V. AISI 2016 CU Nominal Moment Capacity

Determine the nominal moment axial capacity of a simply-supported beam subject to a concentrated load at mid-span.

Reference

Yu, Wei-Wen and Roger A. LaBoube. 2010 Cold-Formed Steel Design, Fourth Edition. Hoboken, NJ: Wiley. Example 4.3.

Details

The member is a 10' simply-supported span subject to a 1 kip load applied at midspan. The section used is a CU_8X2_0.06 channel without lips. The member is laterally supported at the one-fourth points along the span.

Section properties:

Depth: D = 8 in
Width: B = 2 in
Thickness: t = 0.06 in
Inner radius: R = 3/32 in
Area: A = 0.706 in²
Moment of inertia about major axis: I_x = 6.129 in⁴
Moment of inertia about minor axis: I_y = 0.229 in⁴
Major axis section modulus: S_x = 1.532 in³
Minor axis section modulus: S_y = 0.137 in³
Torsional constant: J = 0.00085 in⁴
Distance between centroid and shear center: x₀ = -0.919 in
Radius of gyration about major axis: $r_{x} = \sqrt{\frac{I_{x}}{A}} = 2.946 in.$
Radius of gyration about minor axis: $r_{y} = \sqrt{\frac{I_{t}}{A}} = 0.569 in.$
Polar radius of gyration: $r_{0} = \sqrt{{r_{x}}^{2} + {r_{y}}^{2} + {x_{0}}^{2}} = 3.138 in.$
Warping constant: C_w = 2.568 in⁶

Material properties:

E = 29,500 ksi
G = 11,300 ksi
F_y = 33 ksi
F_u = 58 ksi

Validation

Elastic Critical Lateral-Torsional Stress

Per Cl. F2.1.1 of AISI S100-16:

σ_{e y} = \frac{π^{2} \times E}{{(\frac{K_{y} \times L_{y}}{r_{y}})}^{2}}

where

$K_{y}$	=	Effective length factor for bending about y axis = 1
$L_{y}$	=	Unbraced length of member for bending about y axis = 30 in

So, $σ_{e y} = \frac{π^{2} \times 29,000}{{(\frac{1 \times 30}{0.569})}^{2}} = 103.1 ksi$

C_{b} = \frac{12.5 \times M_{\max}}{2.5 M_{m a x} + 3 M_{A} + 4 M_{B} + 3 M_{C}}

where

$M_{\max}$	=	Absolute value of maximum moment in unbraced segment = 30 kip·in
$M_{A}$	=	Absolute value of moment at quarter point of unbraced segment = 18.75 kip·in
$M_{B}$	=	Absolute value of moment at mid-point of unbraced segment = 22.50 kip·in
$M_{C}$	=	Absolute value of moment at three-quarter point of unbraced segment = 26.25 kip·in

C_{b} = \frac{12.5 \times 30}{2.5 \times 30 + 3 \times 18.75 + 4 \times 22.5 + 3 \times 26.25} = 1.25

(Eq. F2.1.1-2)

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}{0.706 \times {3.138}^{2}} \times [11,300 \times 0.00085 + \frac{π^{2} \times 29,000 \times 2.658}{{(1 \times 30)}^{2}}] = 122.9 ksi

The elastic buckling stress:

F_{c r e} = C_{b} \times r_{0} \times \frac{A}{S_{x}} \times \sqrt{σ_{e y} \times σ_{t}} = 1.25 \times 3.138 \times \frac{0.706}{1.532} \times \sqrt{103.1 \times 122.9} = 203.5 ksi

(Eq. F2.1.1-1)

Calculate the nominal flexural strength, F_n, per Cl. F2.1 of S100-16:

Since $F_{cre} > 2.78 \times F_{y} = 2.78 \times 33 = 97.74 ksi$ , $F_{n} = F_{y} = 33 ksi$

The nominal flexural strength:

M_{n} = S_{x} \times F_{y} = 1.532 \times 33 = 50.57 kip·in

(Eq. F2.1-1)

Allowable flexural strength (ASD):

\frac{M_{n}}{Ω_{b}} = \frac{50.57}{1.67} = 30.278 kip·in

Design flexural strength (LRFD):

M_{n} \times ϕ_{b} = 50.57 \times 0.9 = 45.508 kip·in

Results

Table 1. AISI 2016 Cylindrical Tubular Section comparison
Result Type	Reference	STAAD.Pro	Difference	Comments
Flexural strength, ASD $\frac{M_{n e}}{Ω_{b}}$ (in·kips)	30.278	30.278	none
Flexural strength, LRFD $ϕ_{b} M_{n e}$ (in·kips)	45.508	45.508	none

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 Moment Capacity.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 01-Jun-20
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 10 0 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
TABLE 1
UNIT INCHES KIP
CHANNEL
CU_8X2_0.06
0.706 8 0.06 2 0.06 6.129 0.229 0.00085 0.33 0.462 0.1143
END
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 AMERICAN
1 UPTABLE 1 CU_8X2_0.06
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 2 PINNED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 CON GY -1
PERFORM ANALYSIS
PARAMETER 1
CODE AISI 2016
FYLD 33 ALL
METHOD LRFD
TRACK 2 ALL
LY 30 ALL
UNB 30 ALL
UNT 30 ALL
CHECK CODE ALL
PARAMETER 2
CODE AISI 2016
FYLD 33 ALL
METHOD ASD
TRACK 2 ALL
LY 30 ALL
UNB 30 ALL
UNT 30 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:UPT CU_8X2_0.06     LEN:  120.000     LOC:  60.000 |
| STATUS: PASS       RATIO:  0.895             REF: Sec. F3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:   60.000  Criteria:   Sec. F3              LC:           1       |
|  Fx:(C)        0.000       Fy:         0.500          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:     -30.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.70600       Az:     0.11430            Ay:     0.46200       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         6.12900       Iy:     0.22900             J:     0.00085       |
|  Sz:         1.53225       Sy:     0.13713                                  |
|  Rz:         2.94641       Ry:     0.56953            Cw:     2.65754       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    33.000        Fu:    58.000      E:  29000.000       G:  11300.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:   120.000   Lz:   120.000   Ly:    30.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  210.701   Allowable :  300.000   Ratio :   0.702    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Width To Thickness Ratio     |     32.333   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |    131.333   |    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            |   -   |   -    |     -    |   20.968 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |   20.968 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   30.711 |  -  |Sec. D3    |
| Compression        |   -   |   -    |     -    |    3.623 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |    3.623 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |    4.702 |  -  |Sec. E2    |
| Bending            |     1 |  60.00 |  -30.000 |   33.522 |0.895|Sec. F3    |
|   Major Local Bend |     1 |  60.00 |  -30.000 |   33.522 |0.895|Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    3.998 |  -  |Sec. F3    |
|   Major Sectional  |     1 |  60.00 |  -30.000 |   45.508 |0.659|Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    4.073 |  -  |Sec. F2    |
|   LTB Major        |     1 |  60.00 |  -30.000 |   42.559 |0.705|Sec. F2    |
| Shear              |     1 |   0.00 |    0.500 |    3.562 |0.140|Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    2.150 |  -  |Sec. G2    |
|   Minor Shear (Y)  |     1 |   0.00 |    0.500 |    3.562 |0.140|Sec. G2    |
| Interaction        |     1 |  60.00 |     -    |     -    |0.895|Eq. H1.1-2 |
|   (Bend+Ten) Int1  |     1 |  60.00 |     -    |     -    |0.659|Eq. H1.1-1 |
|   (Bend+Ten) Int2  |     1 |  60.00 |     -    |     -    |0.895|Eq. H1.1-2 |
|   (Bend+Comp)      |     1 |  60.00 |     -    |     -    |0.895|Eq. H1.2-1 |
|   (Bend+Shear) Z   |     1 |  60.00 |     -    |     -    |0.856|Sec. H2    |
|-----------------------------------------------------------------------------|
    48. PARAMETER 2
    49. CODE AISI 2016
    50. FYLD 33 ALL
    51. METHOD ASD
    52. TRACK 2 ALL
    53. LY 30 ALL
    54. UNB 30 ALL
    55. UNT 30 ALL
    56. CHECK CODE ALL
 STEEL DESIGN    
   WARNING # 8 ~ The specified value of FU for member #     1 seems out of range.
                 Reset back to default value.
   NOTE # 1 ~ The assigned section for the member #     1
                  may not be a cold formed section.
      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:UPT CU_8X2_0.06     LEN:  120.000     LOC:  60.000 |
| STATUS: FAIL       RATIO:  1.345             REF: Sec. F3       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:   60.000  Criteria:   Sec. F3              LC:           1       |
|  Fx:(C)        0.000       Fy:         0.500          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:     -30.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.70600       Az:     0.11430            Ay:     0.46200       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         6.12900       Iy:     0.22900             J:     0.00085       |
|  Sz:         1.53225       Sy:     0.13713                                  |
|  Rz:         2.94641       Ry:     0.56953            Cw:     2.65754       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    33.000        Fu:    58.000      E:  29000.000       G:  11300.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:   120.000   Lz:   120.000   Ly:    30.000   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  210.701   Allowable :  300.000   Ratio :   0.702    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Width To Thickness Ratio     |     32.333   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |    131.333   |    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            |   -   |   -    |     -    |   13.951 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |   13.951 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   20.474 |  -  |Sec. D3    |
| Compression        |   -   |   -    |     -    |    2.368 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |    2.368 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |    3.073 |  -  |Sec. E2    |
| Bending            |     1 |  60.00 |  -30.000 |   22.303 |1.345|Sec. F3    |
|   Major Local Bend |     1 |  60.00 |  -30.000 |   22.303 |1.345|Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    2.660 |  -  |Sec. F3    |
|   Major Sectional  |     1 |  60.00 |  -30.000 |   30.278 |0.991|Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    2.710 |  -  |Sec. F2    |
|   LTB Major        |     1 |  60.00 |  -30.000 |   28.316 |1.059|Sec. F2    |
| Shear              |     1 |   0.00 |    0.500 |    2.344 |0.213|Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    1.414 |  -  |Sec. G2    |
|   Minor Shear (Y)  |     1 |   0.00 |    0.500 |    2.344 |0.213|Sec. G2    |
| Interaction        |     1 |  60.00 |     -    |     -    |1.345|Eq. H1.1-2 |
|   (Bend+Ten) Int1  |     1 |  60.00 |     -    |     -    |0.991|Eq. H1.1-1 |
|   (Bend+Ten) Int2  |     1 |  60.00 |     -    |     -    |1.345|Eq. H1.1-2 |
|   (Bend+Comp)      |     1 |  60.00 |     -    |     -    |1.345|Eq. H1.2-1 |
|   (Bend+Shear) Z   |     1 |  60.00 |     -    |     -    |1.287|Sec. H2    |
|-----------------------------------------------------------------------------|