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
 Structure
to be statically sound
– Loads transmitted to foundation through
member stresses
– Members capable of withstanding these
internal stresses

Member Properties

Wide-Flange Shape
Total Flange Area >> Web Area

Weight ≈ Proportional to Flange Area

Full Plastic Moment
Mp ≈ Fyx(Flange Area)xd
Weight ≈ Proportional to Mp
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
Objective Function
 For
a Single Cell Frame
P
P
Mp2
Mp1

Min Weight = 2H x Mp1 + L x Mp2
Objective Function

Frame for Analysis
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
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

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
Use of Virtual Work
 Beam
Mechanism (Typical)
P
θ
θ
2θ
L/2
L/2
-M1θ + 2M2θ – M3θ = P(L/2)θ
or
-M1 + 2M2 – M3 = P(L/2)
IVW = EVW
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
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
Use of Virtual Work
 Sway
Mechanism (Simple Case)
P
H
θ
-M1θ + M2θ – M3θ + M4θ = PHθ
or
-M1 + M2 – M3 + M4 = PH
IVW = EVW
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
Use of Virtual Work
 Joint
Equilibrium (Simple Case)
– Total work done in joint must equal zero for
stability
4
θ
1
2
-M1 + M2 = 0
3
5
6
-M3 – M4 + M5 + M6 = 0
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
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
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