V. AISI 2016 2CU F2F Compressive Strength

Verify the axial compressive strength of a double cold-formed channel section according to the AISI S100 2016 specification using both the LRFD and ASD methods.

References

American Iron and Steel Institute. AISI Manual: Cold-formed Steel Design - Vol. 1.. 2018. Milwaukee, WI.
Young, Warren C. and Richard G. Budynas. Roark's Formulas for Stress and Strain. Seventh Edition. New York, NY: McGraw Hill, 2002. p. 414

Details

The member is 10 ft long cantilever. The section used is a pair of 600T125-54 channels in a front-to-front configuration (no spacing between channel toes).

Section Properties

Properties of a 600T125-54 (listed as a 600CU125-54 in STAAD.Pro):

Depth, D = 6.198 in
Flange width, B = 1.250 in

Thickness, t = 0.0566 in
Inner radius, R = 0.0849 in.
Cross-section area, A = 0.48 in²
Moment of inertia about the Z axis, I_z = 2.34 in⁴
Moment of inertia about the Y axis, I_y = 0.0539 in⁴
Heel to C.G. distance, x = 0.204 in
Torsional constant, J = 0.000513 in⁴
Warping constant, C_w = 0.348 in⁶

Material Properties

E = 29,000 ksi
G = 11,200 ksi
F_y = 36 ksi
F_u = 58 ksi

Calculations

Section Properties

Average radius, r = R + t/2 = 0.0849 + 0.0566 / 2 = 0.113 in.
Clear depth of web, a = D = (2×r + t) = 6.198 - (2× 0.113 + 0.0566) = 5.915 in.
Clear width of the flanges, b = B - (r + t/2) = 1.25 - (0.113 + 0.0556 / 2) = 1.109 in.
Cross-section area, A = 2× 0.48 = 0.96 in²
Moment of inertia about the Z axis, I_z = 2× 2.34 = 4.68 in⁴
Moment of inertia about the Y axis, I_y = 2[0.0539 + 0.48×(1.25 - 0.204)²] = 1.158 in⁴
Radius of gyration about the Z axis, $r_{z} = \sqrt{\frac{I_{z}}{A}} = \sqrt{\frac{4.68}{0.96}} = 2.21 in$
Radius of gyration about the Y axis, $r_{y} = \sqrt{\frac{I_{y}}{A}} = \sqrt{\frac{1.158}{0.96}} = 1.10 in$
Torsional constant, J = 2× 0.000513 = 0.00103 in⁴
Center-to-center depth, $\overset{⎯}{a} = D - t = 6.198 - 0.0566 = 6.141 in$
Center-to-edge flange width, $\overset{⎯}{b} = B - \frac{t}{2} = 1.25 - \frac{0.0566}{2} = 1.222 in$
Warping constant, $C_{w} = \frac{t {\overset{⎯}{a}}^{3}}{24} (56 {\overset{⎯}{b}}^{3} + 4 \overset{⎯}{a} {\overset{⎯}{b}}^{2}) = \frac{0.0566 {(6.141)}^{3}}{24} [56 {(1.222)}^{3} + 4 (6.141) {(1.222)}^{2}] = 12.344 {in}^{6}$ (Taken from Table 10.2, Case 4 of Reference 2; simplified for the case where h = 2×h₁)

Compression Capacity for Yielding and Global Buckling

About the major axis:

The elastic flexural buckling stress

F_{crez} = \frac{π^{2} E}{{(\frac{K_{z} L_{z}}{r_{z}})}^{2}} = \frac{π^{2} \times 29,000}{{(\frac{1.0 \times 120}{2.21})}^{2}} = 96.9 ksi

(Eq. E2.1-1)

λ_{cz} = \sqrt{\frac{F_{y}}{F_{crez}}} = \sqrt{\frac{36}{96.9}} = 0.610

(Eq. E2-4)

Since λ_cz < 1.5, the compressive strength is given by:

F_{nz} = {0.658}^{{λ_{cz}}^{2}} \times F_{y} = {0.658}^{{(0.610)}^{2}} \times 36 = 30.82 ksi

(Eq. 2-2)

About the minor axis:

The elastic flexural buckling stress

F_{crey} = \frac{π^{2} E}{{(\frac{K_{y} L_{y}}{r_{y}})}^{2}} = \frac{π^{2} \times 29,000}{{(\frac{1.0 \times 120}{1.10})}^{2}} = 24.0 ksi

(Eq. E2.1-1)

λ_{cy} = \sqrt{\frac{F_{y}}{F_{crey}}} = \sqrt{\frac{36}{24.0}} = 1.225

(Eq. E2-4)

Since λ_cy < 1.5, the compressive strength is given by:

F_{ny} = {0.658}^{{λ_{cy}}^{2}} \times F_{y} = {0.658}^{{(1.225)}^{2}} \times 36 = 19.20 ksi

(Eq. 2-2)

The controlling compressive stress is the minimum of F_nz and F_ny: F_n = 19.20 ksi

The nominal axial compressive strength, P_ne = A × F_n = 0.96 × 19.20 = 18.43 kips

LRFD: ϕ_cP_ne = 0.85 × 18.43 = 15.67 kips
ASD: P_ne/Ω_c = 18.43 / 1.8 = 10.24 kips

Compression Capacity in Local Buckling

Global column stress about major axis (as defined in Section E2):

F_n = F_ny = 19.20 ksi

Determine the effective area, A_e as per Appendix 1. The flanges of a front-to-front channel are considered stiffened elements.

Compression flange effective width:

w = b = 1.109 in

(AISC Manual 3.3.2)

F_{crℓ} = k \frac{π^{2} E}{12 (1 - μ^{2})} {(\frac{t}{w})}^{2}

(E1. 1.1-4)

where

k	=	Plate bucking coefficient, = 4
μ	=	Poisson's ratio of steel, = 0.3

F_{crℓ} = 4 \frac{π^{2} \times 29,000}{12 [1 - {(0.3)}^{2}]} {(\frac{0.0566}{1.1097})}^{2} = 273.1 ksi

The slenderness factor:

λ = \sqrt{\frac{F_{n}}{F_{crℓ}}} = \sqrt{\frac{19.20}{273.1}} = 0.265

(Eq. 1.1-3)

As λ < 0.673, the local reduction factor, ρ = 1. Therefore, the effective width of the compression flange is b_f = ρ×w = 1.109 in.

Web effective width:

w = a = 5.915 in

(AISC Manual 3.3.2)

F_{crℓ} = 4 \frac{π^{2} \times 29,000}{12 [1 - {(0.3)}^{2}]} {(\frac{0.0566}{5.915})}^{2} = 9.60 ksi

λ = \sqrt{\frac{F_{n}}{F_{crℓ}}} = \sqrt{\frac{19.20}{9.60}} = 1.414

As λ > 0.673, the local reduction factor:

ρ = (1 - \frac{0.22}{λ}) / λ = (1 - \frac{0.22}{1.414}) / 1.414 = 0.597

Therefore, the effective width of the compression flange is b_w = ρ×w = 0.597 × 5.915 = 3.532 in

The total effective area is the gross area minus the difference in nominal widths and effective widths:

A_e = A - 4×(b - b_f)t - 2×(a - b_w)t = 0.96 - 4×(1.109 - 1.109)×0.0566 - 2×(5.915 - 3.532)×0.0566 = 0.69 in.²

P_{nℓ} = A_{e} F_{n} = 0.69 \times 19.20 = 13.25 < P_{ne}

(Eq. E3.1-1)

LRFD: ϕ_cP_nℓ = 0.85 × 11.26 = kips
ASD: P_nℓ/Ω_c = 13.26 / 1.8 = 7.36 kips

Comparison

Table 1. Comparison of results
Result Type		Reference	STAAD.Pro	Difference
LRFD	Axial comp. per section E2 (kips)	15.67	15.671	negligible
LRFD	Local comp. per section E3 (kips)	11.26	11.283	negligible
ASD	Axial comp. per section E2 (kips)	10.24	10.242	negligible
ASD	Local comp. per section E3 (kips)	7.36	7.375	negligible

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISI\AISI 2016 2CU F2F Compressive Strength.STD is typically installed with the program.

The following design parameters are used:

The material properties for design are specified by the parameters FYLD 36 and FU 58.
A member with unreinforced webs is specified using TSA 0

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 11-Jan-22
ENGINEER NAME TK
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 10 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 4.176e+06
POISSON 0.3
DENSITY 0.489024
ALPHA 6.5e-06
DAMP 0.03
G 1.6128e+06
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
UNIT INCHES KIP
MEMBER PROPERTY COLDFORMED AMERICAN
1 TABLE FR 600CU125-54
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FX 10
PERFORM ANALYSIS
PARAMETER 1
CODE AISI 2016
METHOD LRFD
FYLD 36 ALL
FU 58 ALL
FLX 0 ALL
TSA 0 ALL
TRACK 2 ALL
CHECK CODE ALL
PARAMETER 2
CODE AISI 2016
METHOD ASD
FYLD 36 ALL
FU 58 ALL
FLX 0 ALL
TSA 0 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD Output

            STAAD.Pro CODE CHECKING - (AISI S100-16) [LRFD] (v1.0)
           ********************************************************
ALL UNITS ARE IN "INCH" AND "KIP " (U.N.O.)
|-----------------------------------------------------------------------------|
|  MEMBER:     1     SECTION:FR  600CU125-54     LEN:  120.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.322             REF: Sec. D2       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. D2              LC:           1       |
|  Fx:(T)      -10.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.96000       Az:     0.25096            Ay:     0.66958       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         4.68000       Iy:     1.15815             J:     0.00104       |
|  Sz:         1.51016       Sy:     0.92652                                  |
|  Rz:         2.20794       Ry:     1.09837            Cw:     0.00000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    36.000        Fu:    58.000      E:  29000.000       G:  11200.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:   120.000   Lz:   120.000   Ly:   120.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  109.253   Allowable :  300.000   Ratio :   0.364    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.500   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     17.085   |    500.000   |      PASS     |
|  Web Depth To Thickness Ratio |    104.505   |    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            |     1 |   0.00 |  -10.000 |   31.104 |0.322|Sec. D2    |
|   Yielding         |     1 |   0.00 |  -10.000 |   31.104 |0.322|Sec. D2    |
|   Rupture          |     1 |   0.00 |  -10.000 |   41.760 |0.239|Sec. D3    |
| Compression        |   -   |   -    |     -    |   11.283 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |   11.283 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |   15.671 |  -  |Sec. E2    |
| Bending            |   -   |   -    |     -    |   14.721 |  -  |Sec. F2    |
|   Major Local Bend |   -   |   -    |     -    |   14.756 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |   14.846 |  -  |Sec. F3    |
|   Major Sectional  |   -   |   -    |     -    |   48.929 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |   30.019 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |   14.721 |  -  |Sec. F2    |
|   LTB Minor        |   -   |   -    |     -    |   23.956 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    5.150 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    5.150 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    8.154 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.322|Eq. H1.1-1 |
|   (Bend+Ten) Int1  |     1 |   0.00 |     -    |     -    |0.322|Eq. H1.1-1 |
|-----------------------------------------------------------------------------|
    43. PARAMETER 2
    44. CODE AISI 2016
    45. METHOD ASD
    46. FYLD 36 ALL
    47. FU 58 ALL
    48. FLX 0 ALL
    49. TSA 0 ALL
    50. TRACK 2 ALL
    51. CHECK CODE ALL
 STEEL DESIGN    
   WARNING # 14 ~ Channel front to front member #     1 will be considered welded throughout length.
   Spacing will not be considered.
      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:FR  600CU125-54     LEN:  120.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.483             REF: Sec. D2       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. D2              LC:           1       |
|  Fx:(T)      -10.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.96000       Az:     0.25096            Ay:     0.66958       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         4.68000       Iy:     1.15815             J:     0.00104       |
|  Sz:         1.51016       Sy:     0.92652                                  |
|  Rz:         2.20794       Ry:     1.09837            Cw:     0.00000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    36.000        Fu:    58.000      E:  29000.000       G:  11200.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:   120.000   Lz:   120.000   Ly:   120.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  109.253   Allowable :  300.000   Ratio :   0.364    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.500   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     17.085   |    500.000   |      PASS     |
|  Web Depth To Thickness Ratio |    104.505   |    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            |     1 |   0.00 |  -10.000 |   20.695 |0.483|Sec. D2    |
|   Yielding         |     1 |   0.00 |  -10.000 |   20.695 |0.483|Sec. D2    |
|   Rupture          |     1 |   0.00 |  -10.000 |   27.840 |0.359|Sec. D3    |
| Compression        |   -   |   -    |     -    |    7.375 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |    7.375 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |   10.242 |  -  |Sec. E2    |
| Bending            |   -   |   -    |     -    |    9.795 |  -  |Sec. F2    |
|   Major Local Bend |   -   |   -    |     -    |    9.818 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    9.878 |  -  |Sec. F3    |
|   Major Sectional  |   -   |   -    |     -    |   32.554 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |   19.973 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |    9.795 |  -  |Sec. F2    |
|   LTB Minor        |   -   |   -    |     -    |   15.939 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    3.388 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    3.388 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    5.364 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.483|Eq. H1.1-1 |
|   (Bend+Ten) Int1  |     1 |   0.00 |     -    |     -    |0.483|Eq. H1.1-1 |
|-----------------------------------------------------------------------------|