Dynamic Programming Applications Design of Continuous Beam

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D Nagesh Kumar, IISc Optimization Methods: M6L1

1 Dynamic Programming Applications

Design of

Continuous Beam

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Objectives

¾

To discuss the design of continuous beams

¾

To formulate the optimization problem as a dynamic

programming model

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3 Design of Continuous Beam

¾ Consider a continuous beam having n spans with a set of loadings W₁, W₂,…, W_n

¾ Beam rests on n+1 rigid supports

¾ Locations of the supports are assumed to be known

W₁ 0

L₁

W₂ 1

L₂

2

W_i i-1

L_i

W_i+1 i

L_i+1

i+1

W_n n-1

L_n

n

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Design of Continuous Beam …contd.

¾ Objective function: To minimize the sum of the cost of construction of all spans

¾ Assumption: Simple plastic theory of beams is applicable

¾ Let the reactant support moments be represented as m₁, m₂,

…, m_n

¾ Complete bending moment distribution can be determined once these support moments are known

¾ Plastic limit moment for each span and also the cross section of the span can be designed using these support moments

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5 Design of Continuous Beam …contd.

¾ Bending moment at the center of the i^th span is -W_iL_i/4

¾ Thus, the largest bending moment in the i^th span can be computed as

¾ Limit moment for the i^th span, m_lim_i should be greater than or equal to M_i for a beam of uniform cross section

¾ Thus, the cross section of the beam should be selected such that it has the required limit moment

n i

L for W m

m m m

M_i _i _i ⁱ ⁱ ⁱ ⁱ 1,2,...

4 , 2

,

max ₁ ¹ =

⎭⎬

⎫

⎩⎨

⎧ + −

= ₋ ⁻

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Design of Continuous Beam …contd.

¾ The cost of the beam depends on the cross section

¾ And cross section in turn depends on the limit moment

¾ Thus, cost of the beam can be expressed as a function of the limit moments

¾ Let X represents the vector of limit moments

⎪

⎪⎪

⎬

⎫

⎪

⎪⎪

⎨

⎧

= m m

X _lim lim _

2 1

Μ

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7 Design of Continuous Beam …contd.

¾ The sum of the cost of construction of all spans of the beam is

¾ Then, the optimization problem is to find X to Minimize

satisfying the constraints .

¾ This problem has a serial structure and can be solved using dynamic programming

∑

= n

i

i X

C

1

) (

∑

= n

i

i X

C

1

) (

n i

for M

m_ lim_i ≥ _i =1,2,...,

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Dynamic Programming Applications Design of Continuous Beam

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