Transfer Plate

The Seventh China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems Huangshan, June, 18-21, 2012, China

INTEGRATED DESIGN OPTIMIZATION OF TRANSFER PLATE FOR TALL BUILDING STRUCTURES Lam Siu, Kelvin 1, Zou, Xiao Kang2, Wong, Kong Loi3

1

Building Engineering (Structural), AECOM AECOM Asia Co. Ltd, Hong Kong * Corresponding author: [email protected] author: [email protected] 2, 3 Building Engineering (Structural), AECOM AECOM Asia Co. Ltd, Hong Kong

Abstract This paper describes an integrated optimization technique by combining topology optimization and size

optimization for the transfer plate design of tall building structures. The suitable locations and dimensions of the openings in the transfer plate can be predicted by topology optimization, while a set of the optimal thickness solutions and corresponding steel ratios can be obtained by size optimization. A study example has been used to optimize the transfer plate by applying the integrated optimization. Keyword: Size Optimization, Topology Optimization, Transfer Plate, Tall Building, Concrete

1. Introduction Structural optimization has traditional applied in the field of mechanical and aeronautical engineering, in recently, progressively application in structural engineering. There are several examples apply to the high-rise building design [1, 2]; all of them are mainly forced on the entire structural system of the building. Despite a variety of applications in structural engineering field, the focus of this work is forced on the optimal design of transfer system in the high-rise building especially in Hong Kong region. Currently, tall buildings with multi-functional usages become a common trend. Transfer floor is the major component for entire building to allocate change of locations for vertical structural elements due to the need to allow open-layout underneath transfer structure. There are several structural systems to transfer the loading, in forms of beam, plate, truss and box. Mostly, transfer plate system is the most common practical structural form adopted in tall buildings for Hong Kong region. However, in order to transfer the entire loading from the upper floor vertical element to lower columns or walls, a thick reinforced concrete plate is commonly adopted in construction, which is usually very large and heavy in weight, sometime it is quiet difficult and time-consuming to construct. The depth of transfer plate is commonly defined under a preliminary estimation under engineering judgment, structure efficiency efficiency of the transfer transfer plate stills not fully utilized in normal. normal. With diminishing material and increasing the economic competition, the need for efficient and lightweight structure is becoming challenge in structure engineer point of view. These motivate the design engineers to reconsider the transitionally adopted design procedures. By using the optimization methods which become feasible in the design nowadays, the engineers can develop the system from conceptual design to the final sizing of the member though the optimization processes.


In this paper, a fully integrated optimization design procedure is applied to find optimum geometric designs of the reinforced concrete transfer plate complying with the practical design conditions. To achieve this task, objective function and several design constraints are used to quantified, such as the weight of the structure, the stress in the plate. For the parameter that governs the geometry and thickness of the plate were used as variables, such as the minimum and maximum thickness of the plate and several parameters of boundary conditions.

2. Optimization design problem 2.1 Main design optimization procedure The main design optimization consists of five steps, as shown as the flow chat in Figure 1. I.

Study the building structural finite element model, such as ETABS. The existing geometry and load conditions can be applied in the program to evaluate the initial thickness, and its upper bound limits.

II.

Transfer the model into optimization software, such as HyperWork, to do the optimization analysis processes.

III.

Conduct the topology optimization to searching the distribution of material and to identify the potential locations and size of openings.

IV.

Process the size optimization after located the openings, an optimizing thickness based on different steel ratio will be obtained.

V.

Output a set of optimal solutions including the optimal transfer plate thickness and corresponding reinforcement ratio will be used to back-calculate the amount of steel and concrete used. The optimal solution can be identified by apply a corresponding cost function which including material and construction cost. Study building structural finite element analysis model

Define transfer plate optimization model

Topology optimization

Size optimization

Output

Figure 1 Flow chat of optimization process


2.2 Topology optimization Topology optimization is commonly used as an innovative and powerful tool in the design of the structures. The method is by formulated a theoretical and numerical procedure to find the optimal layout by redistributing the material in a fix defined domain. The problem is defined as finding the stiffest possible structure when a certain amount of material a given. A structure with maximum global stiffness provides a minimum for the strain energy. Therefore, the topology optimization problem can be formulated as: Minimize

(1)

subject to

(

Equation (1) defines the design objective function where

is the strain energy, where

(2)

donates

the strains; is the entire volume of element ; is the constitutive matrix of element and is number of finite elements in the discretized domain. Equation (2) defines the strength constraints where is the average stresses of th strips; strips;

is the upper bound of the average axial stresses of th

is the total number of strips in transfer plate.

By considering structural topology optimization as a material distribution problem, the structure can be described by a discrete function

, defined at each point

as (3)

By assuming isotropy for the solid element, the constitutive matrix at a typical point

can be

written as (4) where

is the elasticity matrix of the solid material.

2.3 Size optimization The sizing optimization technique is based on the optimization of variation of thickness throughout the plate; the design variable is the plate thickness of various regions within the plates. Both finite element analysis and the sensitivity analysis are conducted to evacuate the objective function and the constraints and associated derivative with respect to each design variable. The optimal thickness of transfer plate is depend by a series of factors such as the location of concentrated loads, the location of the support columns and the electrical and mechanical openings though the plate , the existence of free edge boundary condition and the global behavior of entire structure. Those factors are strongly depended on the structure form of entire structure due to the change of deflection variation thickness. In this paper, it mainly emphasize on the optimization of the thickness of transfer plate, details analysis of the entire structural behavior will b e done later. Therefore, main objective in the design would be finding an optimal thickness of transfer plate, considering the specified loadings, geometry layout of the structure and properties of the material, satisfies the following design conditions: I.

Stresses enclosed with an interval in both direction of the plate.

II.

Displacement enclosed with an interval in entire plate.

III.

Boundary condition satisfied for all load configurations.

IV.

A final shape that meets the practical design criteria. The aim of the optimal structural design is to obtain a design, a set of design variable, which

minimizes an objective function and complies with the constraints that depend on the variables. The design variables of transfer plate are the thickness of elements and the structural geometry parameters.


The objection function is minimizing the volume of the plate. The constraints are the limit normal stress and shear stress of the plate and the deflection of the plate. The size optimum design problem is formulated as follows: Minimize

(5)

subject to

(6)

(7)

(8) Equation (5) defines the design objective function where where the vectors

represents the volume of transfer plate,

are the design variables of the depth of transfer plate in each strips;

is the

surface area of each strips. Equations (6) and (7) define the strength constraints where

is the

average axial stresses of th strips; is the upper bound of the average axial stresses of th strips; is the average shear stresses of th strips; is the upper bound of the average shear stresses of th strips. Equation (8) defines the deflection constraint of transfer plate, where deflection of th strips;

and

the total number of design variables;

is the

are the lower and upper bound of the deflection of th strips.

is

is the total number of strips in transfer plate.

This problem is solved by mathematical programming using the optimization module in Altair HyperWork [3], which is an integrated analysis and optimization tool for structures analysis. Based on the assumption of plane section remain plan, the behavior of the reinforced concrete material can be equivalent to an equivalence effective stress for the analysis, as shown in Figure. 2 '

σ

h/2

'

σ

(a) Equivalent rectangular stress block

(b) Equivalent effective stress

Figure 2 Stress block and equivalent effective stress The equivalent effective stress can be defined based on following equilibrium equation by assuming no compression bars provided which reference to the HK code 2004 [4]:


where

is the characteristic strength of the concrete;

depth of simplified stress block; compression force;

is the total height of the plate;

is the reinforcement tension force;

is the total amount of tension steel;

characteristic yield stress of the steel and

is the

is the concrete

is the strip width of the plate;

is

is equivalent effective stress.

3. Illustrative example 3.1 Topology optimization In the transfer plate design, there is a need to have openings for electricity and mechanical drainage services passing through the plate vertically. However, feasible location of the void mostly determined by previous engineering experience, which has less scientific based behind. By using the topology optimization technique, it tries to give an insight of the best location for openings under specified geometry of the structure layout. In this example, as transfer plate with covers a surface of 20 m×30 m is used. Young’s module of the material is 26.4 GPa for C45 concrete and Poisson’s modules is 0.2. The structure is subjected to twenty superstructure columns each have a vertical load of 20000 kN and bottom moment of 5000 kN-m in x and y direction, a uniform distribution load of 8 kN/m2, and it is supported by four support columns and a lift core wall, as shown in Figure 3 and 4. Lift core wall Superstructure column

Transfer plate

Support column

(b) Elevation view of transfer plate floor

Lift core wall

Location of transfer plate Support column

Transfer plate Superstructure column

(a) 3-D model of the building stucture

(c) 3-D view of transfer plate floor

Figure 3 Building structure model and transfer plate model


Superstructure column

Lift core wall

Support column

20m

Transfer plate

30m Figure 4 Topology optimization model of transfer plate

Figure 5 Density distribution contour of transfer plate after topology optimization

From the topology analysis, red region is the max density region the lowest density region

and blue region is

. Based on the result, the openings can be located at the region of blue

color where has no significant impact of the total stability of transfer plate, referring to Figure 5. Based on the results, two openings with dimension 2.5m (length) X 1.5m (width) located near the core wall of the structure is determined, as shown in Figure 6.

Figure 6 Dimensions and location of the voids of transfer plate from topology optimization results


3.2 Size optimization The case of size optimization of a transfer plate is studied. As the total thickness of plate is depended on the amount of steel provided within the plate, the larger amount of steel used, the smaller the thickness of plate can sustained under the same condition of loadings. Under this example, the initial thickness of transfer plate is set to be 3000 mm and the definition of material properties and load case are the same as previous one used in topology optimization analysis, the 3-D geometry of the model used in size optimization is shown in Figure 7.

20m

3m

30m Figure 7 Size optimization model of transfer plate Six cases studies with different amount of steel ratios have been conducted to obtain the minimum thickness of transfer plate, which are defined in Table 1. Table 1 Initial and optimal results Steel ratio ρ (%)

Initial thickness

Case

1 (mm)

Optimal thickness and stress hopt(mm)

(N/mm2)

(N/mm2)

1

1.2

3000

3000

25.35

25.40

2

1.4

3000

2768

28.91

28.97

3

1.6

3000

2600

32.34

32.34

4

1.8

3000

2480

35.5

35.53

5

2.0

3000

2380

38.52

38.53

6

2.26

3000

2273

42.13

42.15

where

is the initial thickness of plate;

average stresses per strip and

hopt is the optimal thickness of plate;

is the actual

is the limit average stress per strip.

Case 4 is selected to compare the difference between the optimal and initial results of deflection, axial stresses and shear stresses. It indicates that the optimal values are larger than initial values, the optimal results in deflection and shear stresses are satisfied all the constraints conditions, and the axial stresses is active during the analysis process, the contour of deflection, axial stresses and shear stresses are shown in Figures 8,9,10.


(a) Initial deflection contour

(b) Optimal deflection contour

Figure 8 Initial and optimal deflections for Case 4

(a)Initial axial stresses contour

(b)Optimal axial stresses contour

Figure 9 Initial and optimal axial stress contours for Case 4

(a)Intial shear stresses contour

(b)Optimal shear stresses contour

Figure 10 Initial and optimal shear stress contours for Case 4

The optimization histories of the previous cases have been shown in Figure 11, which are used to measure the efficiencies of the optimization analysis. It is found that all the cases had been coverage within two interactions. It indicates that the optimization methodology developed is very efficient to this design problem. However, Case 1 reaches the optimization result only one design cycle which due to the fact that the design variable (i.e thickness of transfer plate) reaches the lower bound constraint of the thickness of transfer plate.


Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Figure 11 Optimization histories of the cases After comparing the relationship between the steel ratio with the optimal thickness of concrete transfer plate, the relationship between the weight of steel and the weight of concrete is plotted, as shown in Figure 12.

Figure 12 Comparison between amount of steel and the volume of concrete Based on the analysis result, the weight of concrete is inversely proportional to the weight of the total amount of steel used. A feasible design option are be located by defining the cost objective function which included the material cost and certain percentage of construction cost.

4. Conclusions The integrated optimization technique developed can be applied to the transfer plate design of tall building structures by the combination of topology optimization and size optimization. The suitable locations and dimensions of openings can be predicted by topology optimization, while a set of the optimal thickness solutions and corresponding steel reinforcements can be obtained by size optimization. Feasible design solutions can be specified by the adequate cost function in the design. The study results demonstrate that the technique can produce a set of optimal solutions which satisfy all the design constraints and match engineer’s experiences. Further effort will be made to consider more design conditions and constraints to improve the current integrated optimization of transfer plate for tall building structures.


Acknowledgments The authors are grateful to Mr. Lee, Hoi Yuen and Mr. Ng, Tim Yeung, Sammy of the Building Engineering Group, AECOM Asia Company Ltd for their insightful suggestions and support. Also we would like to thank the Building Engineering Group, AECOM Asia Company Ltd for providing research funding.

References 1.

Zou X K and Chan C M. An Optimal Resizing Technique for Seismic Drift Design of Concrete Buildings Subjected to Response Spectrum and Time History Loadings.

Computers and

Structures, 2005, 83: 1689-1704. 2.

Stromberg LL, Beghini A, Baker WF, Paulino GH. Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Struct Multidisc Optim 2011;43(2): 165-80.

3.

HyperWork, Altair Engineering, Inc.

4.

The Code of Practice for Structural Use of Concrete 2004 (Second Edition). Building Department, the Government of the Hong Kong Special Administrative Region, August 2008.

Transfer Plate

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