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V. AISI 2016 Zee Section

Determine the axial strength and flexural strength of a cold-formed zee section without lips according to AISI S100 2016 Specification using the LRFD method.

Details

The member is a simply-supported, 2m span. The member is subject to a uniform load of 60 lbs/ft. The section used is an 8ZU 12.5x105.

Section Properties

  • Depth of section, d = 8 in
  • Width of section, b = 1.25 in
  • Thickness, t = 0.105 in
  • Fillet radius, r = 0.188 in
  • Area, A = 1.059 in2
  • Moment of inertia major axis, Izz = 7.88 in4
  • Moment of inertia minor axis, Iyy = 0.120 in4
  • Section modulus major axis, Zzz = 1.97 in3
  • Section modulus minor axis, Zyy = 0.096 in3
  • Torsion constant, J = 0.004 in4
  • Warping constant, Cw = 1.546 in6
  • Radius of gyration about major axis: r x = I x A = 2.728  in.
  • Radius of gyration about minor axis: r y = I t A = 0.337  in.
  • Polar radius of gyration: r 0 = r x 2 + r y 2 + x 0 2 = 2.749  in.

Material Properties

  • E = 29,000 ksi
  • G = 11,300 ksi
  • Fy = 36 ksi

Design Forces

  • Mz = 3.975 ft·k
  • Vy = 0.197 kips

Validation

Section Dimension Checks

Radius to thickness ratio:

R t = 0.188 0.105 = 1.79 < 10   (OK)

Width to thickness ratio:

W = ( B - R - t ) = 0.957  in
W t = 0.957 0.105 = 9.11 < 60   (OK)

Depth to thickness ratio:

d = ( D - 2 R - 2 t ) = 7.414  in
d t = 7.414 0.105 = 70.60 < 200   (OK)

Calculate Axial Strength

Along Z axis:

K x × L x r x = 1 × 78.74 2.728 = 28.86
σ x = π 2 × E ( K x × L x r x ) 2 = π 2 × 29,000 ( 28.86 ) 2 = 343.6  ksi
σ t = 1 A × r 0 2 × G × J + π 2 × E × C w K t × L t 2 = 1 1.059 ( 2.798 ) 2 × 11,200 × 0.004 + π 2 × 29,000 × 1.546 1 × 78.74 2 = 14.37  ksi

Fcrex is the lesser of σx and σt ; Fcrex = 14.37 ksi

λ cz = F y F c r e x = 1.583 > 1.5
F nz = 0.877 λ cz 2 × F y = 0.877 ( 1.583 ) 2 × 36 = 12.60  ksi
P n z = A × F nz = 1.059 × 12.60 = 13.34  kips

Along Y axis:

K y × L y r y = 1 × 78.74 0.337 = 233.7
σ y = π 2 × E ( K y × L y r y ) 2 = π 2 × 29,000 ( 233.7 ) 2 = 5.244  ksi

Fcrey is the lesser of σy and σt ; Fcrey = 5.244 ksi

λ cy = F y F c r e y = 2.62 > 1.5
F ny = 0.877 λ cy 2 × F y = 0.877 ( 2.62 ) 2 × 36 = 4.599  ksi
P n y = A × F nz = 1.059 × 4.599 = 4.869  kips

Design nominal axial strength (LRFD):

P n y × ϕ c = 4.869 × 0.85 = 4.140  kips

Calculate Bending Strength

Major axis nominal moment capacity:

Mz = Sz × Fy = 1.97 × 36 = 70.92 in·kips

Major axis ultimate moment capacity:

ϕ b M z = 0.90 × 70.92 = 63.83  in·kip

Minor axis nominal moment capacity:

My = Sy × Fy = 0.096 × 36 = 3.456 in·kips

Minor axis ultimate moment capacity:

ϕ b M y = 0.90 × 3.456 = 3.110  in·kip

Lateral-Torsional Buckling Strength

σ ey = σ y = 5.244  ksi
C b = 12.5 M max 2.5 M max + 3 M A + 4 M B + 3 M C = 1.136

For singly symmetric sections bent about the axis of symmetry:

F crez = C b r 0 A 2 S f σ ey σ t = 1.136 ( 2.749 ) ( 1.059 ) 2 ( 1.97 ) 5.244 × 14.37 = 7.283  ksi
where
Sf
=
the elastic section modulus for the unreduced section relative to the extreme compression fiber per AISI S100 2016. Hence, Sf = Sz .

For Fcre < 0.56Fy = 0.56 × 36 = 20.16, Fn = Fcrez .

Mne = Fn × Sf = 7.283 × 1.97 = 14.35 in-k

Design nominal lateral-torsional buckling strength (LRFD):

ϕ b M ne = 0.9 × 14.35 = 12.92  in·kips
Table 1. Comparison of results
Result Type Reference STAAD.Pro Difference Comments
Ultimate axial strength (kips) 4.140 4.130 negligible  
Major axis bending strength (in·kips) 63.83 63.828 negligible  
Minor axis bending strength (in·kips) 3.110 3.110 none  
Lateral-torsional buckling strength (in·kips) 12.92 12.914 negligible  

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 Zee Section.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 08-Mar-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT INCHES KIP
JOINT COORDINATES
1 0 0 0; 2 78.74 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-06
DAMP 0.03
G 11200
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY COLDFORMED AMERICAN
1 TABLE ST 8ZU1.25X105
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -0.004
PERFORM ANALYSIS
PARAMETER 1
CODE AISI 2016
AXIS 1 ALL
METHOD LRFD
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  8ZU1.25X105     LEN:   78.740     LOC:  39.370 |
| STATUS: PASS       RATIO:  0.994             REF: Sec. F2       LC:      1  |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Location:   39.370  Criteria:   Sec. F2              LC:           1       |
|  Fx:(C)        0.000       Fy:         0.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:      -3.100       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                                         |
|  Ag:         1.06000       Az:     0.20107            Ay:     0.77857       |
|  Cz:         0.00000       Cy:     0.00000                                  |
|  Iz:         7.88000       Iy:     0.12000             J:     0.00389       |
|  Sz:         1.97000       Sy:     0.09600                                  |
|  Rz:         2.72653       Ry:     0.33646            Cw:     1.55000       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                              |
|  Fy:    36.000        Fu:    58.000      E:  29000.000       G:  11200.000  |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:    78.740   Lz:    78.740   Ly:    78.740   Lb:       0.000 |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000    Ky:   1.000    NSF:  1.000                                  |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|-----------------------------------------------------------------------------|
|  LC :     1   Actual :  234.023   Allowable :  300.000   Ratio :   0.780    |
|-----------------------------------------------------------------------------|
|  Section Dimension Check      |    Ratio     |    Limit     |     Status    |
|-----------------------------------------------------------------------------|
|  Radius To Thickness Ratio    |      1.786   |     10.000   |      PASS     |
|  Width To Thickness Ratio     |      9.119   |     60.000   |      PASS     |
|  Web Depth To Thickness Ratio |     70.619   |    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            |   -   |   -    |     -    |   34.344 |  -  |Sec. D2    |
|   Yielding         |   -   |   -    |     -    |   34.344 |  -  |Sec. D2    |
|   Rupture          |   -   |   -    |     -    |   46.110 |  -  |Sec. D3    |
| Compression        |   -   |   -    |     -    |    4.127 |  -  |Sec. E3    |
|   Local Comp       |   -   |   -    |     -    |    4.127 |  -  |Sec. E3    |
|   Axial Comp       |   -   |   -    |     -    |    4.130 |  -  |Sec. E2    |
| Bending            |     1 |   0.00 |   -3.091 |    3.110 |0.994|Sec. F2    |
|   Major Local Bend |     1 |  39.37 |   -0.237 |   12.927 |0.018|Sec. F3    |
|   Minor Local Bend |     1 |  39.37 |   -3.091 |    3.275 |0.944|Sec. F3    |
|   Major Sectional  |     1 |  39.37 |   -0.237 |   63.828 |0.004|Sec. F2    |
|   Minor Sectional  |     1 |   0.00 |   -3.091 |    3.110 |0.994|Sec. F2    |
|   LTB Major        |     1 |  39.37 |   -0.237 |   12.914 |0.018|Sec. F2    |
| Shear              |     1 |   0.00 |   -0.157 |    4.126 |0.038|Sec. G2    |
|   Major Shear (Z)  |     1 |   0.00 |   -0.157 |    4.126 |0.038|Sec. G2    |
|   Minor Shear (Y)  |     1 |   0.00 |    0.012 |   14.838 |0.001|Sec. G2    |
| Interaction        |     1 |  39.37 |     -    |     -    |0.240|Eq. H1.1-2 |
|   (Bend+Ten) Int1  |     1 |  39.37 |     -    |     -    |0.049|Eq. H1.1-1 |
|   (Bend+Ten) Int2  |     1 |  39.37 |     -    |     -    |0.240|Eq. H1.1-2 |
|   (Bend+Comp)      |     1 |  39.37 |     -    |     -    |0.240|Eq. H1.2-1 |
|   (Bend+Shear) Z   |     1 |  39.37 |     -    |     -    |0.049|Sec. H2    |
|-----------------------------------------------------------------------------|
Parent topic: V. AISI