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LRFD - Steel Design

LRFD - Steel Design. Dr. Ali I. Tayeh second Semester. Steel Design Dr. Ali I. Tayeh. Chapter 4 -A. Compression Members. Compression Members : Structural elements that are subjected only to axial compressive forces.

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LRFD - Steel Design

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  1. LRFD-Steel Design Dr. Ali I. Tayeh second Semester

  2. Steel DesignDr.Ali I. Tayeh Chapter 4 -A

  3. Compression Members Compression Members: Structural elements that are subjected only to axial compressive forces. the stress can be taken as fa = P/A, where fa is considered to be uniform over the entire cross section. This ideal state is never achieved in reality, because some eccentricity of the load is inevitable. The column: Is the most common type of compression member occurring in buildings and bridges. Sometimes members are also called upon to resist bending, and in these cases the member is a beam column.

  4. Compression Members • Column Theory: • Consider the long, slender compression member. • If the axial load P is slowly applied. it will ultimately become large enough to cause the member to become unstable.

  5. Compression Members • Assume the shape indicated by the dashed line. The member is said to have buckled. • The load at which buckling occurs is a function ofslenderness, and for very slender members this load could be quite small.

  6. Compression Members • Column Theory: • The critical buckling load Pcr: • the load that is just large enough to maintain the deflected shape when the temporary transverse load is removed. • If the member is so slender , the stress just before buckling is below the proportional limit - and the member is still elastic - the critical buckling load is given by

  7. Compression Members • EQUATION DIFFERENTIATION : • The differential equation giving the deflected shape of an elastic member subjected to bending is • The moment in this case equal Pcr × y

  8. Compression Members • EQUATION DIFFERENTIATION : • The differential equation can be written as • The solution of ordinary differential equation with constant coefficients as

  9. Compression Members • To find the constant two boundary condition can be applied. • The last condition requires that sin (cl) be Zero if B is not to be zero ,that obtain if

  10. Compression Members • EQUATION DIFFERENTIATION : • From this equation • We obtain • Values of n larger than 1 are not possible unless the compression member is physically restrained from deflecting at the points where the reversal of curvature would occur.

  11. Compression Members

  12. Compression Members • EQUATION DIFFERENTIATION : • The solution of differential equation is • For usual case n=1. • So this equation can be rewritten as :

  13. Compression Members • Where: • A is the cross-sectional area . • r is the radius of gyration with respect to the axis of buckling. • The ratio L/r is the slenderness ratio . • If the critical load is divided by the cross-sectional area, the critical buckling stress is obtained:

  14. Example:4.1 A WI2 x 50 is used as a column to support an axial compressive load of 145kips. The length is 20 feet, and the ends are pinned. Without regard to load or resistance factors, Investigate this member for stability. Because the applied load of 145 kips is less thanPcrthe column remains stable and has an overall factor of safety against buckling of278.9/145 = 1.92.

  15. Compression Members • EQUATION DIFFERENTIATION : Both 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.

  16. Compression Members • The first two conditions mean that there is no bending moment in the member before budding. • Consider a compression member pinned at one end and fixed against rotation and translation at the other. The Euler equation for this case. derived in the same manner as

  17. 70%L Compression Members • EQUATION DIFFERENTIATION : Equivalent length (pined-pined) actual length (pined-Fixed) 70%100%

  18. Compression Members • AISC REQUIRMENTS : Factored load ≤ factored strength Pu≤ ØcPn Where Slenderness parameter is used instead of Fcr as a function of the slenderness ratio k l / r

  19. Compression Members • AISC REQUIRMENTS : • So critical buckling stress will be summarized as: • Also, Graphically can be summarized as:

  20. ExExample:4.2 ample 4.2: Compute the design compressive strength of a W14 x 74 with a length of 20 feet and pinned ends. A992 steel is used. Compression Members

  21. Compression Members • Local Stability: • The strength corresponding to any buckling mode cannot be developed. • If the elements of the cross section are so thin that local buckling occurs. • This type of instability is a localized buckling or wrinkling at an isolated location. • If it occurs, the cross section is no longer fully effective, and the member has failed.

  22. I-and H-shaped cross sections with thin flanges or webs are susceptible to this phenomenon, and their use should be avoided whenever possible. Otherwise, the compressive strength reduced. • The measure of this susceptibility is the width-thickness ratio λ of each cross-sectional element. • Two types of elements must be considered: unstiffened elements, which are unsupported along one edge parallel to the direction of load, and stiffened elements, which are supported along both edges. • The strength must be reduced if the shape has any slender elements.

  23. Compression Members • Local Stability: • cross-sectional shapes are classified as compact, non compact, or .slender, according to the values of the width-thickness ratios. • If λ.is greater thanthe specified limit, denoted λrthe shape is slender, • For I- and H-shapes, the projecting flange is considered to be an unstiffened element, and its width can be taken as half the full nominal width • wherebfand tfare the width and thickness of the flange. • The upper limit is

  24. Compression Members • Local Stability: • The webs of I- and H-shapes are stiffened elements. and the stiffened width is the distance between the roots of the flanges. • The width thickness parameter is • Where h is the distance between the roots of the flanges, and tw is the web thickness. • The upper limit is

  25. Compression Members • Local Stability:

  26. Compression Members • Local Stability:

  27. Compression Members • Local Stability:

  28. Compression Members • Local Stability: Example 4.3: • Investigate the column in example 4.2 for local stability.

  29. Compression Members • DESIGN: • Example 4.5: A compression member is subjected to service loads of 165 kips dead load and 535 kips live load. The member is 26 feet long and pinned at each end. Use A992 steel • Answer

  30. Compression Members • DESIGN: • Example 4.6:

  31. Compression Members • DESIGN: • Example 4.7: • answer

  32. Compression Members • DESIGN:

  33. Compression Members • DESIGN:

  34. Compression Members

  35. Compression Members -Steel Design End

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