158
CHAPTER 6
COLUMNS
ͳǤ ǣݎൌܦȀͶൌͲǤ͹ͷȀͶൌͲǤͳͺͺǣܭܮȀݎൌͳǤͲሺ͵ʹሻͲǤͳͺͺൌͳ͹ͳ
ʹǤ ȏͳȐ Lαͳͷ
͵Ǥ ȏͳȐ͸Ͳ͸ͳǦͶǤ
ͶǤ ȏͳȐǤ
ͷǤ ȏͳȐǣααͲǤ͸ͷ
159
͸Ǥ ȏͳȐǣ
͹Ǥ ݎൌͲǤͷȀξͳʹൌͲǤͳͶͶ
ͺǤ ݎൌඥሺܦଶ൅݀ଶሻȀͶൌඥሺͳǤ͸Ͳሻଶ൅ሺͳǤ͵ͺʹሻଶȀͶൌͲǤͷʹͻǣܮൌ͸Ǥʹͷቀଵଶ௜௡
௙௧ ቁൌ͹ͷ
ܣൌߨ
ሺܦଶ൅݀ଶሻȀͶൌͲǤͷͳͲ͸ǣܮȀݎൌ͹ͷȀͲǤͷʹͻൌͳͶʹ
ݏ
௬ൌ͵ͲͲͲͲǢܥ௖ൌͳͶͲሺ͸െͷሻ
Ȍ ǣܭܮȀݎൌͳǤͲሺܮȀݎሻൌͳͶʹ൐ܥ௖െ
ࡼ
ࢉ࢘ ൌగమ൫ଷ଴ൈଵ଴ల൯ሺ଴Ǥହଵ଴଺ሻ
ሺଵସଶሻమൌૠ૝ૢૡܔ܊
Ȍ ȂǣܭܮȀݎൌͲǤ͸ͷሺܮȀݎሻൌͻʹǤ͵൏ܥ௖െെ
ࡼ
ࢉ࢘ ൌሺͲǤͷͳͲ͸ሻሺ͵ͲͲͲͲሻቀͳെଷ଴଴଴଴ሺଽଶǤଷሻమ
ସగమሺଷ଴ൈଵ଴లሻቁൌ૚૛૙૙૙ܔ܊
Ȍ ȂǣܭܮȀݎൌͲǤͺሺͳͶʹሻൌͳͳͶ൏ܥ௖െെ
ࡼ
ࢉ࢘ ൌሺͲǤͷͳͲ͸ሻሺ͵ͲͲͲͲሻቀͳെଷ଴଴଴଴ሺଵଵସሻమ
ସగమሺଷ଴ൈଵ଴లሻቁൌ૚૙૜૙૙ܔ܊
0.50in
1.00in
160
ͻǤ ݏ௬ൌͳͷʹǣܥ஼؆͸Ͳሺܨ݅݃ݑݎ݁͸െͷሻǣǣܭൌͳǤͲ
ͳͲǤ ݏ௬ൌ͵Ͳǣܥ௖ൌͳͶͻሺܨ݅݃ݑݎ݁͸െͷሻǣ
ͳͳǤ ʹͲͳͶǦͶǣݏ௬ൌͶʹͲͲͲǢܥ௖؆͸ͻǣ
଺ସሺଷሻሺ଼ହ଴଴ሻሺହ଴ሻమ
161
ͳʹǤ ǡSǣܫൌܵସȀͳʹǢܣൌܵଶǣ͸െͳͲെ
ܫൌܵ
ସȀͳʹൌܰܲሺܭܮሻଶȀߨଶܧ
ܵൌቀ
ଵଶே௉ሺ௄௅ሻమ
గమாቁଵȀସ
ͳ͵Ǥ ǣȏȋ͸ǦͳͲȌȐǢܫൌܰܲሺܭܮሻଶȀߨଶܧǢRαd/D;d=RD
ܫൌ
గ൫஽రିௗర൯
଺ସ ൌగ൫஽రିோర஽ర൯
଺ସ ൌగ஽రሺଵିோరሻ
଺ସ
గ஽ర൫ଵିோర൯
଺ସ ൌே௉ሺ௄௅ሻమ
గమா
଺ସே௉ሺ௄௅ሻమ
d=
D =
OD
162
ͳͶǤ ǣͳʹǣαܵൌቀଵଶே௉ሺ௄௅ሻమ
గమாቁଵȀସ
 ܭܮൌͲǤ͸ͷሺ͸ͶሻൌͶͳǤ͸ሺሻ
ଵଶሺଷሻሺ଺ହ଴଴ሻሺସଵǤ଺ሻమ
ͳͷǤ ܴൌܫܦȀܱܦൌͲǤͺǣሺͳെܴସሻൌͲǤͷͻͲͶ
 Ǣͳ͵
଺ସே௉ሺ௄௅ሻమ
ܹ
௦Ȁ்ܹൌʹǤʹͷȀͲǤ͸͵͸ൌ૜Ǥ૞૝െ܂ܐ܍ܜܝ܊܍ܛܐ܉ܘ܍ܑܛܕܝ܋ܐܕܗܚ܍܍܎܎ܑ܋ܑ܍ܖܜ
ͳ͸Ǥ ǣȏȋ͸ǦͳͳȌȐȏǤȐ
ܦൌቀ
଺ସே௉ሺ௄௅ሻమ
ͳͺǤ Multipledesignspossible.
Cable
Fv =
164
ͳͻǤ Multipledesignspossible.
ʹͲǤ  ܮൌͳͲǤ͹ͷൈଵଶ௜௡
௙௧ ൌͳʹͻǣܭܮൌʹǤͳሺͳʹͻሻൌʹ͹ͳǣݏ௬ൌ͸ͺǢܥ௖ൌͻ͵
ʹ͵Ǥ ǣܽൌͲǤͲͺǢܦൌͲǤ͹ͷǡ ܿൌܦȀʹൌͲǤ͵͹ͷ
 ݎൌܦȀͶൌͲǤͳͺͺǢݎଶൌͲǤͲ͵ͷʹଶǢܣൌͲǤͶͶʹଶǢܲ௖௥ ൌͶͶ͹͵
ʹͶǤ  ǣܽൌͲǤͲͶǢܿൌͲǤͷȀʹൌͲǤʹͷǢܲ௖௥ ൌ
Ͷʹ͸͹ͷሾ͸െ͹ሿ
165
ʹͷǤ  ǣܽൌͲǤͳͷǢܥൌܱܦȀʹൌͲǤͺͲǢܲ஼ோ ൌ
͹Ͷͻͺሺ͸െͺሻ
 ݎൌͲǤͷʹͻǢݎଶൌͲǤʹͺͲଶǢܣൌͲǤͷͳͲ͸ଶǢݏ௬ൌ͵ͲͲͲͲ
ʹ͸Ǥ  ǣܮൌͶʹǢܵൌͳǤʹͷǢܿൌܵȀʹൌͲǤ͸ʹͷ
ݎൌܵȀ
ξͳʹൌͲǤ͵͸ͳǢݎଶൌͲǤͳ͵ͲଶǢܣൌܵଶൌͳǤͷ͸͵ଶǢ݁ൌͲǤ͸Ͳ
ସȀͳʹൌͲǤʹͲ͵ͷସǢݏ௬ൌͳ͵ͲͲͲ͸Ͳ͸͵ǦͶ
ʹ͹Ǥ  ǣܮൌ͵Ǥʹൌ͵ʹͲͲǢܲൌ͵ͲǤͷൌ͵ͲͷͲͲ
 ͵ǦͶͲǢܦ௢ൌ͵ǤͷͲǢܿൌܦ௢ȀʹൌͳǤ͹ͷሺʹͷǤͶ݉݉Ȁ݅݊ሻൌͶͶǤͷ
S
S
ʹͺǤ ǣܮൌͳͶǤ͹ͷǢ݁ൌͲǤ͵ͲǢܵൌͲǤʹͷͲ
ܣൌܵ
ଶൌͲǤͲ͸ʹͷଶǢݎൌܵȀξͳʹൌͲǤʹͷȀξݎൌͲǤͲ͹ʹʹǢݎଶൌͲǤͲͲͷʹͳ
ʹͻǤ ǣܮൌͶͲǢ݁ൌͲǤͷͲǢܲൌ͹ͷͲͲͲ
ͳͷǦͳͶǣܣൌ͵Ǥ͵͹ଶǢݎൌͳǤͷʹǢݎଶൌʹǤ͵ͳǢܿൌͶȀʹൌʹǤͲͲ
S
S
167
͵ͳǤ ǣǦǢܭൌͲǤ͸ͷǢܮ௘ൌͲǤ͸ͷሺ͸͸ሻൌͶʹǤͻ
 Ͷൈ͹Ǥ͹ǣܣൌʹǤʹ͸ଶǢݎ௠௜௡ ൌݎ
௬ൌͲǤͷͺͳǢܮ௘Ȁݎൌ͹͵ǤͺǢܰൌ͵Ǣܲ௔ൌܲ
௖௥Ȁܰ
͵ʹǤ ǣܲൌͳͲͲͲǢ݁ൌͲǤͷͲ൅ͲǤͺͲȀʹൌͲǤͻͲǢܮൌ͹ʹǢ
ܣൌ
ሺͳǤ͸ͲሻሺͲǤͺͲሻൌͳǤʹͺଶǢܿൌͲǤͺͲȀʹൌͲǤͶͲǢݎൌͲǤͺͲȀξͳʹൌͲǤʹ͵Ͳͻ
 ǣܧൌ͵ͲൈͳͲ଺Ǣȏ͸ǦͳͶȐ

168
͵͵Ǥ
ǣǤs
y
α͵͸ͲͲͲǢNα͵
ΦǣP
a
αͷͷͲͲͲȀʹαʹ͹ͷͲͲ
ே௉
ೌ
ሺ௄௅ሻ
మ
͵ͶǤ
ǣͷൈͻǣܮൌͳͳʹǢܣൌʹǤ͸Ͷ
ଶ
ǣݎൌͲǤͶͺͻǢܰൌ͵
͸
ଷሺଶଶଽሻ
మ
͵ͷǤ ǣ͵Ͷǡ
͵ሺͳͶͺǤͻሻ
ଶ
͵͸Ǥ ͵ͶǤǤ
 ݁ൌݔȏͳͷǦͶȐൌͲǤͶ͹ͺܿൌ݁Ǥ
ଶǤ଺ସ
ሺ଴Ǥସ଼ଽሻ
మ
ଶሺ଴Ǥସ଼ଽሻ
ሺଶǤ଺ସሻሺଷ଴ൈଵ଴
ల
ሻ
 ǣࡼࢇൌ૝૚૙૙ܔ܊Ǣߪௗൌ૚૚ૢ૛૙ܘܛܑ
͸Ǧ͵Ǥ
Data from: Example Problem 6-3
CIRCULAR COLUMN ANALYSIS
Cross-section shape: Solid circular section
Refer to Figure 6-7 for analysis logic
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
Diameter for round column: 1.23 in Column is: short
Area, A = 1.188 in
2
Radius of gyration, r = 0.3075 in Critical Buckling Load = 29,679 lb
Design Factor
Design factor on load, N = 3 ——-> Allowable Load = 9,893 lb
͸ǦͶǤ
Ǥ
Data from: Example Problem 6-4
CROOKED COLUMN ANALYSIS
Cross-section shape: Solid circular section
Solves Equation 6-13 for Allowable Load
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
Data To Be Entered: Computed Values:
Length and End Fixity:
Cross Section Properties: Euler buckling load = 4491 lb
[ Note: Enter r or compute r = sqrt(I/A ) ]
[Always enter Area] C
1
in Eqn. 6–13 = –9678
[Enter zero for I or r if not used]C
2
in Eqn. 6-13 = 9.259E+06
2
4
171
Data from: Example Problem 6-6
ECCENTRIC COLUMN ANALYSIS
Cross section shape: Solid circular section
Solves Equation 6-14 for design stress and Equation 6-16 for maximum deflection
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
Data To Be Entered: Computed Values:
Length and End Fixity:
[Always enter Area]
[Enter zero for I or r if not used ] Argument for secant = 0.432 for deflection
Area, A = 0.785 in
2
Value of secant = 1.1014
Moment of Inertia, I = 0 in
4
Radius of Gyration, r = 0.250 in ——> Slenderness ratio, KL/r = 128.0
Values for Eqns. 6-14 and 6-16:
Eccentricity = e = 0.75 in Column is: long
Neutral axis to outside = c = 0.5 in
Allowable load = P
a
= 1075 lb Req’d yield strength = 37,764 psi
Design Factor Must be less than actual yield strength:
Design factor on load, N = 3 ——> s
y
= 42,000 psi
Max. deflection, y
max
= 0.076 in

͸Ǧ͸Ǥ͸Ǥ

172

Data from: Problem 6-37
ECCENTRIC COLUMN ANALYSIS
Cross section shape: Hollow structural section
Solves Equation 6-14 for design stress and Equation 6-16 for maximum deflection
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
[Enter zero for I or r if not used] Argument for secant = 0.449 for deflection
Area, A = 6.020 in
2
Value of secant = 1.1098
Moment of Inertia, I = 0 in
4
173
ǡȋͲǤͶͲȌ
͵ͺǤ ͳǣܣൌሺͳǤͷͲ݅݊ሻሺͲǤͶͲ݅݊ሻൌͲǤ͸ͲͲଶ
ு
Data from: Problem 6-38
ECCENTRIC COLUMN ANALYSIS
Cross section shape: Solid Rectangle
Solves Equation 6-14 for design stress and Equation 6-16 for maximum deflection
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
[Enter zero for I or r if not used] Argument for secant = 0.494 for deflection
Area, A = 0.600 in
2
Value of secant = 1.1355
Moment of Inertia, I = 0 in
4
174
͵ͺǤ ǡ

Data from: Problem 6-38B
RECTANGULAR COLUMN ANALYSIS
Cross-section shape: Rectangular
Refer to Figure 6-7 for analysis logic
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
Data To Be Entered: Computed Values:
Length and End Fixity:
Column length, L = 40 in
End fixity, K = 1 ——> Eq. Length, L
e
= KL = 40.0 in
Area, A = 0.600 in
Radius of Gyration, r = 0.115 in ——> Slender. ratio, KL/r = 347.8
Properties for round column:
Analysis of buckling about the thin dimension
175
Data from: Problem 6-39
ECCENTRIC COLUMN ANALYSIS
Cross section shape: Hollow circular section
Solves Equation 6-14 for design stress and Equation 6-16 for maximum deflection
Enter data for variables in italics in shaded boxes Use consistent SI Metric
[Enter zero for I or r if not used] Argument for secant = 0.468 for deflection
Area, A = 314 mm
2
Value of secant = 1.1206
Moment of Inertia, I = 0 mm
4
176

Data from: Problem 6-40A-Straight
HOLLOW CIRCULAR COLUMN
Cross-section shape: Hollow circular section
Refer to Figure 6-7 for analysis logic
[dimensions for circular cross section]
[in following section of this spreadsheet]
Area, A = 1.075 in
2
Radius of Gyration, r = 0.787 in ——> Slender. ratio, KL/r = 198.2
177

Data from: Problem 6-40B-Crooked
CROOKED COLUMN ANALYSIS
Cross-section shape: Hollow circular section
Solves Equation 6-13 for Allowable Load
Enter data for variables in italics in shaded boxes Use consistent U.S. Customary units.
Data To Be Entered: Computed Values:
Length and End Fixity:
Column length, L = 156 in
Area, A = 1.075 in
2
Moment of Inertia, I = 0 in
4
Problem # 40A 40B
Straight Crooked
Comparisons: Straight (40A) vs. Crooked (40B)