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Minimum Weight Plastic Design For Steel-Frame Structures

Minimum Weight Plastic Design For Steel-Frame Structures. EN 131 Project By James Mahoney. Program. Objective: Minimization of Material Cost Amount of rolled steel required Non-Contributing Cost Factors Fabrication Construction/Labor costs. Program Constraints.

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Minimum Weight Plastic Design For Steel-Frame Structures

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  1. Minimum Weight Plastic DesignFor Steel-Frame Structures EN 131 Project By James Mahoney

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

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

  4. Member Properties • Wide-Flange Shape • Full Plastic Moment Mp ≈ Fyx(Flange Area)xd Weight ≈ Proportional to Mp Total Flange Area >> Web Area Weight ≈ Proportional to Flange Area

  5. Objective Function • Calculating Total Weight • Each member assigned full plastic moment • Weight = member length x “weight per linear foot” • Vertical members: Weight = H x Mp • Horizontal members: Weight = L x Mp

  6. Objective Function • For a Single Cell Frame • Min Weight = 2H x Mp1 + L x Mp2 P P Mp2 Mp1

  7. Objective Function • Frame for Analysis

  8. Objective Function • Minimum Weight Function MIN = H x (Mp1+2xMp2+Mp3+Mp4+2xMp5+Mp6+2xMp13) + L x (Mp7+Mp8+Mp9+Mp10+Mp11+Mp12+Mp14) • Subject to constraints of Static Equilibrium

  9. Equilibrium State • 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

  10. Use of Virtual Work • 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

  11. Use of Virtual Work • Beam Mechanism (Typical) P θ θ 2θ IVW = EVW L/2 L/2 -M1θ + 2M2θ – M3θ = P(L/2)θ or -M1 + 2M2 – M3 = P(L/2)

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

  13. Use of Virtual Work • 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

  14. Use of Virtual Work • Sway Mechanism (Simple Case) P IVW = EVW H θ -M1θ + M2θ – M3θ + M4θ= PHθ or -M1 + M2 – M3 + M4= PH

  15. Use of Virtual Work • 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

  16. Use of Virtual Work • Joint Equilibrium (Simple Case) • Total work done in joint must equal zero for stability θ 4 5 1 2 3 6 -M1 + M2 = 0 -M3 – M4 + M5 + M6 = 0

  17. Use of Virtual Work • 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

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

  19. Design Against Collapse • 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) • Moments to be Safe • Plastic moments set to equal greatest magnitude critical moment in pairing -(Mp)j <= Mi <= (Mp)j for all (i,j) moment pairings

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