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Optimum design of steel frames with tapered members
Compurrrs & krur/ures Vol. 63. No. 4, pp. 797-81 I. 1997 Ltd and Elsevier Science Ltd. All rights reserved. Printed in Great Britain 0045.7949197 $17.00 + 0.00
~c 1997 Civd-Comp
Pergamon
PII: SO0457949(96)00074-O
OPTIMUM
DESIGN OF STEEL FRAMES TAPERED MEMBERS
WITH
M. P. Saka University
of Bahrain,
Civil Engineering
Department,
Bahrain
Abstract-The design of steel tapered members for combined axial and flexural strength is somewhat complex and tedious if no approximation is made. However, recent load and resistance factor design (LRFD) of the AISC code has treated the problem with sufficient accuracy and ease. In this study, an algorithm is developed for the optimum design of steel frames composed of prismatic and,‘or tapered members. The width of an I-section is taken as constant, together with the thickness of web and flange, while the depth is considered to be varying linearly between joints. The depth at each joint in the frame where the lateral restraints are assumed to be provided is treated as a design variable. The objective function which is taken as the volume of the frame is expressed in terms of the depth at each joint. The displacement and combined axial and flexural strength constraints are considered in the formulation of the design problem. The strength constraints, which take into account the lateral torsional bucking resistance of the members between the adjacent lateral restraints, are expressed as an nonlinear function of the depth variables. The optimality criteria method is then used to obtain a recursive relationship for the depth variables under the displacement and strength constraints. These relationships are derived from the Kuhn-Tucker necessary conditions. The algorithm basically consists of two steps. In the first one, the frame is analysed under the external and unit loadings for the current values of the design variables. In the second, I:his response is utilized together with the values of Lagrange multipliers to compute the new values of the depth variables. This process is continued until convergence is obtained. Numerical examples are presented to demonstrate the practical application of the algorithm. 0 1997 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved.
displacement and strength described in Ref. [l].
INTRODUCTION
The frames with talpered members are preferred in the design of steel structures whenever the architectural requirements allow their presence. They not only provide better distribution of strength, but also yield to a lighter design. The methods available for the analysis of such frames are well established. However, it is not as easy to state the same for their design. In most of the practical design codes, approximate procedures are suggested for dimensioning tapered members which are subjected to the combined action of axial force and bending moment. However, in a recent publication of the AISC, titled Load and Resistance Factor, this design problem is well treated [ 11. The optimum (design problem of steel frames composed of prismatic and/or tapered I-section members is formulated by treating the depth of the section at each node as a design variable. Each point along the frame where a lateral restraint is provided is taken as a node. Between the nodes the depth is assumed to vary linearly. The width and the thickness of the flanges and ,web are considered to be constant throughout the frame. Variable linking is also considered in the formulation so that a predetermined shape over a desired length can be achieved in the final design. The design formulation includes
OPTIMUM
constraints
DESIGN
as they are
PROBLEM
The optimum design problem of steel frames with tapered members can be expressed as: m,,
Min W = c
r= I
p,v,
subject to,
g,,,,(Dn)
n=l,Z...,p
1 k= l,L...,ng
Di - DL,,> 0
r = 1,2,
.
, nm,
(1)
where p, and vPare the density and volume of member r, respectively. nm is the total number of members in the frame. DL is the depth variable belonging to group k and ng is the total number of different groups in the frame. DA,, is its lower bound. g,,,,(Dk) represents nth displacement constraints. p is the total number of restricted displacements. g,, (D;, D,) represents the 797
M. P. Saka
798
I-
i
Fig. 1. Linearly tapered beam element with I-section.
combined axial and flexural strength constraint for member r. Di and D, are the depths at the first and second end of member r, as shown in Fig. 1. Objective function
The objective function, which is the total weight of frame, is obtained as a summation of the weights of each member. The volume v, of member r can be expressed in terms of the values of the depth variables at the first and second ends of the member as vr =A,+
2
(2)
’
where X,,, is the vector of virtual displacements of member r due to the virtual loading corresponding to the nth constraint. This is obtained by applying the unit load in the direction of the restricted displacement n. K,(Di, 0,) is the stiffness matrix of member r in the global coordinates. X, is the displacement vector due to applied loads. Combined axial andflexural
The combined axial and flexural strength constraint for member r, which is subjected to axial force and bending moment about its major axis, is given in LRFD as, for
where Ai and A, are the areas of cross-section at nodes i and j. The area of the tapered member at section x can be expressed in terms of depth at i and j as A, = Y, + Y~x,
strength constraints
P”/(4P,,) 2 0.2, (7)
(3) and for
where
PZ,l(W,!) < 0.2,
Y, = tv,(D,- 2tr) + 2brtf Y’z = tw(D,- D,)/l.
(4)
It can be seen that br, tf, and t, are selected to be constant throughout the frame, which leaves only the depths at node i and j as the design variables. It is clear that for x = 0, the area at the first end becomes A, = Y, and for x = I, the area at the second end becomes Ai = Y, + YJ. Displacement constraints
The nth displacement following form:
constraint
g,,“(Dk) has the
&/,,(Dk) = 6, - 6, ,
(5)
where 6, is the displacement at node s and 6, is its upper bound. The displacement ai can be expressed as a function of the depth variables by making use of the virtual work theorem. “,#> 6, = 1 XJK(Di, Dj)X,n, ,=I
(8) where P. is the required strength and P,, is the nominal tensile or compressive strength for the member depending upon whether it is in tension or compression. M,, is the required flexural strength and M,, is the nominal flexural strength about the major axis of the section. The resistance factor 4 is given as 0.90 in the case of tension and as 0.85 in the case of compression in LRFD. The resistance factor for flexure &, is specified as 0.90 by the same code [l]. Since only the nominal strengths are the functions of the depth variables, the strength constraint for member r can be re-written as g,,(Di, Di) = al/P,, + a?lM,,,,
(9)
where aI and a2 are the constants given as, for P”/(+P,,) > 0.2
(6)
aI = P,/O.85,
a2 = 8M.18.1
(10)
199
Design of steel frames with tapered members
0
a Vi ,vi v-4
1
-------
Vj ,vj
____
1%
Pj auj
Mj a@j d
X
I
I-
L
Fig. 2. Member end forces and deformations of a tapered member.
KM = E[C:,gn + 2C,2Cahn + C&En]
and for
K52 =
P”l((bP”) < 0.2 a2 = A4,/0.9.
UI = t:/1.7,
E[C,Kx,gz, + C,,Cd,hn + CaC,,ha +
(11) KS,
=
C4,C4,E211
E[C,,C,2g2,
C,,C’ahm + C’aCxhz,
+
STIFFNESS MATRIX OF TAPERED MEMBER
The derivation of the stiffness matrix of a linearly varying tapered member, made out of a homogenous isotropic linear elastic material of modulus E, is given in detail in Refs i&-11]. The elements of the axial and flexural stiffness matrix of the tapered member derived in this study are based on the displacement functions, which are the exact solutions of pertinent govering differential equations. Thus, these stiffness matrices are exact, not approximate, like some of those given in Refs [12, 131. The stiffness matrix of the tapered member, shown in Fig. 2, is obtained in Ref. [9] using the finite element approach, where the exact displacement functions were employed. Due to lack of space its derivation is omitted here. It has the following form in the local coordinate system: K,(Di, 4) =
0
K22
K,,
I
0 K32 k, -------____I__________
K55 =
G,C42E21]
E[C:,g2,
+
Kc.2= E[C,4C,,ga
2C,,C.,,hn
+ C:,Ea]
+ C,4Cah2, + C’uC,,hn
+ C44C41 E211 K63 = EbC32g2,
+ K65 =
fG6 =
CMCaha + CuC,jhn
+
C,.,Cahz, + CuC,,ha
CuC42E21]
W,&,g2,
+
+
C4&3E21]
E[C&,
+ 2C,Xuh2,
(13)
+ C&En]
in which E is the modulus of elasticity and other terms are given as
I I
k,
+
g21 =g2-gl,
h2, =
h2 -
hi,
’
(12)
K.4,
0
0
I KM
0
K52
J&3
;
0
K55
0
K62
K6,
;
0
Ke.5 K,
Ez, = E2 - E, Gz, = Gz - G,
Hn=H2-H,, C,, = -hx/D,
Cd, = g,,/D
where C,, =
I
c,, = -c,,,
is 0 K22 =
E[C:,g2, -t 2C,,Ci,hn
K32 =
W,2C,11:21
+
C42C,,E2,]
+
(Hz,
-
Ih2)lD,
C42 =
(lgz - Gz,)/D
dx/A
K,, = Ku = --IL, = E
C,&hx
+ C,, E2,] + CaC,,hn
ca = -c,,
CM = (Ih, - Hn)/D,
CM =
(G2, -
g,)/D
D = Gz,hz, - Hnga + g22
g,, = Igh - Ig,h,
(14)
M. P. Saka
800
where
variables D, and D, at the first and second end of the member:
E, =
s
x2 dx/I,
E2 =
,=0
s
+
x2 dx/I.
,=I
+
dw)2(6
+
44x),
(17)
4,
=
r&/12
42 = (0, - Di)/l
+ b,t,/2,
44
=
-
4243
In
$3
t&2/12.
H2 = (434~~4~In 46 - 4~4~~4~- 4444’In & +
dh~2dd4
In
46
-
$2&h
In
dd/@2448)
-6 = (&6,4’In $‘I - 2&&&‘$4 In ‘#‘I-
d’:&dJ3~4
in which 4,) 42r 4, and 4~~are given in eqn (18). & and 4, are given as 46 = 4, + 421, 47 = 43 + 441.
(20)
Since b,, tt, and t,? are considered constant throughout the frame &, &, &, c$~,C#J( and 4, are only the function of the depth variables Di at the first end and Dj at the second end. This makes the stiffness terms in eqn (13) also implicit functions of Di and Dj .
where 41 = D, - t/,
41
+ 4V#J3/(92648)
The area of a tapered member at any cross-section x is given in eqn (3). The moment of inertia of the same section can be expressed in terms of depth at the first and second ends of the member as ($1
h
(15)
It is clear from the above expressions that all that is needed in order to construct the stiffness matrix of a tapered member is to carry out the following six integrations:
I=
$1$2&44
(18)
Hence, it becomes possible to obtain the analytical solutions of the integrations given in eqn (15) by using the relationships of eqns (3) and (17). Once this is carried out, the following expressions are obtained, which are only a function of depth
NOMINAL AXIAL AND FLEXURAL STRENGTH OF TAPERED MEMBER It is shown in Ref. [S] that the combined strength constraint for a tapered member makes it necessary to express the design axial and flexural strength of the
Design of steel frames with tapered members member in terms of depth variables defined at its ends. Nominal
Substituting eqn (28) into eqn (27) and taking Q = 1 gives the effective slenderness as,
tensile stre,qgth
&T = [c,(D,tw -
In the case where the tapered member is in tension, LRFD gives the nominal tensile strength P.
27’Y,
(29
where cl is a constant.
as [II,
Pn = FJ,,
(21)
where F, is specified yield stress and A, is the gross area of the member at the smaller end. Hence, the nomin.al tensile strength P, can be expressed as a function of depth variable D, of the smaller end as P. = F,(Dit,
- 2F, T),
6F,(KI)’
1” = t,tt - brtr.
(30)
cl=-.
As a result, the nominal compressive strength P, of eqn (24) can be expressed in terms of depth variable Di at the smaller end as for &T < 1.5,
(22)
where T is a constant given by
Nominal
801
P. = (Dit, - 2T’)F,0.658’1’~J~-W (23)
(31)
for
compressive strength
& > 1.5,
When the tapered member is in compression, its nominal compressive strength is given by LRFD as,
P, = 0.877F,/[c,(Dit, P, = FuAg,
NominaiJlexural
where F,, is the critical stress computed from one of the following expressions: for
(32)
strength
The nominal flexural strength of tapered flexural member for the limit state of lateral torsional buckling is given in LRFD as M. = (5/3)S.&,
L < 1.5, F,, = (0.658”“R)Fy
- 2T)].
(24)
(25)
(33)
where S, the sectional modulus of the larger end and Fk is the design flexural stress of tapered member,
for J-c,4 > 1.5, 1’o-6B.&!+fi;) F,, = 0.877F,/&
(26)
where ,IeRis called the effective slenderness parameter and is defined as,
1
F, < 0.6F,
unless Fb < F,/3, in which case, Fb: = B,/(Ff_. + F$).
lZc~:= S,/(QFy/nZE)
(34)
(35)
(27)
in which S is equal to Klr,, for weak axis bending and K,.llr,, for strong a.xis bending. K is the effective length factor for the member. Since between the adjacent lateral restraints, buckling about the weak axis governs, S is taken as Kl/r,,. The radius of gyration r,, is defined at the smaller end of the tapered member as,
In eqns (34) and (35)
F.,
=
I,
W’OOAr h,lD,
’
(36)
where factors h, and h, are given as, h, = 1 + O.O23y,/(lDj/A~)
(37)
h, = 1 + O.O0385y,/(l/r~,)
(38)
(28) CAS 63/3-F
M. P. Saka
802
in which rTOis radius of gyration of the section at the smaller end, considering only the compression flange plus l/3 of the compression web area, taken about an axis in the plane of the web, Ar is the area of the compression flange, y is given as ‘i =
(D, - D;)/D,
strained problem into an unconstrained one by using Lagrange multipliers. The necessary Kuhn-Tucker conditions are then written for the unconstrained problem to obtain a recursive relationship for the design variables. The Lagrangian of the design problem (1) is
(39)
7 < the smaller of 0.268 (l/Da) or 6. The relationships listed from eqns (33)-(39) can be expressed in terms of depth variables at the ends of the tapered member as shown in the following. The secttonal modulus S,,
nn,
-
s = t,@, - tr)’ + 66rr@, + tr)’ 60 FV and Fwy of eqn (61) are written
(40)
as
F, = 12,00Ob& s, IhsD, F,; =
where constants
c3 cdh;: + csh;D,’
where Ad,, and A,, are Lagrange multipliers for displacement and strength constraints. The necessary condition for the local constrained optimum is obtained by differentiating this equation with respect to design variable Dk.
(42)
c4 = 3Pbftr
19,
(43)
h, and h, have the form h, = 1 + O.O23y&Di/(brtf)) h, = 1 + O.O0385y,/(l/r~o)
(44)
where mk is the number of depths belonging to group k. The derivative of the volume of a tapered member with respect to depth variable Dk can analytically be obtained from expressions (2H4). The derivative of the displacement constraint from eqn (5) becomes
agdn(Dk ) as, ~=_
(45)
ao,
in which ‘/ is function r,, is
(49)
ao,
of D,, as shown in eqn (39) and which in turn from eqn (6) becomes,
rTo = ,/0.25tfb:/(3trbr
+ t,D,).
(46)
It is clear from eqns (40)-(46) that nominal flexural strength M. can be expressed in terms of depth variables D, and 0,. The value of B in expression (34) is taken as 1.5 for simplicity. Futhermore the magnification factor c,,, of LRFD is assumed to have a value of 1.1. OPTIMALITY
(47)
c2, c4 and c5 are
c, = 425,00Otfb;, c5 =
,T,&rgsr(Di, Dj),
CRlTERlA
FOR DEPTH VARIABLES
It is shown in the previous sections that the displacement and combined axial and flexural constraint, as well as the objective function of the optimum design problem considered in this study, are highly nonlinear functions of design variables. The optimahty criteria approach was found to be an effective method in finding the solution of such design problems [2-51. This technique transforms the con-
2 = ,g,QaKrg; D/lx,,.
(50)
Inspection of expressions (12t(20) reveals the fact that the derivative of the stiffness matrix of the tapered member with respect to depth variable can be obtained analytically. On the other hand, the same can also be achieved for the derivative of strength constraints. From eqn (9), it follows:
ag.r(Dt,D,) = _adaPniaDn) _ a2(aMn,laDA)
aa
Pi
M;,
.
(51)
By using expression (22), (31) or (32) and (33) dP,/aD1 and aM,,,ji?D, can be derived analytically. The expressions for these derivatives are not included in the paper due to the Lack of space.
Design of steel frames with tapered members Hence the optimality criteria for depth variable Da is obtained from eqn (48).
Multiplying both sides by D;, and then taking the cth root, yields 0;”
06
=
1
I
I
I
lllli
(53)
where v and v + 1 represent the current and the following optimum design cycles, c is known as the step size and its value is selected before initiating the design process. In tlhe numerical examples solved, it is found reasonable to use 0.01 for the value of c. To start the design process, eqn (53) requires the value of the Lagrange multipliers to be known. These are obtained from the following simple recursive relationships, as suggested in Ref. [13]. They are derived from the constraints of design problem (1) by taking them as equality, multiplying both sides by I,“:, and 1:; and then taking the mth root.
&+I = &(l/g,(D,,
D,))““’
n=l,2,...,p
(54)
r = 1,2,. . . , nm,
(55)
where m is the step size. Experience obtained from the numerical examples has shown that the selection of a value of 0.5 for WIprovides a reasonable speed of convergence. It is clear that eqns (54) and (55) require the initial values of the Lagrange parameters to be selected. It was found suitable to use 1000 as an initial value for these para.meters. OPTIMUM
DESIGN PROCEDURE
The algorithm developed for the optimum design of steel frames with tapered members consist of the following steps. (1) Input the geometrical properties of the structure, the value of thickness for flange and web width of the flange, as well as initial values of depth variables and Lagrange parameters. (2) Analyze the frame for the current values of design variables and obtain the joint displacements and member forces under the external loads and unit
803
loads acting in the direction of the restricted displacements. (3) Compute the Lagrange parameters from eqns (54) and (55). (4) Take each depth group in the structure respectively and compute the value of the depth variable which is in the cycle from eqn (53). Carry out this computation for all depths in other groups. (5) Check whether convergence is obtained in the weight of the frame and design variables. If it is obtained, then go to step 6, if not go to step 2. (6) Check whether the displacement and combined strength constraints are satisfied. If not, go to step 2, else stop. The design process has the flexibility of allowing the depths to be linked together linearly. This provides practically acceptable final shapes. Any other form of linkage, such as parabolic, is also possible. Furthermore, if uniform elements are required to be used for a certain length, then the depths for the nodes on such elements are made to belong to the same group. It is worthwhile mentioning that the cumbersome design process of frames with tapered members is reduced to the computation of an expression for each depth variable in each design cycle. The design process does not require the separation of active and passive constraints during the design cycles. This adds another feature to its simplicity.
DESIGN EXAMPLES
Four different steel structures with tapered members has been designed successfully by the algorithm presented. Unless stated otherwise the modulus of elasticity and yield strength of steel were taken as 205 kN mm-2 and 275 N mm-*. The density of steel was 7.85 gcmA3. The convergence criteria used for the minimum objective function was O.l%, while it was 1% for the depth variables. The figures within the brackets after the weight of each structure show the total number of iterations for obtaining the final shape. The depths given in the figures are in mm. Fixed end beam
Single span fixed end beam subjected to mid-span point load of 200 kN is shown in Fig. 3a. It is assumed that lateral supports are provided at every 2 m of its span. These points are taken as nodes, and the depths at these nodes are treated as design variables. This introduces five depth variables into the design problem. Depths at nodes 2 and 4 are taken to be equal to those at 3 and 5 to satisfy the symmetry constraint. Furthermore, depths at 1,2 and 4 are linked together so that they vary linearly. The width and thickness of the fiange and thickness of the web are kept constant throughout the beam. Their values are given in Fig. 3a. The minimum depth
804
M. P. Saka
200 kN
@
2n
?
2n I
?
2n I
_?_ I-
2m
Q
8n
(a)
a 626
447 W = 776491
626 gms
(30)
The beam was also designed once more taking into consideration the lateral torsional buckling constraints. The displacement limitation was increased to 50 mm so that lateral instability governs the design. The lateral effective length factors for the parts between the lateral restraints were taken as 1. The optimum beam for this case is shown in Fig. 3d. Finally, the effective length factors were reduced to 0.85, while keeping all other constraints the same. Figure 3e shows the final design for this case. As expected the reduction in the effective length factor results in a reduction in the overall weight of the beam. The effect of the lateral instability on the final shape of the beam can clearly be seen from the comparison of Fig. 3c with Fig. 3d or Fig. 3e. In the absence of lateral instability constraints, maximum flexural stress occurs at the supports, yielding to greater depths at these nodes and a smaller section at the mid-span where the bending stress is small. On the other hand, when the lateral instability constraints are introduced to the design problem, the variation of the taper becomes smooth. It is interesting to note that the minimum weights of both designs Fig. 3c and d are almost the same, while the shapes are quite different. Two-span continuous beam
490
490 w
= 599709
gms
(37)
\I
I#
342
342 W = 603343
gms
(24)
Cd)
323
323 W = 585163
gms
(27)
(e) Fig. 3. Different designs of single span fixed end beam
limitation is taken as 200 mm. The self weight of the beam is not taken into account. First, the beam was designed under the mid-span deflection limitations of 5 mm. The final depths and minimum weight are given in Fig. 3b. Later, the same beam was designed for stress constraints only. The lateral torsional buckling constraints are removed from the problem. Instead, bending stress limitations of 160 N mm-* were imposed on the bending stresses at the modes. The final shape for this design is shown in Fig. 3c.
The simple two-span steel beam shown in Fig. 4a was originally designed in Ref. [14]. The same continuous beam is reconsidered here in order to compare the algorithm presented, with the ones in the literature. The width and thickness of the flanges and the thickness of the web are given in the figure. Seventeen nodes were considered on the beam and at each node the depth is taken as design variable. However, they are linked to each other so that linear variation can be achieved along the spans. In Ref. [14], the beam was designed first for a flexural stress constraint of 160 N mm-*. The minimum depth limitation was taken as 300 mm. The final shape obtained is shown Fig. 4b. The same beam was also designed by the algorithm presented under the same constraints. This is achieved by replacing the combined strength constraints with flexural stress constraints. The optimum design obtained is shown in Fig. 4c. This design is slightly lighter than the beam of Fig. 4b and is obtained in fewer iterations. Later, the same beam was designed considering the deflection limitation of 48 mm for the vertical displacement of node 11, in addition to the flexural stress constraints. The final design obtained for this case in Ref. [14] is given in Fig. 4d. The optimum beam found in this study under the same limitations is shown in Fig. 4e, which is lighter than the beam of Ref. [14]. The difference between the weights may well be due to the fact that the algorithm of Ref. [14] might have been converged to the local optimum. The same beam was once more designed by the algorithm presented by considering the deflection as well as combined strength constraints. It was
Design of steel frames with tapered members 60
kN
4x2n
300
120
n
1032,2
I
300
W = 3583764 (b) 322
kN
12 x 2
I
805
gms (16)
875.8
446
W = 3533677 gms (12) Cc) 1049.3
949.3
W = 4153127 gms (20) (co 973,7
538.9
W = 3717210 gms (33)
868.6
441.6
w = 3503455 gms (5)
assumed that lateral restraints were provided at the nodes. The lateral effective length factor was taken as 1 for all members. First, the vertical displacement limitation was considered to be 150 mm, in which case the combined strength constraints govern the design. The final shape of the beam for this case is shown in Fig. 4f. One bay pitch roof jrrame
The pitch roof fr.ame of Fig. 5a has 19 nodes at which lateral restraints were assumed to be provided. The external loading and dimensions are shown in the
figure, together with the constant dimensions of the i-section adopted for the frame. The minimum size for the depth variables were taken as 300mm. The vertical displacement of node 10 was restricted to 50 mm. The depths are linked to preserve the symmetry as well as linear variation along the columns and rafters. The final design for this case is obtained after 16 iterations, with the depth values shown in Fig. 5b. It was noticed that in this frame the vertical displacement of node 10 was 50 mm, while the combined strength ratios were below the upper bound of 1. Later the displacement restraint is
806
M. P Saka
relaxed by increasing the limitation to 150 mm. This makes the combined strength constraints dominate the design. The optimum frame for this case is shown in Fig. 5c, in which the vertical displacement of node IO was 5 1.98 mm, while the combined strength ratios for members (3 and 4) and (16 and 17) were equal to I. Naturally this frame is lighter than the previous one. To study the effect of support conditions, the fixed supports of the frame were changed to pin supports. The design was repeated as before. The frame shown in Fig. 5d is the final design for the vertical displacement limitation of 50 mm at node 10 and the frame of Fig. 5e was obtained for the displacement limitation of 150 mm, which makes the combined strength constraints dominant in the design problem. Later, the pitch roof frame was rendered into a three hinged frame by introducing another hinge at node 10. The frames shown in Fig. 5f,g are obtained for
vertical displacement limitations of 50 and 150 mm for node 10, respectively. It is clear that frame with fixed supports provides the lightest solution in both cases. The same frame was once more designed under 50 mm deflection limitation and combined strength constraints with different depth linking. The depth variables at nodes 1, 2, 3 and 4 were linked as a first group, the depths at nodes 4 and 5 as a second group, the depths at nodes 5, 6, 7 and 8 were made equal to each other as a third group and finally the depth at nodes 9 and 10 were taken as the fourth group. Preservation of the symmetry was additional restraint to be observed in the design. With this different grouping, the final designs obtained are shown in Fig. 6a for the fixed supports and in Fig. 6b for the pin supports. The depth grouping was changed again by assuming the depths at nodes 1, 2, 3 and 4 as group
i r
6x21-1
580.7
831,9
W = 4732843
gms
(16)
(b>
979.4
427.6 W = 4557160
gms
(23)
Cc) Fig. 5. Different designs of one bay pitch roof frame.
(continued opposite.)
Design of steel frames with tapered members
A&
W = 4789991
gns
(7)
807
&
(d)
355
W = 4722638
355 gms (37)
(e)
1298
378.7
1298
Ai ,
W = 5690543
gms (35)
1116
1116
b
W = 4958766
gms (22)
(9) Fig.
5-continued.
1, the depths at nodes 4, 5 and 6 as group 2, the depths at nodes 6, ‘7and 8 as group 3 having equal value to each other and depths at nodes 8, 9 and 10 as group 4. The final shapes for this grouping under the same limitation of the previous design are given in Fig. 6c,d. It can be noticed that among the three different groupings for a frame with fixed supports, the third type of grouping produces the lightest design. However, for the frame with the pin-support, the first type of depth grouping yields the lightest frame.
Two buy pitch roof frame
The two bay pitch roof frame shown in Fig. 7a has 30 depth variables. These are linked together so that the columns and rafters have linearly tapered members. Furthermore, geometrical symmetry was presented. The vertical displacements of nodes 9 and 22 were limited to 50 mm. The lateral restraints were assumed at the nodes. The lateral effective length factor was taken as 1 for all the members. The minimum depth value was 300 mm. The final
808
M. P. Saka 801 7;26*8 801 825~
409
801d
II
w= :
4705368
808856.7
gms
(15)
I
/////
409
808
848.6
848.6
506.7
506.7 w = 4922342
gms
(7)
W = 4694378
gns
(15)
W =
gms
(3)
(b)
541.2 4867864
Cd) Fig. 6. One bay pitch roof frame with different depth grouping.
design obtained under these constraints is shown in Fig. 7b for the fixed supports. In this frame, both displacement and combined strength constraints were at their bounds. When the supports were changed to pin, the optimum frame became heavier, as shown in Fig. 7c. Later, lateral loads of IO kN were added to nodes 2, 3 and 4. An additional displacement constraint was also introduced to the design problem. The horizontal displacement of node 4 was limited to
20 mm. The optimum shapes for both fixed supported and pin supported frames are shown in Fig. 8a,b. Due to the fact that the lateral loads tend to reduce the effect of the vertical ones, these optimum frames shown in Fig. 8c,d are the final designs for a loading case where the vertical loads in the first bay were doubled, while the horizontal and the vertical loading at the second bay were kept the same. They are naturally heavier than other designs.
Design of steel frames with tapered members
470.6
470.6
w = 6713163
gms
(16)
gms
(34)
(b>
w = 6921973 w
Fig. 7. Optimum designs for two bay pitch roof frame.
CONCLUSIONS
It has been shown that the algorithm presented can effectively be used 1.n the optimum design of steel frames with prismatic and/or tapered members of I-sections. It has the flexibility of taking the depth of I-section as a design variable at each specified node separately, as well as linking them in a desired grouping so that they may vary linearly. It considers displacement limitations, together with combined axial and flexural strength constraints. The latter is included in the algorithm according to the Load and
Resistance Factor Design code of the AISC. The lateral torsional buckling rsistance of the members is taken into account. The optimality criteria method was derived for both displacement and combined strength constraints from Kuhn-Tucker conditions. It is shown that the optimality criteria method can successfully be used in solving highly nonlinear and complex design problem. The number of iterations to reach the optimum design vary depending on the selection of the initial design point. It was noticed that in some design examples, where the initial point was selected far from the optimum, the algorithm
810
M. P. Saka
W
= 6569822
gms
(37)
(a) 353.3
W
353,3
= 6824947
gms
(32)
(b) 439.3
439.3 783.3
518,6 7
W
= 7406318
gms
W
461,4
= 7477988
gms
(30)
(ci) Fig. 8. Optimum designs for two bay pitch roof frame under different loadings.
might produce shapes where some members may have unproportionaly large depths to achieve the required stiffness for the frame. This undesired difficulty can easily be overcome by trying a different initial design point. The numerical examples considered have clearly shown that the design requirements given in the codes can simply be incorporated with optimization algorithms so as to provide efficient tools for the use of designers.
REFERENCES
Load and Resistance Factor Design, Manual of Steel Construction. ABC, U.S.A. (1986). M. R. Khan, Optimality criterion techniques applied to frames having general cross-sectional relationships. AIAA J. 22, 669-676 (1984).
M. P. Saka, Optimum design of steel frames with stability constraints. J. Compu~. Struct. 41, 1365-l 377 (1991).
Design
of steel frames
4. M. P. Saka, Optimum design of steel grillage systems. In: Proc. Third-M. Conf. on-Steel Srru&es~Singapore Steel Societv. Sineaoore. March. vv. 273-290 (1987). 5. M. P. Saks: Optimum geometry’design of rooi trusses by optimality criteria method. J. Comput. Struct. 38, 83-92 (1991). 6. M. P. Saka, Opiimum design of pin-jointed steel structures with practical applications. J. Strucr. Engng ASCE 116, 2599-2620 (1990). 7. M. P. Saka and M. S. Hayalioglu, Optimum design of geometrically nonlinear elastic-plastic steel frames. J. Comouf. Struct. 38. 329-344 (1991). 8. D. J. just, Plane frameworks of tapering box and I-section. Proc. ASCE 103ST1, 71-86 (1977). 9. 0. Hartavi, Elastic-plastic analysis of frames with tapering box and I-sections. MSc dissertation, Karadeniz Technical University, Trabzon, Turkey (1984).
with tapered
members
811
IO. D. L. Karabalis and D. E. Beskos, Static, dynamic and stability analysis of structures composed -of tapered beams. J. Comout. Struct. 18. 731-748 (1983). 1I. T. K. H. Tan and A. Jennings, Optimal‘plasfic design of frames with tapered members. In: Civil-COMP 87, Proc. Third Int. Conf. on Civil and Structural Engineering Computing, Vol. I, pp. 265-271, London (1987). 12. P. H. Gallagher and C. H. Lee, Matrix dynamic and instability analysis with non-uniform elements. In/. J. numer. A?eth. bngng 2, 265-275 (1970). 13. E. Atrek, R. H. Gallagher. K. M. Ragsdell and 0. C. Zienkiewic, New Dirktions in Opt&m Structural Design. Wiley, New York (1984). 14. R. J. Allwood and Y. S. Chung, An optimality criteria method applied to the design of continuous beams of varying depth with stress, deflection and size constraints. J. Comput. Struct. 20, 947-954 (1985).
~c 1997 Civd-Comp
Pergamon
PII: SO0457949(96)00074-O
OPTIMUM
DESIGN OF STEEL FRAMES TAPERED MEMBERS
WITH
M. P. Saka University
of Bahrain,
Civil Engineering
Department,
Bahrain
Abstract-The design of steel tapered members for combined axial and flexural strength is somewhat complex and tedious if no approximation is made. However, recent load and resistance factor design (LRFD) of the AISC code has treated the problem with sufficient accuracy and ease. In this study, an algorithm is developed for the optimum design of steel frames composed of prismatic and,‘or tapered members. The width of an I-section is taken as constant, together with the thickness of web and flange, while the depth is considered to be varying linearly between joints. The depth at each joint in the frame where the lateral restraints are assumed to be provided is treated as a design variable. The objective function which is taken as the volume of the frame is expressed in terms of the depth at each joint. The displacement and combined axial and flexural strength constraints are considered in the formulation of the design problem. The strength constraints, which take into account the lateral torsional bucking resistance of the members between the adjacent lateral restraints, are expressed as an nonlinear function of the depth variables. The optimality criteria method is then used to obtain a recursive relationship for the depth variables under the displacement and strength constraints. These relationships are derived from the Kuhn-Tucker necessary conditions. The algorithm basically consists of two steps. In the first one, the frame is analysed under the external and unit loadings for the current values of the design variables. In the second, I:his response is utilized together with the values of Lagrange multipliers to compute the new values of the depth variables. This process is continued until convergence is obtained. Numerical examples are presented to demonstrate the practical application of the algorithm. 0 1997 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved.
displacement and strength described in Ref. [l].
INTRODUCTION
The frames with talpered members are preferred in the design of steel structures whenever the architectural requirements allow their presence. They not only provide better distribution of strength, but also yield to a lighter design. The methods available for the analysis of such frames are well established. However, it is not as easy to state the same for their design. In most of the practical design codes, approximate procedures are suggested for dimensioning tapered members which are subjected to the combined action of axial force and bending moment. However, in a recent publication of the AISC, titled Load and Resistance Factor, this design problem is well treated [ 11. The optimum (design problem of steel frames composed of prismatic and/or tapered I-section members is formulated by treating the depth of the section at each node as a design variable. Each point along the frame where a lateral restraint is provided is taken as a node. Between the nodes the depth is assumed to vary linearly. The width and the thickness of the flanges and ,web are considered to be constant throughout the frame. Variable linking is also considered in the formulation so that a predetermined shape over a desired length can be achieved in the final design. The design formulation includes
OPTIMUM
constraints
DESIGN
as they are
PROBLEM
The optimum design problem of steel frames with tapered members can be expressed as: m,,
Min W = c
r= I
p,v,
subject to,
g,,,,(Dn)
n=l,Z...,p
1 k= l,L...,ng
Di - DL,,> 0
r = 1,2,
.
, nm,
(1)
where p, and vPare the density and volume of member r, respectively. nm is the total number of members in the frame. DL is the depth variable belonging to group k and ng is the total number of different groups in the frame. DA,, is its lower bound. g,,,,(Dk) represents nth displacement constraints. p is the total number of restricted displacements. g,, (D;, D,) represents the 797
M. P. Saka
798
I-
i
Fig. 1. Linearly tapered beam element with I-section.
combined axial and flexural strength constraint for member r. Di and D, are the depths at the first and second end of member r, as shown in Fig. 1. Objective function
The objective function, which is the total weight of frame, is obtained as a summation of the weights of each member. The volume v, of member r can be expressed in terms of the values of the depth variables at the first and second ends of the member as vr =A,+
2
(2)
’
where X,,, is the vector of virtual displacements of member r due to the virtual loading corresponding to the nth constraint. This is obtained by applying the unit load in the direction of the restricted displacement n. K,(Di, 0,) is the stiffness matrix of member r in the global coordinates. X, is the displacement vector due to applied loads. Combined axial andflexural
The combined axial and flexural strength constraint for member r, which is subjected to axial force and bending moment about its major axis, is given in LRFD as, for
where Ai and A, are the areas of cross-section at nodes i and j. The area of the tapered member at section x can be expressed in terms of depth at i and j as A, = Y, + Y~x,
strength constraints
P”/(4P,,) 2 0.2, (7)
(3) and for
where
PZ,l(W,!) < 0.2,
Y, = tv,(D,- 2tr) + 2brtf Y’z = tw(D,- D,)/l.
(4)
It can be seen that br, tf, and t, are selected to be constant throughout the frame, which leaves only the depths at node i and j as the design variables. It is clear that for x = 0, the area at the first end becomes A, = Y, and for x = I, the area at the second end becomes Ai = Y, + YJ. Displacement constraints
The nth displacement following form:
constraint
g,,“(Dk) has the
&/,,(Dk) = 6, - 6, ,
(5)
where 6, is the displacement at node s and 6, is its upper bound. The displacement ai can be expressed as a function of the depth variables by making use of the virtual work theorem. “,#> 6, = 1 XJK(Di, Dj)X,n, ,=I
(8) where P. is the required strength and P,, is the nominal tensile or compressive strength for the member depending upon whether it is in tension or compression. M,, is the required flexural strength and M,, is the nominal flexural strength about the major axis of the section. The resistance factor 4 is given as 0.90 in the case of tension and as 0.85 in the case of compression in LRFD. The resistance factor for flexure &, is specified as 0.90 by the same code [l]. Since only the nominal strengths are the functions of the depth variables, the strength constraint for member r can be re-written as g,,(Di, Di) = al/P,, + a?lM,,,,
(9)
where aI and a2 are the constants given as, for P”/(+P,,) > 0.2
(6)
aI = P,/O.85,
a2 = 8M.18.1
(10)
199
Design of steel frames with tapered members
0
a Vi ,vi v-4
1
-------
Vj ,vj
____
1%
Pj auj
Mj a@j d
X
I
I-
L
Fig. 2. Member end forces and deformations of a tapered member.
KM = E[C:,gn + 2C,2Cahn + C&En]
and for
K52 =
P”l((bP”) < 0.2 a2 = A4,/0.9.
UI = t:/1.7,
E[C,Kx,gz, + C,,Cd,hn + CaC,,ha +
(11) KS,
=
C4,C4,E211
E[C,,C,2g2,
C,,C’ahm + C’aCxhz,
+
STIFFNESS MATRIX OF TAPERED MEMBER
The derivation of the stiffness matrix of a linearly varying tapered member, made out of a homogenous isotropic linear elastic material of modulus E, is given in detail in Refs i&-11]. The elements of the axial and flexural stiffness matrix of the tapered member derived in this study are based on the displacement functions, which are the exact solutions of pertinent govering differential equations. Thus, these stiffness matrices are exact, not approximate, like some of those given in Refs [12, 131. The stiffness matrix of the tapered member, shown in Fig. 2, is obtained in Ref. [9] using the finite element approach, where the exact displacement functions were employed. Due to lack of space its derivation is omitted here. It has the following form in the local coordinate system: K,(Di, 4) =
0
K22
K,,
I
0 K32 k, -------____I__________
K55 =
G,C42E21]
E[C:,g2,
+
Kc.2= E[C,4C,,ga
2C,,C.,,hn
+ C:,Ea]
+ C,4Cah2, + C’uC,,hn
+ C44C41 E211 K63 = EbC32g2,
+ K65 =
fG6 =
CMCaha + CuC,jhn
+
C,.,Cahz, + CuC,,ha
CuC42E21]
W,&,g2,
+
+
C4&3E21]
E[C&,
+ 2C,Xuh2,
(13)
+ C&En]
in which E is the modulus of elasticity and other terms are given as
I I
k,
+
g21 =g2-gl,
h2, =
h2 -
hi,
’
(12)
K.4,
0
0
I KM
0
K52
J&3
;
0
K55
0
K62
K6,
;
0
Ke.5 K,
Ez, = E2 - E, Gz, = Gz - G,
Hn=H2-H,, C,, = -hx/D,
Cd, = g,,/D
where C,, =
I
c,, = -c,,,
is 0 K22 =
E[C:,g2, -t 2C,,Ci,hn
K32 =
W,2C,11:21
+
C42C,,E2,]
+
(Hz,
-
Ih2)lD,
C42 =
(lgz - Gz,)/D
dx/A
K,, = Ku = --IL, = E
C,&hx
+ C,, E2,] + CaC,,hn
ca = -c,,
CM = (Ih, - Hn)/D,
CM =
(G2, -
g,)/D
D = Gz,hz, - Hnga + g22
g,, = Igh - Ig,h,
(14)
M. P. Saka
800
where
variables D, and D, at the first and second end of the member:
E, =
s
x2 dx/I,
E2 =
,=0
s
+
x2 dx/I.
,=I
+
dw)2(6
+
44x),
(17)
4,
=
r&/12
42 = (0, - Di)/l
+ b,t,/2,
44
=
-
4243
In
$3
t&2/12.
H2 = (434~~4~In 46 - 4~4~~4~- 4444’In & +
dh~2dd4
In
46
-
$2&h
In
dd/@2448)
-6 = (&6,4’In $‘I - 2&&&‘$4 In ‘#‘I-
d’:&dJ3~4
in which 4,) 42r 4, and 4~~are given in eqn (18). & and 4, are given as 46 = 4, + 421, 47 = 43 + 441.
(20)
Since b,, tt, and t,? are considered constant throughout the frame &, &, &, c$~,C#J( and 4, are only the function of the depth variables Di at the first end and Dj at the second end. This makes the stiffness terms in eqn (13) also implicit functions of Di and Dj .
where 41 = D, - t/,
41
+ 4V#J3/(92648)
The area of a tapered member at any cross-section x is given in eqn (3). The moment of inertia of the same section can be expressed in terms of depth at the first and second ends of the member as ($1
h
(15)
It is clear from the above expressions that all that is needed in order to construct the stiffness matrix of a tapered member is to carry out the following six integrations:
I=
$1$2&44
(18)
Hence, it becomes possible to obtain the analytical solutions of the integrations given in eqn (15) by using the relationships of eqns (3) and (17). Once this is carried out, the following expressions are obtained, which are only a function of depth
NOMINAL AXIAL AND FLEXURAL STRENGTH OF TAPERED MEMBER It is shown in Ref. [S] that the combined strength constraint for a tapered member makes it necessary to express the design axial and flexural strength of the
Design of steel frames with tapered members member in terms of depth variables defined at its ends. Nominal
Substituting eqn (28) into eqn (27) and taking Q = 1 gives the effective slenderness as,
tensile stre,qgth
&T = [c,(D,tw -
In the case where the tapered member is in tension, LRFD gives the nominal tensile strength P.
27’Y,
(29
where cl is a constant.
as [II,
Pn = FJ,,
(21)
where F, is specified yield stress and A, is the gross area of the member at the smaller end. Hence, the nomin.al tensile strength P, can be expressed as a function of depth variable D, of the smaller end as P. = F,(Dit,
- 2F, T),
6F,(KI)’
1” = t,tt - brtr.
(30)
cl=-.
As a result, the nominal compressive strength P, of eqn (24) can be expressed in terms of depth variable Di at the smaller end as for &T < 1.5,
(22)
where T is a constant given by
Nominal
801
P. = (Dit, - 2T’)F,0.658’1’~J~-W (23)
(31)
for
compressive strength
& > 1.5,
When the tapered member is in compression, its nominal compressive strength is given by LRFD as,
P, = 0.877F,/[c,(Dit, P, = FuAg,
NominaiJlexural
where F,, is the critical stress computed from one of the following expressions: for
(32)
strength
The nominal flexural strength of tapered flexural member for the limit state of lateral torsional buckling is given in LRFD as M. = (5/3)S.&,
L < 1.5, F,, = (0.658”“R)Fy
- 2T)].
(24)
(25)
(33)
where S, the sectional modulus of the larger end and Fk is the design flexural stress of tapered member,
for J-c,4 > 1.5, 1’o-6B.&!+fi;) F,, = 0.877F,/&
(26)
where ,IeRis called the effective slenderness parameter and is defined as,
1
F, < 0.6F,
unless Fb < F,/3, in which case, Fb: = B,/(Ff_. + F$).
lZc~:= S,/(QFy/nZE)
(34)
(35)
(27)
in which S is equal to Klr,, for weak axis bending and K,.llr,, for strong a.xis bending. K is the effective length factor for the member. Since between the adjacent lateral restraints, buckling about the weak axis governs, S is taken as Kl/r,,. The radius of gyration r,, is defined at the smaller end of the tapered member as,
In eqns (34) and (35)
F.,
=
I,
W’OOAr h,lD,
’
(36)
where factors h, and h, are given as, h, = 1 + O.O23y,/(lDj/A~)
(37)
h, = 1 + O.O0385y,/(l/r~,)
(38)
(28) CAS 63/3-F
M. P. Saka
802
in which rTOis radius of gyration of the section at the smaller end, considering only the compression flange plus l/3 of the compression web area, taken about an axis in the plane of the web, Ar is the area of the compression flange, y is given as ‘i =
(D, - D;)/D,
strained problem into an unconstrained one by using Lagrange multipliers. The necessary Kuhn-Tucker conditions are then written for the unconstrained problem to obtain a recursive relationship for the design variables. The Lagrangian of the design problem (1) is
(39)
7 < the smaller of 0.268 (l/Da) or 6. The relationships listed from eqns (33)-(39) can be expressed in terms of depth variables at the ends of the tapered member as shown in the following. The secttonal modulus S,,
nn,
-
s = t,@, - tr)’ + 66rr@, + tr)’ 60 FV and Fwy of eqn (61) are written
(40)
as
F, = 12,00Ob& s, IhsD, F,; =
where constants
c3 cdh;: + csh;D,’
where Ad,, and A,, are Lagrange multipliers for displacement and strength constraints. The necessary condition for the local constrained optimum is obtained by differentiating this equation with respect to design variable Dk.
(42)
c4 = 3Pbftr
19,
(43)
h, and h, have the form h, = 1 + O.O23y&Di/(brtf)) h, = 1 + O.O0385y,/(l/r~o)
(44)
where mk is the number of depths belonging to group k. The derivative of the volume of a tapered member with respect to depth variable Dk can analytically be obtained from expressions (2H4). The derivative of the displacement constraint from eqn (5) becomes
agdn(Dk ) as, ~=_
(45)
ao,
in which ‘/ is function r,, is
(49)
ao,
of D,, as shown in eqn (39) and which in turn from eqn (6) becomes,
rTo = ,/0.25tfb:/(3trbr
+ t,D,).
(46)
It is clear from eqns (40)-(46) that nominal flexural strength M. can be expressed in terms of depth variables D, and 0,. The value of B in expression (34) is taken as 1.5 for simplicity. Futhermore the magnification factor c,,, of LRFD is assumed to have a value of 1.1. OPTIMALITY
(47)
c2, c4 and c5 are
c, = 425,00Otfb;, c5 =
,T,&rgsr(Di, Dj),
CRlTERlA
FOR DEPTH VARIABLES
It is shown in the previous sections that the displacement and combined axial and flexural constraint, as well as the objective function of the optimum design problem considered in this study, are highly nonlinear functions of design variables. The optimahty criteria approach was found to be an effective method in finding the solution of such design problems [2-51. This technique transforms the con-
2 = ,g,QaKrg; D/lx,,.
(50)
Inspection of expressions (12t(20) reveals the fact that the derivative of the stiffness matrix of the tapered member with respect to depth variable can be obtained analytically. On the other hand, the same can also be achieved for the derivative of strength constraints. From eqn (9), it follows:
ag.r(Dt,D,) = _adaPniaDn) _ a2(aMn,laDA)
aa
Pi
M;,
.
(51)
By using expression (22), (31) or (32) and (33) dP,/aD1 and aM,,,ji?D, can be derived analytically. The expressions for these derivatives are not included in the paper due to the Lack of space.
Design of steel frames with tapered members Hence the optimality criteria for depth variable Da is obtained from eqn (48).
Multiplying both sides by D;, and then taking the cth root, yields 0;”
06
=
1
I
I
I
lllli
(53)
where v and v + 1 represent the current and the following optimum design cycles, c is known as the step size and its value is selected before initiating the design process. In tlhe numerical examples solved, it is found reasonable to use 0.01 for the value of c. To start the design process, eqn (53) requires the value of the Lagrange multipliers to be known. These are obtained from the following simple recursive relationships, as suggested in Ref. [13]. They are derived from the constraints of design problem (1) by taking them as equality, multiplying both sides by I,“:, and 1:; and then taking the mth root.
&+I = &(l/g,(D,,
D,))““’
n=l,2,...,p
(54)
r = 1,2,. . . , nm,
(55)
where m is the step size. Experience obtained from the numerical examples has shown that the selection of a value of 0.5 for WIprovides a reasonable speed of convergence. It is clear that eqns (54) and (55) require the initial values of the Lagrange parameters to be selected. It was found suitable to use 1000 as an initial value for these para.meters. OPTIMUM
DESIGN PROCEDURE
The algorithm developed for the optimum design of steel frames with tapered members consist of the following steps. (1) Input the geometrical properties of the structure, the value of thickness for flange and web width of the flange, as well as initial values of depth variables and Lagrange parameters. (2) Analyze the frame for the current values of design variables and obtain the joint displacements and member forces under the external loads and unit
803
loads acting in the direction of the restricted displacements. (3) Compute the Lagrange parameters from eqns (54) and (55). (4) Take each depth group in the structure respectively and compute the value of the depth variable which is in the cycle from eqn (53). Carry out this computation for all depths in other groups. (5) Check whether convergence is obtained in the weight of the frame and design variables. If it is obtained, then go to step 6, if not go to step 2. (6) Check whether the displacement and combined strength constraints are satisfied. If not, go to step 2, else stop. The design process has the flexibility of allowing the depths to be linked together linearly. This provides practically acceptable final shapes. Any other form of linkage, such as parabolic, is also possible. Furthermore, if uniform elements are required to be used for a certain length, then the depths for the nodes on such elements are made to belong to the same group. It is worthwhile mentioning that the cumbersome design process of frames with tapered members is reduced to the computation of an expression for each depth variable in each design cycle. The design process does not require the separation of active and passive constraints during the design cycles. This adds another feature to its simplicity.
DESIGN EXAMPLES
Four different steel structures with tapered members has been designed successfully by the algorithm presented. Unless stated otherwise the modulus of elasticity and yield strength of steel were taken as 205 kN mm-2 and 275 N mm-*. The density of steel was 7.85 gcmA3. The convergence criteria used for the minimum objective function was O.l%, while it was 1% for the depth variables. The figures within the brackets after the weight of each structure show the total number of iterations for obtaining the final shape. The depths given in the figures are in mm. Fixed end beam
Single span fixed end beam subjected to mid-span point load of 200 kN is shown in Fig. 3a. It is assumed that lateral supports are provided at every 2 m of its span. These points are taken as nodes, and the depths at these nodes are treated as design variables. This introduces five depth variables into the design problem. Depths at nodes 2 and 4 are taken to be equal to those at 3 and 5 to satisfy the symmetry constraint. Furthermore, depths at 1,2 and 4 are linked together so that they vary linearly. The width and thickness of the fiange and thickness of the web are kept constant throughout the beam. Their values are given in Fig. 3a. The minimum depth
804
M. P. Saka
200 kN
@
2n
?
2n I
?
2n I
_?_ I-
2m
Q
8n
(a)
a 626
447 W = 776491
626 gms
(30)
The beam was also designed once more taking into consideration the lateral torsional buckling constraints. The displacement limitation was increased to 50 mm so that lateral instability governs the design. The lateral effective length factors for the parts between the lateral restraints were taken as 1. The optimum beam for this case is shown in Fig. 3d. Finally, the effective length factors were reduced to 0.85, while keeping all other constraints the same. Figure 3e shows the final design for this case. As expected the reduction in the effective length factor results in a reduction in the overall weight of the beam. The effect of the lateral instability on the final shape of the beam can clearly be seen from the comparison of Fig. 3c with Fig. 3d or Fig. 3e. In the absence of lateral instability constraints, maximum flexural stress occurs at the supports, yielding to greater depths at these nodes and a smaller section at the mid-span where the bending stress is small. On the other hand, when the lateral instability constraints are introduced to the design problem, the variation of the taper becomes smooth. It is interesting to note that the minimum weights of both designs Fig. 3c and d are almost the same, while the shapes are quite different. Two-span continuous beam
490
490 w
= 599709
gms
(37)
I#
342
342 W = 603343
gms
(24)
Cd)
323
323 W = 585163
gms
(27)
(e) Fig. 3. Different designs of single span fixed end beam
limitation is taken as 200 mm. The self weight of the beam is not taken into account. First, the beam was designed under the mid-span deflection limitations of 5 mm. The final depths and minimum weight are given in Fig. 3b. Later, the same beam was designed for stress constraints only. The lateral torsional buckling constraints are removed from the problem. Instead, bending stress limitations of 160 N mm-* were imposed on the bending stresses at the modes. The final shape for this design is shown in Fig. 3c.
The simple two-span steel beam shown in Fig. 4a was originally designed in Ref. [14]. The same continuous beam is reconsidered here in order to compare the algorithm presented, with the ones in the literature. The width and thickness of the flanges and the thickness of the web are given in the figure. Seventeen nodes were considered on the beam and at each node the depth is taken as design variable. However, they are linked to each other so that linear variation can be achieved along the spans. In Ref. [14], the beam was designed first for a flexural stress constraint of 160 N mm-*. The minimum depth limitation was taken as 300 mm. The final shape obtained is shown Fig. 4b. The same beam was also designed by the algorithm presented under the same constraints. This is achieved by replacing the combined strength constraints with flexural stress constraints. The optimum design obtained is shown in Fig. 4c. This design is slightly lighter than the beam of Fig. 4b and is obtained in fewer iterations. Later, the same beam was designed considering the deflection limitation of 48 mm for the vertical displacement of node 11, in addition to the flexural stress constraints. The final design obtained for this case in Ref. [14] is given in Fig. 4d. The optimum beam found in this study under the same limitations is shown in Fig. 4e, which is lighter than the beam of Ref. [14]. The difference between the weights may well be due to the fact that the algorithm of Ref. [14] might have been converged to the local optimum. The same beam was once more designed by the algorithm presented by considering the deflection as well as combined strength constraints. It was
Design of steel frames with tapered members 60
kN
4x2n
300
120
n
1032,2
I
300
W = 3583764 (b) 322
kN
12 x 2
I
805
gms (16)
875.8
446
W = 3533677 gms (12) Cc) 1049.3
949.3
W = 4153127 gms (20) (co 973,7
538.9
W = 3717210 gms (33)
868.6
441.6
w = 3503455 gms (5)
assumed that lateral restraints were provided at the nodes. The lateral effective length factor was taken as 1 for all members. First, the vertical displacement limitation was considered to be 150 mm, in which case the combined strength constraints govern the design. The final shape of the beam for this case is shown in Fig. 4f. One bay pitch roof jrrame
The pitch roof fr.ame of Fig. 5a has 19 nodes at which lateral restraints were assumed to be provided. The external loading and dimensions are shown in the
figure, together with the constant dimensions of the i-section adopted for the frame. The minimum size for the depth variables were taken as 300mm. The vertical displacement of node 10 was restricted to 50 mm. The depths are linked to preserve the symmetry as well as linear variation along the columns and rafters. The final design for this case is obtained after 16 iterations, with the depth values shown in Fig. 5b. It was noticed that in this frame the vertical displacement of node 10 was 50 mm, while the combined strength ratios were below the upper bound of 1. Later the displacement restraint is
806
M. P Saka
relaxed by increasing the limitation to 150 mm. This makes the combined strength constraints dominate the design. The optimum frame for this case is shown in Fig. 5c, in which the vertical displacement of node IO was 5 1.98 mm, while the combined strength ratios for members (3 and 4) and (16 and 17) were equal to I. Naturally this frame is lighter than the previous one. To study the effect of support conditions, the fixed supports of the frame were changed to pin supports. The design was repeated as before. The frame shown in Fig. 5d is the final design for the vertical displacement limitation of 50 mm at node 10 and the frame of Fig. 5e was obtained for the displacement limitation of 150 mm, which makes the combined strength constraints dominant in the design problem. Later, the pitch roof frame was rendered into a three hinged frame by introducing another hinge at node 10. The frames shown in Fig. 5f,g are obtained for
vertical displacement limitations of 50 and 150 mm for node 10, respectively. It is clear that frame with fixed supports provides the lightest solution in both cases. The same frame was once more designed under 50 mm deflection limitation and combined strength constraints with different depth linking. The depth variables at nodes 1, 2, 3 and 4 were linked as a first group, the depths at nodes 4 and 5 as a second group, the depths at nodes 5, 6, 7 and 8 were made equal to each other as a third group and finally the depth at nodes 9 and 10 were taken as the fourth group. Preservation of the symmetry was additional restraint to be observed in the design. With this different grouping, the final designs obtained are shown in Fig. 6a for the fixed supports and in Fig. 6b for the pin supports. The depth grouping was changed again by assuming the depths at nodes 1, 2, 3 and 4 as group
i r
6x21-1
580.7
831,9
W = 4732843
gms
(16)
(b>
979.4
427.6 W = 4557160
gms
(23)
Cc) Fig. 5. Different designs of one bay pitch roof frame.
(continued opposite.)
Design of steel frames with tapered members
A&
W = 4789991
gns
(7)
807
&
(d)
355
W = 4722638
355 gms (37)
(e)
1298
378.7
1298
Ai ,
W = 5690543
gms (35)
1116
1116
b
W = 4958766
gms (22)
(9) Fig.
5-continued.
1, the depths at nodes 4, 5 and 6 as group 2, the depths at nodes 6, ‘7and 8 as group 3 having equal value to each other and depths at nodes 8, 9 and 10 as group 4. The final shapes for this grouping under the same limitation of the previous design are given in Fig. 6c,d. It can be noticed that among the three different groupings for a frame with fixed supports, the third type of grouping produces the lightest design. However, for the frame with the pin-support, the first type of depth grouping yields the lightest frame.
Two buy pitch roof frame
The two bay pitch roof frame shown in Fig. 7a has 30 depth variables. These are linked together so that the columns and rafters have linearly tapered members. Furthermore, geometrical symmetry was presented. The vertical displacements of nodes 9 and 22 were limited to 50 mm. The lateral restraints were assumed at the nodes. The lateral effective length factor was taken as 1 for all the members. The minimum depth value was 300 mm. The final
808
M. P. Saka 801 7;26*8 801 825~
409
801d
II
w= :
4705368
808856.7
gms
(15)
I
/////
409
808
848.6
848.6
506.7
506.7 w = 4922342
gms
(7)
W = 4694378
gns
(15)
W =
gms
(3)
(b)
541.2 4867864
Cd) Fig. 6. One bay pitch roof frame with different depth grouping.
design obtained under these constraints is shown in Fig. 7b for the fixed supports. In this frame, both displacement and combined strength constraints were at their bounds. When the supports were changed to pin, the optimum frame became heavier, as shown in Fig. 7c. Later, lateral loads of IO kN were added to nodes 2, 3 and 4. An additional displacement constraint was also introduced to the design problem. The horizontal displacement of node 4 was limited to
20 mm. The optimum shapes for both fixed supported and pin supported frames are shown in Fig. 8a,b. Due to the fact that the lateral loads tend to reduce the effect of the vertical ones, these optimum frames shown in Fig. 8c,d are the final designs for a loading case where the vertical loads in the first bay were doubled, while the horizontal and the vertical loading at the second bay were kept the same. They are naturally heavier than other designs.
Design of steel frames with tapered members
470.6
470.6
w = 6713163
gms
(16)
gms
(34)
(b>
w = 6921973 w
Fig. 7. Optimum designs for two bay pitch roof frame.
CONCLUSIONS
It has been shown that the algorithm presented can effectively be used 1.n the optimum design of steel frames with prismatic and/or tapered members of I-sections. It has the flexibility of taking the depth of I-section as a design variable at each specified node separately, as well as linking them in a desired grouping so that they may vary linearly. It considers displacement limitations, together with combined axial and flexural strength constraints. The latter is included in the algorithm according to the Load and
Resistance Factor Design code of the AISC. The lateral torsional buckling rsistance of the members is taken into account. The optimality criteria method was derived for both displacement and combined strength constraints from Kuhn-Tucker conditions. It is shown that the optimality criteria method can successfully be used in solving highly nonlinear and complex design problem. The number of iterations to reach the optimum design vary depending on the selection of the initial design point. It was noticed that in some design examples, where the initial point was selected far from the optimum, the algorithm
810
M. P. Saka
W
= 6569822
gms
(37)
(a) 353.3
W
353,3
= 6824947
gms
(32)
(b) 439.3
439.3 783.3
518,6 7
W
= 7406318
gms
W
461,4
= 7477988
gms
(30)
(ci) Fig. 8. Optimum designs for two bay pitch roof frame under different loadings.
might produce shapes where some members may have unproportionaly large depths to achieve the required stiffness for the frame. This undesired difficulty can easily be overcome by trying a different initial design point. The numerical examples considered have clearly shown that the design requirements given in the codes can simply be incorporated with optimization algorithms so as to provide efficient tools for the use of designers.
REFERENCES
Load and Resistance Factor Design, Manual of Steel Construction. ABC, U.S.A. (1986). M. R. Khan, Optimality criterion techniques applied to frames having general cross-sectional relationships. AIAA J. 22, 669-676 (1984).
M. P. Saka, Optimum design of steel frames with stability constraints. J. Compu~. Struct. 41, 1365-l 377 (1991).
Design
of steel frames
4. M. P. Saka, Optimum design of steel grillage systems. In: Proc. Third-M. Conf. on-Steel Srru&es~Singapore Steel Societv. Sineaoore. March. vv. 273-290 (1987). 5. M. P. Saks: Optimum geometry’design of rooi trusses by optimality criteria method. J. Comput. Struct. 38, 83-92 (1991). 6. M. P. Saka, Opiimum design of pin-jointed steel structures with practical applications. J. Strucr. Engng ASCE 116, 2599-2620 (1990). 7. M. P. Saka and M. S. Hayalioglu, Optimum design of geometrically nonlinear elastic-plastic steel frames. J. Comouf. Struct. 38. 329-344 (1991). 8. D. J. just, Plane frameworks of tapering box and I-section. Proc. ASCE 103ST1, 71-86 (1977). 9. 0. Hartavi, Elastic-plastic analysis of frames with tapering box and I-sections. MSc dissertation, Karadeniz Technical University, Trabzon, Turkey (1984).
with tapered
members
811
IO. D. L. Karabalis and D. E. Beskos, Static, dynamic and stability analysis of structures composed -of tapered beams. J. Comout. Struct. 18. 731-748 (1983). 1I. T. K. H. Tan and A. Jennings, Optimal‘plasfic design of frames with tapered members. In: Civil-COMP 87, Proc. Third Int. Conf. on Civil and Structural Engineering Computing, Vol. I, pp. 265-271, London (1987). 12. P. H. Gallagher and C. H. Lee, Matrix dynamic and instability analysis with non-uniform elements. In/. J. numer. A?eth. bngng 2, 265-275 (1970). 13. E. Atrek, R. H. Gallagher. K. M. Ragsdell and 0. C. Zienkiewic, New Dirktions in Opt&m Structural Design. Wiley, New York (1984). 14. R. J. Allwood and Y. S. Chung, An optimality criteria method applied to the design of continuous beams of varying depth with stress, deflection and size constraints. J. Comput. Struct. 20, 947-954 (1985).