Calculating Effective Rigidities of a Laminated Composite Beam (Classical Laminate Theory)


The application of fibre-reinforced composite materials in the aerospace industry extends from commercial to military aircraft, such as the Boeing F18, B2 Stealth Bomber, AV-8B Harrier (Jones, 1998). The attractiveness of composites lies in their mechanical properties; such as weight, strength, stiffness, corrosion resistance, fatigue life. Composites are widely used for control surfaces such as ailerons, flaps, stabilizers, rudders, as well as rotary and fixed wings. That is why the analysis of composite structures is imperative for aerospace industry. The main advantage of composites is their flexibility in design. Mechanical properties of the laminate can be altered simply by changing the stacking sequence, fibre lay-up and thickness of each ply which leads to optimization in a design process.


The composite beam is modeled based on the chord-wise bending moment (about the z-axis) being small compared to the span-wise moment (about the y axis, see Figure 2). The chord-wise moment is then neglected. The composite material pertaining to this research is a unidirectional fibre reinforced composite material. The given information of any unidirectional composite material is the elastic modulus in both the longitudinal and transverse axis (see Figures 1 and 2), Poison’s ratio and the shear modulus in the principle directions.

 Effective rigidities for a solid cross-section

The reduced stiffness constants in the material principle directions are:

where T  is the transformation matrix which is used to transform the reduced stiffness constants from the principal material fibre directions to a global (x, y, z) beam coordinates.

Then, the resulting transformed reduced stiffness constants for a unidirectional or orthotropic composite from its principal directions is (Berthelot, 1999):

Both equations (above) can be merged into a single equation commonly known as the “Constitutive Equation”. The constitutive equation describes the stiffness matrix of a laminate plate. The resultant forces and moments are functions of the in-plane strains and curvatures (Berthelot, 1999).

where  is the distance from the mid-plane of the laminate (Figure 3).

For a bending-torsion coupling behaviour the chord wise moment Mx is assumed to be zero so that the kx curvature can be eliminated from (above) and then the matrix  equation (11) reduces to the following form:


The EI, GJ and K represent the effective rigidities of the beam in the global (x, y, z) coordinate system. EI, GJ, and Krepresent, respectively, the bending rigidity, torsion rigidity and bending-torsion coupled rigidity. The effective rigidities are functions of ply angle, thickness, and stacking sequence.



Six Lessons On Innovation From The Wright Brothers

When Wilbur and Orville Wright managed to build and fly an airplane, you might imagine that the world was immediately dazzled by their amazing achievement.

You’d be wrong.

As David McCullough chronicles in his excellent book The Wright Brothers, many of the “most prominent engineers, scientists, and original thinkers of the nineteenth century had been working on the problem of controlled flight,” without success. The endeavor was fraught with hazards that included “humiliating failure, injury, and, of course, death, (but also)… the inevitable prospect of being mocked as a crank, a crackpot, and in many cases with good reason.”


Lesson #1: Don’t worry about failure

John T. Daniels, who witnessed their first successful flights, said later, “It wasn’t luck that made them fly; it was hard work and common sense; they put their whole heart and soul and all their energy into an idea and they had the faith.”

Lesson #2: Have a publicity plan, but don’t expect instant success

The brothers had a preexisting plan to alert the media when they finally achieved success. It involved notifying newspapers and the Associated Press, which they did. A smattering of largely inaccurate accounts appeared in some newspapers, and the story almost instantly disappeared.

Lesson #3: Keep going, no matter what

Fortunately, the brothers were not obsessed with media attention; they were obsessed with their idea. So they kept trying to improve and test their machine.

Locals and local media either ignored them or felt sorry for them. One editor recalled, “They seemed like well-meaning, decent enough young men. Yet there they were, neglecting their business to waste their time day after day on that ridiculous flying machine.”

As they enjoyed one success after another, almost no one believed them or took notice. Today, we understand the immense importance of airplanes to our society and economy, but in those days almost no one had the imagination to take these silly inventors seriously.

Lesson #4: Accept help from strange quarters

After dozens and dozens of successful flights, the brothers were still being ignored by the media. The person who changed this was not at all what you would expect. He didn’t write for The Washington Post, The New York Times, or the AP. He was Amos Root, a deeply religious man who shared his thoughts through his company’s trade journal, Gleanings in Bee Culture.

Yes, Root’s company sold beekeeping supplies, and he had a personal interest in human flight. Root had been persistent in reaching out to the brothers and asking for permission to witness their tests. He was there the first time they flew their machine in a complete circle.

Amos Root wrote about this achievement in his beekeeping publication and sent a copy to the editor of Scientific American, who – you guessed it – ignored the news.

Lesson #5: Don’t be surprised when your breakthrough isn’t your breakthrough

Word was trickling out, but the key word here is trickling. There was some communication with the U.S. and French governments, and with private investors. But the brothers were still pretty much on their own, struggling to keep their endeavor moving forward.

Their tests continued to advance. Instead of flying, say, 1000 feet, the duration of their test flights rose to 11, 12 and then 15 miles.

Lesson #6: Tolerate failure, but avoid disaster

The brothers had one cardinal rule that served them well. They never flew together. In the event of a fatal crash, they did not want their program to die, too. If one of them survived, the initiative could still continue.

In 1906, Scientific American took notice and published a serious article about the brothers’ efforts. The patent they applied for in 1903 was also granted in 1906. The world started to take notice. Crowds gathered to watch their flights. The brothers had succeeded, but only because they kept going when the world was too preoccupied to recognize the wisdom and importance of their efforts.

Bonus lesson: test, test, test

The Wright brothers didn’t sit in an office and dream. They didn’t create a Powerpoint and pitch an unformed idea. They went out into the field (literally) and tested their prototypes, risking their lives in the process. Success eventually came because they built something of value that the world needed, even if the world didn’t recognize it yet.

Test your ideas. Prove your theories. Make people understand.


Sturdy Quadcopter Build

Sturdy Quadcopter Build


Comprehensive guide to build a simple quadcopter for beginners.
With skills like basic soldering and electronics, Chris Schroeder shows you step-by-step how to build from scratch a quadcopter. The guide is intended for beginner users having a plug-and-play construction system and a simple programming code.

All components are purchased from online stores while for programming the designer used a flight control board and a programming card. The values for the flying robot are set by the brushless Electronic Speed Controller (ESC).

Below is an impressive list of the main parts needed to build the quadcopter.