Chapter 5 Deflection and Stiffness A. Aziz Bazoune 10/26/2015Chapter 51

Outline 1.Spring Rates 2.Deflection in Tension, Compression & Torsion 3.Deflection due to Bending 4.Strain Energy 5.Castigliano’s Theorem 6.Statically Indeterminate Problems 7.Compression Members Long Columns with Central Loading Intermediate Length Columns with Central Loading 10/26/2015Chapter 52

 Solved Problems Lec /26/2015Chapter 53

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10/26/2015Chapter 58 A column is pinned at both ends has a length of 4.50 ft and a solid circular cross-section of 1.25 in diameter. If it is made of steel AISI 1020 CD what would be a safe column loading?  Material AISI 1020 CD Steel =  S y = 57 ksi (Table A-20) E = 30 Mpsi (Table A-5)  Ends: Pinned =  Rounded-rounded case =  C=1 (Table 5-2)  Eq. (5-48)  Calculate the slenderness ratio Problem SolutionSolution

10/26/2015Chapter 59  Calculate the slenderness ratio  Since ==  Euler’s Column  Therefore Problem

10/26/2015Chapter 510 Determine the critical load on a steel column having a rectangular cross- section, 12 mm X 18 mm and a length of 280 mm? It is proposed to use AISI 1040 HR Steel. The lower end of the column is inserted into a close-fitting socket and welded securely. The upper end is pinned.  Material AISI 1020 HR Steel  S y = 290 MPa (Table A-20) E = 207 GPa (Table A-5)  Fixed-rounded end Condition  C=1.2 (Table 5-2)  Radii of gyration Problem SolutionSolution

10/26/2015Chapter 511 It is clear that and On the other hand we have from Eq Since  Apply the formula for Johnson’s column to find the critical load on the steel column.  kkg Problem

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10/26/2015Chapter 513  Therefore  kkg Problem

10/26/2015Chapter 514 Design a rectangular section steel column having the following specifications:  Width of the column (w) is three times its thickness (t)  Maximum load that could be carried by the column is 100 kN  Material has a yield strength (S y = 500 MPa) and a modulus of elasticity (E=207 GPa)  Assume end conditions constant C=1 in the weakest direction.  Column length l=150 mm and 400 mm)  Cross-section area is or Problem SolutionSolution

10/26/2015Chapter 515  I xx  I yy or Problem SolutionSolution

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10/26/2015Chapter 517 Leonhard Euler Portrait by Johann Georg Brucker Born April 15April 15, 1707( ) Basel, Switzerland1707 BaselSwitzerland Died September 18September 18 [O.S. September 7] 1783 St Petersburg, RussiaO.S.1783 St PetersburgRussia Residence Prussia Russia Switzerland NationalitySwiss FieldMathematicsMathematics and physicsphysics Institutions Imperial Russian Academy of Sciences Berlin Academy Alma materUniversity of Basel ReligionCalvinist [1] Leonhard Paul Euler (pronounced Oiler; IPA [ ˈɔʏ l ɐ ]) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia andIPAApril September 18O.S.1783SwissmathematicianphysicistRussia GermanyGermany. He published more papers than any other mathematician of his time. [2] [2] Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, s calculusgraph theorymathematical analysis uch as the notion of a mathematical function. [3] He is also renowned for his work in mechanics, optics, and astronomy.mathematical function [3]mechanicsopticsastronomy Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time. He is also one of the most prolific; his collected works fill 60–80 quarto volumes. [4] A statement attributed to Pierre-Simon Laplacequarto [4]Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the teacher (master) of us all". [5] [5] Euler was featured on the sixth series of the Swiss 10-franc banknote [6] and on numerous Swiss, German, and Russianfranc [6] postage stampspostage stamps. The asteroid 2002 Euler was named in his honor. He is also commemorated by the Lutheran Church on their Calendar of Saints on May 24.asteroid2002 EulerLutheran ChurchCalendar of SaintsMay 24

10/26/2015Chapter 518 QUESTIONS ?