Optimal plastic design of frames with tapered members

Compubrs & Strucrures Vol. 30. No. 3. pp. 537-54444,1988 C: 1988 Civil-Comp

Printed in Great Britain.

S3.00 + 0.00 0945-7949/88 Ltd and Pergamon Press plc

OPTIMAL PLASTIC DESIGN OF FRAMES WITH TAPERED MEMBERS THOMAS K. H. TAM and ALAN JENNINGS Department of Civil Engineering, Queen’s University of Belfast, Northern Ireland Abstract-The different formulations for minimum weight plastic design of structural frames are discussed with reference to the inclusion of tapered members in the optimisation process. An equilibrium method is presented in more detail showing how pin-joints can be taken into account and plastic moment constraints imposed. A technique for optimising over several loading cases and a procedure for plastic analysis are both specified. Some comparisons are also made of tableau sizes, computing times and weight savings for three types of frame geometry and loading.

INTRODUCl’ION

EQUILIBRIUM FORMULATION FOR MINIMUM WEIGHT DESIGN

members in frames gives designers scope for producing structural forms which are both more economical and more elegant. In cases where the main design criterion is according to a rigid-plastic theory, the choice of the most economical design according to a minimum weight principle can be obtained directly as a linear programming (LP) exercise and hence is easily computerised. The inclusion of tapered members as well as, or in place of, straight members does, however, require special consideration. This is the subject of the present paper. Many methods have been presented in the past three decades for plastic analysis and design of frames with uniform members, all of which can be classified into four categories according to the ways in which the constraint equations are constructed. These are the method of all conceivable mechanisms, the method of independent mechanisms, the joint equilibrium method (nodal method) and the method of redundant forces (mesh method). Some of these methods have been developed for hand computation whilst others can be expressed in linear programming format for solution on a computer. A comparison of the four approaches and their computational efficiencies is contained in [lf. Absolute minimum weight design methods using continuously varying cross-sections have also been developed [2-6]. For practical reasons, however, tapered rather than continuously varying cross-sections are no~ally more appropriate. Vickery [7] revealed that up to 25% weight reduction can be achieved by using tapered members in pinned base portal frames. He used a graphical analysis based on the method of all conceivable mechanisms which is only suitable for simple frames. The authors are not aware of any publications describing automatic methods for either plastic analysis or minimum weight design of frames with tapered members. The use of tapered

537

In equilibrium formulations of minimum weight plastic design the objective function to be minimised is proportional to the weight. If only uniform members are present in a frame, they may be divided into g groups in which members of the same group have been pre-specified as having the same crosssections (and hence plastic moments). Assuming that a linear relationship exists between the weight per unit length and the plastic moment of each section, the objective function can have the form w=

i

l&m,,

&=I where rnfi is the plastic moment for group k members and 1, is the sum of the lengths of all group k members. Consider now a tapered member of length L to be present and that the variation of plastic moment along it is linear. Such an assumption has been shown to be safe and reasonably accurate [7l. If the two ends of the member are allocated groups ti and v respectively, as shown in Fig. 1, the contribution of the member to the objective function is its length multiplied by its average plastic moment, giving wi = 4 (m, + m&.

(2)

It is also necessary to know the plastic moment of a cross-section at any distance u.E. from the first end mpv

L-J aL

I-

L

Fig. I. Grouping arrangement for a tapered member.

THOMAS K. H. TAMand ALANJENNINGS

538

(6)

subject to

(Fig. 1). This is (mP). = (1 - a )m,, + am,.

(3)

In constructing the problem constraints it is assumed that the material behaviour is rigid-plastic and the presence of axial loads does not have any significant effect on the design weight. It follows from the static theorem of plastic collapse that a design is safe provided that a bending moment distribution in equilibrium with the applied loads can be found such that the plastic moment is nowhere exceeded [8,9]. If the frame is subjected only to concentrated loads, this yield condition needs to be examined only at the various possible plastic hinge locations (called critical sections) comprising the ends of each member together with the full set of load application points. With c critical sections and d redundant forces, equilibrium gives

where r+, r-, mp 2 0 and I is a g x 1 vector of the group lengths. As an example consider the pitched roof portal frame shown in Fig. 2(a) in which all members are to be tapered with group 1 representing the plastic moments at the apex and bases of the columns and group 2 representing the plastic moments at the eaves. The weight function is

m =b + Br,

With the forces and moment acting at the right-hand support considered to be redundant as shown in Fig. 2(b), the following submatrices may be constructed:

(4)

w = 1644m,, + 16.44mp,.

where m is a c x 1 vector of bending moments at the critical sections, r is a d x 1 vector of redundant forces, b is the value of m where r = 0 and B is a c x d force transformation matrix. If mp is a g x 1 vector of group plastic moments, the yield condition applied at each of the critical sections gives Iml

GJm,,

(5)

BY LINEAR PROGRAMMING TECHNIQUES

Static LP formulation In order to cast this problem into standard linear programming format, the unrestricted variables r in eqn (4) need to be replaced by the sum of two non-negative sets of variables r + and r -. Also the modulus in eqn (5) needs to be replaced by a pair of inequality constraints. Equations (1), (4) and (5) can thus be written as minimise

0

20

1’

- 2000

6

20

1

- 1500

$

17f

1

-750

8:

12;

1

9

10

1

8:

7;

1

0

6:

2;

1

0

6

0

1

0

0

0

1

B=

- 500 -250

1

0

0

1

I

;

2 4

!. 4

4

J= SOLUTION

-2180

b=

where J is a c x g matrix containing information about the grouping arrangement. For each crosssection i of a uniform member belonging to group e, Jk = 1 and Jo = 0 (j #e). On the other hand, if cross-section i is located at distance aL along a tapered member of length L for which the two ends are allocated groups u and v, it follows from eqn (3) that Ji. = 1 -a, Ji, =a and Jo=0 (j # u,v). The minimum weight design requires minimisation of the weight function, eqn (l), subject to the equilibrium conditions, eqns (4), and the yield constraints (5). Any non-optimal solution of eqns (4) and constraints (5) gives a safe design.

(7)

1 0 1

1

I 4

1 4

0

1

4

4

,

1=

16.44 [ 16.44

1

(8)

1 0 Standard simplex methods may be used to solve these equations. The solution gives mp = {90,245.4} and w = 5514. This is a 7.8% saving in weight compared with the solution using uniform members and two groups which gives the plastic moment of both the columns and the rafters equal to 182 and w = 5978. The optimal bending moment diagram and plastic moment envelope for the frame with tapered members is shown in Fig. 3.

Optimal design of frames

539

30k N -2.53 1

1 6.32

(a) Foulkes mechanism virtual rotations

(a) Design

showing

specification (b) Alternative

collapse

mechanisms

Fig. 4. Mechanisms for example 1.

(b) Redundant forces and sign convention

Fig. 2. Example 1.

Kinematic LP formulation It is well known that the solution to an LP problem (known as the primal) may be obtained by solving a related LP problem called the dual [IO]. Treating the formulation (6) as the primal problem, the dual is maximise

z=[bT-bTT] [I

;I

)

subject to

equalities at the optimal solution (except in cases where associated “zp values are zero). This derives from the principle of complementary slackness and yields the Foulkes condition requiring that the numerical sum of the plastic hinge rotations in any group should equal the group length [ 11,121. Any non-optimal solution satisfying equations and conditions (9b) corresponds to an unsafe design according to the kinematic theorem of plastic collapse. The kinematic formulation requires tinding the linear combination of all mechanisms which maximises the external virtual work br6 [eqn (9a)]. The solution, 8, to the dual formulation giving the optimum Foulkes mechanism for the frame example, Fig. 2, is shown in Fig. 4(a). The rotations, 6, may alternatively be obtained from the simplex multipliers (also called shadow prices in linear programming terminology) obtained at the optimal solution to the primal formulation. For a frame having g groups there exists g alternative single degree of freedom mechanisms at the optimum. The two alternative collapse modes for the example are shown in Fig. 4(b). An algorithm for the automatic generation of these mechanisms is given in [l].

where 6 +, 6 - 2 0. The first constraint in eqn (9b) can be specified as

SPECIAL FACILITIES

Inclusion of bounds on plastic moments BTO = 0,

(IO)

where 6 = 8 + - 6 - is a c x 1 vector of nodal rotations. This equation represents compatibility conditions which ensure that there are no discontinuities at the position of the redundant forces [see Fig. 2(b)]. The constraints in eqn (9b) will be satisfied as plastic moment envelope

The inclusion of pre-specified maximum and minimum plastic moments may be used to ensure that optimisation yields member sizes within practical limits. They may also be used to ensure that neither excessive deflections nor local or torsional instability occurs where such conditions have been related directly to the size of individual members. The simplest way of imposing such limits is to add constraints of the types

/

and

formulation (6), where rn$ and rni represent the upper and lower limits respectively for

to the primal Fig. 3. Optimal bending moment diagram for example 1.

THOMAS K. H. TAMand ALANJENNINGS

540

the plastic moment of group k. If only lower limits are imposed, the need for extra constraints may be avoided by substituting the relationship (11) into the formulation for each group, where the condition fipk 2 0 replaces the condition rnpk> 0. When upper bounds are also required the above technique will not suffice. In this case the bounded variable algorithm [lo, 111 may be adopted. This is capable of handling variables of the type XFfXi
Fig. 5. Collapse mechanism for example 1 with pinned bases (showing relative hinge rotations).

function, say of the order of 10” greater than other coefficients. The effect of performing the optimisation will then be. to ensure, if at all possible, that these moments will be identically zero. For the above example m, and m, need to ‘.e allocated to a new group, say group 3. The formulation shown in eqn (8) needs to be amended such that

J=

with pinned connections

For frames of arbitrary geometry the construction of equilibrium equations of the form shown in eqn (4) requires a knowledge of the loop topology of the structure, This process can be readily automated if the structure is rigidly jointed throughout [l, 131. When the structure has some pinned connections it is possible to construct the equations automatically ignoring the presence of the pins and then constrain the bending moments acting across the pins to be zero by either of the following two methods. In the first technique the condition that the bending moment is zero at a pin is used to eliminate a redundant force from the equilibrium equations. Thus, for the example of Fig. 2 in which the column bases are assumed to be pinned, two redundant forces (in this case rz and r3) can be eliminated using the moment equations

The solution

0

0

1’

0

1 0

I4

2 4

0

1 4

1 4

0

1 0

0

41

14

0

I4

5

0

0

1 0

0

0

16.44 ,

I=

16.44 10’0

1.

(13)

1

to the revised primal problem yields

m,, = {0,393&O} and w = 6466. This gives a 23.8%

saving in weight compared with the solution using uniform members. This penalty function method is simpler to implement but requires more computation. When the optimum solution is reached the pinconnected nodes appear as plastic hinges for which the plastic moment is zero. The rotations of these hinges in collapse mechanisms can be obtained along with the other plastic hinge rotations and so the collapse mechanisms can be predicted. The collapse mode of the pin-based portal frame example is shown in Fig. 5.

(m, =) - 2180 + Or, + 20r, + r3 = 0 (m,=)O+Or,+Or,+r,=O.

(12)

The nodal equations for m, and m, are then excluded from the minimum weight formulation. Although the resulting tableau is of smaller size, it is necessary to implement the elimination of redundant forces as a separate operation. Also there is no simple method of predicting the rotations of pin-joints in collapse mechanisms once the optima1 solution has been obtained. A simpler procedure is to include all the nodes where pins are present in a separate group and to allocate it an extremely large coefficient in the weight

SOLUTION BY THE MODIFIED SIMPLEX METHOD

The LP tableau can be compacted and the optimisation process made more efficient by using the modified simplex method in which the duplication of variables and constraints required in the primal formulation is avoided [l3-151. This procedure, originally designed for frames with uniform members, has been extended to incorporate tapered members [l]. The duplication of constraints is avoided by retaining equations for bending moments throughout the optimisation process. Thus row i of the tableau is always used to represent the two constraints m, 2 -mPi and

Optimal design of frames

taa) Design speclflcatlwr

( kN and

m units

moment in the group, a feasible solution is immediately obtained. Duplication of columns in the tableau is avoided by retaining the redundant forces as unconstrained rather than constrained variables until such time as they are replaced by slack variables. In other ways the optimisation process is similar to the classical simplex technique. The three examples shown in Figs 2(a), 6(a) and 7(a) have been designed on a VAX8650 mainframe computer using the primal formulation (6), the dual formulation (9) and the modified simplex algorithm. In the first two of these methods the revised simplex algorithm with explicit formation of the inverse has been used [lo, 111.The tableau storage requirements, numbers of iterations and computing times required by each method are shown in Table 1, In general, for a frame with c critical sections, d redundants and g groups, the tableau size for the modified simplex method is (c +g) x (d +g). It can be seen from Table 1 that, using the primal approach, the modified simplex method was approximately 50 times more efhcient than the standard LP formulation and it requires only one-third the storage space for the tableau. The modified simplex method was found to be better also than the dual LP formulation although not by such a large margin. These results suggest that if a dual of the compact technique were to be developed, the efficiency would probably be further improved. The minimum weight solutions for the three examples are shown in Figs 3, 6(b) and 7(b). Savings in weight resulting from the use of tapered members are given in Table 2.

1

9

6

(b)

Minimum weight solution moments in kNm )

(plastic

Fig. 6. Example 2.

mi < mpi where mpi is the plastic moment at node i. The trial solution r = 0 is used to initiate the optimisation process and, by choosing the group plastic moments each to be equal to the maximum bending

I

20 la)Otsign

(b)

20

I_ specification

Minimum weight

s&don

541

(kN

and m units)

(plastic

moments in

Fig. 7. Example 3.

kNm

units)

THOMASK. H. TAMand ALANJENNINGS

542

O&‘TIMBA~ON WITH MULTIPLE LOADING CASES

The storage and computing time requirements for all the methods increase very rapidly as the number of loading cases increases. The various tableau sizes are shown in Table 3.

When a minimum weight design is required for a structure subjected to more than one loading case, a set of redundant forces needs to be defined for each loading case. If there are t loading cases, the t sets of ~uilib~um equations will have the form

MAD

When the plastic moments of the members of a frame are known, a plastic analysis can be carried out to determine the least factor I by which any given set of loads needs to be multiplied before no solution is possible which satisfies the yield condition. This will be the collapse load factor. If fi,, is a c x 1 vector of the plastic moments at the critical sections, the static model can be written as

(m), = (b), + B(r), (m)r = (b)* + B(r)2 . *. (ml, = @I, + Wr),.

FACTOR ANALYSIS

(14)

Applying the yield condition (S), the primal LP formulation for multiple loading cases becomes

maximise 1 subject to

(17a) 12b + Br I< RD.

minimise w = I’m,. subject to -B 3

@),1

(rX

B

(r);

-B -3

(r)t

3 B

6-F

-B .. -E

L

Cri:

B -B

E

@I;

J

MP Similarly the dual fo~ulation can be developed to cover multiple loading cases as follows: maximise (@X (Q; (@),+ z = [(b); - (b):(b);

- (b)T.. . @I: - @ITI

W;

.

(Sji

_w; subject to

‘-BT

BT -BT

BT .. -B’

Jr

JT

JT

Jr..

.

J’

B’ JT

>

f17W

Optimal design of frames

543

Table 1. A comparison of computational Quantity measured

Frame 1

LP format Primal Dual

Tableau size No. of iterations CPU time (set)

18 x 18 23 0.08

5x 18 8 0.01

Tableau size

78 x 23

14 x 78

2

CPU (set) No. oftime iterations

4.12 81

3

Tableau size No. of iterations CPU time (set)

124 x 29 126 18.68

Table 2. Reductions in weight parameter (figures in brackets are for pin-based frame) Weight parameter Tanered Uniform members members

No. of groups

Frame 1

2

2 3

5 4

In linear programming yield b

-b

B

format

B

(2:::) 18.6 11.1

the constraints

1

-B

-B

Saving %

5514 (6466) 31,145 43,621

5978 (8480) 38,258 49,075

dciencies

r+

l[lrl r-

17(b)

0.20 31 17 x 124 28 0.31

Modified simplex format 11 x5 4 0.00 44x 14 0:067 67x 17 32 0.29

may be obtained by solving either the primal or dual formulation. As an example consider the basic frame of Fig. 2 with the members having plastic moments as specific in Fig. 8(a). Treating the haunches as tapered members, an analysis using the primal formulation (18) yields I, = 1.0 with the optimal bending moment diagram and plastic hinge locations as shown in Fig. 8(b). The effective weight of the haunched frame of 5200 compares favourably with 5514 obtained for the similar frame having tapered members but no haunches specified in Table 2. This comparison is valid because the load factors are the same. Therefore

<

mP -

,

(18)

“lP

in which 1, r +, r - 3 0. In the corresponding dual algorithm the internal virtual work is minim&d according to minimise z =

subject

[El,

$1

ef e[

1

(194

(a) frame

specification

to

where 8 +, 0 - 2 0. The first constraint in eqn (19b) is the normalising condition b% = 1, for the external

virtual work. The other constraints comprise the compatibility conditions (10). In common with design problems, the solution to any analysis problem

(b) Bending moment diagram

Fig. 8. Analysis of example 1 with haunches.

Table 3. Tableaux sizes (c = No. of critical sections, d = No. of redundant forces, I = No. of loading cases and g = No. of groups)

Dimension No. of rows ( = No. of constraints) No. of columns ( = No. of variables)

CA..s 30/3--H

LP format Primal Dual

Modified simplex format

2ct

dt +g

et +g

2dt +g

2cr

dt +R

544

THOMAS K. H. TAM and ALAN JENNINGS

the use of haunches can decrease the weight of frames with uniform or tapered members. The relative merits of using tapered or haunched rather than uniform members will depend not only on the weight reductions obtained but also on the fabrication costs. Tests by Vickery [7,16] on pin-based pitched roof portal frames have shown that lateral instability is particularly important where tapered members are used. Although this may be overcome by providing adequate lateral bracing or by the use of closed sections, research in this area would seem to be justified. CONCLUSIONS

The use of tapered members in structures results in significant material savings when designing by rigid-plastic theory. For both design and analysis, solutions can be obtained using standard linear programming techniques. However the use of a nonstandard solution algorithm leads to faster response times which is of importance in an interactive design environment. The inclusion of constraints to simulate the presence of pin-joints and to limit the choice of group plastic moments need not add much to the complexity of the solution process. However, when optimising over multiple loading cases both the tableau size and the computing time are considerably increased. Aeknowledgemenrs-The authors would SERC for support of this project.

like to thank

REFERENCES

1. T. K. H. Tam, Computer methods for optimal plastic design of frames. Ph.D. dissertation, The Queen’s University of Belfast, Northern Ireland (1987).

2. M. R. Home, Determination of the shape of tixedended beams for maximum economy according to the plastic theory. Final report. Int. Assoc. Bridge & Struct. Engng, 4th Cong., Cambridge & London (1953). 3. J. Heyman, On the absolute minimum weight design of framed structures. Quart. J. Mech. appl. Math 12, 314-324 (1959). 4. J. Heyman, Gn the minimum-weight design of a simple portal frame. In?. J. Mech. Sci. 1, 121-134 (1960). -

5. W. Prager and R. T. Shield, A general theory of optimal plastic design. Trans. Am. Sot. Mech. Engrs, J. appl. Me& 34, 184-186 (1967). 6. R. Mayeda and W. Prager, Minimum weight design of beams for multiple loading, Inf. J. Solids Srrucr. 3, 1001-1011 (1967). 7. B. J. Vickery, The behaviour at collapse of simple steel frame with tapered members. Strucr. Eng. 40, 365-376 (1962). 8. H. J. Greenberg and W. Prager, Limit design of beams and frames. Proc. Am. Sot. Civ. Engrs 77, l-12 (1951). 9. M. R. Home, Fundamental propositions in the plastic theory of structures. J. Insr. Civ. Engrs. 34, 174-177. 10. G. B. Dantxig, Linear Programming rmd Extensions. Princeton University Press, Princeton, NJ (1963). 11. K. G. Murty, Linear and Combinatorial Programming. Wiley, NY (1976). 12. J. Foulkes, The minimum weight design of structural frames. Proc. Roy. Sot. A223, 482-494 (1954). 13. A. Jennings and T. K. H. Tam, Automatic plastic design of frames. Engng Srrucr. 8, 138-147 (1986). 14. A. Jennings, Adapting the simplex method to plastic design. Proc. Instability and Plasric Collapse of Sreel Srrucrures, September 1983 (Edited by L. J. Morris), pp. 164-173. University of Manchester, Granada, St Albans (1983). 15. A. Jennings and T. K. H. Tam, The minimum weight concept in interactive plastic design, CIVIL-COMP 85, December 1985 (Edited by B. H. V. Topping), Vol. 1, pp. 363-368. Civil-Comp. Press, Edinburgh (1985). 16. L. J. Morris and A. L. Randall, Plastic Design p. 41. Constrado. Crovdon f 1975).