V. AISI 2016 CS Compressive Strength

Verify the compressive strength of a cold-formed channel section with lips per the AISI S100 2016 specification.

Details

A 10' long, simply supported beam is subjected to a 10 kip axial load (compression). The section used is a 362CS162-54, Grade 50 steel.

Section Properties

Taken from AISI Manual, Cold-Formed Steel Design - Vol. 1, 2017 Table I-2.

Area, A_g = 0.422 in²
Radius of gyration z axis, r_z = 1.44in
Radius of gyration y axis, r_y = 0.605 in
Distance from centroid to shear center, x_o = -1.28 in
Elastic section modulus y axis, S_fz = 0.482 in³
Elastic section modulus z axis, S_fy = 0.142 in³
Torsional constant, J = 0.000451 in⁴
Warping constant, C_w = 0.457 in⁶
j = 2.02 in

Material Properties

E = 29,000 ksi
G = 11,300 ksi
F_y = 50 ksi

Validation

The elastic flexural buckling stress about the z axis:

F_{crez1} = \frac{π^{2} E}{{(K_{z} L_{z} / r_{z})}^{2}} = \frac{π^{2} \times 29,000}{{(1 \times 120 / 1.44)}^{2}} = 41.22 ksi

(Eq. E2.1-1)

Polar radius of gyration of cross-section about shear center:

r_{o} = \sqrt{{r_{z}}^{2} + {r_{y}}^{2} + {x_{o}}^{2}} = 2.019 in

(Eq. E2.2-4)

β = 1 - {(x_{o} / r_{o})}^{2} = 1 - {(\frac{-1.28}{2.019})}^{2} = 0.598

(Eq. E2.2-3)

σ_{t} = \frac{1}{A_{g} {r_{o}}^{2}} [G J + \frac{π^{2} E C_{w}}{{(K_{t} L_{t})}^{2}}] = \frac{1}{0.422 \times {2.019}^{2}} [11,200 \times 0.000451 + \frac{π^{2} \times 29,000 \times 0.457}{{(1 \times 120)}^{2}}] = 8.243 ksi

(Eq. E2.2-5)

σ_{ez} = F_{crez} = 41.22 ksi

(Eq. E2.2-6)

The flexural-torsional buckling stress:

F_{cre} = \frac{1}{2 β} [(σ_{ez} + σ_{t}) - \sqrt{{(σ_{ez} + σ_{t})}^{2} - 4 β σ_{ez} σ_{t}}]

(Eq. E2.2-1)

= \frac{1}{2 \times 0.598} [(41.22 + 8.197) - \sqrt{{(41.22 + 8.197)}^{2} - 4 \times 0.598 \times 41.22 \times 8.197}] = 7.560 ksi

The elastic global buckling stress about the z axis is the minimum of F_cre and F_crez1; so F_crez = 7.560 ksi.

The compressive stress capacity is:

λ_{cz} = \sqrt{\frac{F_{y}}{F_{cre}}} = \sqrt{\frac{50}{7.560}} = 2.572 > 1.5

(Eq. E2-4)

F_{nz} = (\frac{0.877}{{λ_{cz}}^{2}}) F_{y} = \frac{0.877}{{(2.579)}^{2}} 50 = 6.630 ksi

(Eq. E2-2)

The elastic flexural buckling stress about the y axis:

F_{crey1} = \frac{π^{2} E}{{(K_{y} L_{y} / r_{y})}^{2}} = \frac{π^{2} \times 29,000}{{(1 \times 120 / 0.536)}^{2}} = 7.275 ksi

(Eq. E2.1-1)

The elastic global buckling stress about the z axis is the minimum of F_cre and F_crey1; so F_crey = 7.275 ksi.

λ_{cy} = \sqrt{\frac{F_{y}}{F_{crey}}} = \sqrt{\frac{50}{7.275}} = 2.622 > 1.5

(Eq. E2-4)

F_{ny} = (\frac{0.877}{{λ_{cy}}^{2}}) F_{y} = \frac{0.877}{{(2.622)}^{2}} 50 = 6.380 ksi

(Eq. E2-2)

Global buckling about the Y axis governs.

Nominal capacity against global buckling:

P_n = A_gF_n = 0.422 × 6.380 = 2.692 kips

The available axial strength:

LRFD: ϕ_cP_n = 0.85 × 2.692 = 2.289 kips
ASD: P_n/Ω_c = 2.692 / 1.8 = 1.496 kips

Determine the compression capacity for local buckling interacting with yielding and global buckling as per E3:

The effective area, A_e, is equal to the gross area, A_g = 0.422 in². By inspection, the local capacity, P^nℓ is equal to the global buckling capacity in this case.

LRFD: ϕ_cP_nℓ = 0.85 × 2.692 = 2.289 kips
ASD: P_nℓ/Ω_c = 2.692 / 1.8 = 1.496 kips

Determine the compression capacity for distortional buckling as per E4:

The elastic distortional buckling stress, F_crd = 149.8 ksi.

The elastic distortional buckling load, P_crd = A_g × F_crd = 0.422 × 149.8 = 63.21 kips.

P_y = A_g × F_y = 0.422 × 50 = 21.1 kips

(Eq. E4.1-4)

λ_{d} = \sqrt{\frac{P_{y}}{P_{crd}}} = \sqrt{\frac{21.1}{63.21}} = 0.578 > 0.561

(Eq. E4.1-3)

P_{nd} = [1 - 0.25 {(\frac{P_{crd}}{P_{y}})}^{0.6}] {(\frac{P_{crd}}{P_{y}})}^{0.6} P_{y}

(Eq. 4.1-2)

= [1 - 0.25 {(\frac{63.21}{21.1})}^{0.6}] {(\frac{63.21}{21.1})}^{0.6} 21.1 = 21.08 ksi

The available axial strength:

LRFD: ϕ_cP_nd = 0.85 × 21.08 = 17.91 kips
ASD: P_nd/Ω_c = 21.08 / 1.8 = 11.71 kips

Results

Result Type		Reference	STAAD.Pro	Difference
LRFD	Local compression capacity (kips)	2.289	2.282	negligible
	Distorional buckling capacity (kips)	17.91	17.914	negligible
	Axial compression capacity (kips)	2.289	2.282	negligible
ASD	Local compression capacity (kips)	1.496	1.491	negligible
	Distorional buckling capacity (kips)	11.71	11.709	negligible
	Axial compression capacity (kips)	1.496	1.491	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 CS Compressive Strength.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 11-Jan-21
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
MEMBER PROPERTY COLDFORMED AMERICAN
1 TABLE ST 362CS162-54
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FX 10
PERFORM ANALYSIS
UNIT INCHES KIP
PARAMETER 1
CODE AISI 2016
METHOD LRFD
FYLD 50 ALL
FU 58 ALL
FLX 0 ALL
TSA 0 ALL
TRACK 2 ALL
CHECK CODE ALL
PARAMETER 2
CODE AISI 2016
METHOD ASD
FYLD 50 ALL
FU 58 ALL
FLX 0 ALL
TSA 0 ALL
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  362CS162-54     LEN:  120.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.545             REF: Sec. D3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. D3              LC:           1       |
|  Fx:(T)      -10.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.42200       Az:     0.15191            Ay:     0.18916       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         0.87300       Iy:     0.15400             J:     0.00045       |
|  Sz:         0.48166       Sy:     0.14141                                  |
|  Rz:         1.43830       Ry:     0.60409            Cw:     0.45700       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    50.000        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 :  198.645   Allowable :  300.000   Ratio :   0.662    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.500   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     23.710   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |     59.046   |    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 |   18.357 |0.545|Sec. D3    |
|   Yielding         |     1 |   0.00 |  -10.000 |   18.990 |0.527|Sec. D2    |
|   Rupture          |     1 |   0.00 |  -10.000 |   18.357 |0.545|Sec. D3    |
| Compression        |   -   |   -    |     -    |    2.282 |  -  |Sec. E2    |
|   Local Comp       |   -   |   -    |     -    |    2.282 |  -  |Sec. E3    |
|   Distortional     |   -   |   -    |     -    |   17.914 |  -  |Sec. E4    |
|   Axial Comp       |   -   |   -    |     -    |    2.282 |  -  |Sec. E2    |
| Bending            |   -   |   -    |     -    |    5.920 |  -  |Sec. F2    |
|   Major Local Bend |   -   |   -    |     -    |    5.924 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    6.239 |  -  |Sec. F3    |
|   Distortional     |   -   |   -    |     -    |   21.674 |  -  |Sec. F4    |
|   Major Sectional  |   -   |   -    |     -    |   21.674 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    6.364 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |    5.920 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    4.330 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    4.330 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    5.081 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.545|Eq. H1.1-1 |
|   (Bend+Ten) Int1  |     1 |   0.00 |     -    |     -    |0.545|Eq. H1.1-1 |
|-----------------------------------------------------------------------------|
    42. PARAMETER 2
    43. CODE AISI 2016
    44. METHOD ASD
    45. FYLD 50 ALL
    46. FU 58 ALL
    47. FLX 0 ALL
    48. TSA 0 ALL
    49. TRACK 2 ALL
    50. CHECK CODE ALL
 STEEL DESIGN    
      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  362CS162-54     LEN:  120.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.817             REF: Sec. D3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. D3              LC:           1       |
|  Fx:(T)      -10.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.42200       Az:     0.15191            Ay:     0.18916       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         0.87300       Iy:     0.15400             J:     0.00045       |
|  Sz:         0.48166       Sy:     0.14141                                  |
|  Rz:         1.43830       Ry:     0.60409            Cw:     0.45700       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    50.000        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 :  198.645   Allowable :  300.000   Ratio :   0.662    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.500   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     23.710   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |     59.046   |    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 |   12.238 |0.817|Sec. D3    |
|   Yielding         |     1 |   0.00 |  -10.000 |   12.635 |0.791|Sec. D2    |
|   Rupture          |     1 |   0.00 |  -10.000 |   12.238 |0.817|Sec. D3    |
| Compression        |   -   |   -    |     -    |    1.491 |  -  |Sec. E2    |
|   Local Comp       |   -   |   -    |     -    |    1.491 |  -  |Sec. E3    |
|   Distortional     |   -   |   -    |     -    |   11.709 |  -  |Sec. E4    |
|   Axial Comp       |   -   |   -    |     -    |    1.491 |  -  |Sec. E2    |
| Bending            |   -   |   -    |     -    |    3.939 |  -  |Sec. F2    |
|   Major Local Bend |   -   |   -    |     -    |    3.941 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    4.151 |  -  |Sec. F3    |
|   Distortional     |   -   |   -    |     -    |   14.421 |  -  |Sec. F4    |
|   Major Sectional  |   -   |   -    |     -    |   14.421 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    4.234 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |    3.939 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    2.848 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    2.848 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    3.343 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.817|Eq. H1.1-1 |
|   (Bend+Ten) Int1  |     1 |   0.00 |     -    |     -    |0.817|Eq. H1.1-1 |
|-----------------------------------------------------------------------------|