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Secant method for nonlinear semi-rigid frames
J. Construct. Steel Research 14 (1989) 273-299
Secant Method for Nonlinear Semi-Rigid Frames
S. L. L e e & P. K. B a s u * Vanderbilt University, Nashville,Tennessee 37235,USA (Received 24 April 1989;revised version received and accepted 26 June 1989) t
ABSTRACT A large-deformation inelastic formulation based on the secant method is proposed for the limit load analysis of planar frames with partially restrained connections. Four test problems are solved to demonstrate the accuracy and computational efficiency of the proposed algorithm.
INTRODUCTION A n u m b e r of first- and second-order schemes are available for the static and stability analyses of framed structures based on the matrix displacem e n t m e t h o d . In the simplest case of first-order elastic analysis it is assumed that the deformations are Small and the stiffness of a m e m b e r remains unchanged during the complete history of loading. The characteristic feature of a second-order elastic analysis is, however, that the effect of m e m b e r axial force on its stiffness is included. This stiffness can be obtained either analytically or numerically. A typical analytical procedure is the slope--deflection m e t h o d in which the equilibrium equations are written with respect to the deformed shape of the structure, 1 and involves the use of stability functions 2,3 which reflect the effect of m e m b e r axial force exactly. O n the other hand, a c o m m o n approximate numerical m e t h o d is the variational formulation of the finite element method. 4-12 In the case of a perfect frame under ideal loading conditions, which does not *Author to whom correspondence should be addressed. 273 J. Construct. Steel Research O143-974X/90/$03.50 ~ 1990Elsevier Science Publishers Ltd, England. Printed in Great Britain
274
s. L. Lee, P. K. Basu
cause any lateral deflection in the members of the frame until the bifurcation load is reached, these formulations essentially lead to eigenvalue analyses. On the other hand, in the presence of initial imperfections, such as geometric deviations, residual stresses, etc., and/or non-ideal loading conditions, deflection will occur in the frame as soon as the load is applied. The resulting set of nonlinear equations can be solved by either an incremental procedure or an iterative procedure. On the grounds that often the drift criterion rather than stability controls practical frame design, a number of approximate and empirical methods, such as those due to H o m e & Merchant, 3 Roberts 13 and Horne, 14 have been put ~forward to make sure that frame stability does not control the design. In the case of incremental analysis, 15--21 a tangent stiffness matrix is generated according to the current geometry and the state of member forces at each stage of incremental loading. In this manner, the complete load-displacement relationship of the frame for the entire loading history can be obtained. In order to ensure equilibrium at the end of each load increment, an improved Newton-Raphson-type iteration scheme is built into the solution algorithm. 22'23 In the case of iterative analysis, 2~26 the secant stiffness is determined by assuming that the current geometry and the state of forces are known. The new values for the current geometry and the state of force obtained after an iteration cycle are then used in the following cycle. This sequence is repeated until convergence is achieved. W h e n the difference of the forces or the geometric state between two consecutive cycles of calculation are smaller than a prescribed tolerance limit, the calculation is said to have converged. A comparison of these two methods reveals that the tangent stiffness approach controls the error at a local incremental level,27-29 and normally requires the use of a small increment in order to obtain sufficiently accurate results. On the other hand, the secant stiffness approach is not sensitive to the loading history of the frame and a large enough increment can be used to limit the required number of iteration cycles for convergence. Another difference between these two methods is that in the tangent stiffness method an extra matrix called the geometric stiffness matrix, which is sensitive to the local slope of the M--O curve of a member, is Used. The present study is concerned with the secant stiffness approach.
MEMBER NONLINEARITY The factors contributing to member nonlinearity are material (elasticplastic), geometric (large displacement), connection behavior (nonlinear M--O relation), and the P-Delta effect.
Secant method for nonlinear semi-rigid frames
275
The stress-strain relationships of structural carbon steel and highstrength low-alloy steel are commonly approximated by an elasticperfectly-plastic stress-strain curve. The moment-curvature relationship of a section based on such a stress-strain curve is idealized by an elastic-plastic representation which ignores the transition part. A further simplification is sometimes introduced by the rigid-plastic representation, which assumes that there is no strain at a section until the moment equals the plastic moment capacity of the section, after which unrestricted plastic flow is allowed to occur in the section. The rigid-plastic representation can be used when the members are primarily subjected to flexural action. However, if the members carry large compressive forces, the inclusion of elastic deformations and instability considerations become important. This is especially true when high-strength steel is used in the frame. In a general analysis scheme, the need for including elastic-plastic behavior is, thus, quite evident. The results in the elastic stage can be used to check the serviceability limit states. On the other hand, the plastic stage is used to estimate the overload strength, and to find whether the ultimate limit state concerned with structural safety is satisfied. Strain-hardening has considerable influence on the load capacity, especially of small structures. In the case of tall structures, however, the effect of instability on frame behavior is far greater than that of strain-hardening. Therefore, it is reasonable to neglect strain-hardening effects in such structures. In the case of small frames carrying service load, the displacements are small and hence the equilibrium and compatibility equations are written with respect to the undeformed geometry, of the structure, but in the case of tall frames, especially near limit states, the deformations will be large and the state equations should refer to the deformed geometry of the structure. Of the three coordinate systems suitable for this purpose, the total Lagrangian coordinate system9'11'21 is most widely used in the variational finite element formulation of the problem. This is associated with the possible penalty that a large number of elements is often required to achieve a given level of accuracy. Eulerian 19'3°'31 and Updated Lagrangian 25'32'33coordinate systems are convenient in the beam-column approach. The resulting force-displacement relationships, though complex, are more accurate and require fewer elements to model the structure. In the conventional analysis of steel structures, connections are considered either perfectly hinged or rigidly fixed. This classification, though very useful in analysis, is unrealistic because in practice a certain amount of flexibility cannot be avoided in so-called rigid connections, and supposedly pinned connections without a certain amount of moment resistance is rare. In particular, the AISC specification34 designates three types of construction in its 1969, 1978 and 1980 editions. There are Type 1 construction (rigid framing), Type 2 construction (simple framing), and
276
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Type 3 construction (semi-rigid framing). Because of the lack of sufficient knowledge of connection behavior and strength, and the lack of practical methods to analyze flexibly connected frames, Type 3 construction is seldom considered in practical analysis and design. In Type 2 construction, two inconsistent assumptions are made. Firstly, the connections are assumed to have no moment-resisting capacity when the frame is designed for gravity loads only. Secondly, it is assumed that the connections and connected members have adequate capacity to resist the moments induced due to wind loading, and that the connections have adequate inelastic rotation capacity to avoid overstress of the fasteners or welds under combined gravity and wind loading. Recognizing this contradiction, the newly published Load and Resistance Factor Design (LRFD) Specification35 for structural steel buildings explicitly designates only two types of construction: Type FR (Fully Restrained) construction and Type PR (Partially Restrained) construction. It is clear that the behavior of most of the practical connections lies between those of rigid and pinned cases, as shown in Fig. 1. Apart from higher reliability,, considerable economies may be achieved if the true behavior of these joints is taken into account while designing steel frames. Realizing the importance of data on real connection behavior, considerable analytical and experimental research has been carried out to measure the moment-rotation characteristics of various types of commonly used framing connections. During the 1970s and 1980s extensive experimental and analytical data were published for bolted and welded connections both in this country and in Europe. 28The analytical models have been based on semi-empirical considerations as well as finite element analysis and in some cases nonlinear material behavior has been considered. For instance, in 1975 Frye & Morris 36 presented an analysis procedure to account
z
ROTATION Fig. 1. Typical moment-rotation curves of connections.
Secant method for nonlinear semi-rigid frames
277
for the nonlinear behavior of the connection. They have expressed the force-deformation characteristics in a nondimensional form by using polynomial equations for seven commonly used connection types. In addition, a large n u m b e r of papers have been published on this topic, 27'28'33'37-51 some of which include nonlinear behavior. However, for the purpose of the present study, the m o m e n t vs rotation ( M - O ) relationship for a semi-rigid connection will be expressed in the following form:
[ 1 in which O = M = Ci = Mo = a = Mma~ =
(1)
the relative rotation of the connection; the moment acting on the connection; constant coefficients; aMmax; a sensitivity factor ( = 11.4); maximum moment in the experimental M--O data.
S E C O N D - O R D E R S E C A N T STIFFNESS M A T R I X The effect of semi-rigid connections on m e m b e r stiffness can be accounted for in two ways. O n e is to represent the connection stiffness by introducing discrete elements at the ends of the m e m b e r in the form of springs with flexural, shear, and axial stiffnesses. 27,28'aa'as'51-55 In addition, rigid elements may also be introduced to account for the width of the column. 53 The advantage of this approach is that the discretization of the structure into members, spring, and rigid connectors is a relatively easy task. The drawback of this m e t h o d is that the total n u m b e r of degrees of freedom required to model the deformed configuration of the structure increases significantly. This may be of especial concern when all the connections of a large structure are of Type PR. The other way is to modify directly the m e m b e r stiffness relationships to account for the partial rotational restraining effect of the connections. 26'565a As a result, all the displacement components can be referred to the column-line. Also, the relative rotations caused by a Type P R connection do not increase the required n u m b e r of degrees of freedom. The present study is based on this approach. The deformed configuration of a typical m e m b e r with T y p e P R end
278
S. L. Lee, P. K. Basu o
o
°
Ro
X
r
U.~.]o, L-~ .........
o
~",
',
2/02 U2 / Y
Fig. 2. Semi-rigidlyconnected beam-column.
connections is shown in Fig. 2. For the sake of simplicity no m e m b e r loading is shown in the figure. The effect of m e m b e r loading will, however, be expressed in terms of the fixed-end forces and m o m e n t s RS/~,R sv and M/Sv (i = 1, 2). If ki = Elci/L is the current rotational spring constant of the connection at end i of the m e m b e r and Wg is the current connection rotation, the m o m e n t at end i can be expressed as
in which Oi is the joint rotation at the column center line and 0/' is the end rotation of the beam. Therefore, the difference ( e l - 8/) represents the change in end rotation of the beam caused by the flexibility of the connection. As in Galambos, i the end m o m e n t s of the m e m b e r can be given by
in which A = v2 - I;1, and S and C are the so-called stability functions which are functions of L ( P / E D 1/2. Tabulated values of C and S are available
Secant method for nonlinear semi-rigid frames
279
in standard t e x t b o o k s , including Ref. 1. Substituting the expression for 0" b a s e d on e q n (2) into eqns (3a) and (3b), it follows that
MI = _~_[ HIOI+H302-(HI +H3) --~]
(4a)
in which H1 = [Cl(C 2 - S 2) + clc2C]/p
(5a)
/42 = [c2(C 2 - S 2) + cxc2C]/p
(5b)
H3 = Cl c2S/p
(5c)
and p = C 2 - S 2 + (Cl +
c2)C+ClC2
(5d)
B y using the equilibrium equations Fyl = - Fy2 = (M1 + M2 + PA)/L
(6a)
and the axial f o r c e - d i s p l a c e m e n t relationships
-- Fxl = Fx2 = AE(u2 - Ul)/L
(6b)
the six f o r c e - d i s p l a c e m e n t relationships for the m e m b e r , including the effect of fixed e n d forces and m o m e n t s caused by m e m b e r loading, can be written in the matrix form as {F} = [K] {D} + { g ~ in which {F} = {D} = {RF} = [K] =
--a
[rl =
the the the the
(7)
member force vector = {Fxl, Fyl, M1, Fx2, Fyz, ME}; member displacement vector = {ul, vl, 01, u2, vz, 02}; fixed end force vector = {RxVl,RyF1,M~, Rxz,V Ry2,FM2F}; secant stiffness matrix of the member given by
0
0
-a
0
0
0
b
c
0
-b
d
0
c
e
0
-c
f
-a
0
0
a
0
0
0
-b
-c
0
b
-d
0
d
f
0
-d
g
(8)
with a = A E / L , b = ( H , + H2 + 2 H 3 ) E I / L 3 - P / L , c = (H1 + H a ) E I / L 2, d = (HE + H 3 ) E I / L 2 , e = H I E I / L , f = H3EI/L,g = H2EI/L.
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S. L. Lee, P. K. Basu
If one end of the member is PR-conneeted and the other end is rigid (say, End 2), the value of ci corresponding to the rigid end will be infinity and the expressions for H1,//2, and Ha will take the form H1 = cl C / ( c l + C)
/'/2 =
(C 2 -
S 2 -1- Cl
C)/(c1 "~"C )
Ha = Cl S/(Cl + C)
On the other hand, if End 1 of the member is PR-connected and End 2 is pinned, c2 = 0. It means that H2 = / / 3 = 0 and
H1
=
¢1(C 2 -
$ 2 ) / ( C 2 - S 2 -}- Cl C)
Again, if both Cl and c2 are set to infinity, H1 = HE = C and Ha = S and hence the matrix relationship (7) becomes applicable to the case of a rigid jointed frame. Finally, the first-order matrix relationship for a PRconnected frame is obtained by setting C = 4 and S = 2 in the expression for HI, /'/2, and/-/3 given by eqns (Sa)-(Sd) and the resulting matrix relationship can be used for the static analysis of such frames, whereas, if H1 a n d / / 2 are directly set to 4 a n d / / 3 to 2, the first-order linear matrix relationship for a rigid jointed frame is obtained.
F I X E D END M O M E N T Assume that the member shown in Fig. 3 with semi-rigid end connection is subjected to some transverse loading and the area of the resulting static moment diagram is ,4. This area corresponds to the case when the end moments of the member are zero requiring that there be no rotation in the connections and hence the joint and the column rotation be the same as the free end rotations ~ a n d / ~ of the beam given by -
.42
./~ =
EIL
'
.4(L-2) ElL
(9)
in which ~ is the distance of the centroid of the static moment diagram from End 2 of the beam. The fixed end moments M s v and M s v should be of such magnitude that the joint rotations at the column center line caused by these moments alone be - ~ and -0~. In that event MSF = _ E__I_/[H1~ + H30~] L
(10a)
MSF = _ E__I_/[/-/3~ +/-/2 4] L
(10b)
Secant method for nonlinear semi-rigidframes
281
W(x)
(a) Loaded Beam
I_
,
~
'
El
-
(b) Equivalent Beam
(c) Conjugate Beam
Fig. 3. Conjugate beam for semi-rigidly connected member. (a) Loaded beam. (b) Equivalent beam. (c) Conjugate beam. In the case of ideal fixity, the expressions for ~ and 0~ in terms of the fixed end moments M v and M F will be MFL MFL 3 ~ + 6EI
(lla)
=
MFL MFL 6el F 3~
(11b)
#2=
Using eqns (9) and (11), the values of M F and M F for a given transverse loading condition can be obtained. Expressions ( l l a ) and ( l l b ) can be used to replace ~ a n d / ~ appearing in eqns (10a) and (10b). The modified expressions for fixed end moments in the presence of a PR-connected member in terms of the fixed end moments for a rigidly connected m e m b e r will then be obtained as 1 [(2M~ - M~)H1 + (M F - 2M~:)H31
(12a)
= 1 [(2M F _ MF2)H3 + (M F _ 2MF) H2]
(12b)
=
o
282
S. L. Lee, P. K. Basu
In the event of m e m b e r loading, these m o m e n t s can be i n t r o d u c e d into the slope-deflection equations, as in eqn (7). In the case of a first-order analysis or in the absence of axial force, the v a l u e s of C and S in expressions for Ha, H2, a n d / / 3 should be set to 4 and 2, respectively.
LARGE RELATIVE DISPLACEMENT T h e slope-deflection equations (4a) and (4b) are based o n the assumption that the relative displacement A/L is small. If, however, 6JL is not small it is necessary to modify the secant stiffness matrix. By referring to the d e f o r m e d configuration of a m e m b e r , shown as Fig. 2, the axial d e f o r m a t i o n 8 and the rotation R can be expressed as 8 = (u2 - ua) - (v2 - vl) tan (R/2)
R=R'-Ro=tan-a
(1' 2 --
in which
va)/cos(R/2) Lo" A F
AF1 =
)
Y2-Y1
-
o
o
. -a//Y~-Y~'~ -tan k X 2 ~ _ 1: ( v 2 - va) AFa L
L Lo" AF" cos(R/2)
A F = the rotation modification factor = 1.0 - 0.0002R - O.041R2 L0 = original length of the member L = deformed length of the member
Based on the above, the modified slope-deflection equations, w h e n the m e m b e r displacements are referred to the d e f o r m e d configuration, will be
M1
=
H~ 01 +/-/3 02 - (Ha + H 3 ) ~
A/q
(13a)
=EI
Also, F:I-
AE L
--[ul
- u2 + ( v 2 - v a ) t a n ( R / 2 ) ]
(14a)
Secant method for nonlinear semi-rigid frames Fx2 = A E
L [ u 2 - ux - ( v 2 -
Fyl=
283 (14b)
vl)tan(R/2)]
MI+M2
(15a)
L
(lSb)
Fy2 = - f .
These equations can be written in matrix form as {F} = [K'I{D} in which the modified secant stiffness matrix is
[K'] =
a
-h
0
-a
h
0
0
b'
c
0
-b'
d
0
c.AF1
e
0
- c . A F1
f
-a
h
0
a
-h
0
0
-b'
-c
0
b'
-d
o
d.AF1
f
0
-d.AF1
g
(16)
with b' = (H1 + H2 + 2H3) ( E I / L 3 ) ' A F 1 and h = ( A E / L ) t a n ( R / 2 ) . It may be noted that the matrix [K'] is not symmetrical. For the sake of convenience this matrix is expressed as the sum of two matrices: [K] as in eqn (8), and [KL] which contains the correction terms due to large displacements. By including the effect of fixed end forces, the matrix equation takes the form {P'} = [K]{O} + [KL]{O} + {R F}
In this equation [KL]{D} can be combined with the vector {RF} and treated as a pseudo force term, so that {F} = [K]{D} + {R --r} in which {Rv-} {RF} + [KLI{D} If the member curvature effect is considered, then as in Fig. 4,
e = ~ E (A - Ab)
284
S. L. Lee, P. K. Basu
Ab
Fig. 4. Member curvature effect.
and as a result eqns (14a) and (14b) b e c o m e AE FX I
~
Fx2 -
in which
--
AE -~
m
L
[U 2 -- U 1 "b (V 2 -- V l ) t a n ( R / 2 )
(v2 -
[ u l - u2 -
vOtan(R/2)
-
- Ab]
Ab]
Ab = [b1(01 + 02) 2 -~- b2(01 -- ~2) 2] L
(C + S)(S -
bl =
b2 =
2.0)
87r2p
S
8(c + s) P
o = Pe
PL
~EI
Again, the curvature corrections to Fxl and
{
fx2 a s
, 001
can be c o m b i n e d with [R-r] as a p s e u d o force vector so that (R -r} = {RF} + [KLI{D} + {Rb}
ELASTIC-PLASTIC BEHAVIOR
(17)
OF MEMBERS
Based on the elastic-perfectly-plastic material behavior, w h e n e v e r the m o m e n t value at a section reaches the plastic m o m e n t capacity of the
Secant method for nonlinear semi-rigidframes
285
section a hinge is introduced at the section so that the plastic condition including the effect of axial force is always satisfied. To account for the relative rotation between the sections on each side of the hinge (actually a plastic hinge), it is necessary to add a row and a column to the structure stiffness matrix. 24 From a programming point of view, this approach requires complex book-keeping and also may turn out to be uneconomical in terms of storage requirements. The alternative approach in which such augmentation of the structure stiffness matrix can be avoided by modifying the member stiffness matrix due to the presence of a plastic hinge, will be used here. In the process, it will obviate the need for computing the relative rotation of the sections at a plastic hinge.18'59'6° The rotation at the ends of a member with plastic hinges can be computed as discussed below. The end moment eqns (4a) and (4b) can be rewritten in the following form L
E1 M1 = H1 01 +/-/3 02 - (H1 + H3)
L E1
A
(18a)
A
(18b)
M 2 = H3OI + H 2 0 2 - (H~ + H~)--:--
If the first end only has a plastic hinge, then M1 01 =
1 FMpIL
"~i-iL Y
a
"I-(Hi + H3)-~--- H302
= Mpi
and using eqn (18a)
]
Also, if the second end only has a plastic hinge, then M2 = Mr,2 and using eqn (18b)
1
r M~s_, a 02 = "~2"2L y "i"(H 2 -.I-H3) T - H301
]
J
On the other hand, if both the ends have plastic hinges, then M1 and/142 = Mp2, and using eqns (18a) and (18b),
01 -~
=
i-i~ .H~-HI H1 L - - E T - + (H, + H3)
- H3
#I~.H~-~
+ (H~ + H3)
= Mpi
286
S. L, Lee, P. K. Basu
The reduction of the plastic moment capacity of a section (Ml~) due to the presence of axial force can be accounted for in the iteration procedure by using the following standard relationship for W-shapes.
MI'¢ P) <1"0 Mp = 1.18(1.0- ~y in which
Mp¢ = Mp = P = Py =
(19)
reduced plastic moment; full plastic moment of the member; axial force; yield load of the member.
COMPUTER IMPLEMENTATION For the purpose of the present study the computer software LEPPR was developed on the SUN-3 computer using FORTRAN-77 language, based on the flowchart shown in Fig. 5. It uses an iterative algorithm for handling the second-order elastic-plastic stability problem of planar semi-rigid frames. Each cycle of iteration involves updating of the secant stiffness matrix. Convergence is achieved by ensuring that differences of the member foi'ces in two subsequent cycles of iteration are smaller than a specified limit. The iterative process is shown in Fig. 6. An initial load level P1 is chosen and the first cycle of response calculation is done assuming rigid connections and using first-order elastic analysis. The load vector and the linear stiffness matrix K~, which ignores the effects of axial forces and connection flexibility, is assembled. The displacement vector D~ is obtained by solving the structure matrix equation of the form of eqn (7). However, before assembling the structure stiffness matrix and load vector, the member stiffness matrix and vector in the local reference frame are transformed into the global reference frame using standard transformation relationships based on direction cosines of the member axis with respect to the global reference axes. After D~ is obtained, the member forces are calculated, and used to set up the updated stiffness matrix K 2 which is used in the next cycle of iteration. This process is repeated until the convergence criterion is satisfied. At this point the displacement vector D1 and the member forces caused by load P1 are obtained. These member forces are then used to set up the stiffness matrix K~ for first iteration cycle of the next load level P2. In the case of a second-order elastic stability analysis, this process is continued until the critical load level is reached, i.e. when the system becomes unstable.
Secant method for nonlinear semi-rigid frames
I L
287
START ]
LOADINGVECTOR
=1 v]
STIFFNESSMATRIX
I
CORRECTION LOAD VECTOR
I
SOLVESYSTEM
NEWLOADINGLEVEL
ES
DETERMINATEOF STIFFNESSMATRIX
GEOMETRICNONLINEARCOEFF. MEMBERFORCES UPDATECOORDINATE
]
YES
[, MAT-NONLIN. I
T_
I
I
YES~
PRINTOOT
REDUCEDLOADING ] INCREMENT N
O
Fig. 5. Summaryflow chart.
In the computer software, the critical condition is identified in the following three ways. (1) If the determinant of the global stiffness matrix is zero, the structure is said to have reached the critical condition. On the other hand, if the determinant is found to be negative, the current load level is
288
S. L. Lee, P. K. Basu
l
[14
y
Ppz
il.J~~
Pl
Pl
'
h, I
/,p/ I x: a: ~///I
I D' ~-~-Secout ~ l l l i i e hi,,,ge D
~,
plutic
hinge
L
Dz
Dpi
Ds
D#
n.
Displacement Fig. 6. Iterative process.
~ett-1
',~. ",,vnLOAOLE~ R-,~ "', 'Pi "
_
~
Fig. 7. Linear interpolation of load level to attain zero value of DET(K).
Secant method for nonlinear semi-rigid frames
289
reduced to a new value based on linear interpolation using the last two values of the determinant, as per eqn (20), based on Fig. 7:
Pn = P i - 1 +
e l - ei-1 DETi- ~- DETi
DETi-1
(20)
(2) This criterion uses the required number of iterations as an indicator of the unstable state and is based upon the consideration that as the load level approaches the critical value, one or more member forces may blow-up and fail to converge at all. Hence, if the number of iteration cycles exceeds a specified value (say, 300) the load increment is reduced by one-half until the increment is smaller than a specified value when the program will stop further load incrementation. (3) In this criterion, the value of the maximum relative drift, i.e. story drift, is used as an indicator of the critical state. Whenever the relative drift is found to be larger than a specified limit, the program stops further load incrementation. In the case of elastic-plastic member behavior, at the end of each load increment the formation of new plastic hinges is checked, based on the yield condition of eqn (19). For this purpose, a predefined tolerance limit is used. After a load increment, if one or both of the end moments of a member are found to exceed its plastic moment capacity, the load level is reduced by linear interpolation, as in Fig. 8. For End 1, the interpolation equation will take the form
ep = ei
M i - Mp .Mi_ Mi_l ( e i - e l - l )
(21)
The iterative process is continued until the plastic hinge in question is just formed within the predefined tolerance limit. At this point, for additional load increments, the stiffness of the member needs to be modified. Also, the geometric definition of the member gets modified due to the introduction of the plastic hinge, which is treated as a real hinge. After the first plastic hinge is formed, say, at load level Ppl ~a new coordinate system (~,y) with origin located at (Dpl, Ppl) is used for the subsequent load increment(s). This consideration is shown in Fig. 6. It may be noted that based on this new coordinate system P1 = P3 - Ppl Di = D 3 -
Dpl
290
S. L. Lee, P. K. Basu
.~J
--
l2
°1 'II Mi-, Mp
MOMENT Mt
Fig. 8. Linear Interpolation of load level to satisfy the plastic moment value. n
The iteration process is continued for the new path (P1 versus D1), and the load level ~is increased until the next plastic hinge(s) is formed. A corrective load vector is introduced in each cycle of iteration to enforce the force equilibrium when the total load is applied on the system which has been modified due to the formation of the new plastic hinge. This procedure is continued until a failure mechanism is formed, or a critical condition is reached. If, on the other hand, the small-displacement theory is used, no updating of the coordinate system is necessary and the nodal coordinates of the original geometry are used for complete analysis. In contrast, in the case of the large-displacement theory, it is necessary to update the coordinate system at each iteration cycle.
NUMERICAL EXPERIMENTATION In order to determine the reliability and computational efficiency of the proposed algorithm, the following four test problems are considered. All computations are carded out on a time-shared SUN 3/260 workstation.
1 Large-deformation e!~ic analysis of a cantilever beam As shown in Fig. 9, the beam carries a conservative point load at its tip. The size of the beam is taken as W4 x 13 and the length L = 0.3048 m. The theoretical solution of the problem based on inextensional theory was given by Bisshopp & Drucker. 61 Two sets of analyses, using one member model, are carried out with LEPPR, one based on actual cross-sectional
Secant method for nonlinear semi-rigid frames
P i
L
x
¥
X
s
291
~ 1
Fig. 9. Example 1: cantilever beam.
.
Y-I.
x/"°
0
0'2
0.4 0.6 X/L and YIL
0.8
1.0
Fig. 10. Load-deformation curves of cantilever beam. Theoretical (-) axial deformation
(O). No axial deformation (A).
area of the beam (A = 2.47mm x 103 mm), and the other for the inextensional case. The latter case is simulated by assigning a very large number to the area (A = 2.47 mm x 10 7 mm). in Fig. 10, the results of both the analyses are compared with the theoretical results of Bisshopp & Drucker. 61 The close agreement of the inextensional results establishes the accuracy of the model. On the other hand, the disagreement between inextensional and extensional results is found to widen as the load level is increased, signifying that the inextensional theory can be used for low load
292
S. L. Lee, P. K. Basu
×
L Ill IFig. 11. Example 2: fixed-free strut.
levels only. In order to generate the plots in Fig. 10 based on the present analysis, ten load increments were used, requiring 2.88 C P U s. On the other hand, if it is desired to generate data for just one load level with no incremental steps, say, corresponding to pL2/EI = 10, the required C P U time drops to 0.44 s only. The computational efficiency of this m e t h o d is quite evident from this. 2 Elastic post-buckling analysis of a fixed-free strut The strut shown in Fig. 11 is considered. The solution for this elastica problem is available in the publication of Timoshenko & Gere, 62 which assumes inextensional behavior. The size and length of the m e m b e r is taken to be same as in the last example. In order to trace the post-buckling path an initial perturbation in the form of an eccentricity, e, in the load equal to L/IO 000 is introduced in the present analysis, The theoretical results from Ref. 62 based on e = 0 along with the results of the present analysis corresponding to the actual (extensional) and simulated inextensional cases are shown in Figs 12 and 13. The results in Fig. 12 are based on a single-element model and those in Fig. 13 on a four-element model. In each case 12 load increments were used to generate the load-displacement curves. The four-element model shows better agreement with the theoretical results. It is evident from Fig. 13 that the inclusion of the effect of axial deformation modifies the transverse deflection slightly over the range 1 < P/Pcr < 1"6. In the case of 12 load increments, the C P U time usage for one- and four-element cases were 3-52s and l l . 0 2 s , respectively. With one load step only, the corresponding times were 0.6 s and 5.14 s, respectively.
Secant method for nonlinear semi-rigid flames 2.2
2.2
2.C
b
2.0
293'
o
°o °
1.1=
1.e
o
1.d
1..1
1,2
oA
1.--'
I
1.C -0'2
02
0.4
X/L
and
0.6
0.8
1.0
1.0 -0.2
¥/L
Fig. 12. Load-deformation curves for oneelement model of strut of Fig. 11. Theoretical (-). Axial deformation (C)). No axial deformation (A).
0.2
0-4
XIL
and
0.6
0.8
1.0
YIL
Fig. 13. Load--deformation curves for fourelement model of strut of Fig. 11. Theoretical (-). Axial deformation (C)). No axial deformation (A).
3 Large-deflection analysis of fixed arch The arch shown in Fig. 14 is considered. The theoretical and experimental load-deformation responses of this arch were presented by Williams. 63 In the present study, both the two-element model (3.06 C P U s for 14 steps and 0-44 C P U s for one step) and the four-element model (5.96 C P U s for 14 steps and 0.7 s for one step) were used and identical results were obtained. The exact agreement of the present results with the theoretical one is evident from the load vs central deflection plot of Fig. 14. 4 Nonlinear analysis of four-story frame Here the frame of Fig. 15, with r = 0-24, L c = 3.66 m, L o = 9.14 m, E = 201 x 106 kPa, Fy = 236 x 103 kPa, and beams W16 x 40, columns W12 x 79 (bottom story) and W l 0 x 60 (rest), is considered. Kassimali 3° carried out second-order small deformation analysis of the same structure as a rigid frame using the incremental approach in combination with N e w t o n - R a p h s o n iteration. In the case of the rigid frame, the failure mechanism and the load deformation curve are shown in Fig. 16. A l o n g with Kassimali's results, the curves based on present analysis corresponding to large and small deformation approximations are also shown. The results are found to be in good agreement. The same frame was again
294
S. L. Lee, P. K. Basu 300-
a~~ ~.~ o-oo8m
250 -
/
i.-o3286m-~p3_Es_emd
200 -
¢
z
150i11 o .J
100 5G I
0.3 Deflection
I
I
I
I
0.6 0.9 1"2 i.5 o f m i d d l e span ( r a m ) x l O -1
Fig. 14. Example 3: load-displacement curve for a fixed arch. Theoretical ( - ) . Present
(o).
~P/2 r"P/2 t
P
1.2
P/2
1.0
P/2 0.8 i.
£oe
J
'6°i
O.Zl
0.2 gassimali 30 I
!~
L@
Fig. 15. Example 4: four-story frame.
O-
0.'40
I
Small def. Large def. I
0-80 1"~.0 OlHxl o =
I
1"60
.l
2.00
Fig. 16. Load-displacement curves of fourstory frame. Kassimali3° (n). Small deformation. Large deformation (A). Pc, = 151.9 kN, D = horizontal deformation of top right joint and H = 14.63 m.
Secant method for nonlinear semi-rigid frames
295
1.2 6.0
5.O
o.e
~O'E n
0.~ 0.t
0
I
I 1.0
I I I 2.0 3"0 4.0 Rotation (tad) x 10=
I 5.0
// I 0.40
I 6.0
Fig. 17. Moment-rotation curve of end plate connections used in example 4. Test data64( - - - ) . Curve fitting(--).
I '[_ $, I Rigid I, I 0.80 1.20 D I H x 102
OR I 1.00
I 2-00
Fig. 18. Load-displacementcurves of fourstory rigidand PR frames. Rigidconnection (O). PR connection (A). Pcr= 151"9kN, D = horizontal deformation of top right joint and H = 14.63 m.
analyzed with Type PR (end-plate) connections. The M--O curve for the connections used is shown in Fig. 17 based on data from Ref. 64. This curve is the result of the curve fitting with the proposed O--M formula [(eqn (1)]. The load-deformation curves for rigid and semi-rigid frames based on the large-deformation assumption are shown in Fig. 18. The results for the semi-rigid frame based on a small-deformation assumption are found to be the same as for the large-deformation assumption. It is found that the load-deformation curves for all cases agree closely and the limit state occurs at almost the same load level. However, a large lateral drift is obtained in the case of the semi-rigid frame.
CONCLUSION The general formulation of the secant stiffness matrix for a m e m b e r undergoing in-plane large deformations in t h e presence of semi-rigid connections and material nonlinearity is presented. A new nonlinear curve-fitting formula for representing 0 in terms of M for semi-rigid connections is also presented. A set of four examples are considered to demonstrate the accuracy and computational efficiency of the proposed scheme. The scheme can be used either to generate the complete
296
s. L. Lee, P. K. Basu
load--deformation path into the post-buckling range or to locate any point on the path. The former requires load application in an incremental fashion and the latter in one step only. Also, the latter consumes significantly less CPU time than that required to generate the complete load-deformation path. The effect of including the axial deformation at small deformation levels may not be significant, but at large deformation levels it should be included, at least for the problems of the type considered. The proposed method is found to be computationally efficient and reliable in carrying out nonlinear analysis of framed structures. Based on preliminary studies to generate the segment of the load-deformation path for which the stiffness matrix is not positive definite, it appears the proposed method will need to be modified by incorporating either the Control-Displacement Method 6~-68 or the Arc-Length Method. 69-72 Moreover, the proposed method can be extended to a three-dimensional situation, which will b e meaningful only if data on three-dimensional behavior of connections are available.
REFERENCES 1. Galambos, T. V., Structural Members and Frames. Prentice-Hall, New York, 1968. 2. Livesley, R. K. & Chandler, D. B., Stability Functions for Structural Frameworks. Manchester University Press, Manchester, UK, 1956. 3. Horne, M. R. & Merchant, W., The Stability of Frames. Pergamon Press, New York, 1965. 4. Hibbitt, H. D., Marcel, P. V. & Rice, J. R., A finite element formulation for problems of large strain and large displacement. Int. J. Solids Struct., 6 (1970) 1069--86. 5. Bathe, K. J., Ramm, E. & Wilson, E. L., Finite Element Formulation for Large Displacement and Large Strain Analysis, Report No. UC SESM 73-14. University of California, Berkeley, USA, Sept. 1973. 6. Bathe, K. J., Ozdemir, H. & Wilson, E. L., Static and Dynamic Geometric and Material Nonlinear Analysis, Report No. UC SESM 74-4. Structural Engineering Laboratory, University of California, Berkeley, USA, 1974. 7. Bathe, J. K. & Ozdemir, H., Elastic plastic large deformation, static and dynamic analysis. Computer & Structures, 6 (1975) 81-92. 8. Mondkar, D. P. & Powell, G. H., Finite element analysis of nonlinear static and dynamic response. Int. J. Num. Meth. Engng, 11 (1977) 499-520. 9. Kohnke, P. C., Large deflection analysis of frame structures by fictitious forces. Int. J. Num. Meth. Engng, 12 (1978) 1279-94. 10. Milner, H. R., Accurate finite element analysis of large displacement in skeletal frames. Computer & Structures, 14(3-4) (1981) 205-10. 11. EI-Zanaty, M. H. & Murry, D. W., Nonlinear finite element analysis of steel frames. J. Struct. Div. ASCE, 109(2) (Feb. 1983) 353-68.
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12. Robert, K. W. & Rahimzadeh, J., Nonlinear elastic frame analysis by finite element. J. Struct. Div. ASCE, 109(8) (Aug. 1983) 1952-71. 13. Roberts, T, M., Matrix methods of analysis of multi-storeyed sway frame. In Steel Framed Structures, Stability and Strength, ed. R. Narayanan. Elsevier Applied Science, New York, 1985. 14. Horne, M. R. , Frame instability and the plastic design of rigid frames. InSteel Framed Structures, Stability and Strength, ed. R. Narayanan. Elsevier Applied Science, New York, 1985. 15. Halldorsson, O. P. & Wang, C. K., Stability analysis of frameworks by matrix methods. J. Struct. Div. ASCE, 94(ST7) (July 1968) 1745-60. 16. Roberts, T. M., Behavior of nonlinear structures. Ph.D. Thesis, Dept of Civil and Structural Engineering, University College, Cardiff, UK, 1970. 17. Roberts, T. M. & Ashwell, D. G., The use of finite element mid-increment stiffness matrices in the post-buckling of imperfect structures. Int. J. Solids Struct., 7 (1971) 805-23. 18. Mcnamee, B. M. & Lu, L. W., Inelastic multistory frame buckling. J. Struct. Div. ASCE, 98(ST7) (July 1972) 1613-31. 19. Oran, C., Tangent stiffness in plane frames. J. Struct. Div. ASCE, 99(ST6) (June 1973) 973-85. 20. Oran, C., Tangent stiffness in space frames. J. Struct. Div. ASCE, 99(ST6) (June 1973) 987-1001. 21. Bozzo, E. & Gambarotta, L., Inelastic analysis of steel frames for multistory building. Computer & Structures, 20(4) (1985) 707-13. 22. Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill, London, 1977. 23. Crisfield, M. A., Accelerating and damping the modified Newton-Raphson method. Computer & Structures, 18(3) (1984) 395-407. 24. Majid, K. I., Non-Linear Structures, Wiley Interscience, New York, 1972. 25. Goto, Y. & Chen, W. F., Second-order elastic analysis for frame design. J. Struct. Div. ASCE, 113(7) (July 1987) 1501-19. 26. Lee, S. L., Large deformation elastic-plastic stability analysis of plane frames with partially restrained connections. M.S. Thesis, Vanderbilt University, Nashville, TN, USA, Dec. 1987. 27. Ackroyd, M. H., Nonlinear inelastic stability of flexibly-connected plane steel frames. Ph,D. Dissertation, Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, CO, USA, 1979. 28. Goverdhan, A. V., A collection of experimental moment-rotation curves and evaluation of prediction equations for semi-rigid connections. Master's Thesis, Vanderbilt University, Nashville, TN, USA, Dec. 1983. 29. Narayanan, R., ed., Steel Frame Structures, Stability and Strength, Elsevier Applied Science, New York, 1985. 30. Kassimali, A., Large deformation analysis of elastic-plastic frames. J. Struct. Div. ASCE, 109(8) (Aug. 1983) 1869--85. 31. Kam, T. Y., Large deflection analysis of inelastic plane frames, J. Struct. Div. ASCE, 114(1) (Jan. 1988) 184--97. 32. Yang, Y. B. & McGuire, W., Stiffness matrix for geometric nonlinear analysis. J. Struct. Div. ASCE, 112(4) (April 1986) 853-77. 33. Lui, E. M., Effects of connection flexibility and panel zone deformation on
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the behavior of plane steel frames. Ph.D. Thesis, Purdue University, May 1985. 34. AISC, Manual of Steel Construction, 8th edn. America Institute of Steel Construction, Chicago, 1980. 35. AISC, Manual of Steel Construction--Load and Resistance Factor Design, 1st edn. America Institute of Steel Construction, Chicago, 1986. 36. Frye, M. J. & Morris, G. A., Analysis of flexibly connected steel frames. Can. J. Civil Engrs, 2(3) (Sept. 1975) 280--91. 37. Yee, Y. L. & Melchers, R. E., Moment-rotation curves for bolted connections. J. Struct. Div. ASCE, 112(3) (March 1986) 615-35. 38. Shukla, S., Static response of plane steel frames with nonlinear semi-rigid connections. Master's Thesis, Vanderbilt University, Nashville, TN, USA, August 1986. 39. Simitses, G. J. & Vlahinos, A. S., Stability analysis of a semi-rigidly connected simple frame. J. Construct. Steel Res., 2(3) (1982) 29-32. 40. Simitses, G. J., Swisshelm, J. D. & Vlahinos, A. S., Flexibly-jointed unbraced portal frames. J. Construct. Steel Res., 4 (1984) 27-44. 41. Chen, W. F. & Zhou, S. P., Inelastic analysis of steel braced frames with flexible joints. Int. J. Solids Struct., 23(5) (1987) 631-49. 42. Yu, C. H. & Schanmugam, N. E., Stability of frames with semi-rigid joints. Computer & Structures, 23(5) (1986) 639-48. 43. Moncarz, P. D. & Gerstle, K. H., Steel frames with nonlinear connections, J. Struct. Div. ASCE, 107(ST8) (Aug. 1981) 1427-41. 44. Bjorhovde, R., Effect of end restraint on column strength--practical applications. Engng J., AISC, 1 (1984) 1-13. 45. Lindsey, S. D., Ioannides, S. A. & Goverdhan, A. V., LRFD analysis and design of beams with partially restrained connections. Engng J., AISC, 4 (1985) 157-62. 46. Chen, W. F. & Lui, E. M., Columns with end restraint and bending in load and resistance design factor. Engng J., AISC, 3 (1985) 105-32. 47. Arbabi, F., Drift of flexibly connected frames; Computer & Structures, 15(2) (1982) 103-8. 48. Lui, E. M., A practical P-delta analysis method for Type FR and Type PR frames. Engng J., AISC, 3 (1988) 85-98. 49. Nethercot, D. A., Davison, J. B. & Kirby, P. A., Connection flexibility and beam design in non-sway frames. Engng J., AISC, 3 (1988) 99-108. 50. Chen, W. F. & Kishi, N., Semirigid steel beam-to-column connections: data base and modeling. J. Struct. Div. ASCE, 115(1) (Jan. 1989) 105-19. 51. Lui, E. M. & Chen, W. F., Behavior of braced and unbraced semi-rigid frames. Int. J. Solids Struct., 24(9) (1988) 893-913. 52. Galea, Y., Colson, A. & Pilvin, P., Programme d'analyse de structures planes barres avec liaisons semi-rigides ~ comportement non-lineaire. Rev. Construct. Metallique, CTICM No. 2 (1986) 3-16. 53. Cosenza, E., De Luca, A. & Faella, C., Nonlinear behaviour of framed structures with semirigid joints. Construzioni Metalliche, 4 (1984). 54. Fielding, D. J. &Chen, W. F., Steel frame analysis and connection shear deformation. J. Struct. Div. ASCE, 99(1) (Jan. 1973) 1-18. 55. Johnston, B. G. & Mount, E. H., Analysis of building frames with semi-rigid connections. Trans. ASCE, 107 (1942) 993-1019.
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56. Allen, H. G. & Bulson, D. S., Background to buckling. McGraw-Hill, New York, 1980, pp. 269-72. 57. Monforton, G. R. & Wu, T. S., Matrix analysis of semi-rigid connected frames. J. Struct. Div. ASCE, 89(ST6) (Dec. 1963) 13-41. 58. Wang, C. K., Intermediate Structural Analysis, McGraw-Hill, New York, 1983. 59. Harrison, H. B., Structural Analysis and Design, Parts I and II. Pergamon, New York, 1980. 60. Wang, C. K., General computer program for limit analysis. J. Struct. Div. ASCE, 89(ST6) (Dec. 1963) 101-17. 61. Bisshopp, K. E. & Drucker, D. C., Large deflection of cantilever beams. Quart. Appl. Mathe., 111(3) (1945) 272-5. 62. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Stability. McGraw-Hill, New York, 1961. 63. Williams, F. W., An approach to the non-linear behavior of the members of a rigid jointed plate framework with finite deflections. Quart. J. Mech. Applied Math., XVII(4) (1964) 451-69. 64. Surtees, J. O. & Mann, A. P., End plate connections in plastically designed structures. Paper presented at Conference on Joints in Structures, Vol. 1, Paper 5, University of Sheffield, UK, July 1970. 65. Zienkiewicz, O. C., Incremental displacement in nonlinear analysis. Int. J. Num. Meth. Engng, 3 (1971) 587-8. 66. Haisler, W., Stricldin, J. & Key, J., Displacement incrementafion in nonlinear structural analysis by the self-correcting method. Int. J. Num. Meth. Engng, 11 (1977) 3-10. 67. Botoz, J. L. & Dhatt, G., Incremental displacement algorithms for nonlinear problems. Int. J. Num. Meth. Engng., 14 (1979) 1262-7. 68. Rheinboldt, W. C., Numerical analysis of continuation methods for nonlinear structural problems. Computer & Structures, 13 (1981) 103-13. 69. Wempner, G. A., Discrete approximations related to nonlinear theories of solids. Int. J. Solids Struct., 7 (1971) 1581-99. 70. Riks, E., An incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct., 15 (1979) 529-51. 71. Crisfield, M. A., A fast incremental/iteration solution procedure that handles snap-through. Computer & Structures, 13 (1981) 55-62. 72. Crisfield, M. A., An arc-length method including line searches and accelerations. Int. J. Num. Meth. Engng, 19 (1983) 1268-89.
Secant Method for Nonlinear Semi-Rigid Frames
S. L. L e e & P. K. B a s u * Vanderbilt University, Nashville,Tennessee 37235,USA (Received 24 April 1989;revised version received and accepted 26 June 1989) t
ABSTRACT A large-deformation inelastic formulation based on the secant method is proposed for the limit load analysis of planar frames with partially restrained connections. Four test problems are solved to demonstrate the accuracy and computational efficiency of the proposed algorithm.
INTRODUCTION A n u m b e r of first- and second-order schemes are available for the static and stability analyses of framed structures based on the matrix displacem e n t m e t h o d . In the simplest case of first-order elastic analysis it is assumed that the deformations are Small and the stiffness of a m e m b e r remains unchanged during the complete history of loading. The characteristic feature of a second-order elastic analysis is, however, that the effect of m e m b e r axial force on its stiffness is included. This stiffness can be obtained either analytically or numerically. A typical analytical procedure is the slope--deflection m e t h o d in which the equilibrium equations are written with respect to the deformed shape of the structure, 1 and involves the use of stability functions 2,3 which reflect the effect of m e m b e r axial force exactly. O n the other hand, a c o m m o n approximate numerical m e t h o d is the variational formulation of the finite element method. 4-12 In the case of a perfect frame under ideal loading conditions, which does not *Author to whom correspondence should be addressed. 273 J. Construct. Steel Research O143-974X/90/$03.50 ~ 1990Elsevier Science Publishers Ltd, England. Printed in Great Britain
274
s. L. Lee, P. K. Basu
cause any lateral deflection in the members of the frame until the bifurcation load is reached, these formulations essentially lead to eigenvalue analyses. On the other hand, in the presence of initial imperfections, such as geometric deviations, residual stresses, etc., and/or non-ideal loading conditions, deflection will occur in the frame as soon as the load is applied. The resulting set of nonlinear equations can be solved by either an incremental procedure or an iterative procedure. On the grounds that often the drift criterion rather than stability controls practical frame design, a number of approximate and empirical methods, such as those due to H o m e & Merchant, 3 Roberts 13 and Horne, 14 have been put ~forward to make sure that frame stability does not control the design. In the case of incremental analysis, 15--21 a tangent stiffness matrix is generated according to the current geometry and the state of member forces at each stage of incremental loading. In this manner, the complete load-displacement relationship of the frame for the entire loading history can be obtained. In order to ensure equilibrium at the end of each load increment, an improved Newton-Raphson-type iteration scheme is built into the solution algorithm. 22'23 In the case of iterative analysis, 2~26 the secant stiffness is determined by assuming that the current geometry and the state of forces are known. The new values for the current geometry and the state of force obtained after an iteration cycle are then used in the following cycle. This sequence is repeated until convergence is achieved. W h e n the difference of the forces or the geometric state between two consecutive cycles of calculation are smaller than a prescribed tolerance limit, the calculation is said to have converged. A comparison of these two methods reveals that the tangent stiffness approach controls the error at a local incremental level,27-29 and normally requires the use of a small increment in order to obtain sufficiently accurate results. On the other hand, the secant stiffness approach is not sensitive to the loading history of the frame and a large enough increment can be used to limit the required number of iteration cycles for convergence. Another difference between these two methods is that in the tangent stiffness method an extra matrix called the geometric stiffness matrix, which is sensitive to the local slope of the M--O curve of a member, is Used. The present study is concerned with the secant stiffness approach.
MEMBER NONLINEARITY The factors contributing to member nonlinearity are material (elasticplastic), geometric (large displacement), connection behavior (nonlinear M--O relation), and the P-Delta effect.
Secant method for nonlinear semi-rigid frames
275
The stress-strain relationships of structural carbon steel and highstrength low-alloy steel are commonly approximated by an elasticperfectly-plastic stress-strain curve. The moment-curvature relationship of a section based on such a stress-strain curve is idealized by an elastic-plastic representation which ignores the transition part. A further simplification is sometimes introduced by the rigid-plastic representation, which assumes that there is no strain at a section until the moment equals the plastic moment capacity of the section, after which unrestricted plastic flow is allowed to occur in the section. The rigid-plastic representation can be used when the members are primarily subjected to flexural action. However, if the members carry large compressive forces, the inclusion of elastic deformations and instability considerations become important. This is especially true when high-strength steel is used in the frame. In a general analysis scheme, the need for including elastic-plastic behavior is, thus, quite evident. The results in the elastic stage can be used to check the serviceability limit states. On the other hand, the plastic stage is used to estimate the overload strength, and to find whether the ultimate limit state concerned with structural safety is satisfied. Strain-hardening has considerable influence on the load capacity, especially of small structures. In the case of tall structures, however, the effect of instability on frame behavior is far greater than that of strain-hardening. Therefore, it is reasonable to neglect strain-hardening effects in such structures. In the case of small frames carrying service load, the displacements are small and hence the equilibrium and compatibility equations are written with respect to the undeformed geometry, of the structure, but in the case of tall frames, especially near limit states, the deformations will be large and the state equations should refer to the deformed geometry of the structure. Of the three coordinate systems suitable for this purpose, the total Lagrangian coordinate system9'11'21 is most widely used in the variational finite element formulation of the problem. This is associated with the possible penalty that a large number of elements is often required to achieve a given level of accuracy. Eulerian 19'3°'31 and Updated Lagrangian 25'32'33coordinate systems are convenient in the beam-column approach. The resulting force-displacement relationships, though complex, are more accurate and require fewer elements to model the structure. In the conventional analysis of steel structures, connections are considered either perfectly hinged or rigidly fixed. This classification, though very useful in analysis, is unrealistic because in practice a certain amount of flexibility cannot be avoided in so-called rigid connections, and supposedly pinned connections without a certain amount of moment resistance is rare. In particular, the AISC specification34 designates three types of construction in its 1969, 1978 and 1980 editions. There are Type 1 construction (rigid framing), Type 2 construction (simple framing), and
276
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Type 3 construction (semi-rigid framing). Because of the lack of sufficient knowledge of connection behavior and strength, and the lack of practical methods to analyze flexibly connected frames, Type 3 construction is seldom considered in practical analysis and design. In Type 2 construction, two inconsistent assumptions are made. Firstly, the connections are assumed to have no moment-resisting capacity when the frame is designed for gravity loads only. Secondly, it is assumed that the connections and connected members have adequate capacity to resist the moments induced due to wind loading, and that the connections have adequate inelastic rotation capacity to avoid overstress of the fasteners or welds under combined gravity and wind loading. Recognizing this contradiction, the newly published Load and Resistance Factor Design (LRFD) Specification35 for structural steel buildings explicitly designates only two types of construction: Type FR (Fully Restrained) construction and Type PR (Partially Restrained) construction. It is clear that the behavior of most of the practical connections lies between those of rigid and pinned cases, as shown in Fig. 1. Apart from higher reliability,, considerable economies may be achieved if the true behavior of these joints is taken into account while designing steel frames. Realizing the importance of data on real connection behavior, considerable analytical and experimental research has been carried out to measure the moment-rotation characteristics of various types of commonly used framing connections. During the 1970s and 1980s extensive experimental and analytical data were published for bolted and welded connections both in this country and in Europe. 28The analytical models have been based on semi-empirical considerations as well as finite element analysis and in some cases nonlinear material behavior has been considered. For instance, in 1975 Frye & Morris 36 presented an analysis procedure to account
z
ROTATION Fig. 1. Typical moment-rotation curves of connections.
Secant method for nonlinear semi-rigid frames
277
for the nonlinear behavior of the connection. They have expressed the force-deformation characteristics in a nondimensional form by using polynomial equations for seven commonly used connection types. In addition, a large n u m b e r of papers have been published on this topic, 27'28'33'37-51 some of which include nonlinear behavior. However, for the purpose of the present study, the m o m e n t vs rotation ( M - O ) relationship for a semi-rigid connection will be expressed in the following form:
[ 1 in which O = M = Ci = Mo = a = Mma~ =
(1)
the relative rotation of the connection; the moment acting on the connection; constant coefficients; aMmax; a sensitivity factor ( = 11.4); maximum moment in the experimental M--O data.
S E C O N D - O R D E R S E C A N T STIFFNESS M A T R I X The effect of semi-rigid connections on m e m b e r stiffness can be accounted for in two ways. O n e is to represent the connection stiffness by introducing discrete elements at the ends of the m e m b e r in the form of springs with flexural, shear, and axial stiffnesses. 27,28'aa'as'51-55 In addition, rigid elements may also be introduced to account for the width of the column. 53 The advantage of this approach is that the discretization of the structure into members, spring, and rigid connectors is a relatively easy task. The drawback of this m e t h o d is that the total n u m b e r of degrees of freedom required to model the deformed configuration of the structure increases significantly. This may be of especial concern when all the connections of a large structure are of Type PR. The other way is to modify directly the m e m b e r stiffness relationships to account for the partial rotational restraining effect of the connections. 26'565a As a result, all the displacement components can be referred to the column-line. Also, the relative rotations caused by a Type P R connection do not increase the required n u m b e r of degrees of freedom. The present study is based on this approach. The deformed configuration of a typical m e m b e r with T y p e P R end
278
S. L. Lee, P. K. Basu o
o
°
Ro
X
r
U.~.]o, L-~ .........
o
~",
',
2/02 U2 / Y
Fig. 2. Semi-rigidlyconnected beam-column.
connections is shown in Fig. 2. For the sake of simplicity no m e m b e r loading is shown in the figure. The effect of m e m b e r loading will, however, be expressed in terms of the fixed-end forces and m o m e n t s RS/~,R sv and M/Sv (i = 1, 2). If ki = Elci/L is the current rotational spring constant of the connection at end i of the m e m b e r and Wg is the current connection rotation, the m o m e n t at end i can be expressed as
in which Oi is the joint rotation at the column center line and 0/' is the end rotation of the beam. Therefore, the difference ( e l - 8/) represents the change in end rotation of the beam caused by the flexibility of the connection. As in Galambos, i the end m o m e n t s of the m e m b e r can be given by
in which A = v2 - I;1, and S and C are the so-called stability functions which are functions of L ( P / E D 1/2. Tabulated values of C and S are available
Secant method for nonlinear semi-rigid frames
279
in standard t e x t b o o k s , including Ref. 1. Substituting the expression for 0" b a s e d on e q n (2) into eqns (3a) and (3b), it follows that
MI = _~_[ HIOI+H302-(HI +H3) --~]
(4a)
in which H1 = [Cl(C 2 - S 2) + clc2C]/p
(5a)
/42 = [c2(C 2 - S 2) + cxc2C]/p
(5b)
H3 = Cl c2S/p
(5c)
and p = C 2 - S 2 + (Cl +
c2)C+ClC2
(5d)
B y using the equilibrium equations Fyl = - Fy2 = (M1 + M2 + PA)/L
(6a)
and the axial f o r c e - d i s p l a c e m e n t relationships
-- Fxl = Fx2 = AE(u2 - Ul)/L
(6b)
the six f o r c e - d i s p l a c e m e n t relationships for the m e m b e r , including the effect of fixed e n d forces and m o m e n t s caused by m e m b e r loading, can be written in the matrix form as {F} = [K] {D} + { g ~ in which {F} = {D} = {RF} = [K] =
--a
[rl =
the the the the
(7)
member force vector = {Fxl, Fyl, M1, Fx2, Fyz, ME}; member displacement vector = {ul, vl, 01, u2, vz, 02}; fixed end force vector = {RxVl,RyF1,M~, Rxz,V Ry2,FM2F}; secant stiffness matrix of the member given by
0
0
-a
0
0
0
b
c
0
-b
d
0
c
e
0
-c
f
-a
0
0
a
0
0
0
-b
-c
0
b
-d
0
d
f
0
-d
g
(8)
with a = A E / L , b = ( H , + H2 + 2 H 3 ) E I / L 3 - P / L , c = (H1 + H a ) E I / L 2, d = (HE + H 3 ) E I / L 2 , e = H I E I / L , f = H3EI/L,g = H2EI/L.
280
S. L. Lee, P. K. Basu
If one end of the member is PR-conneeted and the other end is rigid (say, End 2), the value of ci corresponding to the rigid end will be infinity and the expressions for H1,//2, and Ha will take the form H1 = cl C / ( c l + C)
/'/2 =
(C 2 -
S 2 -1- Cl
C)/(c1 "~"C )
Ha = Cl S/(Cl + C)
On the other hand, if End 1 of the member is PR-connected and End 2 is pinned, c2 = 0. It means that H2 = / / 3 = 0 and
H1
=
¢1(C 2 -
$ 2 ) / ( C 2 - S 2 -}- Cl C)
Again, if both Cl and c2 are set to infinity, H1 = HE = C and Ha = S and hence the matrix relationship (7) becomes applicable to the case of a rigid jointed frame. Finally, the first-order matrix relationship for a PRconnected frame is obtained by setting C = 4 and S = 2 in the expression for HI, /'/2, and/-/3 given by eqns (Sa)-(Sd) and the resulting matrix relationship can be used for the static analysis of such frames, whereas, if H1 a n d / / 2 are directly set to 4 a n d / / 3 to 2, the first-order linear matrix relationship for a rigid jointed frame is obtained.
F I X E D END M O M E N T Assume that the member shown in Fig. 3 with semi-rigid end connection is subjected to some transverse loading and the area of the resulting static moment diagram is ,4. This area corresponds to the case when the end moments of the member are zero requiring that there be no rotation in the connections and hence the joint and the column rotation be the same as the free end rotations ~ a n d / ~ of the beam given by -
.42
./~ =
EIL
'
.4(L-2) ElL
(9)
in which ~ is the distance of the centroid of the static moment diagram from End 2 of the beam. The fixed end moments M s v and M s v should be of such magnitude that the joint rotations at the column center line caused by these moments alone be - ~ and -0~. In that event MSF = _ E__I_/[H1~ + H30~] L
(10a)
MSF = _ E__I_/[/-/3~ +/-/2 4] L
(10b)
Secant method for nonlinear semi-rigidframes
281
W(x)
(a) Loaded Beam
I_
,
~
'
El
-
(b) Equivalent Beam
(c) Conjugate Beam
Fig. 3. Conjugate beam for semi-rigidly connected member. (a) Loaded beam. (b) Equivalent beam. (c) Conjugate beam. In the case of ideal fixity, the expressions for ~ and 0~ in terms of the fixed end moments M v and M F will be MFL MFL 3 ~ + 6EI
(lla)
=
MFL MFL 6el F 3~
(11b)
#2=
Using eqns (9) and (11), the values of M F and M F for a given transverse loading condition can be obtained. Expressions ( l l a ) and ( l l b ) can be used to replace ~ a n d / ~ appearing in eqns (10a) and (10b). The modified expressions for fixed end moments in the presence of a PR-connected member in terms of the fixed end moments for a rigidly connected m e m b e r will then be obtained as 1 [(2M~ - M~)H1 + (M F - 2M~:)H31
(12a)
= 1 [(2M F _ MF2)H3 + (M F _ 2MF) H2]
(12b)
=
o
282
S. L. Lee, P. K. Basu
In the event of m e m b e r loading, these m o m e n t s can be i n t r o d u c e d into the slope-deflection equations, as in eqn (7). In the case of a first-order analysis or in the absence of axial force, the v a l u e s of C and S in expressions for Ha, H2, a n d / / 3 should be set to 4 and 2, respectively.
LARGE RELATIVE DISPLACEMENT T h e slope-deflection equations (4a) and (4b) are based o n the assumption that the relative displacement A/L is small. If, however, 6JL is not small it is necessary to modify the secant stiffness matrix. By referring to the d e f o r m e d configuration of a m e m b e r , shown as Fig. 2, the axial d e f o r m a t i o n 8 and the rotation R can be expressed as 8 = (u2 - ua) - (v2 - vl) tan (R/2)
R=R'-Ro=tan-a
(1' 2 --
in which
va)/cos(R/2) Lo" A F
AF1 =
)
Y2-Y1
-
o
o
. -a//Y~-Y~'~ -tan k X 2 ~ _ 1: ( v 2 - va) AFa L
L Lo" AF" cos(R/2)
A F = the rotation modification factor = 1.0 - 0.0002R - O.041R2 L0 = original length of the member L = deformed length of the member
Based on the above, the modified slope-deflection equations, w h e n the m e m b e r displacements are referred to the d e f o r m e d configuration, will be
M1
=
H~ 01 +/-/3 02 - (Ha + H 3 ) ~
A/q
(13a)
=EI
Also, F:I-
AE L
--[ul
- u2 + ( v 2 - v a ) t a n ( R / 2 ) ]
(14a)
Secant method for nonlinear semi-rigid frames Fx2 = A E
L [ u 2 - ux - ( v 2 -
Fyl=
283 (14b)
vl)tan(R/2)]
MI+M2
(15a)
L
(lSb)
Fy2 = - f .
These equations can be written in matrix form as {F} = [K'I{D} in which the modified secant stiffness matrix is
[K'] =
a
-h
0
-a
h
0
0
b'
c
0
-b'
d
0
c.AF1
e
0
- c . A F1
f
-a
h
0
a
-h
0
0
-b'
-c
0
b'
-d
o
d.AF1
f
0
-d.AF1
g
(16)
with b' = (H1 + H2 + 2H3) ( E I / L 3 ) ' A F 1 and h = ( A E / L ) t a n ( R / 2 ) . It may be noted that the matrix [K'] is not symmetrical. For the sake of convenience this matrix is expressed as the sum of two matrices: [K] as in eqn (8), and [KL] which contains the correction terms due to large displacements. By including the effect of fixed end forces, the matrix equation takes the form {P'} = [K]{O} + [KL]{O} + {R F}
In this equation [KL]{D} can be combined with the vector {RF} and treated as a pseudo force term, so that {F} = [K]{D} + {R --r} in which {Rv-} {RF} + [KLI{D} If the member curvature effect is considered, then as in Fig. 4,
e = ~ E (A - Ab)
284
S. L. Lee, P. K. Basu
Ab
Fig. 4. Member curvature effect.
and as a result eqns (14a) and (14b) b e c o m e AE FX I
~
Fx2 -
in which
--
AE -~
m
L
[U 2 -- U 1 "b (V 2 -- V l ) t a n ( R / 2 )
(v2 -
[ u l - u2 -
vOtan(R/2)
-
- Ab]
Ab]
Ab = [b1(01 + 02) 2 -~- b2(01 -- ~2) 2] L
(C + S)(S -
bl =
b2 =
2.0)
87r2p
S
8(c + s) P
o = Pe
PL
~EI
Again, the curvature corrections to Fxl and
{
fx2 a s
, 001
can be c o m b i n e d with [R-r] as a p s e u d o force vector so that (R -r} = {RF} + [KLI{D} + {Rb}
ELASTIC-PLASTIC BEHAVIOR
(17)
OF MEMBERS
Based on the elastic-perfectly-plastic material behavior, w h e n e v e r the m o m e n t value at a section reaches the plastic m o m e n t capacity of the
Secant method for nonlinear semi-rigidframes
285
section a hinge is introduced at the section so that the plastic condition including the effect of axial force is always satisfied. To account for the relative rotation between the sections on each side of the hinge (actually a plastic hinge), it is necessary to add a row and a column to the structure stiffness matrix. 24 From a programming point of view, this approach requires complex book-keeping and also may turn out to be uneconomical in terms of storage requirements. The alternative approach in which such augmentation of the structure stiffness matrix can be avoided by modifying the member stiffness matrix due to the presence of a plastic hinge, will be used here. In the process, it will obviate the need for computing the relative rotation of the sections at a plastic hinge.18'59'6° The rotation at the ends of a member with plastic hinges can be computed as discussed below. The end moment eqns (4a) and (4b) can be rewritten in the following form L
E1 M1 = H1 01 +/-/3 02 - (H1 + H3)
L E1
A
(18a)
A
(18b)
M 2 = H3OI + H 2 0 2 - (H~ + H~)--:--
If the first end only has a plastic hinge, then M1 01 =
1 FMpIL
"~i-iL Y
a
"I-(Hi + H3)-~--- H302
= Mpi
and using eqn (18a)
]
Also, if the second end only has a plastic hinge, then M2 = Mr,2 and using eqn (18b)
1
r M~s_, a 02 = "~2"2L y "i"(H 2 -.I-H3) T - H301
]
J
On the other hand, if both the ends have plastic hinges, then M1 and/142 = Mp2, and using eqns (18a) and (18b),
01 -~
=
i-i~ .H~-HI H1 L - - E T - + (H, + H3)
- H3
#I~.H~-~
+ (H~ + H3)
= Mpi
286
S. L, Lee, P. K. Basu
The reduction of the plastic moment capacity of a section (Ml~) due to the presence of axial force can be accounted for in the iteration procedure by using the following standard relationship for W-shapes.
MI'¢ P) <1"0 Mp = 1.18(1.0- ~y in which
Mp¢ = Mp = P = Py =
(19)
reduced plastic moment; full plastic moment of the member; axial force; yield load of the member.
COMPUTER IMPLEMENTATION For the purpose of the present study the computer software LEPPR was developed on the SUN-3 computer using FORTRAN-77 language, based on the flowchart shown in Fig. 5. It uses an iterative algorithm for handling the second-order elastic-plastic stability problem of planar semi-rigid frames. Each cycle of iteration involves updating of the secant stiffness matrix. Convergence is achieved by ensuring that differences of the member foi'ces in two subsequent cycles of iteration are smaller than a specified limit. The iterative process is shown in Fig. 6. An initial load level P1 is chosen and the first cycle of response calculation is done assuming rigid connections and using first-order elastic analysis. The load vector and the linear stiffness matrix K~, which ignores the effects of axial forces and connection flexibility, is assembled. The displacement vector D~ is obtained by solving the structure matrix equation of the form of eqn (7). However, before assembling the structure stiffness matrix and load vector, the member stiffness matrix and vector in the local reference frame are transformed into the global reference frame using standard transformation relationships based on direction cosines of the member axis with respect to the global reference axes. After D~ is obtained, the member forces are calculated, and used to set up the updated stiffness matrix K 2 which is used in the next cycle of iteration. This process is repeated until the convergence criterion is satisfied. At this point the displacement vector D1 and the member forces caused by load P1 are obtained. These member forces are then used to set up the stiffness matrix K~ for first iteration cycle of the next load level P2. In the case of a second-order elastic stability analysis, this process is continued until the critical load level is reached, i.e. when the system becomes unstable.
Secant method for nonlinear semi-rigid frames
I L
287
START ]
LOADINGVECTOR
=1 v]
STIFFNESSMATRIX
I
CORRECTION LOAD VECTOR
I
SOLVESYSTEM
NEWLOADINGLEVEL
ES
DETERMINATEOF STIFFNESSMATRIX
GEOMETRICNONLINEARCOEFF. MEMBERFORCES UPDATECOORDINATE
]
YES
[, MAT-NONLIN. I
T_
I
I
YES~
PRINTOOT
REDUCEDLOADING ] INCREMENT N
O
Fig. 5. Summaryflow chart.
In the computer software, the critical condition is identified in the following three ways. (1) If the determinant of the global stiffness matrix is zero, the structure is said to have reached the critical condition. On the other hand, if the determinant is found to be negative, the current load level is
288
S. L. Lee, P. K. Basu
l
[14
y
Ppz
il.J~~
Pl
Pl
'
h, I
/,p/ I x: a: ~///I
I D' ~-~-Secout ~ l l l i i e hi,,,ge D
~,
plutic
hinge
L
Dz
Dpi
Ds
D#
n.
Displacement Fig. 6. Iterative process.
~ett-1
',~. ",,vnLOAOLE~ R-,~ "', 'Pi "
_
~
Fig. 7. Linear interpolation of load level to attain zero value of DET(K).
Secant method for nonlinear semi-rigid frames
289
reduced to a new value based on linear interpolation using the last two values of the determinant, as per eqn (20), based on Fig. 7:
Pn = P i - 1 +
e l - ei-1 DETi- ~- DETi
DETi-1
(20)
(2) This criterion uses the required number of iterations as an indicator of the unstable state and is based upon the consideration that as the load level approaches the critical value, one or more member forces may blow-up and fail to converge at all. Hence, if the number of iteration cycles exceeds a specified value (say, 300) the load increment is reduced by one-half until the increment is smaller than a specified value when the program will stop further load incrementation. (3) In this criterion, the value of the maximum relative drift, i.e. story drift, is used as an indicator of the critical state. Whenever the relative drift is found to be larger than a specified limit, the program stops further load incrementation. In the case of elastic-plastic member behavior, at the end of each load increment the formation of new plastic hinges is checked, based on the yield condition of eqn (19). For this purpose, a predefined tolerance limit is used. After a load increment, if one or both of the end moments of a member are found to exceed its plastic moment capacity, the load level is reduced by linear interpolation, as in Fig. 8. For End 1, the interpolation equation will take the form
ep = ei
M i - Mp .Mi_ Mi_l ( e i - e l - l )
(21)
The iterative process is continued until the plastic hinge in question is just formed within the predefined tolerance limit. At this point, for additional load increments, the stiffness of the member needs to be modified. Also, the geometric definition of the member gets modified due to the introduction of the plastic hinge, which is treated as a real hinge. After the first plastic hinge is formed, say, at load level Ppl ~a new coordinate system (~,y) with origin located at (Dpl, Ppl) is used for the subsequent load increment(s). This consideration is shown in Fig. 6. It may be noted that based on this new coordinate system P1 = P3 - Ppl Di = D 3 -
Dpl
290
S. L. Lee, P. K. Basu
.~J
--
l2
°1 'II Mi-, Mp
MOMENT Mt
Fig. 8. Linear Interpolation of load level to satisfy the plastic moment value. n
The iteration process is continued for the new path (P1 versus D1), and the load level ~is increased until the next plastic hinge(s) is formed. A corrective load vector is introduced in each cycle of iteration to enforce the force equilibrium when the total load is applied on the system which has been modified due to the formation of the new plastic hinge. This procedure is continued until a failure mechanism is formed, or a critical condition is reached. If, on the other hand, the small-displacement theory is used, no updating of the coordinate system is necessary and the nodal coordinates of the original geometry are used for complete analysis. In contrast, in the case of the large-displacement theory, it is necessary to update the coordinate system at each iteration cycle.
NUMERICAL EXPERIMENTATION In order to determine the reliability and computational efficiency of the proposed algorithm, the following four test problems are considered. All computations are carded out on a time-shared SUN 3/260 workstation.
1 Large-deformation e!~ic analysis of a cantilever beam As shown in Fig. 9, the beam carries a conservative point load at its tip. The size of the beam is taken as W4 x 13 and the length L = 0.3048 m. The theoretical solution of the problem based on inextensional theory was given by Bisshopp & Drucker. 61 Two sets of analyses, using one member model, are carried out with LEPPR, one based on actual cross-sectional
Secant method for nonlinear semi-rigid frames
P i
L
x
¥
X
s
291
~ 1
Fig. 9. Example 1: cantilever beam.
.
Y-I.
x/"°
0
0'2
0.4 0.6 X/L and YIL
0.8
1.0
Fig. 10. Load-deformation curves of cantilever beam. Theoretical (-) axial deformation
(O). No axial deformation (A).
area of the beam (A = 2.47mm x 103 mm), and the other for the inextensional case. The latter case is simulated by assigning a very large number to the area (A = 2.47 mm x 10 7 mm). in Fig. 10, the results of both the analyses are compared with the theoretical results of Bisshopp & Drucker. 61 The close agreement of the inextensional results establishes the accuracy of the model. On the other hand, the disagreement between inextensional and extensional results is found to widen as the load level is increased, signifying that the inextensional theory can be used for low load
292
S. L. Lee, P. K. Basu
×
L Ill IFig. 11. Example 2: fixed-free strut.
levels only. In order to generate the plots in Fig. 10 based on the present analysis, ten load increments were used, requiring 2.88 C P U s. On the other hand, if it is desired to generate data for just one load level with no incremental steps, say, corresponding to pL2/EI = 10, the required C P U time drops to 0.44 s only. The computational efficiency of this m e t h o d is quite evident from this. 2 Elastic post-buckling analysis of a fixed-free strut The strut shown in Fig. 11 is considered. The solution for this elastica problem is available in the publication of Timoshenko & Gere, 62 which assumes inextensional behavior. The size and length of the m e m b e r is taken to be same as in the last example. In order to trace the post-buckling path an initial perturbation in the form of an eccentricity, e, in the load equal to L/IO 000 is introduced in the present analysis, The theoretical results from Ref. 62 based on e = 0 along with the results of the present analysis corresponding to the actual (extensional) and simulated inextensional cases are shown in Figs 12 and 13. The results in Fig. 12 are based on a single-element model and those in Fig. 13 on a four-element model. In each case 12 load increments were used to generate the load-displacement curves. The four-element model shows better agreement with the theoretical results. It is evident from Fig. 13 that the inclusion of the effect of axial deformation modifies the transverse deflection slightly over the range 1 < P/Pcr < 1"6. In the case of 12 load increments, the C P U time usage for one- and four-element cases were 3-52s and l l . 0 2 s , respectively. With one load step only, the corresponding times were 0.6 s and 5.14 s, respectively.
Secant method for nonlinear semi-rigid flames 2.2
2.2
2.C
b
2.0
293'
o
°o °
1.1=
1.e
o
1.d
1..1
1,2
oA
1.--'
I
1.C -0'2
02
0.4
X/L
and
0.6
0.8
1.0
1.0 -0.2
¥/L
Fig. 12. Load-deformation curves for oneelement model of strut of Fig. 11. Theoretical (-). Axial deformation (C)). No axial deformation (A).
0.2
0-4
XIL
and
0.6
0.8
1.0
YIL
Fig. 13. Load--deformation curves for fourelement model of strut of Fig. 11. Theoretical (-). Axial deformation (C)). No axial deformation (A).
3 Large-deflection analysis of fixed arch The arch shown in Fig. 14 is considered. The theoretical and experimental load-deformation responses of this arch were presented by Williams. 63 In the present study, both the two-element model (3.06 C P U s for 14 steps and 0-44 C P U s for one step) and the four-element model (5.96 C P U s for 14 steps and 0.7 s for one step) were used and identical results were obtained. The exact agreement of the present results with the theoretical one is evident from the load vs central deflection plot of Fig. 14. 4 Nonlinear analysis of four-story frame Here the frame of Fig. 15, with r = 0-24, L c = 3.66 m, L o = 9.14 m, E = 201 x 106 kPa, Fy = 236 x 103 kPa, and beams W16 x 40, columns W12 x 79 (bottom story) and W l 0 x 60 (rest), is considered. Kassimali 3° carried out second-order small deformation analysis of the same structure as a rigid frame using the incremental approach in combination with N e w t o n - R a p h s o n iteration. In the case of the rigid frame, the failure mechanism and the load deformation curve are shown in Fig. 16. A l o n g with Kassimali's results, the curves based on present analysis corresponding to large and small deformation approximations are also shown. The results are found to be in good agreement. The same frame was again
294
S. L. Lee, P. K. Basu 300-
a~~ ~.~ o-oo8m
250 -
/
i.-o3286m-~p3_Es_emd
200 -
¢
z
150i11 o .J
100 5G I
0.3 Deflection
I
I
I
I
0.6 0.9 1"2 i.5 o f m i d d l e span ( r a m ) x l O -1
Fig. 14. Example 3: load-displacement curve for a fixed arch. Theoretical ( - ) . Present
(o).
~P/2 r"P/2 t
P
1.2
P/2
1.0
P/2 0.8 i.
£oe
J
'6°i
O.Zl
0.2 gassimali 30 I
!~
L@
Fig. 15. Example 4: four-story frame.
O-
0.'40
I
Small def. Large def. I
0-80 1"~.0 OlHxl o =
I
1"60
.l
2.00
Fig. 16. Load-displacement curves of fourstory frame. Kassimali3° (n). Small deformation. Large deformation (A). Pc, = 151.9 kN, D = horizontal deformation of top right joint and H = 14.63 m.
Secant method for nonlinear semi-rigid frames
295
1.2 6.0
5.O
o.e
~O'E n
0.~ 0.t
0
I
I 1.0
I I I 2.0 3"0 4.0 Rotation (tad) x 10=
I 5.0
// I 0.40
I 6.0
Fig. 17. Moment-rotation curve of end plate connections used in example 4. Test data64( - - - ) . Curve fitting(--).
I '[_ $, I Rigid I, I 0.80 1.20 D I H x 102
OR I 1.00
I 2-00
Fig. 18. Load-displacementcurves of fourstory rigidand PR frames. Rigidconnection (O). PR connection (A). Pcr= 151"9kN, D = horizontal deformation of top right joint and H = 14.63 m.
analyzed with Type PR (end-plate) connections. The M--O curve for the connections used is shown in Fig. 17 based on data from Ref. 64. This curve is the result of the curve fitting with the proposed O--M formula [(eqn (1)]. The load-deformation curves for rigid and semi-rigid frames based on the large-deformation assumption are shown in Fig. 18. The results for the semi-rigid frame based on a small-deformation assumption are found to be the same as for the large-deformation assumption. It is found that the load-deformation curves for all cases agree closely and the limit state occurs at almost the same load level. However, a large lateral drift is obtained in the case of the semi-rigid frame.
CONCLUSION The general formulation of the secant stiffness matrix for a m e m b e r undergoing in-plane large deformations in t h e presence of semi-rigid connections and material nonlinearity is presented. A new nonlinear curve-fitting formula for representing 0 in terms of M for semi-rigid connections is also presented. A set of four examples are considered to demonstrate the accuracy and computational efficiency of the proposed scheme. The scheme can be used either to generate the complete
296
s. L. Lee, P. K. Basu
load--deformation path into the post-buckling range or to locate any point on the path. The former requires load application in an incremental fashion and the latter in one step only. Also, the latter consumes significantly less CPU time than that required to generate the complete load-deformation path. The effect of including the axial deformation at small deformation levels may not be significant, but at large deformation levels it should be included, at least for the problems of the type considered. The proposed method is found to be computationally efficient and reliable in carrying out nonlinear analysis of framed structures. Based on preliminary studies to generate the segment of the load-deformation path for which the stiffness matrix is not positive definite, it appears the proposed method will need to be modified by incorporating either the Control-Displacement Method 6~-68 or the Arc-Length Method. 69-72 Moreover, the proposed method can be extended to a three-dimensional situation, which will b e meaningful only if data on three-dimensional behavior of connections are available.
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