V. AISI 2016 CU Nominal and Local Axial Capacity

Determine the nominal axial capacity and local axial capacity of a 72-Inch-long member subjected to axial compression load.

Details

The member is a 6' simply-supported span subject to a 1 kip axial load. The section used is a 800CU125-68 channel without lips.

Section properties:

Depth: D = 8.25 in
Width: B = 1.25 in
Thickness: t = 0.0713 in
Inner radius: R = 0.1069 in
Area: A = 0.748 in²
Moment of inertia about major axis: I_x = 6.0 in⁴
Moment of inertia about minor axis: I_y = 0.0703 in⁴
Torsional constant: J = 0.00127 in⁴
Distance between centroid and shear center: x₀ = -0.427 in
Radius of gyration about major axis: $r_{x} = \sqrt{\frac{I_{x}}{A}} = 2.832 in.$
Radius of gyration about minor axis: $r_{y} = \sqrt{\frac{I_{t}}{A}} = 0.307 in.$
Polar radius of gyration: $r_{0} = \sqrt{{r_{x}}^{2} + {r_{y}}^{2} + {x_{0}}^{2}} = 2.881 in.$
Warping constant: C_w = 0.92 in⁶

Material properties:

E = 29,500 ksi
G = 11,300 ksi
F_y = 50 ksi

Validation

Section Dimension Checks

Radius to thickness ratio:

\frac{R}{t} = \frac{0.1069}{0.0713} = 1.499 < 10 (OK)

Width to thickness ratio:

W = (B - R - t) = 1.072 in

\frac{W}{t} = \frac{1.072}{0.0713} = 15.03 < 60 (OK)

Depth to thickness ratio:

d = (D - 2 R - 2 t) = 7.849 in

\frac{d}{t} = \frac{7.849}{0.0713} = 110.7 < 200 (OK)

Nominal Strength of Global Buckling

From Cl. E.2 of AISI S100-16:

\frac{K_{y} \times L_{y}}{r_{y}} = \frac{1 \times 72}{0.307} = 234.9

σ_{x} = \frac{π^{2} \times E}{{(\frac{K_{x} \times L_{x}}{r_{x}})}^{2}} = \frac{π^{2} \times 29000}{{(\frac{1 \times 72}{2.832})}^{2}} = 442.9 ksi

F_{c r e y} = \frac{π^{2} \times E}{{(\frac{K_{y} \times L_{y}}{r_{y}})}^{2}} = \frac{π^{2} \times 29000}{{(234.9)}^{2}} = 5.189 ksi

σ_{t} = \frac{1}{A \times {r_{0}}^{2}} \times [G \times J + \frac{π^{2} \times E \times C_{w}}{{(K t \times L t)}^{2}}] = \frac{1}{0.748 {(2.881)}^{2}} \times [11,300 \times 0.001 + \frac{π^{2} \times 29,000 \times 0.92}{{(1 \times 72)}^{2}}] = 10.50 ksi

β = 1 - {\frac{x_{0}}{r_{0}}}^{2} = 1 - ({\frac{- 0.427}{2.881})}^{2} = 0.978

F_{(c r e) T F} = \frac{1}{2 \times β} \times [σ_{e x} + σ_{t} - \sqrt{{(σ}_{e x} + σ_{t)}^{2} - 4 \times β \times σ_{e x} \times σ_{t}})

= \frac{1}{2 \times 0.978} \times [442.9 + 10.50 - \sqrt{{(442.9 + 10.50)}^{2} - 4 \times 0.976 \times 442.9 \times 10.50}) = 10.49 ksi

Since: $F_{(c r e) T F} > F_{c r e y}$ , $F_{(c r e)} = 5.189 ksi$

λ_{c} = \sqrt{\frac{F_{y}}{F_{(c r e)}}} = 3.104 > 1.5

F_{n} = \frac{0.877}{{λ_{c}}^{2}} \times F_{y} = \frac{0.877}{{(3.104)}^{2}} \times 50 = 4.551 ksi

P_{n e} = A \times F_{n} = 0.748 \times 4.551 = 3.404 kips

Allowable nominal axial strength (ASD):

\frac{P_{n e}}{Ω_{c}} = \frac{3.404}{1.80} = 1.891 kips

Design nominal axial strength (LRFD):

P_{n e} \times ϕ_{c} = 3.404 \times 0.85 = 2.893 kips

Local Buckling Interacting with Lateral-Torsional Buckling

From Cl. E.3.1 of AISI S100-16:

Flange elements:

s = 1.28 \times \sqrt{\frac{E}{F_{n}}} = 1.28 \times \sqrt{\frac{29,000}{4.551}} = 102.2

(Eq 1.3-7, Appendix 1 of AISI S100-16)

\frac{W}{t} = 15.03 < 0.328 \times s = 33.51

Therefore, calculation of the effective width is not required.

Web element:

\frac{d}{t} = 110.7 < 500 (OK)

k = 4.0

F_{c r} = \frac{k \times π^{2} \times E}{12 \times (1 - μ^{2})} \times {(\frac{t}{d})}^{2} = \frac{4 \times π^{2} \times 29,000}{12 \times (1 - {0.3}^{2})} \times {(\frac{0.0713}{7.849})}^{2} = 8.553 ksi

λ = \sqrt{\frac{F_{n}}{F_{c r}}} = \sqrt{\frac{4.551}{8.553}} = 0.729

ρ = \frac{1 - \frac{0.22}{λ}}{λ} = 0.958

D_{e =} ρ \times d = 0.958 \times 7.894 = 7.562 in

Effective area, $A_{e} = A - (d - D_{e}) \times t = 0.748 - (7.894 - 7.562) \times 0.0713 = 0.724 {in}^{2}$

P_{nl} = A_{e} \times F_{n} = 0.724 \times 4.551 = 3.295 kips

Allowable local axial strength (ASD):

\frac{P_{n l}}{Ω_{b}} = \frac{3.295}{1.80} = 1.830 kips

Design local axial strength (LRFD):

P_{n l} \times ϕ_{b} = 3.295 \times 0.85 = 2.801 kips

Results

Table 1. AISI 2016 Cylindrical Tubular Section comparison
Result Type	Reference	STAAD.Pro	Difference
Allowable nominal axial strength, ASD $\frac{P_{n e}}{Ω_{c}}$ (kips)	1.891	1.891	none
Allowable local axial strength, ASD $\frac{P_{n l}}{Ω_{c}}$ (kips)	2.893	2.893	none
Design nominal axial strength, LRFD $ϕ_{b} P_{n}$ (kips)	1.830	1.830	none
Design local axial strength, LRFD $ϕ_{b} P_{nl}$ (kips)	2.801	2.800	negligible

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISI\AISI 2016 CU Nominal and Local axial Capacity.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 02-Jun-20
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 6 0 0;
MEMBER INCIDENCES
1 1 2;
UNIT INCHES KIP
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-06
DAMP 0.03
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY COLDFORMED AMERICAN
1 TABLE ST 800CU125-68
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 CON GX -1 0
1 CON GX 1 72
PERFORM ANALYSIS
PARAMETER 1
CODE AISI 2016
FYLD 50 ALL
METHOD LRFD
TRACK 2 ALL
CHECK CODE ALL
PARAMETER 2
CODE AISI 2016
FYLD 50 ALL
METHOD ASD
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

            STAAD.Pro CODE CHECKING - (AISI S100-16) [LRFD] (v1.0)
           ********************************************************
ALL UNITS ARE IN "INCH" AND "KIP " (U.N.O.)
|-----------------------------------------------------------------------------|
|  MEMBER:     1     SECTION:ST  800CU125-68     LEN:   72.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.357             REF: Sec. E3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. E3              LC:           1       |
|  Fx:(C)        1.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.74800       Az:     0.15284            Ay:     0.56281       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         6.00000       Iy:     0.07030             J:     0.00127       |
|  Sz:         1.45455       Sy:     0.06552                                  |
|  Rz:         2.83221       Ry:     0.30657            Cw:     0.92000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    50.000        Fu:    58.000      E:  29000.000       G:  11300.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:    72.000   Lz:    72.000   Ly:    72.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  234.858   Allowable :  200.000   Ratio :   1.174    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.499   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     15.032   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |    110.710   |    200.000   |      PASS     |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    4
            STAAD.Pro CODE CHECKING - (AISI S100-16) [LRFD] (v1.0)
           ********************************************************
ALL UNITS ARE IN "INCH" AND "KIP " (U.N.O.)
|                                              - MEMBER:       1 (Contd.....) |
|                                                                             |
|-----------------------------------------------------------------------------|
| Limit States       | Load# |Location|  Demand  | Capacity |Ratio| Reference |
|--------------------|-------|--------|----------|----------|-----|-----------|
| Tension            |   -   |   -    |     -    |   32.538 |  -  |Sec. D3    |
|   Yielding         |   -   |   -    |     -    |   33.660 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   32.538 |  -  |Sec. D3    |
| Compression        |     1 |   0.00 |   -1.000 |    2.800 |0.357|Sec. E3    |
|   Local Comp       |     1 |   0.00 |   -1.000 |    2.800 |0.357|Sec. E3    |
|   Axial Comp       |     1 |   0.00 |   -1.000 |    2.893 |0.346|Sec. E2    |
| Bending            |   -   |   -    |     -    |    2.913 |  -  |Sec. F3    |
|   Major Local Bend |   -   |   -    |     -    |   14.316 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    2.913 |  -  |Sec. F3    |
|   Major Sectional  |   -   |   -    |     -    |   65.455 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    2.948 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |   14.311 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    4.356 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    4.356 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    6.107 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.357|Eq. H1.2-1 |
|   (Bend+Comp)      |     1 |   0.00 |     -    |     -    |0.357|Eq. H1.2-1 |
|-----------------------------------------------------------------------------|
    40. PARAMETER 2
    41. CODE AISI 2016
    42. FYLD 50 ALL
    43. METHOD ASD
    44. TRACK 2 ALL
    45. CHECK CODE ALL
 STEEL DESIGN    
   WARNING # 8 ~ The specified value of FU for member #     1 seems out of range.
                 Reset back to default value.
      STAAD SPACE                                              -- PAGE NO.    5
            STAAD.Pro CODE CHECKING - (AISI S100-16) [ASD ] (v1.0)
           ********************************************************
ALL UNITS ARE IN "INCH" AND "KIP " (U.N.O.)
|-----------------------------------------------------------------------------|
|  MEMBER:     1     SECTION:ST  800CU125-68     LEN:   72.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.546             REF: Sec. E3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. E3              LC:           1       |
|  Fx:(C)        1.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.74800       Az:     0.15284            Ay:     0.56281       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         6.00000       Iy:     0.07030             J:     0.00127       |
|  Sz:         1.45455       Sy:     0.06552                                  |
|  Rz:         2.83221       Ry:     0.30657            Cw:     0.92000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    50.000        Fu:    58.000      E:  29000.000       G:  11300.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:    72.000   Lz:    72.000   Ly:    72.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  234.858   Allowable :  200.000   Ratio :   1.174    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.499   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     15.032   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |    110.710   |    200.000   |      PASS     |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    6
            STAAD.Pro CODE CHECKING - (AISI S100-16) [ASD ] (v1.0)
           ********************************************************
ALL UNITS ARE IN "INCH" AND "KIP " (U.N.O.)
|                                              - MEMBER:       1 (Contd.....) |
|                                                                             |
|-----------------------------------------------------------------------------|
| Limit States       | Load# |Location|  Demand  | Capacity |Ratio| Reference |
|--------------------|-------|--------|----------|----------|-----|-----------|
| Tension            |   -   |   -    |     -    |   21.692 |  -  |Sec. D3    |
|   Yielding         |   -   |   -    |     -    |   22.395 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   21.692 |  -  |Sec. D3    |
| Compression        |     1 |   0.00 |   -1.000 |    1.830 |0.546|Sec. E3    |
|   Local Comp       |     1 |   0.00 |   -1.000 |    1.830 |0.546|Sec. E3    |
|   Axial Comp       |     1 |   0.00 |   -1.000 |    1.891 |0.529|Sec. E2    |
| Bending            |   -   |   -    |     -    |    1.938 |  -  |Sec. F3    |
|   Major Local Bend |   -   |   -    |     -    |    9.525 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    1.938 |  -  |Sec. F3    |
|   Major Sectional  |   -   |   -    |     -    |   43.549 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    1.962 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |    9.522 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    2.866 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    2.866 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    4.018 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.546|Eq. H1.2-1 |
|   (Bend+Comp)      |     1 |   0.00 |     -    |     -    |0.546|Eq. H1.2-1 |
|-----------------------------------------------------------------------------|
    46. FINISH