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Minimum Weight Plastic Design For Steel-Frame Structures EN 131 Project By James Mahoney

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Program Objective: Minimization of Material Cost Objective: Minimization of Material Cost –Amount of rolled steel required Non-Contributing Cost Factors Non-Contributing Cost Factors –Fabrication –Construction/Labor costs

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Program Constraints Structure to be statically sound Structure to be statically sound –Loads transmitted to foundation through member stresses –Members capable of withstanding these internal stresses

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Member Properties Wide-Flange Shape Wide-Flange Shape Full Plastic Moment Full Plastic Moment M p F y x(Flange Area)xd Weight Proportional to M p Weight Proportional to M p Total Flange Area >> Web Area Weight Proportional to Flange Area Weight Proportional to Flange Area

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Objective Function Calculating Total Weight Calculating Total Weight –Each member assigned full plastic moment –Weight = member length x weight per linear foot linear foot Vertical members: Weight = H x M p Vertical members: Weight = H x M p Horizontal members: Weight = L x M p Horizontal members: Weight = L x M p

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Objective Function For a Single Cell Frame For a Single Cell Frame Min Weight = 2H x M p1 + L x M p2 Min Weight = 2H x M p1 + L x M p2 M p1 M p2 P P

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Objective Function Frame for Analysis Frame for Analysis

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Objective Function Minimum Weight Function Minimum Weight Function MIN = H x (M p1 +2xM p2 +M p3 +M p4 +2xM p5 +M p6 +2xM p13 ) MIN = H x (M p1 +2xM p2 +M p3 +M p4 +2xM p5 +M p6 +2xM p13 ) + L x (M p7 +M p8 +M p9 +M p10 +M p11 +M p12 +M p14 ) + L x (M p7 +M p8 +M p9 +M p10 +M p11 +M p12 +M p14 ) Subject to constraints of Static Equilibrium Subject to constraints of Static Equilibrium

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Equilibrium State Critical Moment Locations in Frame Critical Moment Locations in Frame –Seven critical moment nodes form that are the result of plastic hinging –One hinge develops at each member end (when fixed) and under the point load –Moments causing outward compression are positive while moments producing outward tension are negative Critical moments in each member are paired with an assigned full plastic moment Critical moments in each member are paired with an assigned full plastic moment

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Use of Virtual Work Principle: EVW = IVW Principle: EVW = IVW –The work performed by the external loading during displacement is equal to the internal work absorbed by the plastic hinges –Rotational displacement measured by θ said to be very small

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Use of Virtual Work Beam Mechanism (Typical) Beam Mechanism (Typical) P θ θ 2θ2θ L/2 IVW = EVW -M 1 θ + 2M 2 θ – M 3 θ = P(L/2)θ or -M 1 + 2M 2 – M 3 = P(L/2)

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Use of Virtual Work Loading Schemes Loading Schemes –Point Loads Defined placement along beam Defined placement along beam R (ratio factor) = 0.5 at midspan, etc. R (ratio factor) = 0.5 at midspan, etc. Results in adjustment of beam mechanism equations for correct placement of hinges Results in adjustment of beam mechanism equations for correct placement of hinges –Distributed Load Placed over length of beam Placed over length of beam Result is still a center hinge Result is still a center hinge Change in EVW formula Change in EVW formula EVW = Q*(L^2)/4

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Use of Virtual Work Seven Beam Mechanisms Seven Beam Mechanisms –One for each beam -(1-R1)*VALUE(24)+VALUE(23)-R1*VALUE(22) = P1*R1*(1-R1)*L -(1-R2)*VALUE(21)+VALUE(20)-R2*VALUE(19) = P2*R2*(1-R2)*L -(1-R3)*VALUE(18)+VALUE(17)-R3*VALUE(16) = P3*R3*(1-R3)*L -(1-R4)*VALUE(4)+VALUE(5)-R4*VALUE(6) = P4*R4*(1-R4)*L -(1-R5)*VALUE(7)+VALUE(8)-R5*VALUE(9) = P5*R5*(1-R5)*L -(1-R6)*VALUE(10)+VALUE(11)-R6*VALUE(12) = P6*R6*(1-R6)*L -VALUE(33)+2*VALUE(34)-VALUE(35) = Q1*(L^2)/4

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Use of Virtual Work Sway Mechanism (Simple Case) Sway Mechanism (Simple Case) θ P IVW = EVW -M 1 θ + M 2 θ – M 3 θ + M 4 θ= PHθ -M 1 θ + M 2 θ – M 3 θ + M 4 θ = PHθor -M 1 + M 2 – M 3 + M 4 = PH H

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Use of Virtual Work Three Sway Mechanisms Three Sway Mechanisms –One for each level of framing VALUE(1)-VALUE(25)+VALUE(28)-VALUE(15) = F1*H -VALUE(2)+VALUE(26)-VALUE(29)+VALUE(14)+VALUE(3)- VALUE(27)+VALUE(30)-VALUE(13) = F2*H -VALUE(31)+VALUE(32)-VALUE(36)+VALUE(37) = F3*H

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Use of Virtual Work Joint Equilibrium (Simple Case) Joint Equilibrium (Simple Case) –Total work done in joint must equal zero for stability θ -M 1 + M 2 = M 3 – M 4 + M 5 + M 6 = 0

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Use of Virtual Work Ten Joint Equilibriums Ten Joint Equilibriums –One for each joint VALUE(24)+VALUE(2)-VALUE(1) = 0 VALUE(4)+VALUE(31)-VALUE(3) = 0 VALUE(16)+VALUE(14)-VALUE(15) = 0 VALUE(30)+VALUE(9)-VALUE(10) = 0 VALUE(33)-VALUE(32) = 0 VALUE(36)-VALUE(35) = 0 VALUE(13)-VALUE(12) = 0 VALUE(7)-VALUE(6)+VALUE(37)-VALUE(27) = 0 VALUE(21)-VALUE(22)+VALUE(26)-VALUE(25) = 0 VALUE(19)-VALUE(18)+VALUE(29)-VALUE(28) = 0

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Program Breakdown Solving Critical Moments Solving Critical Moments –37 unknown critical moments –17 levels of structural indeterminacy –Requires 20 indep. equil. equations 7 beam mechanisms 7 beam mechanisms 3 sway mechanisms 3 sway mechanisms 10 joint equations 10 joint equations

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Design Against Collapse Lower Bound Theorem Lower Bound Theorem –Structure will not collapse when found to be in a statically admissible state of stress (in equilibrium) for a given loading (P, F, etc.) Therefore applied loading is less than the load condition at collapse (i.e. P<=Pc and F<=Fc) Therefore applied loading is less than the load condition at collapse (i.e. P<=Pc and F<=Fc) Moments to be Safe Moments to be Safe –Plastic moments set to equal greatest magnitude critical moment in pairing -(M p ) j <= M i <= (M p ) j for all (i,j) moment pairings

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