Key Concepts

The typical approach to engineering analyses of just about anything is to divide the material (or space) into infinitesimally small units, typically cubes, and analyse the forces that the faces of these cubes are subjected to and then integrate over the entire volume to get the net effect of the infinitesimal forces. In beam bending problems you often introduce infinitesimally small spring to hold things together and again integrate up the stretch or compression of the springs or the stored energy in the spring to calculate quantities connected with the elastic deformation of the material.

Forces are vector valued functions meaning that they are defined by both the direction in which they act and their magnitude. In analyzing beam problems this directionality can become confusing because the forces are often named with reference to the particular plane in the beam that you are considering. This reference plance can change without notice!

Shear forces are forces that are act parallel to the plane you are considerind and direct or normal forces  are those acting perpendicular to the plane you are considering. This can get confusing if you lose track of the plane you are considering. 

For example, suppose you have a board laying horizontally. If you place your foot on the middle of the board and stand on it the force acts directly down on the board. In this case the force is a shear force relative to the vertical cross section of the board. However, if you were able to set it up so that you applied a force to the top surface of the board that pulled it to the right and a force to the bottom that pulled it to the left these opposing forces would also be applying shear forces but this time they are shear forces relative to horizontal slice of the board. 

The message is to always check the plane to which the force is being referred or it will become very confusing.

Stress - stress is defined as the force per unit area at a point in or on the material. Its units are force over area. That is N/m^2 (newtons per metre squared) which are also the units of pressure which is calculated as the force per unit area that one substance exerts on another.  N/m^2 is also referred to as a Pascal (Pa). Typical stresses in kite boards are in the order of 10^6 Pa or MPa.

Strain is the fractional change in the length of a material that is subject to a stress. It has no dimensions because it is a ratio of length over length.

Moment – a moment at a point is a quantity equal to the product of the external forces applied to the object by the distance from that force to the point. That is the force times the distance to it. Moments are important because in the same way that all forces must net out to zero if an object is stationery, so too must all the moments net out to zero if the object is not to be rotating (different way of saying the same thing). 

So if an external force is applied at a distance d from a point on the board then we know that to counter that force the board itself must oppose that moment with a moment of equal and opposite value. 

It does this by bending which in turn causes the board material to compress or stretch (ie store energy). These restoring forces at every point in the cross-section of the board create moments about the neutral axis (the axis where the stresses in teh cross-sectino transition from compressive to tensile) which eventually become equal but opposite to the moment bending it. At this point the board stops bending because the internal/ external forces are all balanced. Being able to calculate the moments created in the board as a reaction to the external forces will allow us to determine how much the board will bend.  

Moments have the units of force times distance or Nm (newton-metres)

Elastic Modulus  is an intrinsic property of a material that determines the strain that results from the stress applied to a material. It is defined as the ratio of the stress to the strain and typically represented by the symbol E. E is only a constant over a limited range of stress for any a particular material.  Beyond this ‘elastic limit’ the material undergoes other permanent deformation or failure. Eg snaps of permanently stretches.

The units of E are N/m^2 or Pa, the same as  (stress over strain). 

 

By way of example, e-glass has an E of around 70 GPa (giga pascals or 10^9 Pascals),  PVC foam for cores comes in at around 30 MPa and carbon fibre can be up to 200 GPa. The larger E the smaller the strain (ie the amount it stretches) when subject to a stress. Carbon’s vey high E value means that it does not flex much and so appears to crack without warning. 

It’s important to note that Elastic modulus can vary in different directions in the material. Isotropic materials are ones where E is not dependant on direction and Anisotropic is where is does depend on the direction.

Most of the materials we’ll deal with in board making are isotropic materials.  Even unidirection fiberglass. The actual glass material itself is isotropic but the way that the fibres are held together means that the cloth has properties that depend on direction but this relates to the contruction and not the material properties. 

The elastic modulus, as we shall see, is fundamental to our efforts to model the boards flex. This is because the model of bending (due to moments) that unpins the Bernoulli-Euler equation is one where we think of the face of each cross-section as being covered in infinitely small springs that are either compressed or stretched by the bending action. The elastic modulus tells us how much these springs will stretch or compress when the board is subject to a bending force.

The concept of the elastic modulus is straight forward but there are some complexities introduced when we look for the elastic modulus of structures that use the materials in complex ways. The example of uni-direction glass was mentioned. However, even the case of normal bi-directional glass which has glass fibres running in a woven cloth at 90 degrees to each other makes for some more challenges as the off-axis properties of the structure must be calculated by resolving teh forces into components in the direction of the fibres and also taking account of the shear forces that may be created as a result of these perpendicular forces acting simultaneously.

It should be noted that the elastic modulus is an intrisic property and is not affected by the shape of the cross section. The influence of the shape of the cross section is captured in the Area Moment of Inertia which is often refered to as a 'the measure of a beams resistance to bending'. It takes account of the shape ( demensions, concave etc) and captures it in a single figure. The Area Moment of Inertia (which is sometimes erroneously refered to as the moment of inertia but should not be mixed up with the moment of inertia used in dynamic problems) requires some detailed treatment and so will be defered until a subsequent post.