V. AISI 2016 ZS Compressive Strength

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

Details

A 15' long, simply supported beam is subjected to a 10 kip axial load (compression). The section used is a 12ZS3.25x105, Grade 36 steel.

Section Properties

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

Area, A_g = 2.09 in²
Radius of gyration z axis, r_z = 4.57 in
Radius of gyration y axis, r_y = 1.49 in
Elastic section modulus y axis, S_fy = 1.22 in³
Elastic section modulus z axis, S_fz = 7.29 in³
Torsional constant, J = 0.00769 in⁴
Warping constant, C_w = 123 in⁶

Material Properties

E = 29,000 ksi
G = 11,300 ksi
F_y = 36 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 180 / 4.57)}^{2}} = 184.5 ksi

(Eq. E2.1-1)

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

r_{o} = \sqrt{{r_{z}}^{2} + {r_{y}}^{2}} = 4.81 in

(Eq. E2.2-4)

σ_{t} = \frac{1}{A_{g} {r_{o}}^{2}} [G J + \frac{π^{2} E C_{w}}{{(K_{t} L_{t})}^{2}}] = \frac{1}{2.09 \times {4.81}^{2}} [11,200 \times 0.00769 + \frac{π^{2} \times 29,000 \times 123}{{(1 \times 180)}^{2}}] = 24.14 ksi

(Eq. E2.2-5)

σ_{ez} = F_{crez} = 184.5 ksi

(Eq. E2.2-6)

Per E2.3, the flexural-torsional buckling stress is the lesser of σ_ez and σ_t; so F_crez = 24.14 ksi

The compressive stress capacity is:

λ_{cz} = \sqrt{\frac{F_{y}}{F_{cre}}} = \sqrt{\frac{36}{24.14}} = 1.221 < 1.5

(Eq. E2-4)

F_{nz} = ({0.658}^{{λ_{cz}}^{2}}) F_{y} = {0.658}^{{(1.221)}^{2}} 36 = 19.28 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 180 / 1.49)}^{2}} = 19.61 ksi

(Eq. E2.1-1)

The elastic global buckling stress about the z axis is the minimum of F_crey1 and σ_t; so F_crey = 19.61 ksi.

λ_{cy} = \sqrt{\frac{F_{y}}{F_{crey}}} = \sqrt{\frac{36}{19.61}} = 1.355 < 1.5

(Eq. E2-4)

F_{ny} = ({0.658}^{{λ_{cz}}^{2}}) F_{y} = {0.658}^{{(1.355)}^{2}} 36 = 16.69 ksi

(Eq. E2-2)

Global buckling about the Y axis governs.

Nominal capacity against global buckling:

P_n = A_gF_n = 2.09 × 16.69 = 34.89 kips

The available axial strength:

LRFD: ϕ_cP_n = 0.85 × 34.89 = 29.66 kips
ASD: P_n/Ω_c = 34.89 / 1.8 = 19.38 kips

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

For local buckling about the Z axis, the effective area, A_e = 1.586 in².

P_nℓz = A_e × F_nz = 1.586 × 19.28 = 30.56 kips

For local buckling about the Y axis, the effective area, A_e = 1.627 in².

P_nℓy = A_e × F_nz = 1.627 × 16.69 = 27.15 kips

P_nℓ is taken as the minimum of P_nℓz, P_nℓy, and P_ne; so P_nℓ = 27.15 kips.

LRFD: ϕ_cP_nℓ = 0.85 × 27.15 = 23.08 kips
ASD: P_nℓ/Ω_c = 27.15 / 1.8 = 15.06 kips

Determine the compression capacity for distortional buckling as per E4:

The elastic distortional buckling stress, F^crd = 30.84 ksi.

The elastic distortional buckling load, P_crd = A_g × F^crd = 2.09 × 30.84 = 64.46 kips.

P_y = A_g × F^y = 2.09 × 36 = 75.24 kips

(Eq. E4.1-4)

λ_{d} = \sqrt{\frac{P_{y}}{P_{crd}}} = \sqrt{\frac{75.24}{64.46}} = 1.08 > 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{64.46}{75.24})}^{0.6}] {(\frac{64.46}{75.24})}^{0.6} 75.24 = 52.95 ksi

The available axial strength:

LRFD: ϕ_cP_nd = 0.85 × 52.95 = 45.01 kips
ASD: P_nd/Ω_c = 52.95 / 1.8 = 29.42 kips

Results

Result Type		Reference	STAAD.Pro	Difference
LRFD	Local compression capacity (kips)	23.08	23.212	negligible
	Distorional buckling capacity (kips)	45.01	45.006	negligible
	Axial compression capacity (kips)	29.66	29.809	negligible
ASD	Local compression capacity (kips)	15.06	15.171	negligible
	Distorional buckling capacity (kips)	29.42	29.416	negligible
	Axial compression capacity (kips)	19.38	19.483	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 ZS 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 15 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 12ZS3.25X105
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
AXIS 1 ALL
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
AXIS 1 ALL
FYLD 36 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  12ZS3.25X105    LEN:  180.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.431             REF: Sec. E3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. E3              LC:           1       |
|  Fx:(C)       10.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         2.09000       Az:     0.59243            Ay:     1.19857       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:        43.70000       Iy:     4.67000             J:     0.00769       |
|  Sz:         7.28333       Sy:     1.20164                                  |
|  Rz:         4.57265       Ry:     1.49481            Cw:   123.00000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    36.000        Fu:    58.000      E:  29000.000       G:  11200.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:   180.000   Lz:   180.000   Ly:   180.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  120.417   Allowable :  200.000   Ratio :   0.602    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.786   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     26.868   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |    108.714   |    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            |   -   |   -    |     -    |   67.716 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |   67.716 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   90.915 |  -  |Sec. D3    |
| Compression        |     1 |   0.00 |  -10.000 |   23.212 |0.431|Sec. E3    |
|   Local Comp       |     1 |   0.00 |  -10.000 |   23.212 |0.431|Sec. E3    |
|   Distortional     |     1 |   0.00 |  -10.000 |   45.006 |0.222|Sec. E4    |
|   Axial Comp       |     1 |   0.00 |  -10.000 |   29.809 |0.335|Sec. E2    |
| Bending            |   -   |   -    |     -    |   38.933 |  -  |Sec. F2    |
|   Major Local Bend |   -   |   -    |     -    |   99.087 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |   39.498 |  -  |Sec. F3    |
|   Distortional     |   -   |   -    |     -    |  222.225 |  -  |Sec. F4    |
|   Major Sectional  |   -   |   -    |     -    |  235.980 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |   38.933 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |   98.978 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |   12.157 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |   12.157 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |   13.487 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.431|Eq. H1.2-1 |
|   (Bend+Comp)      |     1 |   0.00 |     -    |     -    |0.431|Eq. H1.2-1 |
|-----------------------------------------------------------------------------|
    43. PARAMETER 2
    44. CODE AISI 2016
    45. METHOD ASD
    46. AXIS 1 ALL
    47. FYLD 36 ALL
    48. FU 58 ALL
    49. FLX 0 ALL
    50. TSA 0 ALL
    51. TRACK 2 ALL
    52. 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  12ZS3.25X105    LEN:  180.000     LOC:   0.000 |
| STATUS: PASS       RATIO:  0.659             REF: Sec. E3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:    0.000  Criteria:   Sec. E3              LC:           1       |
|  Fx:(C)       10.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         2.09000       Az:     0.59243            Ay:     1.19857       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:        43.70000       Iy:     4.67000             J:     0.00769       |
|  Sz:         7.28333       Sy:     1.20164                                  |
|  Rz:         4.57265       Ry:     1.49481            Cw:   123.00000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    36.000        Fu:    58.000      E:  29000.000       G:  11200.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:   180.000   Lz:   180.000   Ly:   180.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  120.417   Allowable :  200.000   Ratio :   0.602    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.786   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     26.868   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |    108.714   |    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            |   -   |   -    |     -    |   45.054 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |   45.054 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   60.610 |  -  |Sec. D3    |
| Compression        |     1 |   0.00 |  -10.000 |   15.171 |0.659|Sec. E3    |
|   Local Comp       |     1 |   0.00 |  -10.000 |   15.171 |0.659|Sec. E3    |
|   Distortional     |     1 |   0.00 |  -10.000 |   29.416 |0.340|Sec. E4    |
|   Axial Comp       |     1 |   0.00 |  -10.000 |   19.483 |0.513|Sec. E2    |
| Bending            |   -   |   -    |     -    |   25.904 |  -  |Sec. F2    |
|   Major Local Bend |   -   |   -    |     -    |   65.926 |  -  |Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |   26.280 |  -  |Sec. F3    |
|   Distortional     |   -   |   -    |     -    |  147.854 |  -  |Sec. F4    |
|   Major Sectional  |   -   |   -    |     -    |  157.006 |  -  |Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |   25.904 |  -  |Sec. F2    |
|   LTB Major        |   -   |   -    |     -    |   65.854 |  -  |Sec. F2    |
| Shear              |   -   |   -    |     -    |    7.998 |  -  |Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    7.998 |  -  |Sec. G2    |
|   Minor Shear (Y)  |   -   |   -    |     -    |    8.873 |  -  |Sec. G2    |
| Interaction        |     1 |   0.00 |     -    |     -    |0.659|Eq. H1.2-1 |
|   (Bend+Comp)      |     1 |   0.00 |     -    |     -    |0.659|Eq. H1.2-1 |
|-----------------------------------------------------------------------------|