Circle in Axonometric Projection

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Apr 6, 2017 - ... Ellipse (Rytz's Construction Method). 13. David Rytz [von Brugg] (1801–1868),. Swiss Mathematician, Prof. at the Technical College Aarau, ...
Mechanics (Strenght) of Materials ‫و ا واد‬ (Lecture Va), 06 April 2017 – Engineering Curves in Mechanics

Dr.mont., Dipl.-Ing., Dipl.-HTL-Ing. Rüdiger B. Richter

Engineering Curves in Mechanics Conic sections – Basics

Conic Sections “Circle – Parabola – Ellipse – Hyperpola”

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Conic Sections THESE CURVES APPEAR ON THE SURFACE OF A CONE WHEN IT IS CUT BY SOME TYPICAL CUTTING PLANES.

Ellipse

Section Plane Through Generators

Section Plane Parallel to Axis.

Section Plane Parallel to end generator. 4

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Hyperbola

Ellipse: Typically area due to radii of gyration

R.B. Richter

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Quadratic Parabula: Bending Moment Diagrams

52.077

75.563 2.792 m 56.23 27.923 74.437

3.74 m 63.77

S. F. D. Mmax=+38.985 kN.m Max=+ 35.59 kN.m 126.704

31.693

35.08

-69.806

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3.74 m

48.307 84.92

-99.985

98.297

2.792 m -96.613

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B. M. D

How to draw an Ellipse or a Parabula? Not so easy as you may think!

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Is this an Ellipse?

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“Burmester Satz” Curve template for drawing Parabolas – Ellipses – Hyperbolas (forgotten tools from yesterday)

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Drawing a proper Ellipse (as often required in Mechanics)

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Drawing a proper Parabula (as frequently required in Mechanics)

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Application: Insertion of parabolic bending moments

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A Circle in Axonometric Projection Conjugate Diameters of an Ellipse (Rytz‘s Construction Method)

David Rytz [von Brugg] (1801–1868), Swiss Mathematician, Prof. at the Technical College Aarau, Switzerland 13

Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Circle in Axonometric Projection

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Other Technical Curves in Mechanics Is a chain line a parabula?

Gottfried Wilhelm Leibniz (1646–1716) 26

Appearance of chain lines Storebælt Bridge (18 km long), Denmark

Festoon, Detail of the Pantheon, Paris

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Chain line function

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Chain line function

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Circular Based Arches –Three-Centered Arches

Adam A. Kochański (1631 - 1700) 30

The classical Three-Centered Arch (Application: Dished bottom pressure vessels)

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Basket Arch: – Drawing a 3-Centered Arch Method of Adam A. Kochański (1631 - 1700)

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Basket Arch: – Drawing a 3-Centered Arch (cont.) Method of Adam A. Kochański (1631 - 1700)

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Dished bottoms – various shapes after DIN

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Mechanics (Strenght) of Materials

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