Column Strength Curve and Euler Modulus Equations
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Questions and Answers

What does the composite curve, called a column strength curve, describe?

  • Strength of any column based only on slenderness ratio (correct)
  • Material properties like 𝑭𝒚, 𝑬, and 𝑬𝒕
  • Strength of any column based only on material properties
  • Relationship between load and deformation of a column

    • What are the assumptions underlying both the Euler and tangent modulus equations?

  • The load is eccentric
  • The column is not pinned at both ends
  • The column has initial crookedness
  • The column is perfectly straight with axial load (correct)

    • Which type of differential equation results when bending moment is a function of 𝑥 in a compression member?

  • Linear differential equation
  • Nonhomogeneous differential equation (correct)
  • Homogeneous differential equation
  • Quadratic differential equation

    • What are the end conditions for the compression member mentioned in Figure 5.7?

    <p>Pinned at one end, fixed against rotation and translation at the other</p> Signup and view all the answers

    What is the equation derived for a compression member pinned at one end and fixed against rotation and translation at the other?

    <p>$𝑷𝒄𝒓 = 2𝑳^2/π^2𝑬𝑰$</p> Signup and view all the answers

    What property does the strength of a column depend on, apart from slenderness ratio?

    <p>$𝑃𝒄𝒓$</p> Signup and view all the answers

    What is the equation that gives the deflected shape of an elastic member subjected to bending?

    <p>$\frac{d^2 y}{dx^2} = -\frac{P_{cr}y}{EI}$</p> Signup and view all the answers

    What is the solution to the differential equation for the deflected shape of an elastic member subjected to bending?

    <p>$y = A\cos cx + B\sin cx$</p> Signup and view all the answers

    What is the condition for the non-trivial solution to the differential equation for the deflected shape of an elastic member subjected to bending?

    <p>$cL = 0, \pi, 2\pi, 3\pi,...$</p> Signup and view all the answers

    What is the expression for the critical buckling load $P_{cr}$ in terms of the flexural rigidity $EI$ and the length $L$ of the elastic member?

    <p>$P_{cr} = \frac{\pi^2 EI}{L^2}$</p> Signup and view all the answers

    What is the physical meaning of the constant $c$ in the solution to the differential equation for the deflected shape of an elastic member subjected to bending?

    <p>$c$ is the square root of the ratio of the critical buckling load to the flexural rigidity, i.e., $c = \sqrt{\frac{P_{cr}}{EI}}$</p> Signup and view all the answers

    What is the physical interpretation of the constants $A$ and $B$ in the solution to the differential equation for the deflected shape of an elastic member subjected to bending?

    <p>$A$ and $B$ are the amplitudes of the cosine and sine terms, respectively, in the solution</p> Signup and view all the answers

    Which equation is used to calculate the critical stress for elastic columns?

    <p>$F_{cr} = 0.877F_e$</p> Signup and view all the answers

    When is the boundary between inelastic and elastic columns reached?

    <p>When $KL/r = 4.71\sqrt{E/F_y}$</p> Signup and view all the answers

    What is the equation used to calculate the critical stress for inelastic columns?

    <p>$F_{cr} = 0.658F_y/F_e$</p> Signup and view all the answers

    What is the relationship between the Euler stress, $F_e$, and the yield stress, $F_y$, at the boundary between inelastic and elastic columns?

    <p>$F_e = 4.71F_y</p> Signup and view all the answers

    Which equation is used to calculate the Euler stress, $F_e$?

    <p>$F_e = \pi^2 E I / (KL/r)^2$</p> Signup and view all the answers

    Study Notes

    Column Strength Curve

    • The column strength curve, also known as the composite curve, completely describes the strength of any column of a given material.
    • The strength of a column is a function only of the slenderness ratio, apart from 𝑭𝒚, 𝑬, and 𝑬𝒕, which are properties of the material.

    Assumptions of Euler and Tangent Modulus Equations

    • The Euler and tangent modulus equations are based on the following assumptions:
      • The column is perfectly straight, with no initial crookedness.
      • The load is axial, with no eccentricity.
      • The column is pinned at both ends.

    Bending Moment and Deflection

    • The bending moment is a function of 𝑥, resulting in a nonhomogeneous differential equation.
    • The boundary conditions will be different from those in the original derivation, but the overall procedure will be the same.
    • The differential equation that gives the deflected shape of an elastic member subjected to bending is 𝒅𝟐 𝒚 𝑴 =− 𝑬𝑰 𝒅𝒙 𝒅𝒙.

    Euler Equation for Pinned-Fixed Column

    • The Euler equation for a compression member pinned at one end and fixed against rotation and translation at the other is 𝑷𝒄𝒓 = 𝟐.𝟎𝟓𝝅𝟐 𝑬𝑰 𝟐.𝟎𝟓𝝅𝟐 𝑬𝑨 𝝅𝟐 𝑬𝑨.

    Critical Stress

    • The Euler stress is 𝑷𝒆 = 𝝅𝟐 𝑬 𝑭𝒆 / 𝑨 𝑲𝑳/𝒓.
    • The critical stress for elastic columns is 𝑭𝒄𝒓 = 𝟎.𝟖𝟕𝟕𝑭𝒆.
    • The critical stress for inelastic columns is 𝑭𝒄𝒓 = 𝑭𝒚 𝟎.𝟔𝟓𝟖𝑭𝒆 / 𝑭𝒚.

    Boundary Between Inelastic and Elastic Columns

    • The boundary between inelastic and elastic columns occurs when 𝑲𝑳/𝒓 is approximately 𝑬 𝟒.𝟕𝟏 𝑭𝒚.
    • At this boundary, Equations 5.8 and 5.9 give the same value of 𝑭𝒄𝒓.

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    Description

    Explore the concept of column strength curve and Euler modulus equations in civil engineering. Learn about the properties of materials such as 𝑭𝒚 , 𝑬, and 𝑬𝒕, and how strength is determined by the slenderness ratio. Delve into the assumptions underlying Euler and tangent modulus equations in column design.

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