STRUCTURAL DESIGN USING FINITE ELEMENTS

STRUCTURAL DESIGN USING FINITE ELEMENTS Nam-Ho Kim

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INTRODUCTION • FEA: determining the response of a given structure for a given set of loads and boundary conditions – Geometry, material properties, BCs and loads are well defined

• Engineering design: a process of synthesis in which parts are put together to build a structure that will perform a given set of functions satisfactorily • Analysis is very systematic and can be taught easily; design is an iterative process

• Creative design: creating a new structure or machine that does not exist • Adaptive design: modifying an existing design (evolutionary process)

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INTRODUCTION – STRUCTURAL DESIGN • Structural design: a procedure to improve or enhance the performance of a structure by changing its parameters • Performance: a measurable quantity (constraint and goal) – the weight, stiffness or compliance; the fatigue life; noise and vibration levels; safety

• Constraint: As long as the performance satisfies the criterion, its level is not important – Ex: the maximum stress should be less than the allowable stress

• Goal: the performance that the engineer wants to improve as much as possible • Design variables: system parameters that can be changed during the design process – Plate thickness, cross-sectional area, shape, etc

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EXAMPLE • Design the height h of cantilevered beam with SF = 1.5 – E = 2.9104 ksi, w = 2.25 in.

1) Allowable tip deflection Dallowable = 2.5 in. (No need SF) –

FE equation after applying BCs

EI L3 –

 12 6L  v2   F     2     6L 4L   2   0 

v2 

FE solution

v2 

3

4FL

Ewh

, 3

2 

2

6FL

Ewh

3



h 

4FL3

Ewh 3

3

 Dallowable

4FL3

EwDallowable

 3.66 in

L = 100 in h w F = 2,000 lb 4

EXAMPLE cont. 2) Failure strength = 40 ksi (Need SF) –

Supporting moment at the wall

C1  –

Maximum stress at the wall

 max –

EI  2 2 6 Lv  4 L   6 Lv  2 L 2   FL 1 1 2 3  L M h2 6FL   I wh 2

Height calculation with the factor of safety 6FL

F  2 SF wh



h 

6FLSF

w F

 4.47 in

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FSD EXAMPLE – CANTILEVERED BEAM • w = 2.25, h = 3.5 in. Determine new height using FSD • Section modulus and max. stress at the initial design wh 2 2.25  3.52 Sold     4.594 in3 h 6 6 M  max   43.537 ksi Sold 2I

• New section modulus using stress ratio resizing Snew  Sold

Snew 

 allowable

wh 2 6

 max 

 4.594 

h

6Snew

w

43.537  7.5 in3 26.667

 4.47 in

L = 100 in h w F = 2,000 lb

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DESIGN PARAMETERS • Selecting design variables – easy for beam and truss, but more complicated for plane or 3D solids • Material property design variable – Varying material properties to find the best material – Not common, but useful for designing composite materials

• Sizing design variable – Geometric parameters as design (parametric design variable) – Appears as a parameter in FEM – Thickness of plate/shell, cross-sectional geometry of truss/beam, etc t b b r b r t h

h

h t

w

w 7

DESIGN PARAMETERS cont. • Shape design variable – Related to the structure’s geometry, which does not appear explicitly as a parameter – Beam cross-section is a geometry, but it appears as a moment of inertia – Cx, Cy, and r determine the size and location of the hole – Shape design variables change FE mesh – Design variables must be limited so that the hole remains inside of the plate

Cx

r Cy (a) Initial design

r

Cx

Cy

(b) Perturbed design 8

PARAMETER STUDY – SENSITIVITY ANALYSIS • Parameter study – Effect of a design variable on performance (gradual change of DV) – Cantilevered beam example:

Allowable stress

Acceptable region

w (in) 2.0 2.0 2.0 2.5 2.5 2.5 3.0 3.0 3.0

h (in) 4.0 4.5 5.0 4.0 4.5 5.0 4.0 4.5 5.0

max (ksi) 37.5 29.6 24.0 30.0 23.7 19.2 25.0 19.8 16.0

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SENSITIVITY ANALYSIS • Parameter study becomes too expensive with many DVs • Unable to capture rapid change in performance locally • Design sensitivity analysis computes the rate of performance change with respect to design variables • Sensitivity analysis calculates gradient of performance for optimization • Explicit dependence – Analytical relationship exists between performance and DVs – Weight of circular cross-section beam

W (r )   r 2L – Sensitivity w.r.t. r :

dW  2 rL dr

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SENSITIVITY ANALYSIS cont. • Implicit dependence – Performance depends on DVs through state variable (displacement) – Sensitivity of stress:

d d d q   dr dq d r Difficult to calculate, time consuming Easy to calculate from given expression of stress

• How to calculate displacement sensitivity?

– Differentiate finite element equation:[K(b )]{ Q }  {F(b )}

 dQ   dF   dK   K     Q db db db       – [dK/db] and {dF/db} can be evaluated using either their analytical expression or numerical differentiation 11

SENSITIVITY ANALYSIS cont. • Sensitivity equation must be solved for each DV • Sensitivity equation uses the same stiffness matrix with the original finite element analysis • Consider RHS as a pseudo-force vector • Similar to finite element analysis with multiple load cases • Thus, solving sensitivity equation is very inexpensive using factorized stiffness matrix • General form of performance H  H  q(b ), b 

– Sensitivity

dH ( q(b ), b ) H  db b

Implicit dependent term

 q  const

H q

Explicit dependent term

 b  const

dq db

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FINITE DIFFERENCE SENSITIVITY • Easiest way to compute sensitivity information of the performance • Calculate performance at two different designs • Forward difference method dH H (b  b )  H (b )  db b

• Central difference method dH H (b  b )  H (b  b )  db 2b

• Consider FEA as a black-box • Sensitivity computation cost becomes high for many design variables – N+1 analyses for forward FDM – 2N+1 analyses for central FDM 13

FINITE DIFFERENCE SENSITIVITY cont. • Accuracy of finite difference sensitivity – Accurate results can be expected when b approaches zero – For nonlinear performances, a large perturbation yields completely inaccurate results – Numerical noise becomes dominant for a too-small perturbation size

H

(dH/db)4 (dH/db) 3

(dH/db)2 (dH/db)1

b0 b1 b2

b3

b4

b 14

EXAMPLE – CANTILEVERED BEAM • At optimum design (w=2.25 in, h=4.47 in), calculate sensitivity of tip displacement w.r.t. h • Exact sensitivity: dv2 dh

 exact

12FL3

Ewh

4



12  2, 000  1003 7

2.9  10  2.25  4.47

3

 4.118

• Differentiate [K]

 12 6L 12 6L   0   12       F  6L 4L2 6L 2L2   0  F  12L   dK       db  { Q }  4Lh  12 6L  4 h 12  6 L 4 L 12          0   6L 2L2 6L 4L2   6 

• Pseudo load vector

 12    F  12L   dF   dK  Q        d b d b 4 h  12        0 

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EXAMPLE – CANTILEVERED BEAM cont. • Sensitivity equation:  12 6L 12 6L   dv1 / db  0   12       EI  6L 4L2 6L 2L2   d1 / db  0  F  12L      3  12 6L  4 h 12  6 L d v / d b  12 L 2        0   6L 2L2 6L 4L2   d2 / db 

• Same way of applying BC EI L3

 12 6L   dv2 / db  F  12      2   d / db  4 h 0  6 L 4 L      2

• Sensitivity of nodal DOFs dv2 d2 12FL3 18FL2  ,  4 db d b Ewh Ewh 3 – Same with the exact sensitivity

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STRUCTURAL OPTIMIZATION • What Is Design Optimization? – To find the best design parameters that meet the design goal and satisfies constraints.

• Design Parameters: Anything the Designer Can Change – Thickness of a plate – Cross-sectional geometry of a beam or truss – Geometric dimensions

• Design Goal: Objective Function – Design criterion that will be minimized (or maximized) – Mass, Stress, Displacement, Natural Frequency, ETC

• Constraint: Conditions that the system must satisfy – Stress, Displacement, ETC

• Note: Design parameters must affect the design goal and constraints. 17

OPTIMIZATION FLOW CHART Physical engineering problem

Structural modeling Design parameterization Performance definition (cost, constraints) Structural model update

Structural analysis (FEM, BEM, CFD…) Design sensitivity analysis Optimized?

No

Yes Stop 18

THREE-STEP PROBLEM FORMULATION 1. Design Parameterization • •

w

Clear identification Independence of designs

t1 t2

h

2. Objective Function • •

Must be a function of design parameters Minimization ( –Maximization)

3. Constraint Functions • • •

Inequality constraints Equality constraints Equality constraints must be less than the number of design parameters 19

STANDARD FORM • Standard form of design optimization minimize f (b) subject to gi (b)  0, i  1,

,N

hj (b)  0, j  1, , M bl L  bl  blU , l  1, , K

b   b1 b2

f (b) g1 (b), , gN (b) h1 (b), , hM (b)

bK

T



bL , bU

: Design parameters : Objective function : Inequality constraints : Equality constraints : Lower and upper bounds

• Feasible set: the set of designs that satisfy constraints S   b gi (b)  0, i  1,

N , hj (b)  0, j  1,

M 20