A variational inequality and quadratic programming approach to the separation problem of steel bolted brackets

compurersd .%ucturesVol. 53. No. 4.

Pergamon

00457949(94)Eo1914

pp. 983-991. 1994 copyright @ 1994 Elsmier Science Ltd Printed in Great Britain. All rights -cd 0045.7949/94 17.00 + 0.00

A VARIATIONAL INEQUALITY AND QUADRATIC PROGRAMMING APPROACH TO THE SEPARATION PROBLEM OF STEEL BOLTED BRACKETS C. C. 3anioto~uI~~

K. M. Abdalla$/ and P. D. Pa~gioto~~~~~

j-Department of Civil Engineering, Institute of Steel Structures, Aristotle University, GR-54006 Thessaloniki, Greece $School of Civil Engineering, Purdue University, West Lafayette, Indiana, U.S.A. (iDepartment of Civil Engineering, Institute of Steel Structures, Aristotle University, GR-54006 Thessaloniki, Greece and Institute for Technical Mechanics, Technical Unjversity, D-5100 Aachen, Germany (Received

20 April 1993)

Abstract-A variational inequality and quadratic programming approach is herein proposed for the investigation of the separation problem of steel bolted brackets. By applying the classic unilateral contact law to describe the separation process along the contact surfaces between the bracket and the column flange, the continuous problem is formulated as a variational inequality or as a quadratic programming problem. By applying an appropriate finite element discretization scheme, the discrete problem is formulated as a quadratic optimization problem with inequality constraints which, in turn, can be effectively treated numerically by means of an appropriate quadratic optimization algorithm. The applicability and the effectiveness of the method is illustrated by means of a numerical application.

1. INTRODUCXTON

steelwork to transmit internal forces from horizontal to vertical structural elements. Such connections, applied for instance as beam-to-column connections, are nowadays used in any possible combination and variation in steel structures (cf. e.g. [l-3]). It is therefore obvious that any improvement in the analysis methods for such steel connections is useful for the improvement of the design principles dictated by modern steel construction codes. Within this framework, the present paper aims to contribute to the research on the advanced methods for the investigation of the structural behaviour of steel baited brackets. It is well known that since loading acts outside the mid-plane of the bolts that join the bracket to the column flange, the contact surfaces between connection members tend to separate each other. However, according to the classical methods used for the analysis of the response of bolted brackets, the previously mentioned detachment phenomenon is in most cases disregarded and complete contact between the adjacent flanges under any type of loading is considered. The consequence of the latter hypothesis Bolted

brackets

are used in structural

1)On leave from the Department of Civil Engineering, Jordan University of Science and Technology, Irbid, Jordan. T To whom correspondence should be addressed.

is that compressive forces between contact fronts are absorbed by the flanges, whereas tension forces are transmitted through the bolts [4,5]. As is obvious, this assumption does not cover numerous cases where the flexibility of the flanges combined with the eccentricity of the loading leads to the development of regions of detachment between the splice members. The evidence of both laboratory experiments and construction practice certifies that steel bolted brackets tend to separate from the column flange under certain operational loading conditions through the development of nonactive contact regions between them. The development of the separation process on a bolted bracket signi~cantly affects the mechanical behaviour of the steel connection under consideration and may lead to its total failure, perhaps at a final stage to the collapse of the whole steel structure. It is therefore obvious that a mode1 to successfully simulate such behaviour would have to take into consideration among others the development of the separation process on the contact flanges. The problem of analysing and calculating the separation process in steel bolted connections attracted the interest of numerous researchers who applied several analytical, numerical and experimental methods. By the appropriate application of the finite element method to numerically simulate the mechanical behaviour of steel brackets, results of considerable importance on the separation phenomenon between the column flanges and the brackets 983

984

C. C. Baniotopoulos et al.

have been recently obtained (cf. e.g. [6-lo]). These numerical investigations led to the ascertainment of the appearance of regions of detachment along the connecting flanges. It is worth noting here that the separation phenomenon is of a geometrical nonlinear nature, since the regions of contact between the connection members and the regions of detachment are not known a priori [ 11, 121.For the latter reasons, the classical structural analysis methods for the analysis and calculation of the problem at hand may lead to erroneous results and must therefore be used with care. Among the models recently proposed to simulate the structural behaviour of steel brackets, those combining the methods of nonsmooth mechanics with finite element discretization schemes are worth mentioning here [13-151. By applying such models, the additional high nonlinearity produced by the development of the separation phenomenon can also be taken into account. The methods of variational inequalities in nonsmooth mechanics which is a recently developed branch of mechanics, have been recently successfully applied to the complete mathematical description of the separation process in bolted steel brackets in a quasi-static way by means of the classic unilateral contact law [16]. Indeed, variational inequalities take into account the exact nature of the unilateral contact nonlinearity. Such a treatment of the problem at hand exhibits among others the advantage of the exact determination of the active contact and detachment zones between the connection contact fronts and of the exact evaluation of the loss of strength of the bolted steel connection due to the development of the separation phenomenon for a given loading without any incremental procedure (cf. e.g. [17-201). Within the aforementioned theoretical framework, the continuous problem which is a typical boundary value problem (BVP) is first formulated as a variational inequality problem with respect to displacements expressing the principle of virtual work of the steel bracket in inequality form at the state of equilibrium. This formulation permits the derivation of the ‘principle’ of minimum potential energy of the steel connection at the state of equilibrium in the form of a quadratic optimization problem which involves a quadratic energy function coupled to the inequality kinematic constraints of the bracket (cf. [13, 19-211). By means of an appropriate finite element discretization scheme, the previously mentioned continuous unilateral contact problem can be put in the form of an equivalent discrete quadratic optimization problem. The latter formulation seems to be very promising because a plethora of quadratic programming algorithms are nowadays available for its effective numerical treatment. It is also worth noting that a dual approach can also be employed for the mathematical formulation of the problem under consideration [13,21]. In the latter case, the formulated variational inequality problem with respect to stresses expresses the ‘principle’ of complementary

virtual work in its inequality form, whereas in the respective quadratic optimization problem, it is the ‘principle’ of minimum complementary energy of the steel splice at the state of equilibrium. The aforementioned quadratic optimization problems can be effectively treated numerically by employing a quadratic optimization algorithm and in particular, the Hildreth-d’Esopo algorithm [22]. This solution method being an iterative Mangasarian-type procedure, seems to be easily programmable and computationally efficient for the numerical treatment of the problem under consideration [17, 181. The range of applicability and the effectiveness of the proposed method is illustrated by means of a numerical application. 2. THE

PROPOSED METHOD

2.1. The continuous problem An elastic body R with boundary F made up of three non-overlapping parts Fr/, FF and Ts is considered in an orthogonal Cartesian system Ox,x2x3. On Ta (respectively F,) the displacements (respectively the surface forces) have given values U, (respectively F,), whereas on Ts, unilateral contact boundary conditions hold (Fig. la). On Fs a frictionless type of contact is assumed and as the positive normal direction the one directed outwards to the boundary is taken into consideration. The unilateral contact conditions with respect to an elastic support are expressed now in the following form: if uN < 0

then S, = 0

(1)

if+>0

thenS,+k(u,)=O,

(2)

where uN (respectively S,) denotes the normal (with respect to the boundary) displacements (respectively reaction forces) on Ts and k(uN) is a nondecreasing function. The previous relations are illustrated in Fig. lb, whereas Figs lc and Id correspond respectively to unilateral contact with a linearly elastic and a rigid support. By assuming that the strains and displacements are small, the problem under consideration consists of the equation of equilibrium, the compatibility relations, the constitutive law relating stresses to strains and the boundary conditions holding on the boundary r. We define a field X* of strains and displacements to be kinematically admissible if it satisfies the compatibility relations, the kinematical boundary conditions on To and on Ts. The volume forces are denoted by pi, and the actual strains and displacements at the position of equilibrium by tii and ui respectively. The differences (E$ - tii) and (UT - u,) represent the kinematically admissible variations of the respective variables. By a; the stress field obtained from 6: by means of the elasticity law is

985

Separation problem of steel bolted brackets

Fi

Fig. 1. On the continuous problem and the forms of k(.).

denoted. By splitting ui into its positive and negative parts defined by the forms: (3) and -24;: + lu$l 2

u$_ = which are non-negative

s

cr~(~~--~~)cu2=

(4)

quantities, the identity

s s

It is proved by applying the method of special variations that variational inequality (7) yields the equation of equilibrium and the boundary conditions on TS and on rr. In this sense, the latter inequality completely characterizes the position of equilibrium of the body R. From the standpoint of mechanics, variational inequality (7) expresses the principle of virtual work in its inequality form for the body under consideration. It has been also proved that at the position of equilibrium any solution of the variational inequality problem (7) minimizes over X* the potential energy of the body given by the form:

p,(u~-ui)clf2

n

0

+

rs

+

I

&,(Ui$- uNi) dr

+

I&+) s rs

(9)

(u f,-“N,)+k(uN+)

(sN,

x(u$+

-uN+))dr

,O

VU$,EX+

(6)

holding on rS, yields by means of the boundary conditions on rF, the variational inequality:

s s n

+

rs

-

kbN+)(ufl+

(8)

(5)

combined with the inequality

s rs

Fiu, dr, s rF

where K(e) is a convex functiondue to the monotonicity of k(.)-defined by the following integral

4(ui+ - ui) dr

rF

vu: E x*

dr -

2.2. The discrete problem

-UN+)dr

F,(u: - u,) dT 2 0 s rF

Vu: E X*.

It has been proved conversely that any solution of the quadratic optimization problem (8) satisfies the variational inequality problem (7) (cf. e.g. [13]). A dual approach with respect to stresses leading to equivalent results can be also employed. In this case, a variational inequality problem expressing from the standpoint of mechanics the principle of complementary virtual work is formulated. The latter gives rise to a quadratic optimization problem of the complementary energy of the body R [13].

(7)

The present section deals with the mathematical description of the separation problem of steel bolted brackets in discrete form by applying the previously presented theory of unilateral contact problems. A

C. C. Baniotopoulos et al.

986

typical steel bolted bracket is first considered (Fig. 2). This eccentrically loaded joint is characterized by the fact that it is subjected to loads acting outside the centre of rotation of the groups of bolts. Due to the eccentricity of the applied loading, the bolts are subjected to tension and shear. In the bracket under consideration, the bolts are arranged symmetrically about the level of action of the force. In such a case, classic structural analysis approaches consider that bolts are uniformly stressed. However, construction practice and laboratory experience certify that bolts in such a bracket are not uniformly stressed since the contact surfaces between joint members tend to separate due to the eccentric loading (Fig. 2). Indeed, the stress distribution along the bolts is very uncertain. It is therefore obvious that more elegant approaches need to be applied so that separation zones on the adjacent fronts of the connection, as well as the stress distribution along the bolts is defined with accuracy. In the present paper, a method for the numerical simulation of the structural response of such steel bolted brackets is presented. As has been previously stated, since on the detached regions between the contact fronts no reaction forces appear, whereas contact reactions do appear on the active contact regions, the development of the separation phenomenon significantly affects the response of the steel connection. The steel connection is discretized by means of an appropriately chosen finite element scheme. In particular, plate elements are used to simulate the behaviour of flanges, whereas the separation conditions holding on the adjacent contact fronts are realized by means of a one-dimensional elastic coupler (fictitious springs of length tending to zero) connecting the adjacent nodes of the contact fronts. The mechanical behaviour of these couplers simulating the possibility of separation of the adjacent nodes of the contact fronts are mathematically described for instance for the ith spring by means of the following law: if [u,(i)] > 0

then R,(i) = 0

(10)

if [u,(i)] = 0

then R,(i) > 0,

(11)

where [u,(i)] = d(i) - up(i) denotes the relative displacements along the z-axis between the column flange (superscript f) and the bracket (superscript b), and R,(i) the respective reaction force. By means of relation (11) it is stated that if the region between the column flange and the bracket connected by the ith spring are in contact, then the reaction forces do appear in the contact region, whereas relation (10) states that if separation phenomena occur, the reaction is equal to zero. It is also assumed that the response of the steel connection under investigation is not affected by any friction effects (frictionless type of contact). By assuming also that the column flange can be considered as rigidified (i.e. exhibiting zero z-axis displacements), then the previous detachment conditions (10) and (11) can be put in the form: if u,(i) > 0

then R(i) = 0

(12)

if u,(i) = 0

then R,(i) 2 0,

(13)

where u,(i) denotes the z-axis displacements of the flexible plate of the bracket in the neighbourhood of the ith spring. By assembling relations (12) and (13) in matrix terms for all the m couplers, the following linear complementarity problem (LCP) is formulated: II: 2 0,

Rr > 0,

R:u: = 0,

(14)

where boldface letters denote vectors and matrices and superscript T denotes transpose vectors or matrices. LCP (14) completely describes in a quasistatic way the separation phenomenon between the column flange and the bracket. We note also that the previously formulated LCP (14) holds on this part of the boundary of the discretized steel connection where unilateral contact conditions hold. By applying now the stiffness method to the simulation of the structural response of the discretized connection, the following matrix equation is obtained: Ku=P

(15)

where K is the stiffness matrix of the discretized connection, u the displacement vector for the whole structure including vector II: and P the load vector. The problem of accurately defining the development of the separation zones between the column flange and the bracket is thus completely described by means of the following quadratic programming problem (QPP) [ 1l-131: II(u) = min{$rTKu - PTu(Au
Fig. 2.

A typical steel bolted bracket.

(If-9

where A (respectively b) is an appropriately chosen transformation matrix (respectively a vector describing the inequality restrictions imposed by the inequalities (12) and (13)). QPP (16) expresses from an engineering point of view the principle of minimum

Separation problem of steel bolted brackets potential energy of the steel connection in a state of equilibrium. The actual displacements of the splice caused by the external loading, as well as the active contact and the regions of separations between the members of the connections can be defined with accuracy by solving the previously formulated problem (16). As has been previously noted for the continuous problem, a dual approach can also be employed for the treatment of the discrete problem. Such a dual approach gives an equivalent rise to a quadratic programming problem of the same type where now stresses are the unknown variables appearing in the quadratic term instead of the displacements and the constraints concern the equilibrium equation and the reaction forces appearing on the splice, i.e., IIC(s) = min{fsTF,s - sTe0]R2> 0, Gs = P},

987

(respectively e,,) the stress (respectively initial strain) vector. QPP (17) expresses from the standpoint of engineering the principle of minimum complementary energy for the steel splice at hand in a state of equilibrium. 3. ON THE NUMERICAL TREATMENT

For the numerical treatment of the previously formulated QPPs, the Hildrethd’Esopo algorithm [22] is applied which is a typical iterative procedure and exhibits the advantage of being easily programmable and computationally efficient [17]. As is well known, the Kuhn-Tucker optimality conditions for the QPP (16) can be written in the following

form [22]:

Au+y=b

(18)

Ku+ATf=P

(19)

(17)

where G (respectively F,) is the equilibrium (respectively flexibility) matrix of the steel connection and s

y>o,

f>O,

yTf=O, 3

01 0’ 03

04 05

t

240

OF THE PROBLEM

klM

i

3

Fig. 3. On the numerical application: geometrical data and cross-sections studied.

(20) 2

1

988

C. C. Baniotopoulos et DBFLXXION

Fig.

OF BRACKET

CONNBCTION

al.

GtOSS SECHON

I- 1

4. Deflections of the bracket connection along cross-section 1-l.

where y is a vector corresponding to the unilateral constraints of the problem and f the vector of reactions on the same constraints. By solving eqn (19) with respect to u, the following relation is obtained:

and F=‘AK-‘AT 2

(23)

,

relations (18)-(20) can be formulated as follows: u = -K-‘(ATf - P),

(21) 2Ff-y=

and then, by putting: h= -AK-‘P+b DEFLEClTON

(22) OF BRACKET

CONNBCTlON

y30,

f>O,

-b

(24)

yY=O.

(25)

SECI’ION 2-2

U FLNQE COL.

Fig. 5. Deflections of the bracket connection along cross-section 2-2.

989

Separation problem of steel bolted brackets DEFLECl-ION OF BRACKET

CONNEtTION

SECI-ION 3-3

t

FLANGE COL

Fig. 6. Deflections of the bracket connection along cross-section 3-3.

The latter relations constitute the Kuhn-Tucker mality conditions for the following QPP: II(I) = min{ff’rFf + hTf]f > 0},

opti-

(26)

where matrix F is a flexibility matrix defined by eqn (23) relating contact forces to the corresponding unilateral contact displacements. When the solution of problem (16) exists, then problem (26) does also have a solution and this is unique [22]. The QPP (26) can be numerically treated by means of the Gauss-Seidel method. During the iterative steps p = 0, 1,2, . , the following iterative values are considered: Is+’ = max{O, wf+i),

where

(27)

i=l,2,...,m

(28)

and m is the number of constraints of the problem. Iterations stop when the computed contact reactions pass the imposed accuracy criteria, i.e., when IlL;-l;+*jj where the symbol defined norm.

<
)I . 1) denotes

(29) an appropriately

DEFLECTIONOF BRACKETCONNECTIONCROSSSECTION4-4

0 BILATERAL

Fig. 7. Deflections of the bracket connection along cross-section 4-4.

990

C. C. Baniotopoulos er al. DEFLECTIONOF BRACKETCONNECTION SECTION5-5

E 2--

0 BILATERAL + TEE * FLNGECOL.

Fig. 8. Deflections of the bracket connection along cross-section 5-5.

4. NUMERICAL APPLICATION

We have developed the code BOLT 1 based on the HiIdreth-d’Esopo quadratic programming algorithm and after an appropriate discretization the following application has been treated. The typical steel bolted bracket shown in Figs 2 and 3 subjected to external loading of P = 34.0 t that acts at a distance of 1Ocm away from the column flange-bracket contact surface is considered. The column section is IPB 240, the bracket fI400 and the bolts are M 27. The previously presented method has been herein applied in order to obtain numerical results concerning the structural response of the steel connection at hand. As has been previously noted, the bolts are directly subjected to tension and also to shear. Under external loading, the contact surfaces simulated here by means of 180 couplers between the adjacent fronts (i.e., column flange and tee) tend to separate, whereas prying forces appear on the active contact zone. For the numerical treatment of the problem at hand, an appropriately chosen finite discretization scheme described in Section 2 and shown in Fig. 3 has been employed. In Fig. 3 the geometric and external loading data, as well as cross-sections l-1, 2-2, 3-3, 4-4 and 5-5 where deflections have been calculated by means of the proposed method and a classic bilateral method are depicted. The solution procedure starts with the assembling of the stiffness matrix of the discretized connection within the theory of small stains and displacements [24]. Then, the potential energy of the structure at hand is formulated in discrete form. Following the proposed method, the potential energy of the discretized steel connection is minimized. This procedure provides the actual deflections of the

bracket and therefore leads to the exact definition the active contact and separation regions along adjacent fronts. The numerical results concerning actual deflections of the bracket in comparison those obtained by means of a classic bilateral proach are depicted in Figs 4-8.

of the the to ap-

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18.

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and

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