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Concept of stress and strain
Published in M.L. Jeremic, Ground Mechanics in Hard Rock Mining, 2020
A state of plane stress is defined as one in which all stress components acting on one of the three orthogonal planes at a point are zero. Such a state of stress exists in photo-elastic or other physical models of excavations in which a perforated plate is loaded by forces applied in the plane of the plate with the material being unconstrained perpendicular to the plane. Plane stress conditions for the x and y planes are defined by the following conditions: σz?=?τzx?=?τyz?0
Special Functions
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A right angle is the angle between two perpendicular lines. It is equal to π/2 radians or 90 degrees. An acute angle is an angle less than π/2 radians. An obtuse angle is one between π/2 and π radians. A convex angle is one between 0 and π radians.
Orientation Controls
Published in James D. Meadows, Geometric Dimensioning and Tolerancing, 2017
We know that these axes are lines. By the definition of perpendicular lines, when these lines cross in space, they must be 90° to one another. One line may actually rotate 360° around the other and not endanger its perpendicularity to that line. Therefore, confinement of one line to the other to preserve perpendicularity is only needed to stop them from leaning in a direction that endangers their 90° relationship. This requires a tolerance zone of two parallel planes. A cylindrical tolerance zone around the controlled feature axis would not aid more in preserving its line-to-line perpendicularity to the datum axis. Perpendicularity will not, however, control the location of one axis to the datum axis. Positional controls may be used to cause the axes to intersect one another. In the case of axis-to-axis perpendicularity, both the controlled size feature’s geometric tolerance and the datum feature of size may have a material condition symbol placed next to them in the feature control frame. Either may use MMC or LMC concepts. However, if no symbol is included, RFS is implied.
Parallel curves
Published in International Journal of Mathematical Education in Science and Technology, 2022
Richard Dexter Sauerheber, Tony Stewart
Figure 1 shows the relationship between parallel lines, where vectors are drawn from one line to corresponding points on the parallel line where the connecting vector is perpendicular to both lines at the same time. Each connecting vector is the same length because the lines are themselves parallel and cannot intersect. Parallel lines must share the same derivative while being shifted in the Euclidean plane. Geometry courses routinely and correctly teach proofs indicating that parallel lines have linear transversals that must form congruent alternate interior angles, congruent alternate exterior angles, and pairs of interior angles and exterior angles on the same side of the transversal that must be supplementary ( = 180°) (Ratti and McWaters, 2010). This also means that any segment or vector Eextending from one line to the other that is perpendicular to one line must also be perpendicular to the other parallel line. All perpendicular segments intersecting any two parallel lines must always form only right angles and must be of equal length. If not, then the lines are not parallel.
Middle school students’ reasoning with regards to parallelism and perpendicularity of line segments
Published in International Journal of Mathematical Education in Science and Technology, 2023
Examples have both critical and non-critical attributes (Hershkowitz, 1989). A concept definition is linked to critical attributes of a mathematical concept (Ulusoy, 2019). Non-critical attributes, on the other hand, can be acquired by reasoning about a concept's formal definition, which contains just necessary and sufficient defining criteria (Tsamir et al., 2008). Non-critical attributes are only found in prototypical examples of the concept (Hershkowitz, 1989; Ulusoy, 2021). For instance, perpendicular lines are two lines that intersect at a right angle. This concept includes a number of critical attributes, some of them are explicit and others are implicit. (Gal & Vinner, 1997) mention these critical attributes as follows: ‘(1) two lines (sometimes two segments or one line and one segment), (2) the lines intersect, (3) a right angle is formed at the point of intersection, (4) there are three more angles at the point of intersection, (5) these three other angles are also right angles, and (6) a right angle is a 90° angle’ (p.282). Therefore, whereas all examples of a concept contain the entire set of critical attributes, only some prototypical examples of the concept contain non-critical attributes of the concept (e.g. Ulusoy, 2019). The prototypes are usually the subset of examples that had the longest list of attributes all the critical attributes of the concept and those specific (noncritical) attributes that had strong visual characteristics’ (Hershkowitz, 1990, p. 82). For example, two parallel line segments are mostly represented in vertical or horizontal orientations in lessons or textbooks. However, when two parallel line segments are represented in an inclined orientation, this example shows parallel line segments in a non-prototypical orientation. In mathematics education, students are expected to use only critical attributes when generating and identifying examples of geometric concepts because it is necessary to recognize critical attributes to provide the correct identification of a figure (van Hiele & van Hiele, 1958). However, according to the findings of relevant studies, students do not have the expected degree of knowledge of geometric ideas (Burger & Shaughnessy, 1986; Gutiérrez & Jaime, 1999; Hannibal, 1999) due to the dominancy of prototypical concept images in students’ minds. As a result, when students encounter prototypical examples of a geometric concept, they make their judgments based on their poorest concept images developed on visual characteristics of prototypes (e.g. Hershkowitz, 1989, 1990; Fujita, 2012). Thus, they can identify some examples of the concept as non-examples or they can identify some non-examples as examples of the concept.