COMPRESSION MEMBER DESIGN

INTRODUCTORY CONCEPTS

·         Compression Members:  Structural elements that are subjected to axial compressive forces only are called columns. Columns are subjected to axial loads through the centroid.

·         Stress: The stress in the column cross-section can be calculated as

                                                                        (2.1)

where f is assumed to be uniform over the entire cross-section.

·         This ideal state is never reached. The stress-state will be non-uniform due to:

­          Accidental eccentricity of loading with respect to the centroid

­          Member out-of –straightness (crookedness), or

­          Residual stresses in the member cross-section due to fabrication processes.

·         Accidental eccentricity and member out-of-straightness can cause bending moments in the member. However, these are secondary and are usually ignored.

·         Bending moments cannot be neglected if they are acting on the member. Members with axial compression and bending moment are called beam-columns.

 

4.2 COLUMN BUCKLING

·         Consider a long slender compression member. If an axial load P is applied and increased slowly, it will ultimately reach a value P_cr that will cause buckling of the column (Figure 1). P_cr is called the critical buckling load of the column.

 

What is buckling?

Buckling occurs when a straight column subjected to axial compression suddenly undergoes bending as shown in the Figure 1(b). Buckling is identified as a failure limit-state for columns.

 

 

Figure 1. Buckling of axially loaded compression members

 

·         The critical buckling load P_cr for columns is theoretically given by Equation (4.1)

P_cr =                                                          (4.1)

where, I = moment of inertia about axis of buckling

           K = effective length factor based on end boundary conditions

·         Effective length factors are given on page 16.1-511(Table C-A-7.1) of the AISC manual.

·         In examples, homeworks, and exams please state clearly whether you are using the theoretical value of K or the recommended design values.

 

Example4.1 Determine the buckling strength of a W 12 x 50 column. Its length is 20 ft. For major axis buckling, it is pinned at both ends. For minor buckling, is it pinned at one end and fixed at the other end.

Solution

Step I. Visualize the problem           

Figure 2. (a) Cross-section; (b) major-axis buckling; (c) minor-axis buckling

 

·         For the W12 x 50 (or any wide flange section), x is the major axis and y is the minor axis. Major axis means axis about which it has greater moment of inertia (I_x>I_y)

Step II. Determine the effective lengths

·         According to Table C-A-7.1 of the AISC Manual (see page 16.1 - 511):

-          For pin-pin end conditions about the major axis

K_x = 1.0 (theoretical value); and K_x = 1.0 (recommended design value)

-          For pin-fix end conditions about the minor axis

K_y = 0.7 (theoretical value); and K_y = 0.8 (recommended design value)

·         According to the problem statement, the unsupported length for buckling about the major (x) axis = L_x = 20 ft.

·         The unsupported length for buckling about the minor (y) axis = L_x = 20 ft.

·         Effective length for major (x) axis buckling = K_x L_x = 1.0 x 20 = 20 ft. = 240 in.

·         Effective length for minor (y) axis buckling = K_y L_y = 0.8 x 20 = 16 ft. = 192 in.

Step III. Determine the relevant section properties

·         For W12 x 50: elastic modulus = E = 29000 ksi (constant for all steels)

·         For W12 x 50:       I_x = 391 in⁴.     I_y = 56.3 in⁴  (see pages 1-26 and 1-27 of the AISC manual)

Step IV. Calculate the buckling strength

·         Critical load for buckling about x - axis = P_cr-x =  =

P_cr-x = 1942.9 kips

·         Critical load for buckling about y-axis = P_cr-y = =

P_cr-y = 437.12 kips

·         Buckling strength of the column = smaller (P_cr-x, P_cr-y) = P_cr = 437.12 kips

Minor (y) axis buckling governs.

 

·         Notes:

-          Minor axis buckling usually governs for all doubly symmetric cross-sections. However, for some cases, major (x) axis buckling can govern.

-          Note that the steel yield stress was irrelevant for calculating this buckling strength.

 

4.3 INELASTIC COLUMN BUCKLING

 

·         Let us consider the previous example. According to our calculations P_cr = 437 kips. This P_cr will cause a uniform stress f = P_cr/A in the cross-section

·         For W12 x 50, A = 14.6 in². Therefore, for P_cr= 437 kips; f = 30 ksi

The calculated value of f is within the elastic range for a 50 ksi yield stress material.

·         However, if the unsupported length was only 10 ft., P_cr =would be calculated as 1748 kips, and f = 119.73 ksi.

·         This value of f is ridiculous because the material will yield at 50 ksi and never develop f = 119.73ksi. The member would yield before buckling.

·         Equation (4.1) is valid only when the material everywhere in the cross-section is in the elastic region. If the material goes inelastic then Equation (4.1) becomes useless and cannot be used.

·         What happens in the inelastic range?

Several other problems appear in the inelastic range.

-          The member out-of-straightness has a significant influence on the buckling strength in the inelastic region. It must be accounted for.

-          The residual stresses in the member due to the fabrication process causes yielding in the cross-section much before the uniform stress f reaches the yield stress F_y.

-          The shape of the cross-section (W, C, etc.) also influences the buckling strength.

-          In the inelastic range, the steel material can undergo strain hardening.

 

All of these are very advanced concepts and beyond the scope of CE470. You are welcome to CE579 to develop a better understanding of these issues.

 

·         So, what should we do? We will directly look at the AISC Specifications for the strength of compression members, i.e., Chapter E (page 16.1-31 of the AISC manual).

 

4.4 AISC SPECIFICATIONS FOR COLUMN STRENGTH

·         The AISC specifications for column design are based on several years of research.

·         These specifications account for the elastic and inelastic buckling of columns including all issues (member crookedness, residual stresses, accidental eccentricity etc.) mentioned above.

·         The specification presented here (AISC Spec E3) will work for all doubly symmetric cross-sections and channel sections.

·         The design strength of columns for the flexural buckling limit state is equal to f_cP_n

Where,       f_c = 0.9                        (Resistance factor for compression members)

                        P_n = A_g F_cr                                                                                           (4.2)

­          When (or )

§  F_cr =  F_y                                                                               (4.3)

­          When             (or )

                                          F_cr =                                                             (4.4)

Where, F_e =                                                                                             (4.5)

      A_g = gross member area;                                 K = effective length factor

      L = unbraced length of the member;              r = corresponding radius of gyration

 

·         Note that the original Euler buckling equation is P_cr = 

·         Note that the AISC equation for  is F_cr = 0.877F_e

­          The 0.877 factor tries to account for initial crookedness.

·         For a given column section:

­          Calculate I, A_g, r

­          Determine effective length K L based on end boundary conditions.

­          Calculate F_e, F_y/F_eor

­          If (KL/r) greater than , elastic buckling occurs and use Equation (4.4)

­          If (KL/r) is less than or equal to , inelastic buckling occurs and use Equation (4.3)

·         Note that the column can develop its yield strength F_y as (KL/r) approaches zero.

 

4.5 COLUMN STRENGTH

·         In order to simplify calculations, the AISC specification includes Tables.

­          Table 4-22 on pages4-322 to 4-326 shows KL/r vs. f_cF_cr for various steels.

­          You can calculate KL/r for the column, then read the value of f_cF_cr from this table

­          The column strength will be equal to f_cF_cr x A_g

 

Example4.2Calculate the design strength of W14 x 74 with length of 20 ft. and pinned ends. A36 steel is used.

Solution

·         Step I. Calculate the effective length and slenderness ratio for the problem

K_x = K_y = 1.0

L_x = L_y = 240 in.

Major axis slenderness ratio = K_xL_x/r_x = 240/6.04 = 39.735

Minor axis slenderness ratio = K_yL_y/r_y = 240/2.48 = 96.77

·         Step II. Calculate the elastic critical buckling stress

The governing slenderness ratio is the larger of (K_xL_x/r_x, K_yL_y/r_y)

= 30.56 ksi

Check the limits

                  ( ) or ()

Since;      Therefore, F_cr =  F_y

Therefore, F_cr = 21.99 ksi

Design column strength = f_cP_n = 0.9 (A_g F_cr) = 0.9 (21.8 in² x 21.99 ksi) = 431.4 kips

Design strength of column = 431 kips

·         Check calculated values with Table 4-22. For KL/r = 97, f_cF_cr = 19.7 ksi
4.6 LOCAL BUCKLING LIMIT STATE

·         The AISC specifications for column strength assume that column buckling is the governing limit state. However, if the column section is made of thin (slender) plate elements, then failure can occur due to localbuckling of the flanges or the web.

Figure 4. Local buckling of columns

·      If localbuckling of the individual plate elements occurs, then the column may not be able to develop its buckling strength.

·      Therefore, the local buckling limit state must be prevented from controlling the column strength.

·      Local buckling depends on the slenderness (width-to-thickness b/t ratio) of the plate element and the yield stress (F_y) of the material.

·      Each plate element must be stocky enough, i.e., have a b/t ratio that prevents local buckling from governing the column strength.

·      The AISC specification B4.1 (Page 16.1-14) provides the slenderness (b/t) limit that the individual plate elements must satisfy so that local buckling does not control.

·      For compression, the AISC specification provides slenderness limit ( l_r) for the local buckling of plate elements.

 Figure 5. Local buckling behavior and classification of plate elements

­          If the slenderness ratio (b/t) of the plate element is greater than l_r then it isslender. It will locally buckle in the elastic range before reaching F_y

­          If the slenderness ratio (b/t) of the plate element is less than l_r, then it is non-slender. It will not locally buckle in elastic range before reaching F_y

­          If any one plate element is slender, then the cross-section is slender.

·      The slenderness limitl_r for various plate elements with different boundary conditions are given in Table B4.1a on page16.1-16 of the AISC Spec.

·      Note that the slenderness limit ( l_r) and the definition of plate slenderness (b/t) ratio depend upon the boundary conditions for the plate.

­          If the plate is supported along two edges parallel to the direction of compression force, then it is a stiffened element. For example, the webs of W shapes

­          If the plate is supported along only one edge parallel to the direction of the compression force, then it is an unstiffened element, e.g., the flanges of W shapes.

 

·      The local buckling limit state can be prevented from controlling the column strength by using sections that are nonslender.

­          If all the elements of the cross-section have calculated slenderness (b/t) ratio less than l_r, then the local buckling limit state will not control.

­          For the definitions of b/t and l_r for various situations see Table B4.1a and Spec B4.1.