Minimum Weight Plastic Design

Urban and Civil

Nov 29, 2013 (4 years and 6 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

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
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

Defined placement along beam

R (ratio factor) = 0.5 at midspan, etc.

Results in adjustment of beam mechanism
equations for correct placement of hinges

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

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