L12

Lecture 12: Elastic Behaviour of Laminates

1. What is a laminated composite?

A laminated composite is made by stacking several thin composite layers on top of each other. Each layer is usually a unidirectional lamina, meaning the fibres in that layer mainly run in one direction.

In earlier lectures, a single unidirectional composite layer was considered. In that case, the fibres were usually analysed in simple directions, such as:

  • along the fibres, called the 0° direction;

  • across the fibres, called the 90° direction;

  • or at an angle to the fibres, called off-axis loading.

A laminate is more complicated because it contains many layers, and each layer can have fibres pointing in different directions. For example, one laminate might contain layers at 0°, 90°, +45° and −45°. This is very common in aerospace structures because real components experience loads from many directions.

The lecture slides show that laminated composites are built from stacked layers, and each layer may have its own fibre direction and thickness. This means the overall elastic behaviour of the laminate depends on the behaviour of every individual layer.

2. Why are laminated composites harder to analyse?

A single unidirectional layer is easier to analyse because the fibre direction is known and fixed. However, in a laminate, every layer can respond differently to the same applied load.

For example, if a tensile load is applied in the x-direction:

  • a 0° layer carries the load mainly along the fibres;

  • a 90° layer carries the load mainly across the fibres;

  • a +45° layer experiences a combination of normal and shear effects;

  • a −45° layer responds differently again.

This means the stresses and strains must be calculated for each layer separately. After that, the effects from all the layers must be combined to predict the behaviour of the whole laminate.

This is why laminate calculations are not simple to do by hand. The lecture explains that computational tools or software are often used because the analysis involves many layers, different fibre angles, transformation matrices, compliance matrices, stiffness matrices, strains, stresses, forces and moments.

3. Fibre angles in laminated composites

In laminated composites, fibres can be placed at many different angles. The common angles include:

  • 0°;

  • 90°;

  • +45°;

  • −45°;

  • +60°;

  • −60°;

  • 30°;

  • 120°;

  • 135°.

The angle chosen depends on the expected loading direction.

If a structure is expected to carry load mainly in one direction, many fibres may be placed in that direction. For example, if the main tensile load acts along the length of a component, 0° fibres are useful because they provide high stiffness and strength along that direction.

If the structure is expected to experience loads from several directions, different fibre angles are used so that the laminate can resist loads more evenly.

4. Normal stress and shear strain in a single layer

For a single composite layer, the fibre angle affects whether normal stresses create shear strain.

A normal stress is a stress that pulls or pushes directly on a material. A shear strain is a distortion where the shape changes angularly, like a rectangle becoming a parallelogram.

The lecture states that there are three special cases where normal stresses do not produce shear strain in a layer:

  1. when the load is applied along the fibres;

  2. when the load is applied perpendicular to the fibres;

  3. at one special intermediate angle.

For most other angles, normal loading causes shear effects in the layer. The slides show this visually using different fibre orientations, where most angled layers deform with shear when loaded.

This is important because shear deformation can affect stiffness, strength and failure behaviour.

5. The 0° direction

The 0° direction means the load is applied along the fibres.

This is usually the strongest and stiffest direction in a continuous fibre composite because the fibres carry most of the load. Fibres such as carbon fibre or glass fibre are usually much stiffer and stronger than the polymer matrix.

When a tensile load is applied along the fibres, the fibres and matrix stretch together. There is normal strain along the fibre direction and some contraction in the transverse direction due to Poisson’s effect. However, there is no significant shear strain generated between the fibres.

6. The 90° direction

The 90° direction means the load is applied across the fibres.

This direction is usually weaker and less stiff than the 0° direction because the matrix plays a larger role in carrying the load. The fibres do not carry the load as efficiently because the load is not aligned with them.

When a tensile load is applied across the fibres, the material extends in the transverse direction and contracts in the other direction. Again, this case does not produce shear strain in the layer, but the stress distribution is less uniform than in the 0° direction.

7. Angled layers and shear strain

When fibres are placed at angles such as 30°, 45° or 60°, the applied load is no longer purely along or across the fibres. The load must be resolved into components relative to the fibre direction.

This means the layer experiences a mixture of:

  • stress along the fibres;

  • stress across the fibres;

  • shear stress between fibres and matrix.

Angled layers are useful because they help the laminate resist shear and off-axis loads. However, they also make the analysis more complex.

For example, a ±45° laminate is often useful for shear-dominated loading because the fibres are well oriented to resist twisting and in-plane shear.

8. Symmetrical laminates

A symmetrical laminate has a mirrored stacking sequence about the mid-plane.

The mid-plane is the central plane through the thickness of the laminate. If the layers above the mid-plane are mirrored below the mid-plane, the laminate is symmetrical.

For example:

[0/45/90/90/45/0][0/45/90/90/45/0][0/45/90/90/45/0]

is symmetrical because the top half mirrors the bottom half.

Another example is:

[90/−45/0/0/−45/90][90/-45/0/0/-45/90][90/−45/0/0/−45/90]

This is also symmetrical if the layers are arranged as a reflection about the centre.

The important rule is that the matching layers must have:

  • the same fibre angle;

  • the same thickness;

  • the same material;

  • the same distance from the mid-plane.

The slides state that symmetrical laminates need reflectional symmetry about the mid-plane.

9. Why symmetrical laminates are important

Symmetrical laminates are important because in-plane stresses do not produce out-of-plane bending or twisting.

This means that if a tensile or compressive load is applied in the plane of the laminate, the laminate should stretch or compress without bending or twisting out of plane.

This is useful in engineering design because unwanted bending or twisting can reduce performance, create assembly problems and increase the chance of damage.

In simple terms, symmetry helps the laminate behave more predictably.

10. Unsymmetrical laminates

An unsymmetrical laminate does not have mirror symmetry about the mid-plane.

For example:

[0/90/0/90][0/90/0/90][0/90/0/90]

is unsymmetrical because the layer sequence is not mirrored about the centre.

Unsymmetrical laminates are often avoided because they can warp, bend or twist due to thermal effects or mechanical loading.

The lecture explains that when an unsymmetrical laminate is heated or cooled, different layers try to expand or contract in different directions. Because the layers are bonded together, they cannot freely deform, so the whole laminate may curve or twist.

The slides show that unsymmetrical layups can warp during curing, while symmetrical layups are more stable.

11. Thermal expansion in laminates

Thermal expansion means a material changes size when temperature changes.

In a composite layer, the resin usually expands more than the fibres. Fibres, especially carbon fibres, often have very low thermal expansion along their length.

This causes a problem in laminates because different layers have fibres in different directions. One layer may resist expansion in the x-direction, while another layer may resist expansion in the y-direction.

If the laminate is unsymmetrical, these different expansions are not balanced through the thickness. This can cause the laminate to bend, twist or warp.

This is especially important during curing, because composites are often cured at elevated temperatures and then cooled down to room temperature.

12. Poisson’s contraction in laminates

Poisson’s contraction occurs when a material is stretched in one direction and contracts in the perpendicular direction.

In a laminate, each layer may contract differently depending on its fibre direction. If the laminate is not properly balanced or symmetrical, these contractions can cause internal stresses, warping and possible microcracking.

The lecture explains that this can lead to damage around the fibre-matrix interface. This is a serious issue because the interface is important for transferring load between the matrix and fibres.

13. Balanced laminates

A balanced laminate is designed so that normal stresses do not create shear strains, and shear stresses do not create normal strains.

In simple terms, a balanced laminate reduces coupling between normal deformation and shear deformation.

The lecture states that balanced laminates require special angular symmetry in the layer directions. The common balanced systems are:

  • 6-fold symmetry, such as 0° / 60° / 120°;

  • 8-fold symmetry, such as 0° / 45° / 90° / 135°;

  • 10-fold symmetry, such as 0° / 30° / 60° / 90° / 120° / 150°.

The slides clearly state that 0° / 60° / 120° and 0° / 45° / 90° / 135° combinations are balanced, but 0° / 90° alone is not balanced.

14. Understanding 120° and 135°

Sometimes angles such as 120° and 135° are written differently.

A 120° layer can also be thought of as a −60° layer because it is 60° in the opposite direction from the 0° axis.

A 135° layer can also be written as −45°.

Therefore:

[0/45/90/135][0/45/90/135][0/45/90/135]

can also be written as:

[0/45/90/−45][0/45/90/-45][0/45/90/−45]

This is why many aerospace laminates are written using ±45° notation.

15. Why 0°/90° is not balanced

A laminate made only from 0° and 90° layers is not considered balanced according to the lecture because it does not satisfy the required 6-fold, 8-fold or 10-fold angular symmetry.

Although 0° and 90° layers can give stiffness in two perpendicular directions, they do not provide the full angular balance needed to eliminate certain normal-shear coupling effects.

This is why balanced laminates usually include angled plies such as ±45°, ±60° or 30° increments.

16. Bending and twisting coupling

Most laminates twist when bending stresses are applied.

Bending and twisting coupling means that when a laminate is bent, it also tends to twist. This is usually undesirable unless the twisting is intentionally designed into the structure.

The lecture explains that it is possible to reduce or eliminate bending-twisting coupling using certain laminate designs. One example is a balanced anti-symmetric laminate.

The slides state that balanced anti-symmetric laminates can eliminate twisting under bending stresses, but it is impossible to eliminate every type of coupling at once.

17. Anti-symmetric laminates

An anti-symmetric laminate does not have normal mirror symmetry. Instead, the stacking sequence is arranged so that fibre angles are repeated in a way that balances certain effects without being symmetrical.

For example, a 6-fold balanced anti-symmetric laminate might be:

[0/60/120/0/60/120][0/60/120/0/60/120][0/60/120/0/60/120]

This laminate is balanced because it uses the 0°, 60° and 120° directions, but it is not symmetrical about the mid-plane.

Balanced anti-symmetric laminates can be useful when the goal is to reduce twisting caused by bending.

18. Four main laminate types

The lecture classifies laminates using two questions:

  1. Is the laminate balanced?

  2. Is the laminate symmetrical?

This gives four main categories.

19. Unbalanced and unsymmetrical laminates

An unbalanced and unsymmetrical laminate is neither balanced nor symmetrical.

Example:

[0/90/0/90][0/90/0/90][0/90/0/90]

This laminate is not balanced because it does not meet the 6-fold, 8-fold or 10-fold angular requirements. It is also not symmetrical because the layers are not mirrored about the mid-plane.

These laminates are generally less desirable because they can show shear coupling, bending, twisting and warping.

The slides give examples such as 0/90/0/90, 30/150/30/150 and 0/45/0/45 as unbalanced unsymmetric laminates.

20. Balanced but unsymmetrical laminates

A balanced but unsymmetrical laminate satisfies the balance requirement, but it is not mirrored about the mid-plane.

Example:

[0/60/120/0/60/120][0/60/120/0/60/120][0/60/120/0/60/120]

Another example:

[0/45/90/135/0/45/90/135][0/45/90/135/0/45/90/135][0/45/90/135/0/45/90/135]

These laminates reduce normal-shear coupling, but they may still experience out-of-plane bending or twisting because they are not symmetrical.

The slides list these as balanced unsymmetric laminates.

21. Unbalanced but symmetrical laminates

An unbalanced but symmetrical laminate is mirrored about the mid-plane but does not meet the balance requirement.

Example:

[0/30/90/90/30/0][0/30/90/90/30/0][0/30/90/90/30/0]

This laminate is symmetrical because the stacking sequence reflects about the middle. However, it is not balanced because it does not satisfy the 6-fold, 8-fold or 10-fold angular symmetry.

This type of laminate avoids bending or twisting from in-plane loading because it is symmetrical, but it may still show normal-shear coupling because it is not balanced.

The slides show examples such as 0/30/90/90/30/0 and refer to this type as unbalanced symmetric.

22. Balanced and symmetrical laminates

A balanced and symmetrical laminate satisfies both conditions.

It has angular balance and mirror symmetry about the mid-plane.

Example:

[0/45/90/135/135/90/45/0][0/45/90/135/135/90/45/0][0/45/90/135/135/90/45/0]

This laminate is balanced because it uses the 8-fold set of 0°, 45°, 90° and 135°. It is also symmetrical because the second half mirrors the first half.

Balanced and symmetrical laminates are very useful because they reduce several unwanted coupling effects. They are especially common in aerospace engineering.

23. Quasi-isotropic laminates

A quasi-isotropic laminate is a special type of balanced and symmetrical laminate.

“Quasi-isotropic” means the laminate behaves almost like an isotropic material in the plane of the laminate. An isotropic material has the same properties in every direction. A quasi-isotropic composite does not become perfectly isotropic, but it has more uniform stiffness in different in-plane directions than a simple 0° or 90° laminate.

A common quasi-isotropic layup is:

[0/45/90/−45/−45/90/45/0][0/45/90/-45/-45/90/45/0][0/45/90/−45/−45/90/45/0]

or equivalently:

[0/45/90/135/135/90/45/0][0/45/90/135/135/90/45/0][0/45/90/135/135/90/45/0]

The slides state that balanced symmetric laminates with 0°, 45°, 90° and 135° are often known as quasi-isotropic laminates.

24. Why quasi-isotropic laminates are common in aerospace

Quasi-isotropic laminates are common in aerospace because aircraft and spacecraft structures often experience loads from many directions.

A wing skin, fuselage panel, fairing or control surface may experience tension, compression, shear, bending and torsion. If fibres were only placed in one direction, the structure would be very strong in that direction but weak in others.

By using 0°, 90°, +45° and −45° layers, the laminate can resist loads more evenly.

This is useful when the exact loading direction changes during service or when the structure must carry complex combined loads.

25. Problems caused by unsymmetrical laminates

Unsymmetrical laminates can suffer from warping.

Warping is when the laminate curves or distorts instead of staying flat.

Warping can happen because of:

  • thermal expansion during heating or cooling;

  • Poisson’s contraction during mechanical loading;

  • mismatch between layers with different fibre angles;

  • residual stresses from curing;

  • different stiffnesses through the thickness.

Warping is a design problem because it can affect dimensional accuracy. It can also cause internal stresses and microcracking.

The slides show that thermal expansion or Poisson’s contraction can lead to warping if the laminate is not properly balanced.

26. Why curvature matters in laminate analysis

Curvature means the laminate bends or curves through its thickness.

For symmetrical laminates, analysis is simpler because the laminate does not develop significant curvature from in-plane loading. Therefore, it is reasonable to assume that the strains in the layers are the same or easier to relate.

For asymmetric laminates, curvature must be included. This means that the strain in each layer depends on both:

  • the mid-plane strain;

  • the curvature through the thickness.

This makes the analysis more complicated.

The slides state that for symmetric laminates, the strains in each layer can be assumed to be the same, while for asymmetric laminates, curvature may modify the strains.

27. How laminate elastic behaviour is calculated

The general process is:

  1. Define the stacking sequence.

  2. Define the material properties of each layer.

  3. Define the thickness of each layer.

  4. Define the fibre angle of each layer.

  5. Calculate strains in each layer.

  6. Use off-axis loading equations to calculate stresses in each layer.

  7. Sum the forces from all layers.

  8. Use layer thickness information to calculate moments.

  9. Predict the overall laminate stiffness and deformation behaviour.

The slides explain that once the strains and curvatures are known, the off-axis loading method can be used to calculate stresses and forces in each layer. The forces are then summed to obtain the total laminate forces, and the thickness information can be used to calculate moments.

28. Why software is used for laminate analysis

Software is commonly used because laminate analysis involves many repeated calculations.

The lecturer mentioned tools such as:

  • The Laminator;

  • E-Laminate;

  • finite element software such as ANSYS.

These tools can calculate laminate properties using classical laminate theory and matrix methods.

A user can usually input:

  • fibre and matrix material;

  • elastic modulus in different directions;

  • shear modulus;

  • Poisson’s ratio;

  • thermal expansion coefficients;

  • strength values;

  • number of layers;

  • fibre angle of each layer;

  • layer thickness;

  • applied loads.

The software can then output properties such as:

  • laminate stiffness in the x-direction;

  • laminate stiffness in the y-direction;

  • shear stiffness;

  • Poisson’s ratios;

  • thermal expansion behaviour;

  • stresses and strains in layers.

The slides show screenshots of laminate analysis software where users can select material properties, input layups and analyse laminate behaviour.

29. Important warning about units

Some laminate analysis tools use imperial units instead of metric units.

This is important because using the wrong units can produce incorrect results. If a software package expects imperial units, values must be converted correctly before inputting data.

For example:

  • stiffness may need to be entered in psi instead of Pa or MPa;

  • thickness may need to be entered in inches instead of millimetres;

  • loads may need to be entered in pounds-force instead of newtons.

Always check the units before using laminate software.

30. Key exam-style points

A laminated composite is made by stacking multiple unidirectional layers with different fibre directions.

The elastic properties of a laminate are harder to calculate than those of a single unidirectional layer because each layer has a different fibre angle and therefore a different stress-strain response.

Normal stresses give no shear strain for three special cases: along the fibres, perpendicular to the fibres and one special intermediate angle. Most other angles create shear.

A symmetrical laminate has reflectional symmetry about the mid-plane. It prevents in-plane stresses from producing out-of-plane bending or twisting.

A balanced laminate reduces coupling between normal stress and shear strain. It requires 6-fold, 8-fold or 10-fold angular symmetry.

A balanced anti-symmetric laminate can reduce twisting when bending stresses are applied.

A quasi-isotropic laminate is balanced and symmetrical, usually using 0°, 90°, +45° and −45° layers. It gives almost uniform in-plane properties.

Unsymmetrical laminates can warp due to thermal expansion, Poisson’s contraction and residual stresses.

For symmetric laminates, analysis is simpler because curvature effects are reduced. For asymmetric laminates, curvature must be included, making the calculations more complex.

31. Quick examples

A laminate written as:

[0/90/0/90][0/90/0/90][0/90/0/90]

is unbalanced and unsymmetrical.

A laminate written as:

[0/60/120/0/60/120][0/60/120/0/60/120][0/60/120/0/60/120]

is balanced but unsymmetrical.

A laminate written as:

[0/30/90/90/30/0][0/30/90/90/30/0][0/30/90/90/30/0]

is symmetrical but unbalanced.

A laminate written as:

[0/45/90/135/135/90/45/0][0/45/90/135/135/90/45/0][0/45/90/135/135/90/45/0]

is balanced and symmetrical. It is also a quasi-isotropic laminate.

32. Final summary

Laminated composites are designed by stacking fibre-reinforced layers at different angles. The fibre directions control stiffness, strength, shear behaviour, bending behaviour and thermal distortion.

The most important design ideas from this lecture are symmetry and balance. Symmetry helps prevent unwanted bending and twisting from in-plane loading. Balance helps prevent coupling between normal stresses and shear strains. A laminate that is both balanced and symmetrical is usually more stable and predictable, which is why these layups are widely used in aerospace applications.