BEAM DESIGN FOR GEOMETRIC NONLINEARITIES

Jordan RadasKantaphat SirisonWendy Zhao

PREMISE

Large deflection

Linear assumptions no longer apply

Is necessary form many real life applications

DESIGN OVERVIEW

Nonlinear Linear

GEOMETRIC NONLINEARITY ASSUMPTIONS Large deformation

Plane cross section remains plane

Linear elastic material

Constant cross section

ux1

ux2

KINEMATICS

Location of particle at deformed configuration relative to displacement and original configuration

KINEMATICS

e

(1 u'x )cos u'y sin 1 1(1 u'x )sin u'y cos

'

Characterize axial strain, shear strain and curvature in terms of the derivatives of the displacement

Green Lagrange Strain Tensor

STRAIN DISPLACEMENT MATRIX

[h]31 [B]36[d]61

e

de

dux1...

de

du 2ddux1

...ddu 2

ddux1

...ddux1

31

ux1...

u 2

61

[B]The components of the strain displacement matrix can be

determined explicitly by differentiation.

B cos sin N1a cos sin N2a

sin cos N1b sin cos N2b

0 0 1 0 0 1

where

a (1 u'x )sin u'y cosb (1 u'x )cos u'y sin

TANGENT STIFFNESS MATRIX

dV ubdx uAtV 0

V 0

(EA GA EI)dx uyqV 0

[K][d]

R

Through discretization and linearization of the weak form

k BTDBdx kgeometricV 0

kmaterial

kg EAL

GAL

sin2 cos2 0 sin2 cos2 0

cos2 sin2 0 cos2 sin2 0

0 0 0 0 0 0

sin2 cos2 0 sin2 cos2 0

cos2 sin2 0 cos2 sin2 0

0 0 0 0 0 0

NEWTON-RAPHSON METHOD

k ttu R tt

F tt R tt

F tt F t ku

k(i 1)u( i) R F ( i 1)

u(i) u(i 1) u( i)

Displacement

Load

F 4 R tt

F 0,u0

F1

F 2

F 3

K1

K 2

u1

u2

u3

u4

NEWTON-RAPHSON METHOD

RESTORING LOAD

dVBF iTii )1()1()1(

Xu

IX

xF

URF

]ln[U

3

1

lni

Tiii ee

Definition of deformation gradient

Spatial Decomposition

Corresponds to element internal loads of current stress state.

From right polar decomposition theorem

INCREMENTAL APPROXIMATION

1 nnn

][][ nDed

2/12/1 RR nT

n

]ln[ U

From

11

nn FFURF

With

With

Xu

IURF

2/12/12/12/1

12/1 2

1 nn uuu

nn uB 2/1~ Wit

h 12/1 2

1 nn XXX

B Evaluated at midpoint geometry

NONLINEAR SOLUTION LEVELS

Load steps:

Adjusting the number of load steps account for: abrupt changes in loading on a structure

specific point in time of response desired

Substeps:

Application of load in incremental substeps to obtain a solution within each load step

Equilibrium Iterations:

Set maximum number of iterations desired

SUBSTEPSEquilibrium iterations performed until convergence

Opportunity cost of accuracy versus time

Automatic time stepping featureChooses the size and number of substeps to optimize

Bisections methodActivates to restart solution from last converged step if a solution does not converge within a substep

MODIFIED NEWTON-RAPHSON

Incremental Newton-Raphson Initial-Stiffness Newton-Raphson

DISPLACEMENT ITERATION

As opposed to residual iteration

ANSYS FEATURES

Predictor

Line Search Option

ANSYS FEATURESAdaptive Descent

DESIGN CHALLENGE: OLYMPIC DIVING BOARD

L = 96in

b = 19.625in

h = 1.625in

P = -2500lbs

Al 2024 – T6 (aircraft alloy)

E = 10500ksi

v = .33

Yield Strength = 50ksi

SOLID BEAM: LINEAR/NONLINEAR

Mesh Size Linear Nonlinear

.125in 8.1043in 2.1671 2.2383 8.1082in 2.1757 2.2442

.25in 8.0255in 2.2170 2.2675 7.9951in 2.2254 2.2749

.5in 8.0566in 2.3051 2.3144 8.0282in 2.3151 2.3195

OPTIMIZATION PROBLEM

ANSYS Goal Driven Optimization is used to create a geometry where hole diameter is the design variable.

Goals include minimizing volume and satisfying yield strength criterion.

OPTIMIZATION AND ELEMENT TECHNOLOGY Optimization samples points in the user specified design space.

The number of sampling points is minimized using statistical methods and an FEA calculation is made for each sample.

Samples are chosen based on goals set for output variables, such as volume and safety factor.

OPTIMIZATION RESULTS

Problem Method Volume Deflection

Max Tensile

Von Mises

Solid Beam

Linear3061.5in3

8.0566in 2.3051 2.3144

Nonlinear 8.0282in 2.3151 2.3195

Optimized

Beam

Linear2719.5in3

8.7993in 1.2754 1.2555

Nonlinear 8.7576in 1.2756 1.2757

CONCLUSION

Analysis serves as a proof of concept that real-world situations involving large structural displacements benefit from nonlinear modeling considerations

Extra computing power and time is worth it

Recommendations/suggestions

QUESTIONS?