Structural Steel Design
CHAPTER
LRFD LRF D Met Method hod
Third Edition
DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS • A. J. Clark School of Engineering •Department of Civil and Environmental Engineering
Part II – Structural Structural Steel Steel Design and Analysis Analysis
6a
FALL 2002
ENCE 355 - Introduc Introduction tion to Structur Structural al Design Design Department of Civil and Environmental Engineering University of Maryland, College Park
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Introduction
Slide No. 1 ENCE 355 ©Assakkaf
The members that can be designed for compression include: – Single shapes – W sections with cover plates – Built-up sections constructed with channels – Sections whose unbraced lengths in the x and y directions. – Lacing and tie plates for built-up sections section s with open sides.
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Slide No. 2
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
Introduction
The Design Process for Columns – It is to be noted that the design of columns wit formulas involves a trial-and-error process. – The design stress φ c F cr is not known until a column size is selected and vice versa. – Once a trial section is assumed, the r value for that section can be obtained and substituted into the appropriate column equation to determine its design stress.
Slide No. 3
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
Introduction
The Design Process for Columns – In the design of columns, the factored load P u is computed for a particular column and then divided by an assumed design stress to give an estimated column area A, that is
Aestimated
=
P u assumed stress
=
P u
φc F cr
(1)
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Introduction
Slide No. 4 ENCE 355 ©Assakkaf
The Design Process for Columns – After an estimated column area is determined, a trial section can be selected with approximately that area. – The design stress for the selected section can be computed and multiplied by the cross sectional area of the section to obtain the member’s design strength. – This design strength is compared with the factored load P u. It must be equal or greater than the load P u.
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Introduction
Slide No. 5 ENCE 355 ©Assakkaf
General Notes on Column Design – The effective slenderness ratio ( KL/r ) for the average column of 10 to 15 ft in length will generally fall between 40 and 60. – A value for KL/r in this range can be assumed and substituted into the appropriate column equation. – Or instead of the column equation, tables in LRFD manual can be consulted to give the design strength for that particular KL/r value. ( KL/r ranges from 1 to 200 in LRFD)
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Introduction
Slide No. 6 ENCE 355 ©Assakkaf
Example 1 Using F y = 50 ksi, select the lightest W14 section available for the service column loads P D = 130 k and P L = 210 k. Assume KL = 10 ft.
= 1.2 P D = 1.2(130) = P u = 1.2 P D + 1.6 P L = 1.2(130) + 1.6(210 ) = 492 k P u
Assume
KL
= 50
r
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Slide No. 7 ENCE 355 ©Assakkaf
Introduction
Governs
Example 1 (cont’d) φc F cr form Table 3.50 (Part 16 of Manual) = 35.4 ksi ∴ Arequired =
P u
φc F cr
=
492 35.4
(
= 13.90 in 2
Try W14 × 48 A = 14.1 in 2 , r x r y
= 1.91
KL r y
=
12 × 10 1.91
= 5.85 in, r y = 1.91 in )
controls
= 62.83
φc F cr form Table 3.50 (Part 16 of Manual) and by iterpolation = 31.85 ksi
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Introduction
LRFD Manual Design Tables (P. 16. I-145)
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Introduction
Slide No. 8 ENCE 355 ©Assakkaf
Slide No. 9 ENCE 355 ©Assakkaf
LRFD Manual Design Tables (P. 16. I-145)
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Slide No. 10 ENCE 355 ©Assakkaf
Introduction
Example 1 (cont’d) φc P n = (φc F cr ) A g = 31.85(14.1) = 449 k < 492 k NG ∴ try next larger W14 Try W14 × 53( A = 15.6 in 2 , r x = 5.89 in, r y = 1.92 in )
= 1.92
r y
KL r y
=
controls
12 × 10 1.92
= 62.5
φc F cr form Table 3.50 (Part 16 of Manual) and by iterpolation = 31.95 ksi
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Slide No. 11 ENCE 355 ©Assakkaf
Introduction Example 1 (cont’d) ∴φc P n = (φc F cr )Ag = 31.95(15.6 ) = 498 k < 492 k
Checking width-thickness ratio for W14
OK ×
53:
b f = 8.060 in , t f = 0.660 in, W14 × 53 k = 1.25 in, d = 13.9 in, t = 0.370 in w b f 2t f h t w
=
=
8.060 2(0.660 )
= 6.11 < 0.56
13.9 − 2(1.25) 0.370
E F y
= 30.81 < 1.49
= 0.56 E F y
29 ×103
= 1.49
50
= 13.49
29 ×103 50
= 35.88
OK OK
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
The LRFD Manual can be used to select various column sections from tables without the need of using a trialand-error procedures.
These tables provide axial design strengths φ c P n for various practical effective lengths of the steel sections commonly used as columns.
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 12 ENCE 355 ©Assakkaf
Slide No. 13 ENCE 355 ©Assakkaf
LRFD Manual Design Tables (P. 4-25)
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Slide No. 14
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
ENCE 355 ©Assakkaf
The values are given with respect to the least radii of gyration for W’s and WT’s with 50 ksi steel.
Other grade steels are commonly used for other types of sections as shown in the Manual and listed there.
These include 35 ksi for steel pipe, 36 ksi for L’s, 42 ksi for round HSS sections, and 46 ksi for square and rectangular HSS sections.
Slide No. 15
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
ENCE 355 ©Assakkaf
For most columns consisting of single steel shapes, the effective slenderness ratio with respect to the y axis ( KL/r ) y is larger than the effective slenderness ratio with respect to the x axis ( KL/r ) x. As a result, the controlling or smaller design stress is for the y axis. Because of this, the LRFD tables provide design strengths of columns with respect to their y axis.
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 16 ENCE 355 ©Assakkaf
Example 2 Using the LRFD column tables with their given yield strengths: a. Select the lightest W section available for the loads, steel, and KL of Example 1. Use F y = 50 ksi. b. Select the lightest satisfactory standard (S), extra strong (XS), and double extra strong (XXS) pipe columns described in part (a) of this example. Use F y = 35 ksi.
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 17 ENCE 355 ©Assakkaf
Example 2 (cont’d) c. Select the lightest satisfactory rectangular and square HSS sections for the situation in part (a). Use F y = 46 ksi. d. Select the lightest round HSS section for part (a). Use F y = 42 ksi. a. Enter LRFD tables with K y L y = 10 ft., P u = 492 k, and F y = 50 ksi.
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Slide No. 18
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
Example 2 (cont’d) Lightest suitable section in each W series:
Lightest Page 4-26 of Manual
W 14 × 53(φc P n
= 498 k ) W 12 × 53(φc P n = 559 k ) W 10 × 49(φ c P n = 520 k ) Therefore, USE W10 × 49
controls
b. Pipe Columns: S : not available Page 4-76 of Manual
XS12 × 0.500(65.5 lb/ft ) = 549 k
Page 4-76 of Manual
XXS8 × 0.875(72.5 lb/ft ) = 575k
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 19 ENCE 355 ©Assakkaf
Example 2 (cont’d) c. Rectangular and square HSS sections:
Page 4-49 of Manual
HSS 14 ×14 ×
Page 4-51 of Manual
HSS 12 ×10 ×
5 16 3 8
(57.3 lb/ft ) = 530 k
(52.9 lb/ft ) = 537 k
d. Round HSS section: Page 4-66 of Manual
HSS 16 × 0.312(52.3 lb/ft ) = 500 k
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Slide No. 20
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
How to handle the situation when ( KL/r ) x is larger than ( KL/r ) y ? – Two methods can be used: • Trial-and error method • Use of LRFD Tables
– An axially loaded column is laterally restrained in its weak direction as shown in Figs. 1 and 2
Slide No. 21
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables Figure 1
P u
This brace must be a section which Prevents lateral movement and twisting Of the column. A rod or bar is not satisfactory.
L 2 L L 2
P u
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Slide No. 22
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables x
y
Figure 2
y
x L 2
L
Bracing
2
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 23 ENCE 355 ©Assakkaf
How to handle the situation when ( KL/r ) x is larger than ( KL/r ) y ? Trial-and-error Procedure: • A trial section can be selected as described previously. • Then the slenderness values ( KL/r ) x and ( KL/r ) x are computed. • Finally, φ c F cr is determined for the larger value of ( KL/r ) x and ( KL/r ) x and multiplied by A g to obtained φ c P n. • Then if necessary, another size can be tried, and so on.
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Slide No. 24
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
How to handle the situation when ( KL/r ) x is larger than ( KL/r ) y ? – It is assumed that K is the same in both directions. Then, if equal strengths about the x and y axis to be obtained, the following relation must hold:
L x r x
=
L y
(2)
r y
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 25 ENCE 355 ©Assakkaf
How to handle the situation when ( KL/r ) x is larger than ( KL/r ) y ? – For L y to be equivalent to L x, the following relation would hold true:
L x
= L y
r x r y
(3)
– If L y (r x/r y) is less than L x, then L x controls. – If L y (r x/r y) is greater than L x, then L y controls.
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 26 ENCE 355 ©Assakkaf
How to handle the situation when ( KL/r ) x is larger than ( KL/r ) y ? Use of LRFD Tables: • Based on the preceding information, the LRFD Manual provides a method with which a section can be selected from tables with little trial and error when the unbraced lengths are different. • The designer enters the appropriate table with K y L y, selects a shape, takes r x /r y value in the table for that shape, and multiplies it by L y. • If the result is larger than K x L x, then K y L y controls and the shape initially selected is the correct one.
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 27 ENCE 355 ©Assakkaf
How to handle the situation when ( KL/r ) x is larger than ( KL/r ) y ? Use of LRFD Tables (cont’d): • If the result of the multiplication is less than K x L x, then K x L x controls and the designer will reenter the tables with a larger K y L y equal to K x L x / (r x /r y) and select the final section.
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Slide No. 28
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
Example 3 Select the lightest satisfactory W12 for the following conditions: F y = 50 ksi, P u = 900 k, K x L x 26 ft, and K y L y 13 ft. =
=
a. By trial and error b. Using LRFD tables a. Using trial and error: Assume
KL r
= 50
Slide No. 29
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
x
Example 3 (cont’d) y
K = 1
P u
KL = L = 26 ft y
L
x
2
L 2
L L
L
2
Bracing
2
P u
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 30 ENCE 355 ©Assakkaf
Example 3 (cont’d)
φc F cr = 35.40 ksi (from Table 3.50 of Manual) Arequired
=
P u
=
900
= 25.42 in 2
φc F cr 35.40 Try W12 × 87( A = 25.6 in 2 , r x = 5.38 in, r y = 3.07 in )
KL = 12 × 26 = 57.99 ≈ 58 controls r x 5.38 KL = 12 ×13 = 50.81 r y 3.07 φc F cr = 33.2 ksi ∴ φc P n = 33.2(25.6) = 850 k < 900 k
NG
A subsequent check of the next larger W section (W12 × 96) shows it will work. Therefore, USE W12 × 96
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
LRFD Design Tables
Slide No. 31 ENCE 355 ©Assakkaf
LRFD Manual Design Tables (P. 16. I-145)
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Slide No. 32
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
LRFD Manual Design Tables (P. 16. I-145)
Slide No. 33
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
ENCE 355 ©Assakkaf
LRFD Design Tables
Example 3 (cont’d) See P. 4-25 of Manual
b. Using LRFD tables:
Enter tables with K y L y = 13 ft, F y = 50 ksi, and P u = 900 k. r x = 1.75 with φc P n based on K y L y r y
Try W12 × 87
Equivalent K y L y
=
K x L x r x / r y
= 13(1.75) = 22.75 ft < K x L x
Therefore, K x L x controls. Reenter ta bles with K y L y USE W12 × 96
=
K x L x r x / r y
=
φc P n = P u = 935 k
26 1.75
= 14.86
OK
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Built-up Columns
Slide No. 34 ENCE 355 ©Assakkaf
Compression members may be constructed with more shapes built-up into a single member.
They may consist of parts in contact with each other, such as cover-plated sections:
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Built-up Columns
Slide No. 35 ENCE 355 ©Assakkaf
Or they may consist of parts in near contact with each other, such as pair of angles:
These pairs of angles may be separated by a small distance from each other equal the thickness of the end connection or gusset plates between them.
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Built-up Columns
Slide No. 36 ENCE 355 ©Assakkaf
They may consist of parts that are spread well apart, such as pairs of channels:
Or four angles, and so on.
CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Built-up Columns
Slide No. 37 ENCE 355 ©Assakkaf
Two-angle sections probably are the most common type of built-up members. They are frequently used as the members of light trusses.
When a pair angles are used as a compression member, they need to be fastened together so they will act as a unit.
Welds may be used at intervals or they may be connected with bolts.
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CHAPTER 6a. DESIGN OF AXIALLY LOADED COMPRESSION MEMBERS
Built-up Columns
Slide No. 38 ENCE 355 ©Assakkaf
For long columns, it may be suitable to use built-up sections where the parts of the columns are spread out or widely separated from each other.
These types of built-up columns are commonly used for crane booms and for compression members of various kinds of towers.
The widely spaced parts of these types must be carefully laced or tied together.
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