Advances in Steel Structures - part 21 pdf

STABILITY ANALYSIS OF MULTISTORY FRAMEWORK UNDER UNIFORMLY DISTRIBUTED LOAD Chen Haojun and Wang Jiqing Department of Construction Engineering, Changsha Communications University 45 Chi

180 J.Y.R Liew et al

with 'EEP' connections and the frame with 'DWA' connections It can be concluded that if proper semi-rigid connections are used, the frame can be constructed much faster and cheaper than the rigid frame, at the same time satisfying the strength and serviceability limit states

CONCLUSIONS

The basic principles of the proposed advanced inelastic analysis program have been presented Inelastic analysis has been applied to study the roof truss system and emphasises the importance of lateral brace to assure the system's stability, which is important for the safe erection of such structure Inelastic analyses on core-braced frame with semi-rigid connections show that construction with proper selection of connections can satisfy limit states design and achieve fastrack construction When properly formulated and executed, the advanced analysis can be used to assess the interdependence of member and system strength and stability, the actual failure mode and the maximum strength of the overall framework, and, hence, efficient and cost-effective design solutions can be obtained This is in line with the modem design codes such as Eructed, which allows the use of advanced analysis for designing steel structures

REFERENCES

Chen, W.F., Goto, Y., and, Liew, J.Y.R (1996), Stability Design of Semi-Rigid Frames, John

Wiley& Sons, NY

Hsieh, S.H (1990), Analysis of three-dimensional steel frames with semi-rigid connections, Structural Eng Report 90-1, School of Civil and Environmental Eng., Comell University, NY Liew, J.Y.R., White, D.W., and Chen, W.F (1993), Limit-states design of semi-rigid frames using

advanced analysis: Part 1: Connection modelling and classification, J Construct Steel Res., 26,

1-27

Liew, J.Y.R., Chen, H., Yu, C.H., Shanmugam, N.E., and Tang, L.K (1997), Second-order inelastic analysis of three-dimensional core-braced frames, Research Report No: CE024/97, Dept of Civil Eng., National University of Singapore

Liew, J.Y.R., Chen, H., Yu, C.H., and Shanmugam, N.E (1998), Advanced inelastic analysis of

thin-walled core-braced frames, Proc of the 2nd International Conference on Thin-Walled Structures, Dec 2-4, 1998, Singapore

Liew, J.Y.R., Chen, H., and Shanmugam, N.E (1999), Stability functions for second-order inelastic

analysis of space frames, Proc of 4th International Conference on Steel and Aluminium Structures, June 20-23, 1999, Espoo, Finland

Liew, J.Y.R., and Tang, L.K (1998), Nonlinear refined plastic hinge analysis of space frame structures, Research Report No: CE029/99, Dept of Civil Eng., National University of Singapore

Table 1 Parameters and Mn/M values for connections under in-plane bending moment

Connection

type M0/Mn ~ n Kp/Mn

DWA: Double web-angle connection

TSAW: Top- and seat-angle connections with double web angles

EEP: Extended end-plate connection without column stiffeners

At the beam framing about the major-axis

of column (see Fig 3) 0.05

At the beam framing about the minor-axis

of column (see Fig 3) 0.025 0.2 0.5

Advanced Inelastic Analysis of Spatial Structures 181

Fig 2 Four-parameter power model

Fig 1 Thin-walled beam-column element

Fig 3 Beam-to-column connections

Fig 4 Plan view of roof truss system Fig 5 Elevation view of truss

Fig 6 Load-lateral displacement curve under

gravity load

Fig 7 Deformed shape of roof truss system at collapse under gravity load

182 J Y R Liew et al

Fig 8 Horizontal load-lateral displacement curve

Fig 9 Deformed shape of roof truss system at collapse under the horizontal

surge forces

Fig 10 Plan view of core-braced frame

Fig 12 Top-storey load-deflection curves

Fig 11 Elevation view of core-braced frame: (a) at axes 1, 2, 5, 6 (b) at axes 3, 4 Table 2 Comparison of limit loads and initial lateral stiffness Connection

types Pin connection

Limit load 36%

Initial lateral stiffness 21%

All % values are compared with the core- braced frame with rigid connections

STABILITY ANALYSIS OF MULTISTORY FRAMEWORK

UNDER UNIFORMLY DISTRIBUTED LOAD

Chen Haojun and Wang Jiqing Department of Construction Engineering, Changsha Communications University

45 Chiling Road, Changsha 410076 China

ABSTRACT

Problems of overall stability in a multistory framework become significant with the increase in its height This paper presents the stability analysis to a one-bay multistory framework under uniformly distributed load by means of continuum model Continuum model is a substituting column converted from multistory framework So, the analysis to multistory frame, which is an indeterminate structure,

is reduced to that to a determinate one The formula of critical load is developed by Galerkin method The effect of the axial compressive deformation of framework column is taken into consideration

KEYWORDS

Multistory framework, overall stability, continuum model, uniformly distributed load, critical load, substituting column

In the analysis to a multistory framework structure, one pays more attention to analysis to internal forces of a multistory framework at vertical and horizontal loads, than to analysis to overall stability However, the problems of overall stability in a multistory framework become significant with the increase in height Generally, the exact stability analysis of multistory frames can be solved by finite element method This is an extremely complex procedure, even with the help of computer The higher the structure is, the more complicated the problem is to handle The critical load is usually obtained by determination of effective length factor of each framework column In this paper, the framework structure will be taken as a whole for determination of the critical load A critical load for

183

184 C Haojun and IV Jiqing

a one-bay multistory framework subjected to uniformly distributed load at floor level is developed by Galerkin method The effect of axial compressive deformation on the critical load is taken into account in following analysis

1 BASIC ASSUMPTIONS

During the analysis, following assumptions will be used

A) The material of the structure is homogeneous, isotropic and obeys Hook's law

B) The loads are applied statically and maintain their direction during buckling

C) The structure develops small deformation and the axial deformation in the beam is negligible when the axial framework buckles

D) All stories have the same height and the structure are at least four story high

E) The structure has a rectangular net work with elements attached by rigid joints to each other F) The stiffness (El/l) of beams is the same

G) The inflection point is on the middle of the beam when the framework buckles

2 SUBSTITUTING COLUMN

The continuum model of multistory framework is a substituting column converted from the framework The substituting column is obtained from the original framework (Fig 2.1a) in several steps First, the UDL on the beam is transferred to the columns at floor levels (Fig 2.1b) in the form of concentrated forces (the reactions on the beams) These concentrated forces are then distributed along story height (Fig 2.1 c), in fact along the height of the framework The beams are cut through

at inflection points (Fig 2.1 d) and finally the columns are added up into a single substitute cantilever (Fig 2.1 e)

P

46464+444

p

4~4~4+4~4

P

l

4)

t ~ q= q l ) m

4)

1 ,' l

Fig 2.1 Continuum Model The bending stiffness of the substituting column is the sum of the bending stiffness of columns of the framework The load on the substituting column equals the total load on the original framework The distributed force along the height of substituting column is converted from the uniformly

Stability Analysis of Multistory Framework 185 distributed load at floor levels The distributed moments along the substituting column are induced from deformation of the framework during buckling In doing so, the framework is converted into a fixed-free column on which a distributed force and a distributed moment act It should be noted that the difference of axial compressive deformation between two framework columns makes the framework have sway This phenomenon is not shown in substituting column Comparing actual column with substituting column, it is known that the restraint moment acting at floor level due to beam bending makes the column double-curvature between two beams for an actual framework But for a substituting column, the restraint moment due to beam bending is distributed along the substituting column and does not make the substituting column double-curvature

3 CRITICAL LOAD OF A ONE-BAY MULTISTORY FRAMEWORK

There is a one-bay multistory framework as shown in Fig 3.1 The stiffness of beam of each floor level is Eblb except the top one of Eblb/2; and the stiffness of framework columns is Eclc There are uniformly distributed loads at each floor level According to preceding procedure, the substituting column is shown in Fig 3 lb When the framework buckles, it can be in equilibrium both in original configuration (undeformed configuration) and in slightly deformed configuration Now, let us consider the equilibrium of framework in slightly deformed configuration

EcI~/2

P

I I I I I I I I I Edj2 Edb

I I I I I I I I I Edb

I I I I I I I I I Edb

I I I I I I I I I E~b

EcIc/2 h

h

h

h

~y

I

l

l

ql

I

I

t

I

) ) m

)

~y

H=eh

Fig.3.1 Substituting Column

3.1 Distributed Moment When Framework Bends

The separated body for analysis may be taken as shown in Fig 3.2 when the framework bends It is cut at the middle point of beams (inflection points) and replaces with a shear force T This shear force T can be obtained by the condition that the deformation at the middle point of beams (inflection points) is equal to zero,

y, l TM (t/2) 3

in which TM is the shear force in beams due to framework bending; l is the distance between the axes

of columns; Eb is the elastic modulus of beams; Ib is the moment of inertia of beam; Yb ~ is the first

186

derivative of framework

C Haojun and W Jiqing

Eqn 3.1 gives

12EbIb ,

The distributed force along the framework column due to bending is

= - ~ - = h ~yM

Transfer of the shear forces at inflection points TM to the axis of the columns produces the concentrated moments acting on the column at floor level,

M M = T M x - = ~ YM

Distribution of the concentrated moment M i along the column height leads to

-if = hl YM

3.2 Consideration of Axial Deformation of Column

Shear force TM at beams makes the axial forces in two columns different The axial force increases in

TM in right column and decreases in left column This variation of axial force causes an additional axial deformation in left and right columns It is denoted by the sign AN This deformation consists

of two parts (Fig 3.3) One (denoted by AN1) makes the beams bend and the other (denoted by Am) makes the columns bend Hence,

The bending moment at beam end due to AN1 is

MN1 = /2 AN1

Fig 3.2 Separated Body When Buckling Fig 3.3 Compressive Deformation

Letting 2AN1/I=y'N1, one obtains the distributed moment along column

Stability Analysis o f Multistory Framework

MN1 6Eblb , mN1 = ~ = - - ~ Y N 1

187

(3.8)

Variation of axial force in column due to AN~ is

TN1 = MN A_I = _ 12Eblb

l

Distribution of the force TN1 along the column leads to

TNI 12EbI b ,

= - - ~ Y N 1

tN1 = h hl 2

(3.10) According to Figs 3.3 and 3.4, the compressive deformation distant to z from original O is

AN(Z) ~ ~(tM-~'tN1)d(dz

E cAc

(3.11)

where A c is the cross-section area of column Substitution of Eqns 3.3 and 3.10 into Eqn 3.11 leads

to

AN(z ) = f ~ 12Eblb (y~ y'N,)d(dz

Making use of Y'N =2AN/l, Ir=2Ac(l/2) 2, and differentiating twice, Eqn 3.12 may be written in the form

,, 12Eblb

Integrating Eqn 3.13 once and making use of the boundary condition, y"N(0)=0 and yN~(0)=yN2(0)=0, one obtains

" 12Eblb(YNl_yg2 )

The Equilibrium differential equation is

Eclcy" + ~ q ( y - rl)d ~ - ~m(~:)d~: = 0

It is known that m=2(mM+mN0, and making use of Eqns 3.5 and 3.8, one obtains

(3.15)

m 12Eblb (Y~ - Y'~I)

hl

(3.16)

Substituting Eqn 3.16 into Eqn 3.15, one obtains

Ec lc Y " + f q (Y - rl )d ~ - ~ 12 E b l b (Y " - Y 'N1 )dz = 0

188 C Haojun and W Jiqing

The bending deformation of the substituting column is

Y = YM + YJv2

(3.18) 3.4 Solution of Differential Equations

I

I

I

r

i

Y

Z

Fig 3.4 Coordinate for Calculation of

Compressive Deformation

Z

1

q

I

Fig 3.5 Coordinate and Separated Body

of Substituting Column Combination of Eqns (3.17), (3.13), (3.6) and (3.18) gives

Eclcy" + ~ q(y - ~7)dr - ~ 12Eblb (Y'M - Y N 1 ) d ~ = 0

12EbIb ,

Arrangement of above equations and letting Kb=12Eblb/hl leads to the equilibrium differential equation

Eci~ ] " Kbq

Eqn (3.20) is solved by Galerkin method Letting the approximate deflection curve be

(3.20)

7De

y = 6 s i n ~

which satisfies the geometric and mechanic boundary conditions

Stability Analysis of Multistory Framework

y(O)= y'q): y"(O)= y"(l)=O

one obtains the Galerkin equation

f L ( y ) s i n az dz=O

2H

in which

L(y)= EclcY'V + qz- Kb - Kb ~cI~ ) y + qY'-~l~ ~(y- rl)dr

Substituting Eqns (3.24) and (3.21) into Eqn (3.23) and making use of the integration

~ sin 2 az d z = H

f z sin 2 nz dz = ( 1 I _]H 2

2H 4 + r e 2 )

f c o s ~ s i n 7tz 7tz dz = H

2H 2H n"

~ =

1 - cos ~ sin

H

d z = m

one obtains

Letting

F c =rc2EcI~//(2H) 2

F o =rc2Ej,.(2H) 2

and substituting these into Eqn 3.25, one obtains

189

(3.22)

(3.23)

(3.24)

(3.25)

= 0.279(1+ Kb/Fo)

Eqn 3.26 is the critical load of the one-bay multistory framework according to Galerkin method by