me 582 f06 - chapter 4 macro mechanical analysis of laminates part 1
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ME 582 Advanced Materials Science Chapter 4 Macromechanical Analysis of Laminates
(Part 1)Dr. Jan Gou Composite Materials Research Laboratory Department of Mechanical Engineering University of South Alabama, Mobile, AL 36688
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
HW #44.1 4.6 4.8 Due Day: 6:00 PM, 10/11/2006, Wednesday.
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Laminate Code
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Laminate Code
[0/-45/90/60/30]T
[0/-45/902/60/0]T
[0/-45/60]S
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Laminate Code
[0/-45/60]s
[0Gr / 45 B ]s
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Special Types of LaminatesSymmetric laminate: for every ply above the laminate midplane, there is an identical ply (meterial and orientation) an equal distance below the midplane Balanced laminate: for every ply at a + orientation, there is another ply at the orientation somewhere in the laminate Cross-ply laminate: composed of plies of either 0 or 90 (no other ply orientation) Quansi-isotropic laminate: produced using at least three different ply orientations, all with equal angles between them. Exhitbits isotropic extensional stiffness properties
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Quiz: Laminate CodeWrite the ply orientation code for the laminates shown below 90 0 -45 45 90 0 -45 45 0 45 0 45 90 45 0 45 0 [(02/90)2/45]S Draw the laminate corresponding to the ply orientation codes listed below
[0 2 / 45 / m 45]s
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Quiz: Special LaminatesFor each laminate shown, determine whether it exhibits the special qualities printed below 0 90 45 90 0 0 0 90 90 0 0 90 90Symmetric Balanced Quasi-isotropic Crossply Symmetric Balanced Quasi-isotropic Crossply Symmetric Balanced Quasi-isotropic CrossplyDr. Jan Gou
0 45 90 -45 -45 90 45 0
ME 582 Advanced Materials Science Department of Mechanical Engineering
1D Isotropic Beam Stress-Strain Relation
1
=
M EI
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Strain-Displacement Equations
Nx = normal force resultant in the x direction (per unit length) Ny = normal force resultant in the y direction (per unit length) Nxy = shear force resultant (per unit length)
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Strain-Displacement Equations
Mx = bending moment resultant in the yz plane (per unit length) My = bending moment resultant in the xz plane (per unit length) Mxy = twisting moment resultant (per unit length)ME 582 Advanced Materials Science Department of Mechanical Engineering Dr. Jan Gou
Strain-Displacement EquationsAssumptions:
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Strain-Displacement Equations
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Laminate Strains
k k0 0 x , y = Midplane normal strains in the laminate
0 xy =
Midplane shear strain in the laminate
k x , k y = Bending curvatures in the laminate k xy =z=Twisting curvature in the laminate Distance from the midplane in the thickness directionDr. Jan Gou
ME 582 Advanced Materials Science Department of Mechanical Engineering
Strain and Stress in a Laminate
k
k
k k k k
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Strain and Stress in a Laminate
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Coordinate Locations of Plies in a Laminate
Ply 1:
Ply k:
Ply n:
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Force and Moment Resultant
[A] extensional stiffness matrix relating the resultant in-plane forces to the in-plane strains. [B] coupling stiffness matrix coupling the force and moment terms to the midplane strains and midplane curvatures.ME 582 Advanced Materials Science Department of Mechanical Engineering Dr. Jan Gou
Force and Moment Resultant
[B] coupling stiffness matrix coupling the force and moment terms to the midplane strains and midplane curvatures. [D] bending stiffness matrix relating the resultant bending moments to the plate curvatures.ME 582 Advanced Materials Science Department of Mechanical Engineering Dr. Jan Gou
Force and Moment Resultant
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated Composites
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 1: Find the reduced stiffness matrix [Q] for each ply
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 2: Find the transformed stiffness matrix [Q] using the reduced stiffness matrix [Q] and the angle of the ply
c = cos() s = sin()
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 3: Find the coordinate of the top and bottom surface of each ply
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 4: Find three stiffness matrices [A], [B], and [D]
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 5: Substitute the three stiffness matrices [A], [B], and [D] and the applied forces and moments
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 6: Solve the six simultaneous equations to find the midplane strains and curvatures
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 7: Find the global strains in each ply
k k
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 8: Find the global stresses using the stress-strain equation
k
k
k
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 9: Find the local strains using the transformation equation
k
k
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Analysis Procedures for Laminated CompositesStep 10: Find the local stresses using the transformation equation
k
k
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Example 4.2
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.2Find the reduced stiffness matrix [Q]:
Find the transformed reduced stiffness matrix [Q] for each ply:
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.2
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.2
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.2
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.2
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou
Solution to Example 4.3
ME 582 Advanced Materials Science Department of Mechanical Engineering
Dr. Jan Gou