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

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 modeled as 1D elements IFEM Ch 5 – Slide 2

Introduction to FEM MoM Members Tend to Look Alike ... cross section z y ¯ x ¯ n o i t c e r i d l a n d i u t g i n o l x z ¯ y One dimension (longitudinal) much larger than the other two (transverse) IFEM Ch 5 – Slide 3

Introduction to FEM 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 IFEM Ch 5 – Slide 4

Introduction to FEM Common Features of MoM Finite Element Models End quantities are defined at the joints z y ¯ x ¯ e j i z ¯ x y Internal quantities are defined in the member IFEM Ch 5 – Slide 5

Introduction to FEM Governing Matrix Equations for Simplex MoM Element From node displacements to internal deformations (strains) _ Kinematic v = B u From deformations to internal forces Constitutive p = S v From internal forces to node forces _ f = A p Equilibrium ¯ If f and u are PVW (Virtual Work) conjugate, B = A ¯ IFEM Ch 5 – Slide 6

Introduction to FEM Tonti Diagram of Governing Matrix Equations for Simplex MoM Element Stiffness T ¯ ¯ ¯ f = A S B u = K u ¯ u ¯ f ¯ ¯ T Kinematic v = B u f = A p Equilibrium p = S v p v Constitutive IFEM Ch 5 – Slide 7

Introduction to FEM Elimination of the Internal Quantities v and p gives the Element Stiffness Equations through Simple Matrix Multiplications f = A T S B ¯ ¯ u = ¯ K ¯ u K = A T S B ¯ B = A If K = B T S B ¯ symmetric if S is IFEM Ch 5 – Slide 8

Introduction to FEM The Bar Element Revisited (a) Axial rigidity EA , length L y ¯ z − F F z ¯ x ¯ x y (b) y ¯ EA ¯ ¯ f xi , ¯ u f , ¯ u xj j i xi xj x ¯ L IFEM Ch 5 – Slide 9

Introduction to FEM The Bar Element Revisited (cont'd) � ¯ � u xi d = [ − 1 1 ] = B ¯ u u x j ¯ F = E A L d = S d , � ¯ � − 1 � � f xi F = A T F ¯ f = = ¯ 1 f x j � 1 � K = A T S B = S B T B = E A − 1 ¯ − 1 1 L Can be expanded to the 4 x 4 of Chapter 2 by adding two _ _ zero rows and columns to accomodate u and u y j yi IFEM Ch 5 – Slide 10

Introduction to FEM Discrete Tonti Diagram for Bar Element Stiffness _ _ 1 −1 = E A u f −1 1 L _ _ u f Kinematic Equilibrium ¯ −1 f xi u xi ¯ −1 1 ¯ F = A T F d = = B ¯ u f = = 1 ¯ u x j ¯ f x j F = E A L d = S d d F Constitutive IFEM Ch 5 – Slide 11

Introduction to FEM The Spar (a.k.a. Shear-Web) Element V (a) y ¯ x ¯ z z ¯ − V Shear rigidity GA , length L s y x y ¯ (b) ¯ ¯ f , ¯ u yj f , ¯ u yi yi yj GA s j i ( e ) x ¯ L IFEM Ch 5 – Slide 12

Introduction to FEM Spars used in Wing Structure (Piper Cherokee) COVER PLATES SPAR RIB IFEM Ch 5 – Slide 13

Introduction to FEM The Spar Element (cont'd) � ¯ � γ = 1 u yi L [ − 1 1 ] = B ¯ u u yj ¯ V = G A s γ = S γ � ¯ � − 1 � � f yi V = A T V ¯ f = = ¯ f yj 1 � 1 � ¯ � � ¯ � � u = G A s f yi − 1 u yi = A T S B ¯ ¯ = ¯ f = K ¯ u ¯ f yj − 1 1 u yj ¯ L � 1 � − 1 G A s ¯ K = − 1 1 L IFEM Ch 5 – Slide 14

Introduction to FEM The Shaft Element Torsional rigidity GJ , length L (a) x ¯ y ¯ _ _ m , θ xj xj z T T _ _ m , θ xi xi z ¯ x y y ¯ GJ (b) m , θ m , θ ¯ ¯ ¯ j ( e ) x ¯ i xi xi xj xj L For stiffness derivation details see Notes IFEM Ch 5 – Slide 15

Introduction to FEM Matrix Equations for Non-Simplex MoM Element From node displacements to internal deformations at each section Kinematic u v = B ¯ From deformations to internal forces at each section Constitutive p = Rv From internal forces to node forces T d p ¯ d f = A Equilibrium IFEM Ch 5 – Slide 16

Introduction to FEM Tonti Diagram of Matrix Equations for Non-Simplex MoM Element (with A=B) Stiffness � L ¯ T f = B R B dx u ¯ ¯ ¯ u 0 ¯ f Kinematic ¯ T ¯ v = B u d f = B d p Equilibrium (at each section) p = R v v p Constitutive (at each section) IFEM Ch 5 – Slide 17

Introduction to FEM High-Aspect Wing, Constellation (1952) IFEM Ch 5 – Slide 18

Introduction to FEM Low-Aspect Delta Wing, F-117 (1975) IFEM Ch 5 – Slide 19

Introduction to FEM Low-Aspect Delta Wing, Blackhawk (1972) IFEM Ch 5 – Slide 20