Minimum Weight Plastic Design

aboundingdriprockUrban and Civil

Nov 29, 2013 (3 years and 4 months ago)

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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










Full Plastic Moment

M
p

≈ F
y
x(Flange Area)xd


Weight ≈ Proportional to M
p

Total Flange Area >> Web Area




Weight ≈ Proportional to Flange Area










Objective Function


Calculating Total Weight


Each member assigned full plastic moment


Weight = member length x “weight per


linear foot”



Vertical members: Weight = H x M
p


Horizontal members: Weight = L x M
p

Objective Function


For a Single Cell Frame








Min Weight = 2H x M
p1

+ L x M
p2

M
p1

M
p2

P

P

Objective Function


Frame for Analysis

Objective Function


Minimum Weight Function




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
)



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

2
θ

L/2

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)

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)

θ

P

IVW = EVW

-
M
1
θ

+ M
2
θ



M
3
θ

+ M
4
θ

= PH
θ

or

-
M
1
+ M
2



M
3 +
M
4

= PH

H

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

θ

-
M
1

+ M
2

= 0

1

2

4

5

6

3

-
M
3



M
4
+ M
5

+ M
6

= 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

-
(M
p
)
j

<= M
i

<= (M
p
)
j

for all (i,j) moment pairings