by equation:

N N

D = ~/. --~-~1 = C ~ i (A6pi)C (6)

This equation is the simplest that can be proposed for the consideration of damage in structural steel elements. Miner's rule does not account for the order effects of the plastic deformations sequence. Nevertheless, as


t. P ,Asp

Fig. 6. Plastic deformation range.

will be shown, this is not relevant for loading histories such as the seismic ones.

The damage accumulation model expressed by eqn (6), shows that the damage in structural elements depends on the amplitude of the plastic deformations for each cycle, ASpi, on the number of inelastic cycles N, and on two parameters C and c that are related to the behavior of the element. These parameters should be determined by experiment results or sophisti- cated analytical models. The numerical models previously presented for the simulation of the cyclic behavior of steel cantilever beams or bracing systems can be used for that purpose, given the fact that they accurately simulate the behavior of such elements. Krawinkler et al. ~'1° suggest that the parameter c must be within 1.5 and 2.0, given that it is much more stable than C. This last parameter, according to the same author, depends strongly on the type of failure and can be determined through constant amplitude loading tests.

If the plastic deformations are made non-dimensional with respect to the elastic limit, eqn (6) can be written in the following non-dimensional form, to obtain a damage parameter:

~i ~A~pi~ c D = C . \ ASy / (7)

This way, a characteristic value for the accumulated damage in a structural element can be obtained. Parametric studies carried out by the authors have shown that if c is assumed to be equal to 1, the values obtained for the accumulated damage in a given structural element are


almost independent of the loading history. It is also suggested that C should be made equal to 1, given that the damage model is to be utilized within the already described models (beams subjected to bending and bracing systems) that already include the effects of local buckling, fracture and fatigue when modeling the cross-section. Thus, damage accumulation up to failure depends only on the sum of the plastic deformations and is independent of their sequence.

If the sequence of the cycles were to be taken into account, eqn (7) could be written in an empirical form as:

( 3c D - - C i a

• \ / (8)

i is the order number of the cycle and a a coefficient related to the importance of the order of the cycle (<--1.00). Assuming a small number in absolute value for a, the first cycles would have a larger contribution for the overall damage. Oppositely, assuming a large value for a would make almost uniform the contribution of any cycle to the damage value.

To predict the failure of structural elements with this damage accumulation model, it is necessary to associate a criteria for failure that can be used both in static and dynamic loading conditions. Among the possible criterion based on the loss of stiffness, strength or hysteretic energy dissipation capacity the last one was adopted. Failure is thus assumed to occur when the normalized hysteretic energy (7/) falls below a certain level. This measure of non-dimensional energy is the ratio between the dissipated hysteretic energy for a given cycle (Ai) and the energy that would be dissipated if the element was considered to have an elasto-plastic behavior (Ay/) (Fig. 7). One should note that it is possible to have values of the normalized hysteretic energy greater than 1. This is true especially for the first cycles, for which the phenomena of global and local buckling are not present. In these first cycles and mainly due to the hardening of the steel, the stress values are larger than the elastic limit stress and so the normalized hysteretic energy is greater than one. As soon as the above-mentioned phenomena arise the value of the normalized hysteretic energy becomes small than one. The failure criteria can then be expressed by the following equation:

A i < ), (9) I~i : A y i

where A/represents the hysteretic energy dissipated in the ith cycle, Ayi the energy that the element would dissipate if it had an elasto-plastic


F I

I r ' ' " . . . . . . " ' " " ' ~ ' " " " " " " " ~

• ::::::[: :i: : :i:i: .~ ff :::::::::::::::::::::::::::::::::::::: ::::: ::: :.::::::~ • ...................... +

i / F i - i: :i: iii i ::::::::: :::::::::::iiiiiill i i i !!!iiii! i[ ii ii~ii!i :

V :?i:i:?[:i:i:[:~: ii i :i j :]; t v ~ : : : : : : : : : : : : : : : : : : ; : : : : . ::.: . : . : : : :i:i: ::::::::::: ;:: : : ::::::: :::::::::::::::::::::::. :::: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :::::::::::::::::::::::::::::::::::::::::::::::::::::::: .-~

x . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : . . . . . . . : : . : : . : : : : : . : . : . : , , : : . : : ' : : : : : . : . : : - :R . : : . : : , I t : : : : : : : : : : : : : : : : : : : ; : : : : :i:!::: : : : : : : : : : ::i: :i:::i:i: :~: : : :i:i: ::i: : : : : : : : : : : : : : : : : : : : : : : : V .

- : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :::::::::::::::::::::::::::::::::::::::::::::::::::: 1 F : : N :+: : ' : : ' : : : . : : : : : : : : : : : : : : : : : ::::: ::: : : ::: : : : : : :::::::::::::::::::::::::: : : : : : ::::::::::::::::dr::

i : . ......................................................... ::: i ::::::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::: ....

~ ~ i ~ i ~:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~:::::::::::::::::::::::::::::::::::::::::

Fig. 7. F-v diagram in elasto-plast ic range.

1.75

1.50

1.25

1.00

0.75

0.50

0.9..5

0.00

IHI IiD lal IM lurid ~ P I M I N I I In1 PLIPIPLI~dJ~ Julpq.lPU I nn n i l I i •

I l l £1 • D

[ ]

+ • . _ Lus

a

• ÷ D

0 l0 20

H i

i •

• Vmax = 3 Vy tn Vn'u~ = 4 Vy + Vmax = 5 Vy • Vrmm = 6 Vy

C y c l e

30 40 50

Fig. 8. Results o f constant ampli tude s imulations.

behavior and 3' the percentage of the normalized energy corresponding to failure.

In Fig. 8 the normalized hysteretic energy (r/), is represented against the number of loading cycles (N) , for a steel cantilever beam subjected to cyclic bending. These results were obtained using numerical models that simulate the behavior of the beam and the cross-section. The beam, an IPE300, was subjected to several loading histories with a constant ampli- tube (Vmax), proportional to the limit elastic displacement (vy).


In Fig. 8 two distinct behaviors can be observed: one where the dissipated energy is approximately constant and other where it decreases abruptly. It can also be observed that in both cases a small number of cycles is enough to change the energy value from a unit value to a value close to zero. This shows that it is indifferent to consider that the element failure rises for r/i values equal to 20%, 30% or any other percentage values of the dissipated energy, as sometimes a single cycle is enough to diminish 7/i by 30 or 40%. Due to the stochastic nature of all the loading histories originated by earthquakes, it is not advisable to admit that the failure occurs for extremely low values of the dissipated energy. The available numerical models can simulate the hysteretic behavior up to very low values of the percentage of the dissipated hysteretic energy. Neverthe- less, to ensure structural safety, it is advisable to admit that failure occurs for rh values approximately equal to 50%.

5 A P P L I C A T I O N OF TH E D A M A G E A C C U M U L A T I O N M O D E L IN STATIC ANALYSIS

The above-mentioned damage models were inserted in the numerical models that simulate the behavior of bracing systems or cantilever beams under cyclic loading, along with the model for the behavior of the cross-section. The objective is to verify if the models can predict the failure of structural steel elements, and if this prediction is possible based on a single loading history.

5.1 Generation of loading histories

The utilized loading histories were generated by means of a stochastic process and intend to re-create those corresponding to seismic loading. According to some studies 1~ the distribution of the accelerogram peaks belongs to the exponential family and can be represented by a cumulative function similar to the one shown in Fig. 9. In that figure it can be seen that the higher peaks have a lower probability of occurrence than the peaks of lower intensity. Using the above distribution, several loading histories were generated, as shown in Fig. 10. These histories are normalized with respect to the elastic limit displacement and have a maximum value equal to one. To obtain the failure of structural elements using these loading histories it is necessary to have them amplified.

At this stage of the research, the use of the loading history suggested by ECCS to characterize the seismic behavior of structural steel elements, was regarded as convenient. This history is shown in Fig. 11. It corres-


1.2

1.0

0.8

0.6 "

I / ~ z E i 0.4 ~ ~ ....................

I / i 1 ~ i : 1 ~ i ] 0 . 2 . . . . . . . . ~ . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . .

: z | i x

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Fig. 9. Cumulative function to represent the distribution of the accelerogram peaks.

(a) 2"0

1 . 0

V/Vy 0.0

-I.0

- 2 . 0

I I Displacement 07

.̂J.L.,L ̂- 'V" , r , - I , vv~

i , , i

5 ]o ]5 20 25 30 35 40

Cycle

45

(b) 2.o

1 . 0

V/Vy 0.0

-1.0

-2.0

Dl~placcmJat e9 I I

_ e . l t A A A A . A e _ _ t L,LAA _ ~ _A ,.AA ~ y ' ~ " - ~ " I t " V 'ar v - , v , , - ~ , • V - 1 v-v V-

0 5 10 15 20 25 30 35 40 45

Cycle

Fig. 10. Typical loading histories.

54

Vy

L. Calado, J. Azevedo

20. I I ' ' ' t EccsTwo,3i i

~o] I t i l i , l l l l l l l

J__..[.,,k,AA AAA/ AAAAnAAA ,ot .... l'"vlvvvvlvvvvlvflflVVVI

i ot .... 1 .... 1 .... I .... l','.','.l!!!l . . . . .

0 5 10 15 20 25 30

Cycle

Fig. 11. ECCS loading history.

35 40 45

ponds to a series of elastic cycles with amplitude + Vy/4; +2Vy/4; +3vy/4 and ___vy, followed by groups of three cycles with amplitude equal to + ( 2 + 2 n ) Vy, wi thn = 0 , 1 , 2 , . . . .

5.2 Influence of the loading history

The use of different loading histories is intended to verify if the damage models presented are independent of the loading history, and so if it is possible to predict the failure of a structural element based on any loading history. For such purpose, several steel cantilever beams were studied, with different cross-sections (IPE, HEA, HEB), different slendernesses (h) (25 to 100) and different steel grades (Fe360 and Fe510). They were all subjected to the previously referred loading histories, applied at the free edge. These histories were amplified until failure was reached. In Fig. 12 one of the studied cases is presented, including graphics of the normalized dissipated energy (r/) against damage, according to the three damage models presented. Each graphic presents the accumulated damage value (D) versus the normalized hysteretic energy (77) obtained throughout the loading processes that include all the previously mentioned loading histories. The numerical simulations were carried out until a 20% value of the normalized hysteretic energy was reached. At this stage of the research project the use of a very low value of the normalized dissipated energy was regarded as convenient in order to enable the study of the model up to the full failure. It should be remembered that from the point of view of limit state, failure should be related to a percentage value approximately equal to 50%.

From these studies it is possible to extract the following conclusions:

(1) The use of the damage accumulation model D = ~,(A~pi/A~y), 'Damage 1' Fig. 12(a), originates the smallest coefficient of variation

A model for predicting the failure of structural steel elements 5 5

(a) 175

1.50

1.25

1.00

0.75

0.50

0.25

0.00

11 i i I

e • ~ 0

0 50 I00

i "

failure

"IPE 300 Fe 360 ~ -

~. = 100 b/t = 14.0 m

v

D a m a g e 1

150 200 250

(b) 1.75

1.5O

1.25

1.00 1

0.75 "

0.50 "

0.25

0 . 0 0 0

r ~ . r - ."%'~',,

Io ..... •

" ~ - . - _ - . ~ .~ . . ' .~ ." .

- "3.:* ":.

e • • o ° °

15 3O

IPE 300 Fe 3 6 0 ~

~. = I00 b/t = 14.0

v

• •

• D a m a g e 2

4 5 6 0 7 5

(C) 1.75

1.50

1.25

1.00

0.75

0.50

0.25

0.00 I

0

: ' : . . . : . . • _ . _ : , . t _ ~ N ~ h

0 8 • •

f i g ; • F . . • • . .

500 1000

"IPE 300 Fe 360 m

~, = 100 b/t = 14.0

v .[~ -

• D a m a g e 3

1500 2000 2500

F i g . 12 . I n f l u e n c e o f t h e l o a d i n g h i s t o r y and damage characterization.


for the failure (tr/x = 20%). The average value of the coefficient of variation obtained by using this model is approximately equal to 10%, increasing to 21% using D = Ei -°5 (A~pi/Ay), 'Damage 2' Fig. 12(b), to consider the influence of the order number of the cycle and increasing to 50% using D = S(ASp/ASy) 19, 'Damage 3' Fig. 12(c), as proposed by Krawinkler.l'10

(2) These results (Fig. 12(a)) suggest that damage accumulation in a structural element depends only on the sum of the plastic deformations defined according to Fig. 6, and is independent of their sequence.

(3) The accumulated damage (Fig. 12(a)) obtained by means of the loading history suggested by ECCS is within the values obtained by means of the other loading histories, suggesting that it can be used to predict the failure of structural elements.

5.3 Influence of the slenderness of the beams

In Fig. 13, numerical results for a steel cantilever beam analogous to the one referred in Fig. 12(a), but with a different slenderness (A = 25) are presented. From this figure it can be concluded that damage accumulation, using damage model 'Damage 1', is independent of the beam slenderness. The values presented in this figure are analogous to the ones of Fig. 12(a), which refer to the same beam with a slenderness equal to 100. As the plastic deformation is normalized with respect to the elastic limit deformation, values of the accumulated damage, which seem to be independent of the element's slenderness, can be obtained from the previous figure. This conclusion is important both from the static and the dynamic points of view. In this way the static failure of steel cantilever beams without axial loading seems to be independent of their longitudinal dimensions, but dependent on the slenderness of the flanges of the cross-sections. The results presented in Figs 12(a) and 13 represent a huge advantage, because to assess damage accumulation for a given type of element, it is only necessary to test a single specimen.

From the dynamic point of view, it implies that damage accumulation is independent of the stiffness and frequency, thus reducing the number of numerical simulations necessary to predict the dynamic failure of such elements.

5.4 Influence of the slenderness of the flanges and of the steel grade

The analytical studies performed on the steel beams (IPE, HEA, HEB- Fe360, Fe510) to assess the influence of the b/t ratio of the cross-section


1.75 '] ' T I

1.50 "

1.25 ~

1.00"

0.75 '=

0.50

0.25

0.00

05

I _ eas_

.~ -@~.

i! ~ e °

°"

IPE 300 Fe 360

~ ,=25 b/t=14.0 i

V

~ | Dsmsge 1

0 50 100 150 200 250 failure

F i g . 13. I n f l u e n c e o f t h e s l e n d e r n e s s o f t h e b e a m .

and the steel grade on the damage accumulation were all performed on beams with a slenderness equal to 25, given that damage accumulation is independent of the element's slenderness. The specimens had cross- sections with slendernesses between 11.4 and 20.0. They belong to class 1 as defined in Eurocode 3.12 This class includes all the cross-sections which can develop a plastic hinge with sufficient rotational capacity to allow redistribution of bending moments in the structure. The use of cross- sections belonging to this class allows for a design based on the plastic hinge theory. The utilized loading histories were the ones already mentioned.

In Figs 14 and 15 the results obtained according to the damage model 'Damage 1' are presented. From these figures the following qualitative conclusions can be made:

(1) Increasing the slenderness of the cross-sectix)n (case of Fig. 14) causes a reduction in the values of accumulated damage and simultaneously shortens the length of the horizontal zone of the graphs. These results are due to the fact that the failure of such beam is basically due to local buckling phenomena. Increasing the flanges' slendernesses makes the cross-section more susceptible to local buckling reducing the value of damage accumulation and the zone for which there is no reduction in the element stiffness.

(2) The steel grade influences the values of accumulated damage slightly (Fig. 15). Increasing the yield limit decreases the damage accumulated values, given that the specimens are more susceptible to local buckling. It can be observed that the horizontal zones of the graphs are almost independent on the steel grade, although there is a tendency for a quicker reduction on the stiffness for the higher steel grades after the occurrence of local buckling.

58 L. Calado, I. A z e v e d o

1.75

1.50

1.25

1.00

0.75

0.50

0.25

0.00

.-.,.,. '_,:

w o ~ T

• q

:.~v-_._ " • w " d r - -

• : 'e • . o . ;

e " o ,

H E A 2 2 0 F e 3 6 0

) , = 2 5 b i t = 2 0 . 0 m

v ¢ : ~

k____- - " ~"

Damage 1

50 100 failure 150 200 250

Fig. 14. Influence of the slenderness of the flanges.

1 . 7 5 '

• ~1 1.50

1.25

• k Loo,. ~ . . . ; . . . ~ .

0.75

0.50

0.25 . . . .

0.00 0

• • o •

)

o" o • "

• ol,

• , -7" -, ~,,~.

50 100 failure 150

I ~IPE 300 FeSI0

~ . = 2 5 bit = 14.0 .j Damage 1

200 250

Fig. 15. Influence of the steel grade.

(3) With the presence of local buckling and fracture (end of the horizontal zone of the graph), the energy dissipation capacity rapidly decreases, suggesting, once again, the use of the 50% energy value to represent the failure of the element. It should also be noted that increasing the value of the percentage of the normalized hysteretic energy results in decreasing the uncertainty of the damage values (smaller standard deviation).

6 APPLICATION OF THE D A M A G E A C C U M U L A T I O N MODEL IN THE DYNAMIC ANALYSIS

To check the validity of the damage model ' D a m a g e 1', single degree of freedom systems formed by the previously mentioned cantilever beams were used in a dynamic analysis. The purpose of this simulation was to


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Time (sec)

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Fig. 16. Typical accelerograms.

examine if the values of accumulated damage obtained by means of static analyses could be used to predict the failure of the same specimens when subjected to seismic loading. The main objective is the evaluation of q factors to be used in a reliable seismic design.

Several accelerograms with different characteristics, both referring to duration and sequence of the peaks, were used. The values that were used for damage accumulation are the ones corresponding to the equation D = ~ , (A~p i /A~y ) , 'Damage 1', adopting 7 equal to 50%.

6.1 Accelerograms used

Two of the accelerograms that were used in this study are represented in Fig. 16. The accelerograms correspond to several earthquakes recorded between 1980 and 1985 in some seismographic stations located Northeast of Taiwan, near the city of Lotung and commonly referred as Array Smart ('Strong Motion Array of Taiwan'). The accelerograms, despite being different in what concerns duration and sequence of the peaks, are also


different with respect to epicentral distance, hypocental depth, magnitude and hypocentral azimuth. In this way it was intended to take into account several variables that characterize the accelerograms, and to investigate if the damage accumulated values obtained for the different accelerograms were similar to the ones obtained by means of the static loadings.

6.2 Influence of the accelerograms and evaluation of the q factors

Several dynamic analyses were performed for different columns (IPE, HEA, HEB-Fe360, Fe510) with natural frequencies ( f ) varying from 0-50 Hz to 2-00 Hz and damping coefficient (~:) equal to 3%. At this stage of the studies the influence of the axial loading (N) was not taken into account. It was assumed that the columns bent around the axis of larger inertia.

The accelerograms were progressively amplified until the elastic limit was reached and then up to failure. It should be noted that according to the previously mentioned damage accumulation model, there is no damage accumulation until the elastic limit is reached.

Figures 17 and 18 present two numerical results obtained for different

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Time (sec)

Fig. 17. In f luence o f the a c c e l e r o g r a m o n the d a m a g e accumula t ion .


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F i g . 18 . I n f l u e n c e o f t h e a c c e l e r o g r a m o n t h e d a m a g e a c c u m u l a t i o n .

specimens and accelerograms (peak ground acceleration = PGA) for simulations conducted up to failure. In each figure, the evolution with time of the structural response by means of the normalized force (F/Fy), as well as the evolution of damage accumulation, using 'Damage 1', are presented.

From the analytical studies and from the presented figures the following qualitative conclusions can be drawn:

(1) For the same cross-section, and independently of its dynamic characteristics (natural frequency), the damage accumulated values obtained from different dynamic loadings are identical to the values previously obtained for the static analyses. The type of accelerogram does not have any influence on the value of accumulated damage evaluated by means of the dynamic analyses.

(2) Based on the obtained results, it seems that the value of accumulated damage is a characteristic of the type of cross-section and respective slenderness and is independent of the type of loading (static or dynamic).


(3) If the q factor is defined as the ratio between the peak value of the accelerogram that causes failure, and the peak value that causes the reaching of the elastic limit 13 it can be verified that as the value of the accumulated damage increases, there is an increase of the q factor. This shows that a structural element has a more ductile response the larger the charcteristic value of the accumulated damage.

(4) Structural elements with a very low characteristic value of acumu- lated damage cannot be considered as dissipative elements to be used in the seismic design of steel structures. Such elements should be considered with q factors equal to one, thus, as non dissipative elements.

(5) The amplification necessary to reach the elastic limit and failure is not independent of the earthquake. For a given structural system, the q factor value is independent of the earthquake. In reality, for any earthquake the ratio between the amplification factor necessary to reach failure and that to reach the elastic limit is approximately constant.

7 CONCLUSIONS AND C URRENT STUDIES

The mentioned studies can already give some qualitative guidelines with respect to the seismic behavior of structural steel elements. Nevertheless, they should be considered as research in progress and so should not be used as design criteria. Some general considerations can nevertheless be presented:

(1) The numerical model for the simulation of the behavior of the cross-section considering local buckling, fracture and fatigue, gives a good representation of several aspects of the experimental behavior up to failure.

(2) The accumulated damage in the structural elements seems to depend only on the sum of the plastic deformations and is independent of its sequence and maximum amplitude.

(3) It seems that the equation D = Y(A~pi/A~$y) is the most adequate to model the accumulated damage, originating the smallest dispersion of values.

(4) Damage accumulation up to failure for structural elements subjected to dynamic loading can be predicted based on static results.

(5) Damage accumulation obtained by means of the loading history suggested by ECCS is within the average values obtained by means of other loadings histories, suggesting that it can be used to predict damage accumulation up to failure.


(6) Damage accumulation is very sensitive to the b/t ratio of the cross- section and not very sensitive to the steel grade. The higher the b/t ratio the smaller the damage accumulation. Damage accumulation slightly diminishes when the resisting characteristics of the steel increase.

(7) It seems reasonable to consider that failure corresponds to a 50% value of the normalized dissipated hysteretic .energy, given that it is convenient to guarantee a certain reserve of structural safety.

(8) The presented methodology can be used in the dynamic analyses of structural elements, to evaluate q factors to be used in seismic design.

(9) Structural elements with high damage accumulation can be considered as dissipative elements in the structure and thus with a q factor larger than one.

The use of this methodology for bracing systems and beam-column connections is currently under development. Afterwards it will be im- plemented in a general-purpose dynamic analysis program. If the accumulated damage in the different structural elements can be determined, it is possible, according to the presented methodology,to evaluate the q factors to be used in the design of steel structures.

ACKNOWLEDGMENTS

The experimental studies were carried out at the Laboratory for Materials Testing of the Structural Engineering Department of the Polytechnic of Milan under the supervision of Professor Giulio Ballio, to whom the authors wish to express their gratitude.

The authors also wish to thank Professor Ant6nio Lamas from Instituto Superior T6cnico of Lisbon for his interest and comments during the formulation of this paper.

Part of the numerical simulation was conducted by Ms Diana Aires Barros.

The financial support from JNICT (Junta Nacional de Invest iga~o Cientifica e Tecnol6gica) is gratefully acknowledged.

REFERENCES

1. Krawinkler, H., Performance assessment of steel components. Earthquake Spectra, 3(1) (1987) 27-41.

2. Study on design of steel building in earthquake zone, ECCS - Technical Committee 1 - Structural Safety and Loadings, Technical Working Group 1.3

- Seismic Design, n.47/86, 1986.


3. Ballio, G. & Calado, L., Sezioni inflesse in acciaio sottoposte a carichi ciclici- sperimentazione e simulazione numerica. Costruzioni Metalliche, XXXVIII(1) (1986) 1-23.

4. Ballio, G. & Perotti, F., Cyclic behavior of axially loaded members: numerical simulation and experimental verification. J. Constructional Steel Research, 7(1) (1987) 3-41.

5. Ballio, G. & Mazzolani, F., Theory and Design of Steel Structures. Chapman and Hall Ltd, London, 1983.

6. Johnston, B. Guide to Stability Design Criteria for Metal Structures, 3rd Edition. John Wiley & Sons, New York, 1976.

7. DIN 4114 English Translation, Column Research Council, New York, 1952. 8. Sih, G. & Madenci, E., Crack growth resistance characterized by the strain

energy density function. Engineering Fracture Mechanics, 18(6) (1983) 1159- 71.

9. Miner, M., Cumulative damage in fatigue. J. Applied Mechanics, ASCE, 67 (September 1945) A159-A164.

10. Krawinkler, H., Zohrei, M., Irvani, B., Cofie, N. & Tamjed, H., Recom- mendations for experimental studies on the seismic behavior of steel components and materials. Report N.61, The John A. Blume Earthquake Engineer- ing Center, September 1983.

11. Azevedo, J., Characterization of structural response to earthquake motion. PhD thesis, Stanford University, August 1984.

12. Eurocode N. 3 - Common unified rules for steel structures. EUR 8849 DE, 1984.

13. Draft Eurocode N. 8- Common unified rules for structures in seismic regions. EUR 8850 DE, 1984.

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A model for predicting the failure of structural steel elements