Tangent and Secant

Diagram to illustrate isothermal tangent and secant bulk... 

Figure 1-9. Stress-strain curve showing tangent and secant modulus.

Figure 1.1(b) Idealized load/deflection characteristic for a fibre which behaves in a non-linear manner. Note definitions of tangent and secant modulus. 

Although most types of fibre do not exhibit fully plastic behaviour, the stress/strain characteristic of carbon and organic fibres is not linear to failure. The modulus of carbon fibres may increase with increasing extension and hence ideally one should distinguish between tangent and secant modulus (section 1.4). 

Some fibre properties, assembled from manufacturers trade data are listed in Table 3.1. Carbon fibres have been divided up, somewhat arbitrarily, following Lovell (1991). This is necessary because of the number of different fibre types available. Frequently the data sheets do not distinguish between tangent and secant modulus, though for Kevlar the modulus values are 1% secant ones. Properties do vary between sources and all those in Table 3.1 are indicative rather than absolute. 

For tensile and compressive loading, the slope of the linear elastic region of the stress-strain curve is the modulus of elasticity ( ), per Hooke s law (Equation 6.5). For a material that exhibits nonlinear elastic behavior, tangent and secant moduli are used. 

Fig. 2-2 (a) Example of the modulus of elasticity determined on the initial straight portion of the stress-strain curve and secant modulus and (b) secant modulus for two different plastics that are 85% of the initial tangent modulus. 

Given the following figure with one tangent and one secant drawn to the circle, what is the measure of angle ADB ... 

Figure 7.20 Examples of tangent moduli and secant moduli...

FIGURE 11.21 Reinforced concrete pile arrangements (a) tangent, (b) secant, (c) staggered, (d) spaced pile walls. (After Cornforth, D., Landslides in Practice, Investigation, Analysis, and Remedial/Preventative Options in Soils, Wiley, 2005.)... 

The trigonometric functions illustrate a general property of the functions that we deal with. They are single-valued for each value of the angle a, there is one and only one value of the sine, one and only one value of the cosine, and so on. The sine and cosine functions are continuous everywhere. The tangent, cotangent, secant, and cosecant functions are piecewise continuous (discontinuous only at isolated points, where they diverge). 

There are some materials (i.e., gray cast iron, concrete, and many polymers) for which this elastic portion of the stress-strain curve is not linear (Figure 6.6) hence, it is not possible to determine a modulus of elasticity as described previously. For this nonlinear behavior, either the tangent or secant modulus is normally used. The tangent modulus is taken as the slope of the stress-strain curve at some specified level of stress, whereas the secant modulus represents the slope of a secant drawn from the origin to some given point of the cr-e curve. The determination of these moduli is illustrated in Figure 6.6. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

Tangent-Secant Theorem— f given an angle with its vertex on a circle, formed by a secant ray and a tangent ray, then the measure of the angle is half the measure of the intercepted arc. 

Tangent-Secant Power Theorem— f given a tangent segment QT to a circle and a secant line through Q, intersecting the circle at R and S, then... 

For the past century one successful approach is to plot a secant modulus that is at 1% strain or 0.85% of the initial tangent modulus and noting where they intersect the stress-strain curve (Fig. 2-2). However for many plastics, particularly the crystalline thermoplastics, this method is too restrictive. So in most practical applications the limiting strain is decided based on experience and/or in consultation between the designer and the plastic material manufacturer. Once the limiting strain is known, design methods based on its creep curves become rather straightforward (additional information to follow). 

Secant modulus The secant modulus is the ratio of stress to the corresponding strain at any specific point on the stress-strain curve. As shown in Fig. 2-2(a), the secant modulus is the slope of the line joining the origin and a selected point C on the stress-strain curve this could represent a vertical line at the usual 1 % strain. The secant modulus line is plotted from the initial tangent modulus and where it intersects the stress-strain curve. The plotted line location is also based on the angle used in relation to the initial tangent line from the ab-... 

If a vector it is a function of a single scalar quantity s, the curve traced as a function of s by its terminus, with respect to a fixed origin, can be represented as shown in Fig. 8. Within the interval As the vector AR = R2 - Ri is in the direction of the secant to the curve, which approaches the tangent in the limit as As - 0. Tins argument corresponds to that presented in Section 2.3 and illustrated in Fig. 4 of that section. In terms of unit vectors in a Cartesian coordinate system... 

A graphical interpretation of the derivative is introduced here, as it is extremely important in practical applications. The quantities Ajc and Ay are identified in Fig. 4a. It should be obvious that the ratio, as given by Eq. (8) represents the tangent of the angle 0 and that in the Unfit (Fig. 4b), the slope of the line segment A B (the secant) becomes equal to the derivative given by... 

The instantaneous rate of a reaction is the rate of the reaction at a particular time. To find the instantaneous rate of a reaction using a concentration-time graph, draw a tangent line to the curve and find the slope of the tangent. A tangent line is like a secant line, but it touches the curve at only one point. It does not intersect the curve. 

In principle, Curtis-Godson pressures and temperatures have to be computed for each gas, each layer and each limb view of the scan. In practice, only a sub-set of paths (combination of layer and limb view) requires a customised calculation, because, except for the tangent path, the secant law approximation can be applied and consequently the corresponding equivalent quantities are independent on the limb view angle. Therefore equivalent quantities are computed for the paths corresponding to the lowest geometry and only the tangent layers of the other limb views. This is a very effective optimisation because it reduces the number of paths for which cross-sections have to be computed. 

This equation states that the angle of the tangent taken at the midpoint (n + An/2) of a volume interval (not midpoint with respect to time) equals the angle of the secant. The rule is valid regardless of the size of An and is generally best applied to smoothed data. 

The advantages of this method are that it offers rapid convergence without requiring the first derivative. Convergence is between linear and quadratic, i.e., a power ranging from 1 to 2. The value is an approximation of the tangent of the secant. The value depends upon the steepness of the curve. Because the denominator always approaches zero near the root, this method is prone to instability. It also requires two initial estimates to start. 

The ordinary trigonometric functions include the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. These are sometimes called the circular trigonometric Junctions to distinguish them from the hyperbolic trigonometric functions discussed briefly in the next section of this chapter. 

The other hyperbolic trigonometric functions are the hyperbolic tangent, denoted by tanh(x) the hyperbolic cotangent, denoted by coth(x) the hyperbolic secant, denoted by sech(x) and the hyperbolic cosecant, denoted by csch(x). These functions are given by the equations... 

The standard test methods, calculate the tensile modulus by drawing a tangent to the initial linear part of the stress-strain curve and calculating the slope of the line. In cases where no clearly defined linear portion exists, the secant modulus should be determined. 

Figure 10.7 The (a) secant modulus and (b) tangent modulus of a material...

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