Classical Laminated Plate Theory

Combining Constituents Micro Mechanics Experimental Data CONSTITUENTS STRUCTURE COMPOSITE STRUCTURAL ELEMENT ELEMENTARY STRUCTURE Ey Finite Element Analysis Ex G 

Material Forms

Plane-Stress Assumption Fiber-reinforced materials are utilized in beams, plates, cylinders and other structures Typically one characteristic geometric dimension is an order of magnitude less than the other two Three of the six components of stress are generally much smaller than the other three

Plane Stress Inaccuracies Errors in analysis near edges The stresses s3, t23, t13 lead to delaminations Bonded joints can not be modeled Adhesive or cocured interface can not be evaluated The stress components equated to zero are forgotten and no attempt is made to estimate their magnitude erroneously assumed that e3 is zero

Stress Transformation

Compliance Transformation Equations

Reduced Stiffness Transformation

Classical Lamination Theory The influence of fiber direction, stacking arrangements, material properties, and more on structural response

Laminate Coordinate System Laminate thickness H Layer thickness h not all layers same h kth layer - hk z-axis downward from geometric midplane can be between layers can be within a layer Fiber angles identified relative to x axis

Laminate Nomenclature Layer 1 is the most -z Layer N is the most +z To catergorize a laminate as symmetric a mirror about the geometric midplane material properties fiber orientation thickness of layer

The Kirchhoff Hypothesis Mid 1800’s, simplified analysis Beams, plates, shells metal, wood, concrete, and other materials

Initially Flat Laminated Plate Acted Upon by Various Loads applied moments, M distributed loads, q inplane loads, N point loads, P Multiple layers of fiber reinforced material Fibers parallel to the plane of the plate Layers are perfectly bonded

Deformation of Lines Normal to Geometric Midplane Before deformation are straight Despite the deformations caused by the applied loads, line AA’ remains straight and normal to the deformed geometric midplane and does not change length

Consequences of Kirchhoff Hypothesis

Implications of the Kichhoff Hypothesis in X-Z Plane No through-thickness strain Small deformations Two components of translation uo in x direction, horizontal translation wo in z direction, vertical translation One component of rotation about y-axis

Resulting Displacement Field in XZ plane

Resulting Displacement Field in YZ plane

Strain-Displacement Relations from Theory of Elasticity

Laminate Strains

Laminate Strains Composed of Two Parts Extensional Stain of the Reference Surface Curvature of the Reference Surface inverse of the radius of curvature involves more than just second derivative For small strains second derivative and curvature identical

Strain Notation

Laminate Strains using Revised Notation The gyz, and gxz are zero because the Kirchhoff hypothesis assumes that lines perpendicular to the reference surface before deformation remain perpendicular after the deformation; right angles in the thickness direction do not change when the laminate deforms

Laminate Stresses

[0/90]s Laminate, Axial 1000me Laminate Stress & Strain

[0/90]s Laminate, Axial 1000me Material Stress & Strain

Aluminum, Axial 1000me

[0/90]s Laminate, kxo 3.33 m-1 Laminate Stress & Strain

[0/90]s Laminate, kxo 3.33 m-1 Material Stress & Strain

Aluminum, kxo 3.33 m-1

Definitions of Stress Resultants Stress in each ply varies through the thickness It is convenient to define stresses in terms of equivalent forces acting at the middle surface Stresses at the edge can be broken into increments and summed The resulting integral is defined as the stress resultant, Ni [force per length]

Stress Resultant in X direction

Stress and Moment Resultants bend bend twist

Putting the Resultants in Matrix Form and Summing

Relating Stress to Strain

Performing the Integration

Defining Laminate Stiffness Terms

Constitutive Equations in Matrix Form

Symmetric Laminates For every layer to one side of the laminate reference surface with a specific thickness, material properties, and fiber orientation, there is another layer an identical distance on the opposite side All components of [B] are zero 6x6 set of equations decouples into two 3x3 sets of equations

Balanced Laminates For every layer with a specified thickness, material properties, and fiber orientation, there is another layer with the identical thickness, material properties, but opposite fiber orientation somewhere in the laminate If a laminate is balanced, A16 and A26 are always zero Q16 & Q26 from opposite orientation have opposite signs

Effective Engineering Properties of a Laminate