Deflection and Stiffness

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Chapter 5
Deflection and Stiffness
A. Aziz Bazoune
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Outline

• Spring Rates
• Deflection in Tension, Compression Torsion
• Deflection due to Bending
• Strain Energy
• Castiglianos Theorem
• Statically Indeterminate Problems
• Compression Members
• Long Columns with Central Loading
• Intermediate Length Columns with Central Loading

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Lec. 14

• Solved Problems

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Problem

• 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 ? Sy 57 ksi
  (Table A-20)
• E 30 Mpsi (Table A-5)
• Ends Pinned ? Rounded-rounded case ? C1
  (Table 5-2)
• Eq. (5-48)
• Calculate the slenderness ratio

Solution
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Problem

• Calculate the slenderness ratio
• Since ? Eulers
  Column
• Therefore

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Problem

• 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 ?
• Sy 290 MPa (Table A-20)
• E 207 GPa (Table A-5)
• Fixed-rounded end Condition ?
• C1.2 (Table 5-2)
• Radii of gyration

Solution
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Problem

• It is clear that and
• On the other hand we have from Eq. 5-48
• Since
• Apply the formula for Johnsons column to find
  the critical load on the steel column.

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Problem

• Therefore

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Problem

• 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 (Sy 500 MPa) and a
  modulus of elasticity (E207 GPa)
• Assume end conditions constant C1 in the weakest
  direction.
• Column length l150 mm and 400 mm)
• Cross-section area is
• or

Solution
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Problem

• Ixx
• Iyy
• or

Solution
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Problem
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http//en.wikipedia.org/wiki/Leonhard_Euler
Leonhard Paul Euler (pronounced Oiler IPA
'??l?) (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 and Germany. He published
more papers than any other mathematician of his
time.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 uch as
the notion of a mathematical function.3 He is
also renowned for his work in mechanics, optics,
and astronomy. 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
6080 quarto volumes.4 A statement attributed
to Pierre-Simon Laplace expresses Euler's
influence on mathematics "Read Euler, read
Euler, he is the teacher (master) of us
all".5 Euler was featured on the sixth series
of the Swiss 10-franc banknote6 and on numerous
Swiss, German, and Russian postage 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.
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