V. AISI 2016 Channel section

Determine the flexural strength and shear strength of a cold-formed channel section without lips according to AISI S100 2016 Specification using both LRFD and ASD methods.

Details

The member is a simply-supported, 6 ft span. The member is subject to a uniform load of 194.4 lbs/ft. The section used is a 550CU125-54.

Section Properties

Depth of section, d = 5.698 in
Width of section, b = 1.25 in
Thickness, t = 0.0566 in
Fillet radius, r = 0.0849 in
Area, A = 0.452 in²
Moment of inertia major axis, I_zz = 1.9 in⁴
Moment of inertia minor axis, I_yy = 0.053 in⁴
Section modulus major axis, Z_zz = 0.667 in³
Section modulus minor axis, Z_yy = 0.0512 in³
Distance between centroid and shear center: x₀ = -0.539 in
Radius of gyration about major axis: $r_{x} = \sqrt{\frac{I_{x}}{A}} = 2.053 in.$
Radius of gyration about minor axis: $r_{y} = \sqrt{\frac{I_{t}}{A}} = 0.342 in.$
Polar radius of gyration: $r_{0} = \sqrt{{r_{x}}^{2} + {r_{y}}^{2} + {x_{0}}^{2}} = 2.15 in.$
Torsion constant, J = 0.00048 in⁴

Material Properties

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

Design Forces

M_z = 10.5 ft·k
V_y = 0.583 kips

Validation

Lateral-Torsional Buckling Strength

σ_{ey} = \frac{π^{2} \times E}{{(\frac{K_{y} \times L_{y}}{r_{y}})}^{2}} = \frac{π^{2} \times 29,000}{{(\frac{1 \times 72}{0.342})}^{2}} = 6.586 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}{0.452 {(2.15)}^{2}} \times [11,300 \times 0.000483 + \frac{π^{2} \times 29,000 \times 0.315}{{(1 \times 72)}^{2}}] = 11.12 ksi

C_b = 1.136

For singly symmetric sections bent about the axis of symmetry:

F_{cre} = \frac{C_{b} r_{0} A}{S_{f}} \sqrt{σ_{ey} σ_{t}} = \frac{1.136 (2.15) (0.452)}{0.668} \sqrt{6.586 \times 11.12} = 14.15 ksi

where

S_f

the elastic section modulus for the unreduced section relative to the extreme compression fiber per AISI S100 2016. Hence, S_f = S_z .

For F_cre < 0.56F_y = 0.56 × 33 = 18.58, F_n = F_cre .

M_ne = F_n × S_f = 14.15 × 0.667 = 9.433 in-k

Allowable nominal lateral-torsional buckling strength (ASD):

\frac{M_{ne}}{Ω_{b}} = \frac{9.433}{1.67} = 5.649 in·kips

Design nominal lateral-torsional buckling strength (LRFD):

ϕ_{b} M_{ne} = 0.9 \times 9.433 = 8.49 in·kips

Bending About Major Axis

M_y = F_y × S_f = 33 × 0.667 = 22.01 in-k

Allowable nominal bending strength (ASD):

\frac{M_{y}}{Ω_{b}} = \frac{21.01}{1.67} = 13.18 in·kips

Design nominal bending strength (LRFD):

ϕ_{b} M_{y} = 0.9 \times 22.01 = 19.81 in·kips

Local Buckling Interaction with Yield and Global Buckling

From Section F3:

S_e = 0.665 in³

M_nl = S_e × F_n = 0.665 × 14.15 = 9.412 in-k

Allowable nominal bending strength (ASD):

\frac{M_{nl}}{Ω_{b}} = \frac{9.412}{1.67} = 5.636 in·kips

Design nominal bending strength (LRFD):

ϕ_{b} M_{nl} = 0.9 \times 9.412 = 8.471 in·kips

Governing strength ratio:

LRFD: 10.498 / 8.471 = 1.239

ASD: 10.498 / 5.636 = 1.862

Shear Strength along Minor Axis

h = D - 2× (t - r) = 5.698 - 2× (0.0566 - 0.0849) = 5.145 in

h /t = 5.415 / 0.0566 = 95.67

A_w = h × t = 5.415 × 0.0566 = 0.307 in²

V_y = 0.6 × A_w × F_y = 0.6 × 0.307 × 33 = 6.069 kip

F_{cr} = \frac{π^{2} \times E K_{v}}{12 (1 - μ^{2}) {(\frac{h}{t})}^{2}} = \frac{π^{2} \times 29,000 \times 5.34}{12 (1 - {0.3}^{2}) {(95.67)}^{2}} = 15.56 ksi

V_cr = A_w × F_cr = 0.307 × 15.56 = 4.75 kip

λ_{v} = \sqrt{\frac{V_{y}}{V_{cr}}} = \sqrt{\frac{6.069}{4.75}} = 1.128

For 0.815 < λ_v ≤ 1.227,

V_{n} = 0.815 \sqrt{V_{cr} \times V_{y}} = 0.815 \sqrt{4.75 \times 6.069} \times 6.069 = 4.384 kip

Allowable nominal shear strength (ASD):

\frac{V_{n}}{Ω_{v}} = \frac{4.384}{1.60} = 2.739 kips

Design nominal shear strength (LRFD):

ϕ_{v} V_{n} = 0.95 \times 4.384 = 4.615 kips

Governing strength ratio:

LRFD: 0.583 / 4.165 = 0.14

ASD: 0.583 / 2.74 = 0.213

Table 1. Comparison of results
Result Type		Reference	STAAD.Pro	Difference
LRFD	Major bending (in-k)	19.81	19.807	negligible
	LTB bending (in-k)	8.49	8.49	none
	Major local bending (in-k)	8.471	8.477	negligible
	Bending ratio	1.239	1.238	negligible
	Shear strength (kips)	4.165	4.163	negligible
	Shear ratio	0.14	0.14	none
ASD	Major bending (in-k)	13.18	13.178	negligible
	LTB bending (in-k)	5.649	5.649	none
	Major local bending (in-k)	5.636	5.64	negligible
	Bending ratio	1.862	1.861	negligible
	Shear strength (kips)	2.739	2.739	none
	Shear ratio	0.213	0.213	none

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISI\AISI 2016 Channel section.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
JOB NAME AISI C-Section Without Lips Braced at Mid-Span
JOB CLIENT BENTLEY SYSTEMS Inc.
ENGINEER DATE 04-Mar-20
JOB REV 1.0
JOB REF AISI Manual Cold-Formed Steel Design - Vol. 1
END JOB INFORMATION
INPUT WIDTH 79
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 72 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29500
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 550CU125-54
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 2 LOADTYPE Live  TITLE LL
MEMBER LOAD
1 UNI GY -0.0162
PERFORM ANALYSIS
PARAMETER 1
CODE AISI 2016
METHOD LRFD
BEAM 1 ALL
FYLD 33 ALL
TRACK 2 ALL
CHECK CODE ALL
PARAMETER 1
CODE AISI 2016
METHOD ASD
BEAM 1 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  550CU125-54     LEN:   72.000     LOC:  36.000 |
| STATUS: FAIL       RATIO:  1.238             REF: Sec. F3       LC:      2  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:   36.000  Criteria:   Sec. F3              LC:           2       |
|  Fx:(C)        0.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:     -10.498       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.45200       Az:     0.12548            Ay:     0.30649       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         1.90000       Iy:     0.05300             J:     0.00048       |
|  Sz:         0.66690       Sy:     0.05121                                  |
|  Rz:         2.05025       Ry:     0.34243            Cw:     0.31500       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    33.000        Fu:    58.000      E:  29500.000       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 :     2   Actual :  210.263   Allowable :  300.000   Ratio :   0.701    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.500   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     19.585   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |     95.671   |    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            |   -   |   -    |     -    |   13.424 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |   13.424 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   19.662 |  -  |Sec. D3    |
| Compression        |   -   |   -    |     -    |    2.185 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |    2.185 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |    2.219 |  -  |Sec. E2    |
| Bending            |     2 |  36.00 |  -10.498 |    8.477 |1.238|Sec. F3    |
|   Major Local Bend |     2 |  36.00 |  -10.498 |    8.477 |1.238|Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    1.518 |  -  |Sec. F3    |
|   Major Sectional  |     2 |  36.00 |  -10.498 |   19.807 |0.530|Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    1.521 |  -  |Sec. F2    |
|   LTB Major        |     2 |  36.00 |  -10.498 |    8.490 |1.236|Sec. F2    |
| Shear              |     2 |   0.00 |    0.583 |    4.163 |0.140|Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    2.360 |  -  |Sec. G2    |
|   Minor Shear (Y)  |     2 |   0.00 |    0.583 |    4.163 |0.140|Sec. G2    |
| Interaction        |     2 |  36.00 |     -    |     -    |1.238|Eq. H1.1-2 |
|   (Bend+Ten) Int1  |     2 |  36.00 |     -    |     -    |0.530|Eq. H1.1-1 |
|   (Bend+Ten) Int2  |     2 |  36.00 |     -    |     -    |1.238|Eq. H1.1-2 |
|   (Bend+Comp)      |     2 |  36.00 |     -    |     -    |1.238|Eq. H1.2-1 |
|   (Bend+Shear) Z   |     2 |  36.00 |     -    |     -    |0.591|Sec. H2    |
|-----------------------------------------------------------------------------|
    43. PARAMETER 1
    44. CODE AISI 2016
    45. METHOD ASD
    46. BEAM 1 ALL
    47. TRACK 2 ALL
    48. 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  550CU125-54     LEN:   72.000     LOC:  36.000 |
| STATUS: FAIL       RATIO:  1.861             REF: Sec. F3       LC:      2  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:   36.000  Criteria:   Sec. F3              LC:           2       |
|  Fx:(C)        0.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:     -10.498       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         0.45200       Az:     0.12548            Ay:     0.30649       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         1.90000       Iy:     0.05300             J:     0.00048       |
|  Sz:         0.66690       Sy:     0.05121                                  |
|  Rz:         2.05025       Ry:     0.34243            Cw:     0.31500       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    33.000        Fu:    58.000      E:  29500.000       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 :     2   Actual :  210.263   Allowable :  300.000   Ratio :   0.701    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.500   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |     19.585   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |     95.671   |    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            |   -   |   -    |     -    |    8.932 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |    8.932 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   13.108 |  -  |Sec. D3    |
| Compression        |   -   |   -    |     -    |    1.428 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |    1.428 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |    1.450 |  -  |Sec. E2    |
| Bending            |     2 |  36.00 |  -10.498 |    5.640 |1.861|Sec. F3    |
|   Major Local Bend |     2 |  36.00 |  -10.498 |    5.640 |1.861|Sec. F3    |
|   Minor Local Bend |   -   |   -    |     -    |    1.010 |  -  |Sec. F3    |
|   Major Sectional  |     2 |  36.00 |  -10.498 |   13.178 |0.797|Sec. F2    |
|   Minor Sectional  |   -   |   -    |     -    |    1.012 |  -  |Sec. F2    |
|   LTB Major        |     2 |  36.00 |  -10.498 |    5.649 |1.858|Sec. F2    |
| Shear              |     2 |   0.00 |    0.583 |    2.739 |0.213|Sec. G2    |
|   Major Shear (Z)  |   -   |   -    |     -    |    1.553 |  -  |Sec. G2    |
|   Minor Shear (Y)  |     2 |   0.00 |    0.583 |    2.739 |0.213|Sec. G2    |
| Interaction        |     2 |  36.00 |     -    |     -    |1.861|Eq. H1.1-2 |
|   (Bend+Ten) Int1  |     2 |  36.00 |     -    |     -    |0.797|Eq. H1.1-1 |
|   (Bend+Ten) Int2  |     2 |  36.00 |     -    |     -    |1.861|Eq. H1.1-2 |
|   (Bend+Comp)      |     2 |  36.00 |     -    |     -    |1.861|Eq. H1.2-1 |
|   (Bend+Shear) Z   |     2 |  36.00 |     -    |     -    |0.888|Sec. H2    |
|-----------------------------------------------------------------------------|
    49. FINISH
             *********** END OF THE STAAD.Pro RUN ***********