5 Constructing MoM Members IFEM Ch 5 Slide 1 Introduction to FEM - - PDF document

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Introduction to FEM 5 Constructing MoM Members IFEM Ch 5 Slide 1 Introduction to FEM What Are MoM Members? Skeletal structural members whose stiffness equations can be constructed by Mechanics of Materials (MoM) methods Can be locally

SLIDE 1

Introduction to FEM

5 Constructing MoM Members

IFEM Ch 5 – Slide 1

SLIDE 2

What Are MoM Members?

Skeletal structural members whose stiffness equations can be constructed by Mechanics of Materials (MoM) methods Can be locally modeled as 1D elements

Introduction to FEM

IFEM Ch 5 – Slide 2

SLIDE 3

MoM Members Tend to Look Alike ...

¯ x ¯ z ¯ y z y x

l

g i t u d i n a l d i r e c t i

cross section

One dimension (longitudinal) much larger than the other two (transverse)

Introduction to FEM

IFEM Ch 5 – Slide 3

SLIDE 4

But Receive Different Names According to Structural Function

Bars: transmit axial forces Beams: transmit bending Shafts: transmit torque Spars (aka Webs): transmit shear Beam-columns: transmit bending + axial force

Introduction to FEM

IFEM Ch 5 – Slide 4

SLIDE 5

Common Features of MoM Finite Element Models

i j ¯ x ¯ z ¯ y z y x Internal quantities are defined in the member End quantities are defined at the joints

Introduction to FEM

e

IFEM Ch 5 – Slide 5

SLIDE 6

Governing Matrix Equations for Simplex MoM Element

¯ ¯

From node displacements to internal deformations (strains) From deformations to internal forces From internal forces to node forces

Equilibrium Constitutive Kinematic

If f and u are PVW (Virtual Work) conjugate, B = A

Introduction to FEM

v = B u p = S v f = A p _ _

IFEM Ch 5 – Slide 6

SLIDE 7

Tonti Diagram of Governing Matrix Equations for Simplex MoM Element

Introduction to FEM

v ¯ u ¯ f p ¯ f = A p

Equilibrium

Stiffness

Constitutive Kinematic

p = S v ¯ v = B u ¯

¯ ¯ f = A S B u = K u

IFEM Ch 5 – Slide 7

SLIDE 8

¯ f = AT S B ¯ u = ¯ K¯ u ¯ K = AT S B ¯ K = BT S B B = A

If

Introduction to FEM

Elimination of the Internal Quantities v and p gives the Element Stiffness Equations through Simple Matrix Multiplications

symmetric if S is

IFEM Ch 5 – Slide 8

SLIDE 9

The Bar Element Revisited

Introduction to FEM

i j ¯ x ¯ y ¯ y L ¯ z ¯ fxi

, ¯

u ¯ f , ¯ uxj

−F (a) (b) z y x EA F ¯ x Axial rigidity EA, length L

IFEM Ch 5 – Slide 9

SLIDE 10

The Bar Element Revisited (cont'd)

d = [ −1 1 ] ¯ uxi ¯ ux j

= B¯

u F = E A L d = S d, ¯ f = ¯ fxi ¯ fx j

−1 1

F = AT F

¯ K = AT S B = S BT B = E A L 1 −1 −1 1

y j

Can be expanded to the 4 x 4 of Chapter 2 by adding two zero rows and columns to accomodate u and u

Introduction to FEM

_ _

IFEM Ch 5 – Slide 10

SLIDE 11

Discrete Tonti Diagram for Bar Element

Introduction to FEM

d F

Equilibrium Stiffness Constitutive Kinematic

d = −1 1 ¯ uxi ¯ ux j = B¯ u F = E A L d = S d ¯ f = ¯ fxi ¯ fx j = −1 1 F = ATF = E A L 1 −1 −1 1 u

IFEM Ch 5 – Slide 11

SLIDE 12

The Spar (a.k.a. Shear-Web) Element

Introduction to FEM

i j ¯ x ¯ x ¯ y ¯ y L ¯ z (b) V (a)

z y x GA ¯ f , ¯ uyj

¯ f , ¯ uyi

Shear rigidity GA , length L (e) −V

IFEM Ch 5 – Slide 12

SLIDE 13

Spars used in Wing Structure

(Piper Cherokee)

Introduction to FEM

SPAR COVER PLATES RIB

IFEM Ch 5 – Slide 13

SLIDE 14

The Spar Element (cont'd)

γ = 1 L [ −1 1 ] ¯ uyi ¯ uyj

= B¯

u V = G Asγ = S γ ¯ f = ¯ fyi ¯ fyj

−1 1

V = AT V

¯ f = ¯ fyi ¯ fyj

= AT S B¯

u = G As L 1 −1 −1 1 ¯ uyi ¯ uyj

= ¯

K¯ u

¯ K = 1 −1 −1 1

G As

L

Introduction to FEM

IFEM Ch 5 – Slide 14

SLIDE 15

The Shaft Element

Introduction to FEM

i j ¯ x ¯ y ¯ y L ¯ z (a) (b) z y x GJ ¯ x T T Torsional rigidity GJ, length L ¯ ¯ m , θ

xi xi

¯ m , θ

xj xj

(e) m , θ

xi xi

_ _ m , θ

xj xj

_ _

For stiffness derivation details see Notes

IFEM Ch 5 – Slide 15

SLIDE 16

Matrix Equations for Non-Simplex MoM Element

v = B ¯ u p = Rv ¯ d f = A

T dp

From node displacements to internal deformations at each section From deformations to internal forces at each section From internal forces to node forces

Equilibrium Constitutive Kinematic

Introduction to FEM

IFEM Ch 5 – Slide 16

SLIDE 17

v ¯ u ¯ f p

Equilibrium Constitutive (at each section) Kinematic (at each section)

p = R v ¯

d f = B dp ¯ v = B u ¯ f = B R B dx u

¯ ¯

L

Tonti Diagram of Matrix Equations for Non-Simplex MoM Element (with A=B)

Introduction to FEM

Stiffness

IFEM Ch 5 – Slide 17

SLIDE 18

High-Aspect Wing, Constellation (1952)

Introduction to FEM

IFEM Ch 5 – Slide 18

SLIDE 19

Low-Aspect Delta Wing, F-117 (1975)

Introduction to FEM

IFEM Ch 5 – Slide 19

SLIDE 20

Low-Aspect Delta Wing, Blackhawk (1972)

Introduction to FEM

IFEM Ch 5 – Slide 20

SLIDE 21