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1. 4.1 Introduction
• In Chapter 2, stress–strain equations were developed for a single lamina. A real
structure, however, will not consist of a single lamina but a laminate consisting
of more than one lamina bonded together through their thickness.
• Why? First, lamina thicknesses are on the order of 0.005 in. (0.125 mm),
implying that several laminae will be required to take realistic loads (a typical
glass/epoxy lamina will fail at about only 750 lb per inch [131,350 N/m] width
of a normal load along the fibers). Second, the mechanical properties of a
typical unidirectional lamina are severely limited in the transverse direction

2. If one stacks several unidirectional layers, this may be an
optimum laminate for unidirectional loads. However, for complex
loading and stiffness requirements, this would not be desirable.
This problem can be overcome by making a laminate with layers
stacked at different angles for given loading and stiffness
requirements.
This approach increases the cost and weight of the laminate and
thus it is necessary to optimize the ply angles. Moreover, layers of
different composite material systems may be used to develop a
more optimum laminate.

3. • Similar to what was done in Chapter 2, the macromechanical analysis
will be developed for a laminate. Based on applied in-plane loads of
extension, shear, bending, and torsion, stresses and strains will be
found in the local and global axes of each ply. Stiffnesses of whole
laminates will also be calculated. Because laminates can also be
subjected to hygrothermal loads of temperature change and moisture
absorption during processing and servicing, stresses and strains in each
ply will also be calculated due to these loads.

4. Behavior of laminate
• Elastic moduli
• Stacking position
• Thickness
• Angle of orientation

5. Types of Laminates
Based on the stacking sequence, composite laminates can be categorized as:
• Angle-ply Laminate
A laminate is called an angle-ply laminate if it has layers of the same
thickness and material, and are oriented at θ and −θ.
• Cross-ply Laminate
A laminate is called a cross-ply laminate if all the layers have the
orientation 0° and 90°.
• Balanced Laminate
A laminate is called a balanced laminate when it has pairs of layers of the
same thickness and material, and the angles of the layers have opposite signs.
Balanced laminates can also have layers at 0° and 90°.

6. Symmetric Laminate
A laminate is called symmetric when the material, angle, and thickness of
the layers are the same above and below the midplane.
Antisymmetric Laminate
A laminate is called antisymmetric when the material and thickness of the
layers are the same above and below the midplane, but the orientation of the
layers have opposite signs above and below the midplane.

7. 4.2 Laminate Code
• A laminate is made of a group of single layers bonded to each other.
Each layer can be identified by its location in the laminate, its
material, and its angle of orientation with a reference axis (Figure
4.1). Each lamina is represented by the angle of ply and separated
from other plies by a slash sign.
• The first ply is the top ply of the laminate. Special notations are used
for symmetric laminates, laminates with adjacent lamina of the same
orientation or of opposite angles, and hybrid laminates.

13. 4.3 Stress–Strain Relations for a Laminate
• 4.3.1 One–Dimensional Isotropic Beam Stress–Strain Relation
• Consider a prismatic beam of cross-section A (Figure 4.2a) under a
simple load P; the normal stress at any cross-section is given by
• The corresponding normal strain for a linearly elastic isotropic beam is
• Where E is the Young’s modulus of the beam. Note the assumption that
the normal stress and strain are uniform and constant in the beam and
are dependent on the load P being applied at the centroid of the cross
section.

14. Now consider the same prismatic beam in a pure bending moment M
(Figure 4.2b). The beam is assumed to be initially straight and the applied
loads pass through a plane of symmetry to avoid twisting. Based on the
elementary strength of material assumptions,
• The transverse shear is neglected
• Cross-sections retain their original shape
• The yz-plane before and after bending stays the same and normal to the
x-axis.

15. FIGURE 4.2
A beam under (a) axial load, (b) bending moment, and (c) combined axial and bending moment.

17. 4.3.2 Strain-Displacement Equations
In the previous section, the axial strain in a beam was related to the midplane strain and
curvature of the beam under a uniaxial load and bending. In this section, similar relationships
will be developed for a plate under in-plane loads such as shear and axial forces, and bending
and twisting moments (Figure 4.3). The classical lamination theory is used to develop these
relationships.
The following assumptions are made in the classical lamination theory to develop the
relationships:
• Each lamina is orthotropic.
• Each lamina is homogeneous.
• A line straight and perpendicular to the middle surface remains
straight and perpendicular to the middle surface during deformation

19. WHAT ARE THE USES OF LAMINATES?
• Laminates can be used for a variety of purposes in both residential and
commercial establishments. Decorative sheets, especially those used for
furniture and indoor elements, are widely used to improve the appeal of
furniture pieces while giving them a certain degree of protection from
damage.
• Decorative laminates are available in a wide variety of colors, textures, and
designs, allowing property owners and designers to manifest their creative
vision.
• In commercial and industrial settings, laminates are used for functional
purposes. These laminates are thicker, stronger and incredibly resistant. In
addition, many of the laminates used in the industry are designed to provide
antibacterial properties and even fire retardation.

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