17 Questions
What does the composite curve, called a column strength curve, describe?
Strength of any column based only on slenderness ratio
What are the assumptions underlying both the Euler and tangent modulus equations?
The column is perfectly straight with axial load
Which type of differential equation results when bending moment is a function of 𝑥 in a compression member?
Nonhomogeneous differential equation
What are the end conditions for the compression member mentioned in Figure 5.7?
Pinned at one end, fixed against rotation and translation at the other
What is the equation derived for a compression member pinned at one end and fixed against rotation and translation at the other?
$𝑷𝒄𝒓 = 2𝑳^2/π^2𝑬𝑰$
What property does the strength of a column depend on, apart from slenderness ratio?
$𝑃𝒄𝒓$
What is the equation that gives the deflected shape of an elastic member subjected to bending?
$\frac{d^2 y}{dx^2} = -\frac{P_{cr}y}{EI}$
What is the solution to the differential equation for the deflected shape of an elastic member subjected to bending?
$y = A\cos cx + B\sin cx$
What is the condition for the non-trivial solution to the differential equation for the deflected shape of an elastic member subjected to bending?
$cL = 0, \pi, 2\pi, 3\pi,...$
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?
$P_{cr} = \frac{\pi^2 EI}{L^2}$
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?
$c$ is the square root of the ratio of the critical buckling load to the flexural rigidity, i.e., $c = \sqrt{\frac{P_{cr}}{EI}}$
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?
$A$ and $B$ are the amplitudes of the cosine and sine terms, respectively, in the solution
Which equation is used to calculate the critical stress for elastic columns?
$F_{cr} = 0.877F_e$
When is the boundary between inelastic and elastic columns reached?
When $KL/r = 4.71\sqrt{E/F_y}$
What is the equation used to calculate the critical stress for inelastic columns?
$F_{cr} = 0.658F_y/F_e$
What is the relationship between the Euler stress, $F_e$, and the yield stress, $F_y$, at the boundary between inelastic and elastic columns?
$F_e = 4.71F_y
Which equation is used to calculate the Euler stress, $F_e$?
$F_e = \pi^2 E I / (KL/r)^2$
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.
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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