Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 109 OPTIMUM DESIGN OF STEEL FRAMES COMPOSED OF TAPERED MEMBERS USING STRENGTH AND DISPLACEMENT CONSTRAINTS WITH GEOMETRICALLY NONLINEAR ELASTIC ANALYSIS Haitham Ali Bady University of Qadisiya \College of Engineering Abstract Design of steel tapered member under combined axial and flexural strength is somewhat complex if no approximations are made. However, recent Load Resistance Factor Design (LRFD) of the AISC code has treated the problem with sufficient accuracy and ease. The aim of this study is to present an algorithm for the optimum design of steel frames composed of tapered beams and columns with I-section in which the width is taken as constant, together with the thickness of web and flange, while the depth is considered to be varying linearly between joints .The objective function which is taken as the weight of the steel frame is expressed in terms of the depth at each joint. Both the displacement and combined axial and flexural strength constraints are considered in the formulation of the design problem .The strength constraints are expressed as a nonlinear function of the depth variables. The optimality criteria method is then used to obtain a recursive relationship for the depth variable under the displacement and strength constraints. Numerical examples are presented to demonstrate the practical application of the algorithm. Keywords: Design, tapered, steel, axial, flexural, strength, constraints , optimum, nonlinear, stability و اإلزاحة غير موشورية و باستخدام محددي أعضاء للمنشات الحديدية المرآبة من األمثلالتصميم المرن للمنشأو السلوك غير الخطي تأثيرالمقاومة تحت باديهيثم علي آليَّة هندسة\ جامعة القادسية \ تدريسي في الخالصة المعقدة األمور االنثناء من وأحمال لمحورية المركبة ااألحمال أثيرتالموشوري تحت غير اإلنشائييعتبر تصميم العضو التابعة LRFD األمريكيةان المدونة , صياغة المعادالت التصميمية التقريبية في الحلول لم تستخدم بعض إذا الشيءبعض ان الهدف من هذا ابحث هو . بهاألبأسة للمنشات الحديدية تعاملت مع هذه المسالة بطريقة بسيطة نسبيا وبدقاألمريكيهد عللم ومكونة ) جسور , أعمدة( الموشورية أعضاء للهياكل الحديدية المؤلفة من األمثلتقديم طريقة ومن ثم مخطط انسيابي للتصميم ون متغيرا كبحيث يكون عرض المقطع مع سمك الشفة و الوتر ثابتين بينما عمق المقطع ي) I (حرف من مقاطع على شكل ولذلك تم التعبير عن هذه الدالة بداللة سمك للمنشأان الدالة المطلوبة في هذه الدراسة تمثل الوزن الكلي . لى طول ذلكع و المقاومة في كتابة المعادالت اإلزاحة بنضر االعتبار محددي األخذ تم . هالمقطع العرضي للعضو في كل نهاية من نهايتي ومن ثم وباستخدام محدد , أيضاحدد المقاومة بصيغة غير خطية وبداللة سمك المقطع تم التعبير عن دالة م . ةالتصميمي Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 110 و اإلزاحةتحت محددي ) اإلنشائيسمك المقطع ( عالقات تكرارية متتالية للمتغير التصميمي ى عدةة تم الحصول علياالمثل . ة التقنية المتبعة والبرنامج المستخدم فعاليح لتوضياألمثلةعلى مجموعة من ح المقتراألسلوبتم تطبيق . المقاومة Nomenclature li is the length of the tapered member i. tf, tw, thickness of flange and web of the I-section of the tapered member respectively. bf is the width of the flange ρ and iν are the density and volume of typical tapered member i shown in Fig.(1) , respectively . nm is the total number of tapered members in the frame . Di is the depth variable belonging to member i, D1i is the lower bound of the depth variable . gdj (D1i , D2i) represents jth displacement constrains . gsri (D1i , D2i) represents strength constrains for member i. k Is the total number of restricted displacement. jδ is the displacement at node where constraints is wanted . juδ is its upper bound. Fy is specified yield stress Ag is the gross area of· the member at the smaller end. effλ is called the effective slenderness parameter. Sx, the sectional modulus of the larger end Fb is the design flexural stress of tapered member, Introduction Steel frames with tapered members were preferred in the design of structure whenever the architectural requirements allow their presence .They provide better distribution of strength as well as yield lighter design. The methods available for the analysis of such frames are well established (Haitham, 2000), (Oran, 1974) .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 . In this study, an optimum design algorithm is presented which takes into account the geometrical nonlinearity for steel frames with tapered members. This is achieved by coupling optimality criteria approach with large deformation analysis method of elastic tapered steel frame develops in Ref (2). Optimum Design Problem The optimum design problem a nonlinear steel frames composed of tapered members subjected to displacement and strength constraints can be expressed as follows :- Min W= ∑ = nm i ii v 1 ρ i=1,…,nm (1) Subjected to gdj (D1i, , D2i) ≤ 0 j=1,…, k gsri (D1i, , D2j) ≤ 1 i=1,…,nm D1i-D1il ≥ 0 i=1… nm (2) D2i-D1il ≥ 0 i=1… nm Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 111 Objective Displacement The objective function, which is the total volume of frame, is obtained as a summation of weights of all members. The volume vi of member i as shown in Fig. (1) can be expressed in terms of the values of the depth variables (D1) and (D2) , as follow:- Vi = ( ) ( ) iwffwii ltbtt DD ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ + 2 2 21 (3) where D1i, D2i is the depth variables of the smallest and largest end respectively of tapered member i. of the I-section of the tapered member. It can be seen that bf, tf and tw are selected to be constant throughout the frame, which leaves only the depths at nodes (1) and (2) as the design variables. The elastic sectional modulus for symmetrical sections are calculated easily when the values of D1i and D2i are known. Combined Axial and Flexural Strength Constraints The combined axial and flexural strength constraint for member i , which is subjected to axial force and bending moment about its major axis, is given in LRFD(4) as, for Pu/(φ Pn) ≥ 0.2 (4) 1 9 8 ≤+ nxb uxu M M nP P φφ (5) and for Pu/(φ Pn) < 0.2, (6) 1 2 ≤+ nxb uxu M M nP P φφ , (7) where Pu is the required axial strength and Pn is the nominal tensile or compressive strength for the member depending upon whether it is in tension or compression. Mux is the required flexural strength and Mnx is the nominal flexural strength about the major axis of the section. The resistance factor (φ ) 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 bφ is specified as 0.90 by the same code(Hayalioglu, Saka,1992). Since only the nominal strengths are the functions of the depth variables, the strength constraint for member i can be re-written as:- gsr(Di,Dj)=a1/Pn+ a2/Mnx, (8) Where a1 and a 2 are the constants given as, for Pu/(φ Pn) ≥ 0.2 a1=Pu/0.85 a2=8Mu/8.1 (9) Pu/(φ Pn) < 0.2, a1=Pu/1.7 a2=Mu/0.9 (10) Displacement Constraint The jth displacement constraints gdj (D1i, , D2i) has the following form:- gdj (D1i, , D2i) = juj δδ − (11) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 112 The displacement jδ can be expressed as a function of the depth variable by making use of the virtual work theorem ( ) iji nm i T ij XiDiDKX 21 1 ,∑ = =δ (12) Where Xi is the vector of virtual displacements of member i due to the virtual loading corresponding to the jth constrains. This is obtained by applying the unit load in the direction of the restricted displacement j. K (D1i, D2i) is the stiffness matrix of member i in the global coordinate. Xi is the displacement vector e due to applied load. Nominal Axial And Flexural Strength of Tapered Member It is shown from Eqs. (8) to (10) that the combined strength constraint for a tapered member makes it necessary to express the design axial and flexural strength of the member in terms of depth variables defined at its ends. Nominal tensile strength In the case where the tapered member is in tension, LRFD gives the nominal tensile strength Pn as , gyn AFP ×= (13) Hence, the nominal tensile strength Pn can be expressed as a function of depth variable D1, of the smaller end as Pn=Fy(D1tw+2T) (14) where T is a constant given by T = (tf bf – tw tf) (15) Nominal Compressive Strength When the tapered member is in compression, its nominal compressive strength is given by LRFD as; gcrn AFP ×= (16) where Fcr is the critical stress computed from one of the following expressions: 5.1≤effλ Fcr= )658.0.( 2 eff yF λ (17) 5.1>effλ Fcr= 2 ).877.0( eff yF λ , (18) )/( 2 EQFS yeff πλ = Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 113 (19) in which S is equal to Kl/roy for weak axis bending and Kl/rox for strong axis 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/roy. The approximate radius of gyration roy is defined at the smaller end of the tapered member as, 2/1 1 3 )2(6 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + = TtD bt r w ff oy (20) Substituting Eq. (20) into Eq. (19) and taking Q = 1 gives the effective slenderness as, ( )[ ] 2/111 2TtDc weff +=λ (21) where c1 is a constant. ( ) ff bEt KlFy c 32 2 1 6 π = (22) Hence, the nominal compressive strength Pn of eq. (16) can be expressed in terms of depth variable Di, at the smaller end as for :- Nominal Flexural Strength The nominal flexural strength of tapered flexural member for the limit state of lateral torsional buckling is given in LRFD(1) as Mn = (5/3) SxFb (25) ( ) yyws y b FF FF F F 6.0 12 13/2 22 ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + −= (26) Unless Fb ≤ Fy/3, in which case, ( )wsb FFF 225.1 += (27) In Eq (26) and (27) 1 12000 lDh A F s f s = , 2170000 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = TO w w rlh F (28) Where factors hs and hw are given as, x 5.1≤effλ ( ) ( )TtDcwn wFyTtDP 21 11658.02 −××+= (23) 5.1>effλ 1/)877.0( cFyPn ×= (24) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 114 hs = 1+ 0.023 f i A lDγ (29) hw = 1 + 0.00385 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ TOr lγ (30) in which rTo is radius of gyration of the section at the smaller end, considering only the compression flange plus 1/3 of the compression web area, taken about an axis in the plane of the web, Af is the area of the compression flange, γ is given as 1 12 D DD − =γ 268.0≤ , (l/D1) or 6. (31) The relationships listed in eqns. (25)-(31) can be expressed in terms of depth variables at the ends of the tapered member as shown in the following. The sectional modulus Sx ( ) 1 2 2 3 1 6 )(6 D tDtbtDt S fff fw x ++− = (32) Fs and Fw of Eq. (28) are written as 1 12000 lDh tb F s ff s = , 1 2 4 2 3 2 Dhchc cF ww w + = (33) where constants c2, c3 and c4 are c2 = 425,000tfb3f , c3 = 3l2 tf bf , c4=l2tw (34) In which γ is function of Di as shown in Eq. (31), and rTo is ( )1 3 34 Dtbt bt r wff ff TO +× = (35) It is clear from Eqs. (32) to (35) that nominal flexural strength Mn can be expressed in terms of depth variables D1, and D2 . Figures (2) to (5) shows the relationships between the Nominal axial force and flexural moment strength and their derivatives with the design variables D1 & D2 .Figures (6) and (7) shows the relationship between the strength constrain with the design variables D1 & D2. From these figures we can conclude that :- 1-The nominal axial tension force strength for the cross section is greater that the nominal axial compression force strength in specified depth variables D1 & D2. This is due to the material properties for the steel which is included empirically in the equation pg the nominal Strength. 2-The flexural moment strength for the section decreases with increasing the depth variable D1 but it is increases with increasing the depth variable D2. This is because that the flexural moment strength equation depends basically on the depth variable D2. 3-The strength constrain function is slightly affected by the design variable D1 but it is strongly affected by the design variable D2. Optimality Criteria for Depth variables Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 115 It is shown previously that displacement and strength constraints in addition to the objective function of the optimum design problem considered is highly nonlinear function of design variables. The optimality criteria approach was found to be an effective method in finding the solution of such design problem (1, 5and 8).This technique transformation the constrained problem into an unconstrained one by using Lagrange multipliers. The Lagrangian of design problem is:- ),(.),(),,,( 21 1 21 11 21 iisti nm i sriiidj j dji nm i isridjii DDgDDgvDDL ∑∑∑ === ++= λλρλλ ρ (36) where djλ and sriλ are the Lagrange multipliers for the displacement and strength constraints respectively . The necessary condition for the local constraint optimum is obtained by differentiating this equation with respect to design variables (D1 D2 as the follows:- 0 ),( . ),(),,,( 1 21 11 21 1111 21 = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∑∑∑ === i iisti nm i sri i iidj j dj i nm i i i sridjii D DDg D DDg D vi D DDL λλρ λλ ρ (37) 0 ),( . ),(),,,( 2 21 12 21 1212 = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∑∑∑ === i iisti nm i sri i iidj j dj i i nm i i i sridjji D DDg D DDg D v D DDL λλρ λλ ρ (38) The derivative of the volume of tapered member with respect to depth variables can analytically obtained as follows :- iw ii wffi lt DD tbtv ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + +−= ) 2 )( )(2 21 (39) iw i i i i lt D v D v . 2 1 21 = ∂ ∂ = ∂ ∂ (40) The derivative of the displacement constraint from Eq. (11) becomes:- i j i iidj DD DDg 11 21 ),( ∂ ∂ = ∂ ∂ δ (41) i j i iidj DD DDg 22 21 ),( ∂ ∂ = ∂ ∂ δ (42) Which in turn from Eq. (12) becomes:- ij i iii nm i T i i j X D DDK X D 1 21 11 ),( ∂ ∂ = ∂ ∂ ∑ = δ (43) ij i iii nm i T i i j X D DDK X D 2 21 12 ),( ∂ ∂ = ∂ ∂ ∑ = δ (44) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 116 The derivatives of stiffness matrix of the tapered member can be achieved analytically in Ref.(3 ). On the other hand the , the same can also be achieved for the derivative of the strength constraints with respect to design variables(D1 , D2) as follows:- n i n n i n i iisri M D Ma P D Pa D DDg 2 1 1 2 1 1 1 21 )()(),( ∂ ∂ − ∂ ∂ −= ∂ ∂ (45) n i n n i n i iisri M D Ma P D Pa D DDg 2 2 2 2 2 1 2 21 )()(),( ∂ ∂ − ∂ ∂ −= ∂ ∂ (46) The derivatives )( J n D P ∂ ∂ and )( J n D M ∂ ∂ can be achieved using numerical technique (finite difference technique) as follows;- 1121 12 11 ii nn i n i n DD PP D P D P − − = ∆ ∆ = ∂ ∂ , 1121 12 11 ii nn i n i n DD MM D M D M − − = ∆ ∆ = ∂ ∂ (47) 1222 12 22 ii nn i n i n DD PP D P D P − − = ∆ ∆ = ∂ ∂ , 1222 12 12 ii nn i n i n DD MM D M D M − − = ∆ ∆ = ∂ ∂ (48) Hence the optimality criteria for depth variables are obtained from Equation (37) and (38) as follows:- 0) )()( ( ),(),,,( 2 1 1 2 1 1 11 21 11111 21 = ∂ ∂ − ∂ ∂ −×+ ∂ ∂ ×+ ∂ ∂ = ∂ ∂ ∑∑∑∑ ==== n i n n i n nm i sriji i iii nm i T i j dj ii i nm i i i sridj M D Ma P D Pa X D DDK X D v D DDL λλρ λλ ρ (49) which lead to:- ∑ ∑∑∑ = === ∂ ∂ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − ∂ ∂ −×+ ∂ ∂ × − nm i i i i n i n n i n nm i sriji i iii nm i T i j dj D v M D Ma P D Pa X D DDK X 1 1 2 1 1 2 1 1 11 21 11 ) )()( ( ),( ρ λλ ρ And 0) )()( ( ),(),,,( 2 2 2 2 2 1 12 22 11212 21 = ∂ ∂ − ∂ ∂ −×+ ∂ ∂ ×+ ∂ ∂ = ∂ ∂ ∑∑∑∑ ==== n i n n i n nm i sriij i iii nm i T i j dj i i nm i i i sridj M D Ma P D Pa X D DDK X D v D DDL λλρ λλ ρ (51) which lead to:- 1 ) )()( ( ),( 1 2 2 2 2 2 2 1 12 21 11 = ∂ ∂ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − ∂ ∂ −×+ ∂ ∂ × − ∑ ∑∑∑ = === nm i i i i n i n n i n nm i sriij i iii nm i T i j dj D v M D Ma P D Pa X D DDK X ρ λλ ρ =1 (50) (52) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 117 Multiplying both sides of Equations (52) and (54) by Dc1i and Dc 2 i , respectively, and then taking the cth root yields:- ×=+ i tt i DD 1 1 1 c t nm i i i i n i n n i n nm i sriji i iii nm i T i j dj D v M D Ma P D Pa X D DDK X 1 1 1 2 1 1 2 1 1 11 11 11 ) )()( ( ),( ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − ∂ ∂ −×+ ∂ ∂ × − ∑ ∑∑∑ = === ρ λλ ρ ×=+ t i t i DD 2 1 2 c t nm i i i i n i n n i n nm i sriij i iii nm i T i j dj D v M D Ma P D Pa X D DDK X 1 1 2 2 2 2 2 2 1 12 21 11 ) )()( ( ),( ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − ∂ ∂ −×+ ∂ ∂ × − ∑ ∑∑∑ = === ρ λλ ρ where t and t+1 represent the current and the following optimum design cycles, and the c is known as the step size process. It is apparent that the use of Equation. (53) and (54) require that values of Lagrange multipliers to be known .There are several methods to obtain their values .One simple and effective way used in Ref.(1).This method takes the constraint equality and multiplies both sides by m djλ and then takes the mth root .This leads to the following recursive relationship :- m ju jr j r dj 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅= + δ δ λλ j=1, 2,eq. ρ (55) where m is the step size and its value form the numerical examples is between 0.8 and 0.7 for m 1 .It is clear that Eqs. (54) and (55) require the initial values of the Lagrange parameters to be selected .It was found suitable to use (10000) as an initial value for these parameters (multipliers). Figures (8) and (9) shows the relationships between the derivatives of the strength constraints with the design variables D1 & D2 . From these figures we can conclude that :- 1-for the value of 2.0< n u P P , the derivative of strength constraints for the cross section is slightly greater in compression that in the tension force with respect to design variables D1 & D2. 2- for the value of 2.0≥ n u P P the derivative of strength constraints for the cross section is slightly greater in tension that in the compression force with respect to design variables D1 . (53) (54) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 118 Nonlinear Elastic Analysis of Steel Frames Composed of Tapered Members. The nonlinear elastic analysis of frames composed from tapered members is obtained by the method reported in Ref.(3) .This method improved from the nonlinear elastic analysis of frames composed of prismatic members described in Ref.(7) which takes into account both the geometrical and material nonlinearities . Design Convergence Criteria:- Two types of design criteria are use in this study to insurance the satisfaction of the convergence in design, these are :- 1-weigh criteria: This criterion depends on comparison of the weight of the frame for the current design cycle and the weight of the frame for the previous design cycle, and convergence is assumed to have occurred when the inequality :- 12 2 1 tolW WW r rr ≤−+ )( is satisfied (56) Where: - Wr+1: represents the total weight of the structure in the current design cycle Wr : represents the total weight of the structure in previous design cycle 2-depth criteria :- This criterion depends on comparison of the depth at both ends of each design group of the frame for the current design cycle and for the previous design cycle, and convergence is assumed to have occurred when the inequality when the inequality 21 )( tolDDD ttt ≤−+ (57) In Eq. (56 ) and (57 ), the dimensionless quantity, tol. represents a prescribed tolerance each criteria In this study the tolerance used as indication for satisfied the convergence is as follows tol1= 0.005. tol2= .01 Flow chart and computer program:- The algorithm developed for optimum design of geometrical nonlinear elastic –frame composed of tapered members can be described by the following chart of the program with a brief description for each subroutine, Fig.(10) and a computer program (EDTS) is developed using QBasic language. . Design Examples Two examples are used her to demonstrate the capability of the algorithm developed in this study to achieve the optimum design of tapered steel frame under elastic nonlinear behavior , the values of modulus of elasticity and yield strength of the steel used to fabricate the structure were taken as 205 kN/mm2 and 275N/mm2 respectively .The density of the steel was 7850kg/m3.Th convergence criteria used for the minimum objective function was 0.1% while it was 1% for the depth variables 1-Fixed Ends Tapered Beam. In this example a single span beam was designed using the algorithm developed in this study , the dimensions of the beam ,member cross section and loading condition is shows in Fig. (11) the beam was divided into two linearly tapered beams which introduced two design variables in each beam (1,2)in beam No.1 and (2,3)in beam No.2 , due to symmetry the depths and nodes 1 & 3 was assumed to be the same . This would eliminate the design variables into two variables (D1& D2), the frames was designed under three cases of constraints 1-Displacement constraints 2-Strength constraints. 3-Both displacements & strength constraints. The results of both studies is shown in Figs.(12 to14) It shows from these figures that the reduction in depth variable D2 is more faster than in depth variable D1 which mean that the value of depth variable D2 more effective in the optimum design processes from the value of D1 this may be caused by the including of the geometrical nonlinearity in the analysis and taking in the account the large deformation, bowing effect and stability behavior of the structure which lead to increasing the Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 119 effect of design components relating to depth variable D2 on the other hand the depth variable D2 usually used in maximum flexural moment zone which is usually near the support so that the deformations and displacement is being at their maximum value and then cause that increase in optimum design components. From figures (12-14) we can note that when excluding the displacement constraints from the optimum design processes the decreasing in depth variables D1 & D2 become more that when using both displacements and strength constraints and which lead to lighter structure and subsequently more economic and more save in cost without increase in the constrained displacement on its upper bound. This mean that including the strength constraints in the optimum design processes will improve the design efficiently. On the other hand the optimum design reached after design cycle No.8 we using strength constraints only in the optimum design comparing with design cycle No.9 when including both displacement and strength constraints. 2-Pitched roof tapered steel frame. In the example a one bay pitch roof frame is designed using the optimum design algorithm developed her, the frame is divided to 15 node at the point of application the external loads and 14 tapered member , the dimension of the frame ,member cross section and loading condition is shows in Fig.(15 ). This frame was designed by Ref. (5 ) using linear elastic analysis , in this study the frame is designed three constraints cases :- 1-Displacement constraints 2-Strength constraints. 3-Both displacements & strength constraints. The results of our study is shown in Figures (16 to 19) , from these figures we reach to the same view obtained from the previous example in addition to noting that in this example we have two deign groups the rafters (Beams ) and the columns , each group treated separately in design processes but at the joins the developed program takes into account the effect of changing in each depth variable on the connected members which help in giving more reliable design .T he results of design for each group are shown separately in figures ( 16 ) and ( 17 ) we can note the similarity in behavior for each group . the effect of nonlinear analysis is shown obviously in Fig.( 19 ) hence from this Figure we can observe that the displacement reached to its upper limit faster than the former example. Conclusions:- Depending on the design results obtained from the present study, one can draw several conclusions, concerning the optimum design of the tapered steel frames with I –section these may be summarized as follows: - 1-The optimum design components represented in this study by the strength and displacement constraints equations is affected by the design variable D2, specially when including the geometrical nonlinearity and stability behaviors in the analysis of steel frame. This is required to choose the value of depth variable D2 carefully in design of such frames. 2- The excluding of displacement constraints in the optimum design processes( using strength constraints only )lead to faster design and more economic and saving in cost design . 3-In frames composed from different structural members, the behavior of the optimum design results will be slightly different. 4- This study may be improved by including more restrains in to design processes that make the design more economic and more saving in time and cost such as (buckling constraint, plasticity constraint, creep constraint) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 120 tf tw bf D2 D1 L tw bf tf References Fig. (1): Typical Tapered Member With Linear Variation 1- Hayalioglu, M. S. and Saka, M. P.,” Optimum Design of Geometrically Nonlinear Elastic- Plastic Steel Frames With Tapered Members”, Inter. Jour. of Compu. And Struc. Vol.44, No.4,1992, pp.915-924. 2- Haitham A. Bady, " Nonlinear Elastic-Plastic Instability Analysis of Frames with Nonprismatic Members", MSc. Thesis, University of Technology/building and Construction Department, 2000. 3-Haitham A. Bady," Optimum Design of Nonlinear Elastic-Plastic Steel Frames Composed From Non-prismatic Members with Displacement Constraints"", Jour. of Babylon University, Vol. 13, No.5, 2005. 4-Load and Resisting Factored Design ,Specification for Structural Steel Build, AISC, U.S.A, 2000. 5- M.P.Saka," Optimum Design Of Steel Frame with Tapered Members" J. Computer &Structures, Vol.63, No.4, PP.797-811, 1997. 6-M.P.Saka."Optimum Design Of Steel Frames With Stability Constraints" J. Computer & Structures, Vol.41, PP.1365-1277, 1991. 7- Oran, C., “Geometric Nonlinearity in Nonprismatic Members”, Jour. of the Stru. Div., ASCE, Vol. 100, No. ST7, July, 1974, pp.1473-1487. 8- Raphael.T.Haftka and Zfer Gurdal "Element Structural Optimizations" , Solid Mechanics and its Applications , mKluwer Academic Puplishers,1993 9-Tam, T. K. H. and Jennings, A., “Optimal Elastic Design of Frames with Tapered Members” , Inter. Jour. of Computers and Structures, Vol. 30, No.3, 1988, pp.537-544. Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 121 Fig.(2): Relationship between Design variables D1 & D2 and the Nominal Flexural Strength Mn 400 600 800 1000 1200 1400 Design Variables D1,D2 (mm) 8E+8 1E+9 1E+9 1E+9 2E+9 2E+9 N om in al F le ct ur al S tr en gt h (k N .m ) D1 D2 400 600 800 1000 1200 1400 Design Variables D1,D2 (mm) 2E+6 3E+6 3E+6 4E+6 4E+6 5E+6 5E+6 N om in al A ix al F or ce S tr en gt h (N ) D1, Compression Force D2, Compression Force D1, Tension Force D2, Tension Force Fig.(3): Relationship between Design variables D1 & D2 and the Nominal Axial Force Strength Pn 500 700 900400 600 800 1000 Design Variables D1 (mm) 1500 2500 3500 1000 2000 3000 4000 D er iv at iv e of A xi al F or ce S tr en gt h w ith r es pe ct to D 1 D1, Compression D1, Tension 500 700 900 1100 1300400 600 800 1000 1200 1400 Design Variables D1, D2 (mm) -5E+5 5E+5 2E+6 3E+6 4E+6 -1E+6 0E+0 1E+6 2E+6 3E+6 4E+6 D er iv at iv e of F le ct ur al M om en t S tr en gt h w ith r es pe ct to D 1, D 2 D1, Tension ,Compression D2, Tension ,Compression 400 600 800 1000 1200 1400 Design Variables D1,D2 (mm) 0.55 0.65 0.75 0.85 0.95 0.50 0.60 0.70 0.80 0.90 1.00 S tr en gt h C on st ra in ts D1,Compression D2, Compression D1,Tension D2,Tension Pu/Pn>=0.2 400 600 800 1000 1200 1400 Design Variables D1,D2 (mm) 0.50 0.70 0.90 0.40 0.60 0.80 1.00 S tr en gt h C on st ra in ts D1,Compression D2, Compression D1,Tension D2,Tension Pu/Pn<0.2 Fig.(4): Relationship between Design variable D1 and the Derivatives of Nominal Axial Force Strength Pn with respect to D1 Fig.(5): Relationship between Design variables D1 & D2 and the Derivatives of Nominal Flexural Strength Mn with respect to D1&D2 Fig.(6): Relationship between Design variable D1 and the Strength Constraints Fig.(7): Relationship between Design variable D1 and the Strength Constraints Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 122 Fig.(9): Relationship between Design variable D1 &D2 and the Derivatives of Strength Constraints in the case of compression force Fig.(8): Relationship between Design variable D1 &D2 and the Derivatives of Strength Constraints in the case of tension force 400 600 800 1000 1200 1400 Design Variables D1,D2 (mm) -0.00 -0.00 -0.00 -0.00 -0.00 -2.5E-3 -2.0E-3 -1.5E-3 -1.0E-3 -5.0E-4 0.0E+0 D er iv at iv e of S tr en gt h C on st ra in ts D1,Tension, Pu/Pn >=0.2 D2, Tension, Pu/Pn>= 0.2 D1, Tension, Pu/Pn < 0.2 D2, Tension, Pu/Pn < 0.2 400 600 800 1000 1200 1400 Design Variables D1,D2 (mm) -0.00 -0.00 -0.00 -0.00 0.00 -2.0E-3 -1.5E-3 -1.0E-3 -5.0E-4 0.0E+0 5.0E-4 D er iv at iv e of S tr en gt h C on st ra in ts D1,Compression, Pu/Pn <0.2 D2, Compression, Pu/Pn <0.2 D1, Compression, Pu/Pn >=0.2 D2, Compression, Pu/Pn >=0.2 Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 123 Select initial values of design variables (D1,D2 ) and other sectional properties for each group in addition to the geometrical and material properties for each member. Select the initial values of the design components (Lagrange multiplier, step size, constraints displacements, tolerance, upper bound displacement , )carry out the nonlinear elastic analysis of the frame and calculate the displacement vector [Xi] carry out the linear elastic analysis of the steel frame using the original coordinates of the frame due to a unite load and then obtain the joint displacement vector [Xij] of Eq.(12 ). Optimum Design Cycles tt DD ,1+ Compare these values with their lower bound , take whichever greater as a new values of depth variables I-1 TO NG NG=Number of design groups in frame No Yes Continue Equal Compare Set the design variables equal to their lower bound Weight Convergence Yes No Print the results Start Select Design Parameters Nonlinear Linear calculate all the components of Eqs. (55) & (56) and the new values of Lagrange multipliers using Eq. ( 57 ). Determine the new values of design variables (D1,D2) for each group in the frame using Eqs.(55 ) and (56 ). Calculate the new weight of the frame ,cheak the convergence of both weights an depths , if it is obtain , terminate the design and printout the results otherwise go to the next step Fig. (10) Flow Chart of the Computer Program (EDTS) Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 124 Fig.(11): Fixed Ends Tapered Beam. Fig.(13):Relationships Between the Iteration Cycles and Constrained Displacement 1 3 5 7 90 2 4 6 8 10 Iteration Cycle 0.75 0.85 0.95 1.05 0.70 0.80 0.90 1.00 1.10 To ta l W ie gh t E 8 (k g) Displacement and Strength Con. Displacement Con. Only Strength Con. Only 1 3 5 7 90 2 4 6 8 10 Iteration Cycle 300 500 700 900 200 400 600 800 D es ig n V ar ia bl es D 1& D 2 (m m ) Design Variables D1,Both Displacement and Strength Con D2,Both Displacement and Strength Con D1, Displacement Con. Only D2, Displacement Con. Only D1, Strength Con. Only D1, Strength Con. Only 0 2 4 6 8 Iteration Cycle 2 6 10 0 4 8 12 C on st ra in te d D is pl ac em en t( m m ) Both Displacement and Strength Con Displacement Con.Only Strength Con.Only Fig.(12):Relationships Between the Iteration Cycles and the Design variables D1 &D2 Fig.(14):-Relationships Between the Iteration Cycles and Total Weight of the Beam Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 1 Year 2008 125 Fig(15): Pitched Roof Fixed Ends Steel Frame . ` 1 3 5 70 2 4 6 8 Iteration Cycle 500 700 900 400 600 800 1000 D ep th V ar ia bl es D 1 & D 2 D2 , Displacemet and Strength constraints D1 , Displacemet and Strength constraints D2 , Strength constraint only D1 , Strength constraint only Group (1) 1 3 5 70 2 4 6 8 Iteration Cycle 500 700 900 400 600 800 1000 D ep th V ar ia bl es D 1 & D 2 Group (2) Fig.(16):Relationships Between the Iteration Cycles and Depth Variables D1&D2 for Rafters Fig.(17):Relationships Between the Iteration Cycles and Depth Variables D1&D2 D2 for Columns 1 3 5 70 2 4 6 8 Iteration Cycle 5.300 5.500 5.700 5.900 5.200 5.400 5.600 5.800 6.000 To ta l s tr uc tu re w ei gh t Gruop (1) Gruop (2) 1 3 5 70 2 4 6 8 Iteration Cycle 38 42 46 50 54 36 40 44 48 52 56 C on st ra in te d D is pl ac em en t (m m ) Displacemnt and Strength Constraints Strength Constraint Only Fig.(18):Relationships Between the Iteration Cycles and Total Weight of the Frame Fig.(19):Relationships Between the Iteration Cycles and the Constrained Displacements