# End Restraint and Effective Lengths of Columns

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1 CHAPTER Structural Steel Design LRFD Method Third Edition INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II Structural Steel Design and Analysis 5c FALL 2002 By Dr. Ibrahim. Assakkaf ENCE Introduction to Structural Design Department of Civil and Environmental Engineering University of Maryland, College Park CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 1 Limitations of Basic Euler Formula The basic Euler formula is only useful if the end support conditions are carefully considered. The results obtained by application of the formula to specific examples compare very well with test results for centrally loaded, long, slender columns with rounded ends. 1

2 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 2 Limitations of Basic Euler Formula In real life, these types of columns barely exists. For example, the columns with which one works do not have rounded ends and are not free to rotate because their ends are bolted or welded to other members. Furthermore, the axial load applied to these columns are not centric in most cases. CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 3 Effect of End Restraint on Column Load Capacity End restraint and its effect on the loadcarrying capacity of columns is very important subject. Columns with appreciable rotational and transnational end restraint can support considerably more load than those with little rotational end restraint as at hinged end. 2

3 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 4 General Notes On Column Buckling 1. Boundary conditions other than simplysupported will result in different critical loads and mode shapes. 2. The buckling mode shape is valid only for small deflections, where the material is still within its elastic limit. 3. The critical load will cause buckling for slender, long columns. In contrast, failure will occur in short columns when the strength of material is exceeded. Between the long and short column CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 5 General Notes On Column Buckling limits, there is a region where buckling occurs after the stress exceeds the proportional limit but is still below the ultimate strength. These columns are classified as intermediate and their failure is called inelastic buckling. 4. Whether a column is short, intermediate, or long depends on its geometry as well as the stiffness and strength of its material. This concept is addressed in the columns introduction page. 3

4 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 6 The Concept of Effective Length The Euler buckling formula, namely Eqs. 9 or 11 of Chapter 5b, were derived for a column with pivoted ends. The Euler equation changes for columns with different end conditions, such as the four common ones found in Figs.1and 2. While it is possible to set up the differential equation with appropriate boundary conditions to determine the Euler buckling CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 7 Figure 1 4

5 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 8 Figure 2 (Beer and Johnston 1992) CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 9 The Concept of Effective Length formula for each case, a more common approach makes use of the concept of an effective length. The pivoted ended column, by definition, has zero bending moments at each end. The length L in the Euler equation, therefore, is the distance between successive points of zero bending moment. 5

6 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 10 The Concept of Effective Length All that is needed to modify the Euler column formula for use with other end conditions is to replace L by L. L is defined as the effective length of the column. where L = Le = KL (1) K = effective length factor as defined by LRFD CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 11 The Effective Length Concept Definition: The effective length L (or L e or KL) of a column is defined as the distance between successive inflection points or points of zero moment. 6

7 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 12 The Effective Length Concept Based on the effective length concept, the Euler buckling load and stress formulas become, respectively 2 π EI Pcr = (2) 2 KL or F KL = L = L = effective length e e = ( ) π 2 E ( KL / r) 2 (3) CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 13 AISC LRFD Specifications for End Restraint Table C-C2.1 of the Commentary on the LRFD Specification gives recommended values of the effective length factors K when ideal conditions are approximated. This table is reproduced here as Table 1 (Table 5.1, Textbook). Two sets of K values are provided in the table. 7

8 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 14 Table 1 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 15 AISC LRFD Specifications for End Restraint One being the theoretical values and the other being the recommended design values. The recommended values are based on the fact that perfectly pinned and fixed conditions are not always possible. For continuous frames, the LRFD provides special K-value Charts for that purpose. 8

9 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 16 Effect of Braced and Unbraced Structural Frames on Columns Strength Structural steel columns can be parts of structural frames. These frames are sometimes braced and sometimes unbraced. A braced frame is one for which sideway (joint translation) by means of bracing, shear walls, or lateral support from adjoining structure (see Fig. 2a). CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 17 Effect of Braced and Unbraced Structural Frames on Columns Strength An unbraced frame does not have any of these types of bracing provided, and must depend on the stiffness of its own members and rotational rigidity of the joints between the frames members to prevent lateral buckling (see Fig. 2b) 9

10 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 18 Figure 2 L B Lateral deflection Column is in this Position after sideway And joint rotation B L A A Lateral L deflection (a) (b) CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 19 Effect of Braced and Unbraced Structural Frames on Columns Strength Examination of Fig 2a will show that the effective length will exceed the actual length of the column as the elastic curve will theoretically take the shape of the curve of a pinned-end column of twice its length and K will theoretically equal 2.0. Notice in Fig 2b how much smaller the lateral deflection of column AB would be if it were pinned at both ends to prevent sideway. 10

11 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 20 Effect of Braced and Unbraced Structural Frames on Columns Strength For braced frames, K values can never be greater than 1.0, but for unbraced frames the K values will always be greater than 1.0 because of sideway. The smaller the effective length (i.e., braced) of a particular column, the smaller its danger of lateral buckling and the greater its load-carrying capacity. CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 21 Local Buckling Up to this point, the overall stability of a particular column has been considered. Yet, it is entirely possible for thin flanges or webs of a column or beam to buckle locally in compression well before the calculated buckling strength of the whole member is reached. When thin plates are used to carry compressive stresses they are particularly susceptible to buckling about their weak axes due to small moment of inertia. 11

12 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 22 LRFD Specification (Section B5) The LRFD Specification provides limiting values for the width-thickness ratios of the individual parts of compression members and for the parts of beams in their compression regions. Two categories are listed in the LRFD Manual: Stiffened elements Unstiffened elements CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 23 LRFD Specification (Section B5) An unstiffened element is a projecting piece with one free edge parallel to the direction of the compression force. A stiffened element is supported along two edges in that direction. These types of elements are shown in Figure 3. In each case, the width b and the thickness t of the elements in question are shown. 12

13 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 24 Figure 3. Unstiffened and Stiffened Elements CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 25 LRFD Specification (Section B5) For establishing width-thickness ratio limits for elements of compression members, the LRFD Specification divides members into three distinct classifications as follows: 1. Compact sections 2. Noncompact sections 3. Slender compression elements 13

14 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 26 Compact Sections A compact section is one that has a sufficiently stocky profile so that it is capable of developing a fully plastic stress distribution before buckling. For a section to be compact, it has to have a width-thickness ratios equal to or less than the limiting values provided in Table 4 (Table 5.2, Text, or Table B5.1, LRFD Maual). CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 27 Figure 4. Limiting Width-Thickness Ratios for Compression Elements 14

15 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 28 Figure 4. (cont d) Limiting Width-Thickness Ratios for Compression Elements CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 29 Figure 4. (cont d) Limiting Width-Thickness Ratios for Compression Elements Stiffened Elements 15

16 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 30 Figure 4. (cont d) Limiting Width-Thickness Ratios for Compression Elements CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 31 Noncompact Sections A noncompact section is one for which the yield stress can be reached in some but not all of its compression elements before buckling occurs. It is not capable of reaching fully plastic stress distribution. For a section to be noncompact, it has to have a width-thickness ratios greater than λ p but less than λ r as provided in Table 4 (Table 5.2, Text, or Table B5.1, LRFD Maual). 16

17 CHAPTER 5c. INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS Slide No. 32 Slender Compression Elements A slender element with a cross section that does not satisfy the width-thickness ratio requirements of Table 4 (Table 5.2, Text, or Table B5.1, LRFD Maual). For a section to be slender, it has to have a width-thickness ratios greater than λ r as provided in Table 4 (Table 5.2, Text, or Table B5.1, LRFD Maual). 17

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