Advances in Steel Structures - part 4 potx

A CUMULATIVE DAMAGE MODEL FOR THE ANALYSIS OF STEEL FRAMES UNDER SEISMIC ACTIONS* Zu-Yan Shen I ~State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Sha

10 D.A Nethercot and J.S Hensman

STEP 1

Define frame geometry

STEP 2

Define load types and magnitude

(1) Gravity

(2) wind

NOTE: This flow chart is not a design procedure

It should be used only as a 'first check', to determine if the wind-moment method outlined

in this document is a suitable design method for the frame in question

STEP 3

Design composite beams as

simply supported with a capacity

of 0.9Mp

STEP 4

Estimate required column

sections

I

I

,

I

The frame design is likely to be controlled

by SLS sway However, a suitable frame design may still be achieved using the wind-moment method

Consider increasing the member sizes

l

STEP 5

Predict the SLS sway using

the method in Section 5 Design the frame as rigid, or include

vertical bracing

~ " Is the t~alframe ~ Is the to!al frame ~ design this frame for ULS is likely sway <.n:.~uu: / ~ sway <h:200? J " to result in unsatisfactory SLS

~ Yes

~ ] Using the wind-moment method to

~ ~ ~ _ _ ~ _ [design this frame for the ULS is likely]

to result in unsatisfactory inter-storey I sway behaviour Design frame as I rigid at the 1st storey level, or include I

/

Design the frame using the I The bottom storey SLS sway is likely

wind-moment method,as ] to control the frame design

detailed in this document Increasing the column section sizes

may resolve this problem; if not then

it may be appropriate to use an alternative method

Figure 4 Steps in design approach

Unbraced Composite Frames." Application of the Wind Moment Method 11

critical for the design of internal columns when heavy imposed loads are present on long span beams

For beam design under gravity loading an end restraint moment of 10% of the maximum sagging moment in the beam should be assumed For horizontal loading, frame analysis should be by the

"portal method"

Composite beams should be Class 1 designed for 90% of their plastic moment of resistance at mid- span This provision has been introduced so as to ensure that adequate rotation capacity is present

in the composite connections to develop the required span moment Previous studies, Li, Choo and Nethercot, (1995), Nethercot, Li and Choo, (1995), have shown both that the available rotation capacity of composite connections is limited and that the non-linear relationship between beam span design moment and the amount of moment redistribution necessary to achieve this substantially reduces the rotational demands on the connections

Columns, which are assumed to be of bare steel, should be designed by the usual interaction formula approach Effective lengths for in-plane and out of plane checks should be taken as 1.5L and 1.0L respectively Column end moments should allow for both the end restraint moment due

to partial fixity when considering gravity loading and the moments calculated due to horizontal loading

Connections must be designed for both maximum hogging and minimum sagging loads in recognition of the fact that wind loading can reverse

The parametric study indicated deflections under serviceability loading to be of the order of 30% greater than those calculated assuming rigid joints due to the greater overall flexibility of the frames with semi-rigid connections Rather than permit the use of any method for the determination of sway deflections, a development of that proposed by

Wood and Roberts, (1975), that employed a simple graphical technique is proposed In this way the common drift limit of h/300 recommended by BS5950: Part 1 and EC3 may be achieved In addition to considering the behaviour of the complete frame, it is important to check each individual story The first story is likely to be the most critical, typically accounting for the percentage of total frame sway indicated in Table 3

CONCLUSIONS

Based on the findings of a careful numerical study that employed a synthesis of the best currently available scientific evidence, proposals for the application of the Wind Moment Method to the design of a restricted range of unbraced composite frames have been made In application, these closely follow the established procedure of the SCI Design Guide for bare steel frames The background study has, however, recognised the need to properly consider the behaviour of both the composite beams and the composite connections; it has also recognised the desirability of using less than fully rigid column bases In deriving the design procedure, account has been taken of industry wishes, important practical differences in the likely configuration of composite frames from steel frames and recent changes in the general structural design climate in the UK

12

REFERENCES

D.A Nethercot and J.S Hensman

Ackroyd, M (1987), Design of flexibly connected unbraced steel building frames, Journal Constructional Steel Research, No 8, pp 261-286

Ackroyd, M H and Gerstle, K H (1982), Behaviour of type II steel frames, Journal of the Structural Division ASCE, Vol 108, No 7, pp 1541-1556

Ahmed, B and Nethercot, D A (1997), Design of composite flush end-plate connections, The Structural Engineer, Volume 75, No 14, pp 233-244

Anderson, D, Reading S, J and Kavianpour, K (1991), Wind-moment design for unbraced frames,

The Steel Construction Institute, Publication No 082

Hensman, J S (1998), Investigation of the wind moment method for unbraced composite frames,

MPhil thesis, Department of Civil Engineering, University of Nottingham

Hensman, J S and Nethercot, D A, (2000a) Utilisation of test data for column bases when addressing overall frame response: a review, Advances in Structural Engineering, to be published

Hensman, J S and Nethercot, D A, (2000b) "Numerical studies of composite sway frames: Generation of Data to Validate Wind Moment Method of design", to be published

Kavianpour, K (1990), Design and analysis of unbraced steel frames, PhD thesis, University of

Warwick

Li, T Q, Choo, B S and Nethercot, D A (1995), Determination of Rotation Capacity Requirements for Steel and Composite Beams, Journal of Constructional Steel Research, Vol 32, pp 303-332

Li, T Q, Nethercot, D A and Choo, B S (1993), Moment curvature relations for steel and composite beams, Steel Structures, Journal of Singapore Structural Steel Society, Volume 4, No 1, pp 35-51

Nethercot, D A, Li, T Q and Choo, B S (1995), Required Rotations and Moment-Redistribution for Composite Frames and Continuous Beams, Journal of Constructional Steel Research, Vol 35 No

2, pp 121-164

Reading, S J (1989), Kavianpour, K (1990), Anderson, D, Reading S, J and Kavianpour, K (1991)

Investigation of the wind connection method, MSc thesis, University of Warwick,

Wood, R H and Roberts, E H (1975), A graphical method of predicting sidesway in multistorey buildings, Proceedings Institution of Civil Engineers, Part 2, Vol 59, pp 353-272

Ye Mei-Xin, Nethercot, D A and Li, T Q (1996), Non-linear finite element analysis of composite frames, Proceedings Institution of Civil Engineers, Structures and Buildings, Volume 116, pp 244-

247

A CUMULATIVE DAMAGE MODEL FOR THE ANALYSIS OF STEEL FRAMES UNDER SEISMIC ACTIONS*

Zu-Yan Shen I

~State Key Laboratory for Disaster Reduction in Civil Engineering,

Tongji University, Shanghai, 200092, China

ABSTRACT

Based on a series of experiments and theoretical analysis, a cumulative damage mechanics model of steel under cyclic loading, hysteresis models for plane and space steel members with damage cumulative effects and a cumulative damage model for steel frames under seismic actions are presented Using these cumulative damage models, the elasto-plastic response of steel framed structures sustained the major shock and successive aftershocks of an earthquake can be calculated and the performance of the structures can be predicted more precisely and more realistically In order

to verify the theoretical results and the acceptability of the cumulative damage models, shaking table tests were conducted by the author in the State Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University

KEYWORDS

Cumulative damage model, Steel framed structures, Seismic response, Shaking table test

INTRODUCTION

The collapse of structures is usually due to the cumulation of damage to certain extent In order to take the cumulation of damage into account in analysis, damage mechanics has been developed, Kachnov(1986) But up to now few of the research results have been used in the seismic analysis of steel framed structures Recently Shen and Dong (1997) suggested an experiment-based cumulative damage mechanics model for steel subjected to cyclic loading Shen et al (1998) derived a hysteresis model for plane steel members with damage cumulative effects Since these research results can take the damage cumulative effects into consideration, it becomes possible to establish an analysis approach for calculating the elasto-plastic response of steel framed structures sustained the major shock and successive aftershocks of an earthquake

~ National Key Projects on Basic Research and Applied Research: Applied Research on Safety and Durability of Major Construction Projects

13

14 Z - Y Shen

In the paper, a cumulative damage model for steel framed structures under seismic actions is established The performance of the structures subjected to a major shock and successive aftershocks

of an earthquake can be analyzed and the extent of damage including the damage cumulation of the structures after each shock can be calculated Shaking table tests were conducted by the author in the State Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University for verifying the theoretical analysis

CUMUALTIVE DAMAGE MECHANICS MODEL OF STEEL UNDER CYCLIC LOADING

A cumulative damage mechanics model of steel under cyclic loading was established by the author, Shen and Dong (1997) The model can be expressed as follows

Damage index D is calculated by

" ~ C - 6g

where N is the number of half cycles which cause plastic strain, fl is the weight value, 6 p is the

P is the plastic strain during the ith half cycle, 6 p is the ultimate plastic strain of the material and 6 largest plastic strain during all halfcycles

The effects of damage on elastic modulus, yield strength and strain hardening coefficient are

E o = ( 1 - 4 D ) E

where D is the damage index, D = 0 means no damage and D = 1 means complete failure of the material, E and E n are the elastic modulus in respect of D = 0 and D, respectively, tr s is the initial yield stress when D = O, try is the yield stress in respect of D, k 0 is the original strain hardening Cbefficient and k (") is the strain hardening coefficient after the nth halfcycle ~:~, ~:2 and ~3 are three material parameters

And the cumulative damage mechanics model of the steel under cyclic loading is

for first half cycle

(3)

from the second half cycle,

Cumulative Damage Model for the Analysis of Steel

f ! = erA.+, + ED(')(C CA.+, )

- - a c 2 -~- b g + c

O'Cn+l -I- k(')ED(')(c - Cc.+,)

?'or ~ < [era,,+ , -o" I < (2 + r/)cr~ <')

IOta.,+,- or] > (2 + r/)cr~ ~

(4)

15

where ~s and es are the initial yield stress and strain, CA~+, and eAn+~ axe the stress and strain at unloading point A of the (n+ 1)th halfcycle, ~cn+, and ecn+, are the stress and strain at stress hardening point C of the (n+l)th halfcycle, y and r/ are two material parameters and a, band c are the

constants of the parabolic curve connecting the yielding point B and the stress hardening point C The cumulative damage mechanics model can be illustrated by Figure 1

Figure 1: Hysteretic model of steel considering damage cumulation

For steel Q235, all the material parameters can be adopted according to Table 1

TABLE 1 MATERIAL PARAMETERS FOR DAMAGE MECHANIC MODEL OF STEEL Q235

0.0081 0.227 0.119 Eqn 5 0.000073 1.44 0.041

PRECISE HYSTERESIS MODEL OF STEEL MEMBERS WITH DAMAGE CUMULATION EFFECTS

Shen and Lu have developed a powerful integration method to calculate the behavior of steel members, Shen and Lu (1983) The method can analyze strength problems and stability problems as well, taking into account the effects of initial geometrical and physical imperfections including residual stresses of the steel member and can give the complete load-deflection relationship of the member including both the ascending and descending branches

16 Z.- Y Shen

Since the basic required input of the integration method is the stress-strain relationship of material, the method can be used to obtain the hysteresis model of steel members with damage cumulation effects,

if we take the cumulative damage mechanics model of steel under cyclic loading as the input of the method

Two experiments conducted by the author, Li et al (1999), were calculated for verifying the proposed method The material properties and the sectional dimension of the H section columns are shown in Table 2 or, and 6, denote yield stress and yield strain, respectively, crb and 6,, are ultimate stress and ultimate strain, respectively, b and h are the width and height of the section and t, and tf are

the thickness of the web and the flange, respectively

TABLE 2 MATERIAL PROPERTIES AND SECTIONAL DIMENSION

OF H-SECTION COLUMNS

The column specimens are cantilevers with length 1100mm For specimen A, there are only two horizontal forces acting at the top, and for specimen B, there are two horizontal forces and one constant vertical force with magnitude 300kN acting at the top The loading path of the specimens are shown in Figure 2 and the comparison between calculation and tests can be seen in Figure 3, where

Px and Py are the horizontal forces applied on the top ends of the tested column specimens, and Dx

solid lines, and the calculated results by dashed lines The comparison shows that using the method proposed, a precise hysteresis curve can be obtained by theoretical analysis

Figure 2: Loading path of the tested specimens

Cumulative Damage Model for the Analysis of Steel 17

Figure 3" Comparison of measured and calculated results

CUMULATION EFFECTS

In order to put the model into practice, the author has established a simplified hysteresis model for plane steel members with damage cumulation effects, Shen et al (1998) The model can be extended

to spatial steel members with damage cumulation effects

The hysteresis parameter for a spatial member at nth loading can be expressed as the same as the plane member, Shen at al (1998), if the ~o denotes the yield function of the spatial member

[k(1-~,D)

(for (P < (Ps,n)

1

~o - fps,, + ( k - l ) ] (for ~os., <(p < ~Op.,) (6)

J

(for q) > ~Op, n )

where D is called equivalent damage index of the cross-section to be substituted for the actual damage,

.~ IDidA~

A

is the area of the ith subsection, D, is the damage index of the ith subsection ~o is the yield function for the spatial member ~os, and ~Op,, denote the value of initial yield and the perfect yield during the nth loading, respectively, and k is the strain hardening coefficient The yield function for the spatial member has been developed by many authors, Chen and Austra (1976), Duan and Chen (1990), Kitipomchai et al (1991 )

When damage and plastic yield occur at both ends of the spatial member, the elasto-plastic tangent stiffness matrix can be expressed by the following equation

18 Z.- Y Shen

[KpD] = [Ke]-[K,][G][E][L][E]r[G]r[K,] (8)

where [Ke] is the elastic stiffness matrix of the spatial member element, [G],[E] and [L] can be obtained from the reference, Li et al (1999)

The same experiments mentioned in the above section can be used for verifying the simplified hysteresis model for spatial members with damage cumulation effects Figure 4 shows the comparison

of the test results and the calculated results using the simplified hysteresis model, in which the test results are shown by solid lines and the calculated results by the dashed lines

Figure 4: Comparison of tested results and calculated results The comparison illustrates that the accuracy of the simplified hysteresis model is quite good and the model is acceptable for practical analysis

After every half cycle of loading the strain of every subsection can be calculated, the damage index D~ of the subsection is obtained by using Eqn 1 and hence the equivalent damage index /9 of the member is also obtained by Eqn 7 Summing up the equivalent damage of all previous half cycles of loading, the cumulated damage index of the member can be obtained

At the end of loading the cumulated damage index of specimen A and B are 0.121 and 0.305, respectively

The simplified hysteresis model not only can be used to imitate the actual hysteresis curves of spatial members but also has several unique advantages

First, it can be used to analyze the hysteresis behavior of a member with initial damage, if the initial damage index is known Second, it can get the damage index and emulated damage index of a member after loading And third, it can be used to analyze the behavior of members subjected to the seismic loading more than one times

NONLINEAR DYNAMIC ANALYSIS OF STEEL FRAMES WITH CUMULATIVE DAMAGE EFFEXTS UNDER SEISMIC ACTION

Cumulative Damage Model for the Analysis of Steel 19 From the ealsto-plastic tangent stiffness matrix, Eqn 8, for spatial steel member with damage cttmualtion effects, the global tangent stiffness matrix [Kpo ] of the spatial steel frames can be easily assembled Then the deferential equation of dynamic equilibrium for the frames can be built as follows

[M] {A6} + [C] {AS} + [Kpo] {A6} = [MI[E]{A6g } (9) where [M] is the mass matrix of the structure, [C] is the damping matrix of the structure, {6} is the displacement vector of the structure and {6g } is the acceleration vector of ground movement Solving the above equation, the displacement response of the structure can be determined Through the displacement, the damage index of each member can be calculated

In order to verify the proposed method, a shaking table test was conducted by the author Figure 5 shows the sketch of the spatial steel frame model used for shaking table test All of the members are H-typed cross-sections Node numbers are given in Figure 5

A3 ,1 ?/~ //~x,2

5

77

2400

Figure 5 Sketch of the frame model

TABLE 3 MATERIAL PROPERTIES

TABLE 4 SECTIONAL DIMENSION OF H-SECTION BEAMS AND COLUMNS

The material properties and the sectional dimension of the H-section members are shown in Table 3