# V. AISC 360-16 W Flexural Check LRFD

Verify the flexural yielding strength and lateral torsional buckling of an I section using the LRFD method per AISC 360-16.

## References

1. Steel Construction Manual, 15th Edition, American Institute of Steel Construction, 2017

## Details

Verify the flexural yielding strength and lateral torsional buckling of an I section (W12X90) gr. ASTM A992 material, with a member length of 41'-8", and a concentrated moment of 35,000 in-kip applied at midspan. The member is unbraced along its length.

## Validation

As the section is compact, nominal flexural yielding strength about X axis is:

 Mnx = fy × Zxx = 65 ksi × 311 in3 = 20,215 in·kip (F2-1, ref. 1)

Mx = ϕc×Mnx = 0.9(20,215 in·kip) = 18,193.5 in·kip

As this is the minor axis, nominal flexural yielding strength about Y axis is:

 Mny = fy × Zyy = 65 ksi × 143 in3 = 9,295 in·kip (F6-1, ref. 1)

My = ϕc×Mny = 0.9(9,295 in·kip) = 8,365.5 in·kip

Lateral torsional buckling about X axis

$r y = I y y / A g = 3.243 in 2$
limiting laterally unbraced length for the yielding limit state:
 $L p = 1.76 × r y E f y = 120.56 in$ (F2-5, ref. 1)
 $r t s = I y y × C w S x x = 3.769 in$ (F2-7, ref. 1)
For an I section, C = 1 (per F2-28a in ref. 1). The limiting unbraced length for the inelastic lateral-torsional buckling limit state is:

h0 = d - tf = 12.66 in

$L r = 1.95 × r t s E 0.7 F y J c S x x h 0 + ( J c S x x h 0 ) 2 + 6.76 ( 0.7 F y E ) 2 = 810.27 in$

Lp < Lb = 500 in < Lr

 $C b = 12.5 M max ⁡ 2.5 M max ⁡ + 3 M A + 4 M B + 3 M c = 12.5 ( 20 , 000 ) 2.5 ( 20 , 000 ) + 3 ( 10 , 000 ) + 4 ( 20 , 000 ) + 3 ( 10 , 000 ) = 1.316$ (F1-1, ref. 1)

Mp = fy × Zxx = 65 ksi × 311 in3 = 20,215 in·kip

Nominal LTB strength:

$M n = C b [ M p − ( M p − 0.7 F y S x ) ( L b − L p L r − L p ) ] ≤ M p$

Mn = 20,611.4 in·kip > Mp

ϕcMnx = 0.9(20,215 in·kip) = 18,193.5 in·kip

## Results

Table 1. Comparison of results
Flexural yielding capacity, X axis (in·kip) 18,194 18,190 negligible
Flexural buckling capacity, Y axis (in·kip) 8366 8366 none
Lateral-torsional buckling about X axis (in·kip) 18,194 18,190 negligible

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC 360-16 W Flexural Check LRFD.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 05-Feb-18
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 41.67 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
TYPE STEEL
STRENGTH FY 5184 FU 8352 RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TABLE ST W12X190
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT MY
UNIT INCHES KIP
1 CMOM GZ 35000 250
UNIT FEET KIP
PERFORM ANALYSIS
PARAMETER 1
CODE AISC UNIFIED 2016
LY 41.6667 ALL
LZ 41.6667 ALL
CB 0 ALL
FU 9360 ALL
FYLD 9360 ALL
METHOD LRFD
PROFILE W12 ALL
STP 1 ALL
TRACK 2 ALL
UNIT KIP INCHES
CHECK CODE ALL
FINISH


                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
*****************************************************
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
***NOTE : AISC 360-16 Design Statement for STAAD.Pro.
*** AXIS CONVENTION ***:
========================
The capacity results and intermediate results in the report follow the notations
and axes labels as defined in the AISC 360-16 code.
The analysis results are reported in STAAD.Pro axis convention and the AISC 360:16
design results are reported in AISC 360-16 code axis convention.
------------    -----------     -------------
X               Z          Axis typically parallel to the sections principal major axis.
Y               Y          Axis typically parallel to the sections principal minor axis.
Z               X          Longitudinal axis perpendicular to the cross section.
SECTION FORCES AXIS MAPPING: -
------------    -----------     -------------
Pz              FX         Axial force.
Vy              FY         Shear force along minor axis.
Vx              FZ         Shear force along major axis.
Tz              MX         Torsional moment.
My              MY         Bending moment about minor axis.
Mx              MZ         Bending moment about major axis.
*** DESIGN MESSAGES ***:
=======================
1. Section classification reported is for the cross section and loadcase that
produced the worst case design ratio for flexure/compression Capacity results.
2. Results for any Capacity/Check that is not relevant for a section/loadcase
based on the code clause in AISC 360-16 will not be shown in the report.
3. Bending results are reported as being �about� the relevant axis (X/Y), while
the results for shear are reported as being for shear forces �along� the axis.
E.g : Mx indicates bending about the X axis, while Vx indicates shear along
the X axis.
*** ABBREVIATIONS ***:
======================
F-T-B = Flexural-Torsional Buckling
L-T-B = Lateral-Torsional Buckling
F-L-B = Flange Local Buckling
W-L-B = Web Local Buckling
L-L-B = Leg Local Buckling
C-F-Y = Compression Flange Yielding
T-F-Y = Tension Flange Yielding
STAAD SPACE                                              -- PAGE NO.    4
STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
*****************************************************
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
- Member :     1
|-----------------------------------------------------------------------------|
|  Member No:        1       Profile:  ST  W12X190             (AISC SECTIONS)|
|  Status:        PASS       Ratio:         0.903       Loadcase:        2    |
|  Location:    208.35       Ref:      Cl.F2.1                                |
|  Pz:       0.000     T     Vy:        78.85           Vx:      0.000        |
|  Tz:       0.000           My:        0.000           Mx:     -.1643E+05    |
|-----------------------------------------------------------------------------|
| SLENDERNESS                                                                 |
| Actual Slenderness Ratio    :    154.172                                    |
| Allowable Slenderness Ratio :    300.000            LOC  :     0.00         |
|-----------------------------------------------------------------------------|
| STRENGTH CHECKS                                                             |
| Critical L/C  :      2             Ratio     :        0.903(PASS)           |
|          Loc  :  208.35            Condition :    Cl.F2.1                   |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES  (LOC:   208.35, PROPERTIES UNIT: IN  )                  |
| Ag  :   5.600E+01     Axx :   4.420E+01     Ayy :   1.526E+01               |
| Ixx :   1.890E+03     Iyy :   5.890E+02     J   :   4.880E+01               |
| Sxx+:   2.625E+02     Sxx-:   2.625E+02     Zxx :   3.110E+02               |
| Syy+:   9.276E+01     Syy-:   9.276E+01     Zyy :   1.430E+02               |
| Cw  :   2.380E+04     x0  :   0.000E+00     y0  :   8.882E-16               |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                         |
| Fyld:          65.000              Fu:          65.000                      |
|-----------------------------------------------------------------------------|
| Actual Member Length:       500.040                                         |
| Design Parameters                                            (Rolled)       |
| Kx:    1.00  Ky:    1.00  NSF:    1.00  SLF:    1.00  CSP:   12.00          |
|-----------------------------------------------------------------------------|
| COMPRESSION CLASSIFICATION (L/C:      2 LOC:   500.04)                      |
|                            λ         λp        λr       CASE                |
| Flange: NonSlender       3.65       N/A      11.83     Table.4.1a.Case1     |
| Web   : NonSlender       9.19       N/A      31.47     Table.4.1a.Case5     |
|                                                                             |
| FLEXURE CLASSIFICATION     (L/C:      1 LOC:   500.04)                      |
|                            λ         λp        λr       CASE                |
| Flange: Compact          3.65       8.03     21.12     Table.4.1b.Case10    |
| Web   : Compact          9.19      79.42    120.40     Table.4.1b.Case15    |
|-----------------------------------------------------------------------------|
STAAD SPACE                                              -- PAGE NO.    5
STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
*****************************************************
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
- Member :     1 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL TENSION                            |
|-----------------------------------------------------------------------------|
| TENSILE YIELDING                                                            |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       3276.       0.000     Cl.D2          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom. Ten. Yld Cap         : Pn     =  3640.0     kip        Eq.D2-1        |
|-----------------------------------------------------------------------------|
| TENSILE RUPTURE                                                             |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       2730.       0.000     Cl.D2          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective area            : Ae     =  56.000     in2        Eq.D3-1        |
|  Nom. Ten. Rpt Cap         : Pn     =  3640.0     kip        Eq.D2-2        |
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL COMPRESSION                        |
|-----------------------------------------------------------------------------|
| FLEXURAL BUCKLING X                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       1620.       0.000     Cl.E3          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective Slenderness     : Lcx/rx =  86.066                Cl.E2          |
|  Elastic Buckling Stress   : Fex    =  38.639     ksi        Eq.E3-4        |
|  Crit. Buckling Stress     : Fcrx   =  32.146     ksi        Eq.E3-2        |
|  Nom. Flexural Buckling    : Pnx    =  1800.2     kip        Eq.E3-1        |
|-----------------------------------------------------------------------------|
| FLEXURAL BUCKLING Y                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       532.2       0.000     Cl.E3          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective Slenderness     : Lcy/ry =  154.17                Cl.E2          |
|  Elastic Buckling Stress   : Fey    =  12.042     ksi        Eq.E3-4        |
|  Crit. Buckling Stress     : Fcry   =  10.560     ksi        Eq.E3-3        |
|  Nom. Flexural Buckling    : Pny    =  591.39     kip        Eq.E3-1        |
|-----------------------------------------------------------------------------|
| FLEX-TOR-BUCKLING                                                           |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       2911.       0.000     Cl.E4          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Elastic F-T-B Stress      : Fe     =  230.56     ksi        Eq.E4-2        |
|  Crit. F-T-B Stress        : Fcr    =  57.765     ksi        Eq.E3-2        |
|  Nom. Flex-tor Buckling    : Pn     =  3234.9     kip        Eq.E4-1        |
|-----------------------------------------------------------------------------|
STAAD SPACE                                              -- PAGE NO.    6
STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
*****************************************************
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
- Member :     1 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR SHEAR                                    |
|-----------------------------------------------------------------------------|
| SHEAR ALONG X                                                               |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       1551.       0.000     Cl.G1          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Coefficient Cv Along X    : Cv     =  1.0000                Eq.G2-9        |
|  Coefficient Kv Along X    : Kv     =  1.2000                Cl.G6          |
|  Nom. Shear Along X        : Vnx    =  1723.6     kip        Eq.G6-1        |
|-----------------------------------------------------------------------------|
| SHEAR ALONG Y                                                               |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              78.85       595.3       0.132     Cl.G1          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Coefficient Cv Along Y    : Cv     =  1.0000                -              |
|  Coefficient Kv Along Y    : Kv     =  5.3400                Eq.G2-5        |
|  Nom. Shear Along Y        : Vny    =  595.30     kip        Eq.G2-1        |
|-----------------------------------------------------------------------------|
STAAD SPACE                                              -- PAGE NO.    7
STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
*****************************************************
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
- Member :     1 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR BENDING                                  |
|-----------------------------------------------------------------------------|
| FLEX YIELDING (X)                                                           |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -.1643E+05  0.1819E+05   0.903     Cl.F2.1        2    208.35   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom Flex Yielding Along X : Mnx    =  20215.     kip-in     Eq.F2-1        |
|-----------------------------------------------------------------------------|
| FLEX. YIELDING (Y)                                                          |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       8366.       0.000     Cl.F6.1        2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom Flex Yielding Along Y : Mny    =  9295.0     kip-in     Eq.F6-1        |
|-----------------------------------------------------------------------------|
| LAT TOR BUCK ABOUT X                                                        |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -.1643E+05  0.1819E+05   0.903     Cl.F2.2        2    208.35   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom L-T-B Cap             : Mnx    =  20215.     kip-in     Eq.F2-2        |
|  Mom. Distr. factor        : CbX    =  1.3868                Eq.F1-1        |
|  Limiting Unbraced Length  : LpX    =  120.56     in         Eq.F2-5        |
|  coefficient C             : Cx     =  1.0000                Eq.F2-8a       |
|  Effective Rad. of Gyr.    : Rts    =  3.7768     in         Eq.F2-7        |
|  Limiting Unbraced Length  : LrX    =  812.00     in         Eq.F2-6        |
|-----------------------------------------------------------------------------|
STAAD SPACE                                              -- PAGE NO.    8
STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
*****************************************************
ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
- Member :     1 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL BEND INTERACTION                   |
|-----------------------------------------------------------------------------|
| COMBINED FORCES CLAU                                                        |
|                            RATIO      CRITERIA           L/C      LOC       |
|                            0.903      Eq.H1-1b            2      208.35     |
|                                                                             |
| Intermediate Results :                                                      |
|  Modification Fact. for Cb : Cbf    =  1.0000                Eq.H1-2        |
|  Axial Capacity            : Pc     =  2730.0     kip        Cl.H1.1        |
|  Moment Capacity           : Mcx    =  18194.     kip-in     Cl.H1.1        |
|  Moment Capacity           : Mcy    =  8365.5     kip-in     Cl.H1.1        |
|-----------------------------------------------------------------------------|