V. AISC 360-16 Tapered I Section

Verify the axial compression capacity, flexure capacity, and interaction ratio of a tapered I section member per both the LRFD and ASD methods of the AISC 360-16 code.

Details

A 5' × 5' portal frame consists of W12x72 columns and a steel tapered member for the beam.

The following load cases are evaluated:

A uniformly distributed load of -3.5 k/ft along the beam in the global Y direction.
Lateral loads at the top of the left column of 50 kips in the global X direction and 25 kips in the global Z direction (out of plane).
Lateral loads at the top of the right column of -50 kips in the global X direction and 45 kips in the global Z direction (out of plane).
A concreted load of -100 kips in the global Y direction and a concentrated torque of 9 in·kips at mid-span of the beam.

Material Properties

E = 29,000 ksi
Fy = 50 ksi

Validation

Design Forces

By inspection, the left end of the beam (shallowest overall depth) governs under the symmetric loads present. The following design loads are used from the STAAD.Pro analysis are then used:

M_z = 32.26 in·kips, F_y = 8.755 kips
M_z = 677.6 in·kips, M_y = 2.942 in·kips, M_x = 5.106 in·kips, F_y = 22.98 kips, F_z = 0.098 kips, F_x = 24.70 kips
M_z = 275.9 in·kips, M_x = 4.497 in·kips, F_y = 50.23 kips, F_x = 5.473 kips
M_z = 654.5 in·kips, M_y = 5.298 in·kips, M_x = 9.192 in·kips, F_y = 22.50 kips, F_z = 0.177 kips, F_x = 24.35 kips

Section Properties

The section at the left end (D = 18 in.) has the following properties:

Web height, d_w = 18 - 2×0.84 = 16.32 in.
Area, A_g = (0.84)(9 + 7.5) + 16.32 (0.36) = 19.74 in²
Shear area in local y, A_vy = 18(0.36) = 6.48 in²
Shear area in local x, A_vx = 0.84(9 + 7.5) = 13.86 in²
Dist to centroid y, $c = \frac{7.5 (0.84) (\frac{0.84}{2}) + 0.36 (16.32) (0.84 + \frac{16.32}{2}) + 9 (0.84) (18 - \frac{0.84}{2})}{19.74} = 9.548 in.$
Moment of inertia about major axis,
$I_{zz} = \begin{matrix} \frac{7.5 {(0.84)}^{3}}{12} + 7.5 (0.84) {(9.548 - \frac{0.84}{2})}^{2} \\ + \frac{0.36 {(16.32)}^{3}}{12} + 0.36 (16.32) {(9.548 - \frac{16.32}{2} - 0.84)}^{2} \\ + \frac{9 {(0.84)}^{3}}{12} + 9 (0.84) {(18 - \frac{0.84}{2} - 9.548)}^{2} & = 1, 147 {in}^{4} \end{matrix}$
Moment of inertia about minor axis,
$I_{y y} = \frac{0.84 {(9)}^{3}}{12} + \frac{16.32 {(0.36)}^{3}}{12} + \frac{0.84 {(7.5)}^{3}}{12} = 80.62 {in}^{4}$
Elastic section modulus about major axis top flange fiber, $S_{x x, t o p} = \frac{I_{z z}}{D - c} = \frac{1, 146}{18 - 9.548} = 135.5 {in}^{3}$
Elastic section modulus about major axis bottom flange fiber, $S_{x x, b o t t o m} = \frac{I_{z z}}{c} = \frac{1, 146}{9.548} = 120.0 {in}^{3}$
Elastic section modulus about minor axis, $S_{y y} = \frac{I_{y y}}{b_{f 1} / 2} = \frac{80.62}{9 / 2} = 17.92 {in}^{3}$
Radius of gyration about major axis, $r_{x} = \sqrt{\frac{I_{z z}}{A_{g}}} = \sqrt{\frac{1, 147}{19.74}} = 7.619 in$
Radius of gyration about minor axis, $r_{y} = \sqrt{\frac{I_{y y}}{A_{g}}} = \sqrt{\frac{80.62}{19.74}} = 2.021 in$
Location of the plastic neutral axis, $y_{PNA} = 0.84 + \frac{19.74 - 2 (7.5) (0.84)}{2 \times 0.36} = 10.75 in.$
Plastic section modulus about major axis,
$Z_{z t} = \begin{matrix} \frac{7.5 {(10.75)}^{2}}{2} - \frac{(7.5 - 0.36) {(10.75 - 0.84)}^{2}}{2} \\ + \frac{9 {(18 - 10.75)}^{2}}{2} - \frac{(9 - 0.36) {(18 - 0.84 - 10.75)}^{2}}{2} & = 141.8 {in}^{3} \end{matrix}$
Plastic section modulus about minor axis, $Z_{y} = \frac{0.84 {(9)}^{2}}{4} + \frac{16.32 {(0.36)}^{2}}{4} + \frac{0.84 {(7.5)}^{2}}{4} = 29.35 {in}^{3}$
Torsional constant, $J = \frac{9 {(0.84)}^{3} + 16.32 {(0.36)}^{3} + 7.5 {(0.84)}^{3}}{3} = 3.514 {in.}^{4}$

Section Classification

Flange in compression

λ = \frac{b}{t} = \frac{1 / 2 \times 7.5}{0.84} = 4.464

k_{c} = \frac{4}{\sqrt{\frac{d_{w}}{t_{w}}}} = \frac{4}{\sqrt{\frac{16.32}{0.36}}} = 0.594

Per Table B4.1a, $λ < λ_{r} = 0.64 \sqrt{\frac{k_{c} E}{F_{y}}} = 0.64 \sqrt{\frac{0.594 (29,000)}{50}} = 11.88$ , therefore flanges are non-slender for compression.

Web in compression

λ = \frac{d_{w}}{t_{w}} = \frac{16.32}{0.36} = 45.33

Per Table B4.1a, Case 5, $λ > λ_{r} = 1.49 \sqrt{\frac{E}{F_{y}}} = 1.49 \sqrt{\frac{29,000}{50}} = 35.88$ , therefore the web is slender for compression.

Flange in bending (use half the longer flange width as the outstanding width):

λ = \frac{b_{f}}{t_{f}} = \frac{9}{2 (0.84)} = 5.36

Per Table 1.b, Case 11, $λ < λ_{p} = 0.38 \sqrt{\frac{E}{F_{y}}} = 0.38 \sqrt{\frac{29,000}{50}} = 9.15$ , therefore flange is compact for bending.

Web in bending

λ = \frac{d_{w}}{t_{w}} = \frac{16.32}{0.36} = 45.33

Per Table 1.b, Case 15, $λ < λ_{p} = 3.76 \sqrt{\frac{E}{F_{y}}} = 3.76 \sqrt{\frac{29,000}{50}} = 90.55$ , therefore web is compact for bending.

Compression Capacity

About X Axis:

\frac{k_{x} L_{x}}{r_{x}} = \frac{1.0 (60)}{7.619} = 7.875 < 4.71 \sqrt{\frac{E}{F_{y}}} = 4.71 \sqrt{\frac{29, 000}{50}} = 113.4

F_{ex} = \frac{π^{2} E}{{(\frac{k_{x} L_{x}}{r_{x}})}^{2}} = \frac{π^{2} \times 29, 000}{{(7.875)}^{2}} = 4,615 ksi

(Eq. E3-4)

\frac{F_{y}}{F_{ex}} = \frac{50}{4,615} = 0.011 < 2.25

F_{crx} = {0.658}^{\frac{F_{y}}{F_{ex}}} F_{y} = {0.658}^{(0.011)} 50 = 49.77 ksi

(Eq. E3-2)

Since the web is slender, use the effective area. The effective width imperfection adjustment factors (Table E7.1):

c₁ = 0.18

c₂ = 1.31

F_{el} = {(c_{2} \frac{λ_{r}}{λ})}^{2} F_{y} = {(1.31 \frac{35.88}{45.33})}^{2} = 53.76 ksi

(Eq. E7-5)

Therefore, the effective depth of the web in compression for the effective area:

d_{e} = d_{w} (1 - c_{1} \sqrt{\frac{F_{el}}{F_{crx}}}) \sqrt{\frac{F_{el}}{F_{crx}}} = 16.32 (1 - 0.18 \sqrt{\frac{53.76}{49.77}}) \sqrt{\frac{53.76}{49.77}} = 13.78 in.

(Eq. E7-3)

The reduced area of the section then becomes:

A_{e} = 19.75 - (16.32 - 13.78) 0.36 = 18.84 {in.}^{2}

P_nx = A_e× F_crx = 18.84 × 49.77 = 937.4 kips

(Eq. E7-1)

About Y Axis:

\frac{k_{y} L_{y}}{r_{y}} = \frac{1.0 (60)}{2.021} = 29.69 < 4.71 \sqrt{\frac{E}{F_{y}}} = 113.4

F_{ey} = \frac{π^{2} E}{{(\frac{k_{y} L_{y}}{r_{y}})}^{2}} = \frac{π^{2} \times 29, 000}{{(29.69)}^{2}} = 324.7 ksi

(Eq. E3-4)

\frac{F_{y}}{F_{ey}} = \frac{50}{324.7} = 0.154 < 2.25

F_{cry} = {0.658}^{(F_{y} / F_{ey})} F_{y} = {0.658}^{(0.154)} 50 = 46.88 ksi

(Eq. E3-2)

Therefore, the effective depth of the web in compression for the effective area:

d_{e} = d_{w} (1 - c_{1} \sqrt{\frac{F_{el}}{F_{cry}}}) \sqrt{\frac{F_{el}}{F_{cry}}} = 16.32 (1 - 0.18 \sqrt{\frac{53.76}{46.88}}) \sqrt{\frac{53.76}{46.88}} = 14.11 in.

(Eq. E7-3)

The reduced area of the section then becomes:

A_{e} = 19.75 - (16.32 - 14.11) 0.36 = 18.93 {in.}^{2}

P_ny = A_e× F_cry = 18.93 × 46.88 = 887.4 kips

(Eq. E7-1)

The Y axis governs; P_n = P_ny. The axial capacity:

The ultimate compression capacity (LRFD) = ϕ_cP_n = 0.9 × 887.4 = 798.3 kips
The allowable compression capacity (ASD) = P_n/Ω_c = 887.4 / 1.67 = 531.4 kips

Calculate Shear Capacity

Shear capacity along major axis (Y axis)

Assume no transverse stiffeners are present. Therefore, k_v = 5.34

V_{n} = 0.6 F_{y} A_{w} C_{v1}

(Eq. G2-1)

where

C_v1	=	1.0, since: $\frac{D}{t_{w}} = \frac{18}{0.36} = 50 < 1.10 \sqrt{k_{v} E / F_{y}} = 1.10 \sqrt{\frac{5.34 (29,000)}{50}} = 61.2$
A_w	=	A_vy = 6.48 in²

V_{n} = 0.6 (50) (6.48) (1.0) = 194 kips

The shear capacity along the Y axis:

The ultimate shear capacity (LRFD) = ϕ_vV_n = 0.9 × 1,434 = 1,290 in·kips
The allowable shear capacity (ASD) = V_n /Ω_v = 1,434 / 1.67 = 858.4 in·kips

Calculate Bending Capacity

Bending capacity in plastic yielding about Y axis:

M_y = F_y × Z_y = 50 × 29.35 = 1,468 in·kips > 1.6F_y × S_y = 1.6 (50) (17.92) = 1,434 in·kips

Use M_y =1,434 in·kips

The bending capacity for flexural yielding about the Y axis:

The ultimate bending capacity (LRFD) = ϕ_bM_y = 0.9 × 194 = 175 kips
The allowable bending capacity (ASD) = M_y/Ω_b = 194 / 1.67 = 116.4 kips

Bending capacity in plastic yielding about X axis:

Note: In the critical load case (LC4), the bending at the left end is reverse (i.e., "hogging"). Thus, the bottom flange is in compression; which has a lower elastic section modulus.

Lateral-torsional buckling about the X axis:

L_{p} = 1.1 r_{t} \sqrt{\frac{E}{F_{y}}}

(Eq. F4-7)

where

r_t

the effective radius of gyration for lateral-torsional buckling. For I-shapes with rectangular compression flanges:

= \frac{b_{fc}}{\sqrt{12 (1 + \frac{1}{6} a_{w})}}

Thus, $r_{t} = \frac{7.5}{\sqrt{12 (1 + \frac{1}{6} 0.995)}} = 2.005$ and $L_{p} = 1.1 (2.005) \sqrt{\frac{29,000}{50}} = 53.12 in.$ . Therefore, lateral-torsional buckling must be checked.

L_{r} = 1.95 r_{t} \frac{E}{F_{L}} \sqrt{\frac{J}{S_{x c} h_{o}} + \sqrt{{(\frac{J}{S_{x c} h_{o}})}^{2} + 6.76 (\frac{F_{L}}{E})}}

(Eq. F4-8)

where

F_L	=	0.7F_y = 0.7 (50) = 35 ksi for $\frac{S_{x t}}{S_{x c}} = \frac{135.5}{120.0} = 1.13 > 0.7$
h_o	=	distance between flange centroids = 18 - 0.84 = 17.16 in.

L_{r} = 1.95 (2.005) \frac{29, 000}{35} \sqrt{\frac{3.514}{120 \times 17.16} + \sqrt{{(\frac{3.514}{120 \times 17.16})}^{2} + 6.76 {(\frac{35}{29, 000})}^{2}}} = 235.4 in.

As L_r < L_b = 60 < L_p, the nominal moment capacity is:

M_{n} = C_{b} [R_{p c} M_{y c} - (R_{p c} M_{y c} - F_{L} S_{x c}) (\frac{L_{b} - L_{p}}{L_{r} - L_{p}})] \leq R_{p c} M_{y c}

(Eq. F4-2)

where

C_b	=	1.0
M_yc	=	F_yS_yc = 50 × 120 = 6,000 ksi
M_p	=	F_yZ_x = 50 × 141.8 = 7,090 ksi ≤ 1.6 F_yS_x = 9,600 ksi

M_{n} = 1.0 {1.182 (6, 000) - [1.182 (6, 000) - 35 (120)] (\frac{60 - 53.12}{235.4 - 53.12})} = 6, 983 ksi

The bending capacity for lateral-torsional buckling:

The ultimate bending capacity (LRFD) = ϕ_bM_y = 0.9 × 6,983 = 6,285 in·kips
The allowable bending capacity (ASD) = M_y/Ω_b = 6,983 / 1.67 = 4,181 in·kips

Compression Flange Yielding

M_n = R_pc × M_yc

(Eq. F4-1)

where

I_yc	=	moment of inertia of the compression flange about the Y axis $= \frac{0.84 {(7.5)}^{3}}{12} = 29.53 {in.}^{4}$
M_yc	=	M_y × S_xc = M_y × S_xx,top = 50 × 135.5 = 6,775 in·kips
R_pc	=	the web plastification factor, $= \frac{M_{p}}{M_{yc}} = \frac{7,090}{6,775} = 1.046$ (Eq. F4-9a) when $I_{y c} / I_{y} = \frac{29.53}{80.62} = 0.366 > 0.23$ and $\frac{h_{c}}{t_{w}} = \frac{17.42}{0.36} = 48.38 < λ_{p w} = 90.55$ (limiting slenderness for a compact web per Table B4.1b case 15)

M_n = 1.046 × 6,775 = 7,087 in·kips

The bending capacity for compression flange yielding:

The ultimate bending capacity (LRFD) = ϕ_bM_n = 0.9 × 7,087 = 6,378 in·kips
The allowable bending capacity (ASD) = M_n/Ω_b = 7,087 / 1.67 = 4,244 in·kips

Interaction Ratio for Bending and Compression

Check the interaction ratio for Load Case 2:

\frac{P_{r}}{P_{c}} = \frac{24.98}{798.3} = 0.031 < 0.2 (LRFD)

\frac{P_{r}}{P_{c}} = \frac{24.98}{531.4} = 0.046 < 0.2 (ASD)

So use Eq. H1-1b for both methods: $\frac{P_{r}}{2 P_{c}} + (\frac{M_{r x}}{M_{c x}} + \frac{M_{r y}}{M_{c y}}) \leq 1.0$ . Note that these equations use the compression flange capacity.

\frac{0.031}{2} + (\frac{677.6}{6,378} + \frac{2.942}{1,321}) = 0.124 (LRFD)

\frac{0.046}{2} + (\frac{677.6}{4,244} + \frac{2.942}{878.7}) = 0.186 (ASD)

Results

Table 1.
Result Type		Reference	STAAD.Pro	Difference
LRFD	Compression capacity (kips)	798.3	798.8	negligible
	Major axis bending capacity (in·kips)	6,285	6,282	negligible
	Minor axis bending capacity (in·kips)	1,290	1,290	none
	Shear capacity along major axis (kips)	175	175.0	none
	Interaction ratio	0.124	0.124	none
ASD	Compression capacity (kips)	531.4	531.48	negligible
	Major axis bending capacity (in·kips)	4,181	4,180	negligible
	Minor axis bending capacity (in·kips)	858.4	858.3	negligible
	Shear capacity along major axis (kips)	116.4	116.4	none
	Interaction ratio	0.186	0.186	none

STAAD.Pro Input

The following design parameters are used:

The welded tapered member is specified using STP 2

The remaining parameters all use their default values.

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

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 14-Jan-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 0 5 0; 3 5 0 0; 4 5 5 0;
MEMBER INCIDENCES
1 1 2; 2 2 4; 3 3 4;
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 RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY
2 TAPERED 1.5 0.03 1.65 0.75 0.07 0.625 0.07
MEMBER PROPERTY AMERICAN
1 3 TABLE ST W12X72
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 3 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
2 UNI GY -3.5
LOAD 2 LOADTYPE None  TITLE LOAD CASE 2
JOINT LOAD
2 FX 50 FZ 25
LOAD 4 LOADTYPE None  TITLE LOAD CASE 4
JOINT LOAD
4 FX -50 FZ 45
LOAD 3 LOADTYPE None  TITLE LOAD CASE 3
MEMBER LOAD
2 CON GY -100
2 CMOM GX 0.75
PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
UNIT INCHES KIP
PARAMETER 1
CODE AISC UNIFIED 2016
BEAM 1 MEMB 2
FYLD 50 ALL
FU 60 ALL
MAIN 200 MEMB 2
METHOD LRFD
STP 2 MEMB 2
TMAIN 300 MEMB 2
TRACK 2 MEMB 2
CHECK CODE MEMB 2
PARAMETER 2
CODE AISC UNIFIED 2016
BEAM 1 MEMB 2
FYLD 50 ALL
FU 60 ALL
MAIN 200 MEMB 2
METHOD ASD
STP 2 MEMB 2
TMAIN 300 MEMB 2
TRACK 2 MEMB 2
CHECK CODE MEMB 2
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
                     *****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
      STAAD SPACE                                              -- PAGE NO.    8
                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
                     *****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
 - Member :     2 
|-----------------------------------------------------------------------------|
|  Member No:        2       Profile:      TAPERED             (AISC SECTIONS)|
|  Status:        PASS       Ratio:         0.286       Loadcase:        3    |
|  Location:      0.00       Ref:      Cl.G1                                  |
|  Pz:       5.473     C     Vy:        50.02           Vx:     -.5769E-06    |
|  Tz:      -4.497           My:       0.1731E-04       Mx:      275.9        |
|-----------------------------------------------------------------------------|
| COMPRESSION SLENDERNESS                                                     |
| Actual Slenderness Ratio    :     29.685                                    |
| Allowable Slenderness Ratio :    200.000            LOC  :     0.00         |
|-----------------------------------------------------------------------------|
| STRENGTH CHECKS                                                             |
| Critical L/C  :      3             Ratio     :        0.286(PASS)           |
|          Loc  :    0.00            Condition :    Cl.G1                     |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES  (LOC:     0.00, PROPERTIES UNIT: IN  )                  |
| Ag  :   1.974E+01     Axx :   1.386E+01     Ayy :   6.480E+00               |
| Ixx :   1.146E+03     Iyy :   8.062E+01     J   :   3.514E+00               |
| Sxx+:   1.355E+02     Sxx-:   1.200E+02     Zxx :   1.418E+02               |
| Syy+:   1.792E+01     Syy-:   1.792E+01     Zyy :   2.935E+01               |
| Cw  :   5.508E+03     x0  :   0.000E+00     y0  :   0.000E+00               |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                         |
| Fyld:          50.000              Fu:          60.000                      |
|-----------------------------------------------------------------------------|
| Actual Member Length:        60.000                                         |
| Design Parameters                                            (Built-Up)     |
| Kx:    1.00  Ky:    1.00  NSF:    1.00  SLF:    1.00  CSP:   12.00          |
|-----------------------------------------------------------------------------|
| COMPRESSION CLASSIFICATION (L/C:      2 LOC:     0.00)                      |
|                            λ         λp        λr       CASE                |
| Flange: NonSlender       5.36       N/A      11.88     Table.B4.1a.Case2    |
| Web   : Slender         45.33       N/A      35.88     Table.B4.1a.Case5    |
|                                                                             |
| FLEXURE CLASSIFICATION     (L/C:      2 LOC:     0.00)                      |
|                            λ         λp        λr       CASE                |
| Flange: Compact          5.36       9.15     21.08     Table.B4.1b.Case11   |
| Web   : Compact         42.29     126.82    137.27     Table.B4.1b.Case15   |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    9
                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
                     *****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL TENSION                            |
|-----------------------------------------------------------------------------|
| TENSILE YIELDING                                                            |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       888.1       0.000     Cl.D2          1      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom. Ten. Yld Cap         : Pn     =  986.76     kip        Eq.D2-1        |
|-----------------------------------------------------------------------------|
| TENSILE RUPTURE                                                             |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       888.1       0.000     Cl.D2          1      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective area            : Ae     =  19.735     in2        Eq.D3-1        |
|  Nom. Ten. Rpt Cap         : Pn     =  1184.1     kip        Eq.D2-2        |
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL COMPRESSION                        |
|-----------------------------------------------------------------------------|
| FLEXURAL BUCKLING X                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              24.70       843.0       0.029     Cl.E3          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective Slenderness     : Lcx/rx =  7.8750                Cl.E2          |
|  Elastic Buckling Stress   : Fex    =  4615.2     ksi        Eq.E3-4        |
|  Crit. Buckling Stress     : Fcrx   =  49.774     ksi        Eq.E3-2        |
|  Nom. Flexural Buckling    : Pnx    =  936.63     kip        Eq.E7-1        |
|-----------------------------------------------------------------------------|
| FLEXURAL BUCKLING Y                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              24.70       798.8       0.031     Cl.E3          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective Slenderness     : Lcy/ry =  29.685                Cl.E2          |
|  Elastic Buckling Stress   : Fey    =  324.80     ksi        Eq.E3-4        |
|  Crit. Buckling Stress     : Fcry   =  46.880     ksi        Eq.E3-2        |
|  Nom. Flexural Buckling    : Pny    =  887.57     kip        Eq.E7-1        |
|-----------------------------------------------------------------------------|
| FLEXURAL-TORSIONAL-BUCKLING                                                 |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              24.70       798.8       0.031     Cl.E4          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Constant                  : H      =  1.0000                Eq.E4-8        |
|  Euler Buckling Stress     : Fez    =  389.10     ksi        Eq.E4-7        |
|  Elastic F-T-B Stress      : Fe     =  324.80     ksi        Eq.E4-3        |
|  Crit. F-T-B Stress        : Fcr    =  46.880     ksi        Eq.E3-2        |
|  Nom. Flex-tor Buckling    : Pn     =  887.57     kip        Eq.E7-1        |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   10
                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
                     *****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR SHEAR                                    |
|-----------------------------------------------------------------------------|
| SHEAR ALONG X                                                               |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             0.1766       374.2       0.000     Cl.G1          4      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    =  415.80     kip        Eq.G6-1        |
|-----------------------------------------------------------------------------|
| SHEAR ALONG Y                                                               |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              50.02       175.0       0.286     Cl.G1          3      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    =  194.40     kip        Eq.G2-1        |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   11
                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
                     *****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR BENDING                                  |
|-----------------------------------------------------------------------------|
| FLEXURAL YIELDING (Y)                                                       |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -5.298       1290.       0.004     Cl.F6.1        4      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom Flex Yielding Along Y : Mny    =  1433.3     kip-in     Eq.F6-1        |
|-----------------------------------------------------------------------------|
| LAT TOR BUCK ABOUT X                                                        |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              654.5       6282.       0.104     Cl.F4.2        4      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom L-T-B Cap             : Mnx    =  6980.3     kip-in     Eq.F4-2        |
|  Mom. Distr. factor        : CbX    =  1.0000                Custom         |
|  Limiting Unbraced Length  : LpX    =  53.119     in         Eq.F4-7        |
|  Limiting Unbraced Length  : LrX    =  235.38     in         Eq.F4-8        |
|  Aw Ratio                  : AwX    = 0.99518                Eq.F4-12       |
|  Nom. Compr. Flange Stress : FLX    =  35.000     ksi        Eq.F4-6a       |
|  Compr. Flange Yield Mom.  : MycX   =  5999.4     kip-in     Eq.F4-4        |
|  Web Plastification Factor : RpcX   =  1.1817                Eq.F4-9a       |
|-----------------------------------------------------------------------------|
| COM FLANGE YIELDING                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -677.6       6380.       0.106     Cl.F4.1        2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom comp. Flange Yld      : Mny    =  7089.4     kip-in     Eq.F4-1        |
|  Compr Flange Yld Mom.     : Myc    =  6777.0     kip-in     Eq.F4-4        |
|  Web Plastification Factor : Rpc    =  1.0461                Eq.F4-9a       |
|-----------------------------------------------------------------------------|
| TEN FLANGE YIELDING                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -677.6       6380.       0.106     Cl.F4.4        2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom Tens. Flange Yielding : Mny    =  7089.4     kip-in     Eq.F4-15       |
|  Tens. Flange Yield Moment : Myt    =  5999.4     kip-in     Eq.F4-4        |
|  Web Plastification Factor : Rpt    =  1.1817                Eq.F4-16a      |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   12
                       STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2)
                     *****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL BEND INTERACTION                   |
|-----------------------------------------------------------------------------|
| COMBINED FORCES CLAUSE H1                                                   |
|                            RATIO      CRITERIA           L/C      LOC       |
|                            0.124      Eq.H1-1b            2        0.00     |
|                                                                             |
| Intermediate Results :                                                      |
|  Axial Capacity            : Pc     =  798.81     kip        Cl.H1.1        |
|  Moment Capacity           : Mcx    =  6380.4     kip-in     Cl.H1.1        |
|  Moment Capacity           : Mcy    =  1290.0     kip-in     Cl.H1.1        |
|-----------------------------------------------------------------------------|
 ***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.
   AISC Spec.      STAAD.Pro      Description
  ------------    -----------     -------------
       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: -
   AISC Spec.      STAAD.Pro      Description
  ------------    -----------     -------------
       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
    56. PARAMETER 2
    57. CODE AISC UNIFIED 2016
    58. BEAM 1 MEMB 2
    59. FYLD 50 ALL
    60. FU 60 ALL
    61. MAIN 200 MEMB 2
    62. METHOD ASD
    63. STP 2 MEMB 2
    64. TMAIN 300 MEMB 2
    65. TRACK 2 MEMB 2
    66. CHECK CODE MEMB 2
 PARAMETER 2       
      STAAD SPACE                                              -- PAGE NO.   13
                       STAAD.PRO CODE CHECKING - AISC 360-16 ASD (V1.2)
                     ****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
      STAAD SPACE                                              -- PAGE NO.   14
                       STAAD.PRO CODE CHECKING - AISC 360-16 ASD (V1.2)
                     ****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
 - Member :     2 
|-----------------------------------------------------------------------------|
|  Member No:        2       Profile:      TAPERED             (AISC SECTIONS)|
|  Status:        PASS       Ratio:         0.430       Loadcase:        3    |
|  Location:      0.00       Ref:      Cl.G1                                  |
|  Pz:       5.473     C     Vy:        50.02           Vx:     -.5769E-06    |
|  Tz:      -4.497           My:       0.1731E-04       Mx:      275.9        |
|-----------------------------------------------------------------------------|
| COMPRESSION SLENDERNESS                                                     |
| Actual Slenderness Ratio    :     29.685                                    |
| Allowable Slenderness Ratio :    200.000            LOC  :     0.00         |
|-----------------------------------------------------------------------------|
| STRENGTH CHECKS                                                             |
| Critical L/C  :      3             Ratio     :        0.430(PASS)           |
|          Loc  :    0.00            Condition :    Cl.G1                     |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES  (LOC:     0.00, PROPERTIES UNIT: IN  )                  |
| Ag  :   1.974E+01     Axx :   1.386E+01     Ayy :   6.480E+00               |
| Ixx :   1.146E+03     Iyy :   8.062E+01     J   :   3.514E+00               |
| Sxx+:   1.355E+02     Sxx-:   1.200E+02     Zxx :   1.418E+02               |
| Syy+:   1.792E+01     Syy-:   1.792E+01     Zyy :   2.935E+01               |
| Cw  :   5.508E+03     x0  :   0.000E+00     y0  :   0.000E+00               |
|-----------------------------------------------------------------------------|
| MATERIAL PROPERTIES                                                         |
| Fyld:          50.000              Fu:          60.000                      |
|-----------------------------------------------------------------------------|
| Actual Member Length:        60.000                                         |
| Design Parameters                                            (Built-Up)     |
| Kx:    1.00  Ky:    1.00  NSF:    1.00  SLF:    1.00  CSP:   12.00          |
|-----------------------------------------------------------------------------|
| COMPRESSION CLASSIFICATION (L/C:      2 LOC:     0.00)                      |
|                            λ         λp        λr       CASE                |
| Flange: NonSlender       5.36       N/A      11.88     Table.B4.1a.Case2    |
| Web   : Slender         45.33       N/A      35.88     Table.B4.1a.Case5    |
|                                                                             |
| FLEXURE CLASSIFICATION     (L/C:      2 LOC:     0.00)                      |
|                            λ         λp        λr       CASE                |
| Flange: Compact          5.36       9.15     21.08     Table.B4.1b.Case11   |
| Web   : Compact         42.29     126.82    137.27     Table.B4.1b.Case15   |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   15
                       STAAD.PRO CODE CHECKING - AISC 360-16 ASD (V1.2)
                     ****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL TENSION                            |
|-----------------------------------------------------------------------------|
| TENSILE YIELDING                                                            |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       590.9       0.000     Cl.D2          1      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom. Ten. Yld Cap         : Pn     =  986.76     kip        Eq.D2-1        |
|-----------------------------------------------------------------------------|
| TENSILE RUPTURE                                                             |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              0.000       592.1       0.000     Cl.D2          1      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective area            : Ae     =  19.735     in2        Eq.D3-1        |
|  Nom. Ten. Rpt Cap         : Pn     =  1184.1     kip        Eq.D2-2        |
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL COMPRESSION                        |
|-----------------------------------------------------------------------------|
| FLEXURAL BUCKLING X                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              24.70       560.9       0.044     Cl.E3          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective Slenderness     : Lcx/rx =  7.8750                Cl.E2          |
|  Elastic Buckling Stress   : Fex    =  4615.2     ksi        Eq.E3-4        |
|  Crit. Buckling Stress     : Fcrx   =  49.774     ksi        Eq.E3-2        |
|  Nom. Flexural Buckling    : Pnx    =  936.63     kip        Eq.E7-1        |
|-----------------------------------------------------------------------------|
| FLEXURAL BUCKLING Y                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              24.70       531.5       0.046     Cl.E3          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Effective Slenderness     : Lcy/ry =  29.685                Cl.E2          |
|  Elastic Buckling Stress   : Fey    =  324.80     ksi        Eq.E3-4        |
|  Crit. Buckling Stress     : Fcry   =  46.880     ksi        Eq.E3-2        |
|  Nom. Flexural Buckling    : Pny    =  887.57     kip        Eq.E7-1        |
|-----------------------------------------------------------------------------|
| FLEXURAL-TORSIONAL-BUCKLING                                                 |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              24.70       531.5       0.046     Cl.E4          2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Constant                  : H      =  1.0000                Eq.E4-8        |
|  Euler Buckling Stress     : Fez    =  389.10     ksi        Eq.E4-7        |
|  Elastic F-T-B Stress      : Fe     =  324.80     ksi        Eq.E4-3        |
|  Crit. F-T-B Stress        : Fcr    =  46.880     ksi        Eq.E3-2        |
|  Nom. Flex-tor Buckling    : Pn     =  887.57     kip        Eq.E7-1        |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   16
                       STAAD.PRO CODE CHECKING - AISC 360-16 ASD (V1.2)
                     ****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR SHEAR                                    |
|-----------------------------------------------------------------------------|
| SHEAR ALONG X                                                               |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             0.1766       249.0       0.001     Cl.G1          4      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    =  415.80     kip        Eq.G6-1        |
|-----------------------------------------------------------------------------|
| SHEAR ALONG Y                                                               |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              50.02       116.4       0.430     Cl.G1          3      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    =  194.40     kip        Eq.G2-1        |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   17
                       STAAD.PRO CODE CHECKING - AISC 360-16 ASD (V1.2)
                     ****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR BENDING                                  |
|-----------------------------------------------------------------------------|
| FLEXURAL YIELDING (Y)                                                       |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -5.298       858.3       0.006     Cl.F6.1        4      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom Flex Yielding Along Y : Mny    =  1433.3     kip-in     Eq.F6-1        |
|-----------------------------------------------------------------------------|
| LAT TOR BUCK ABOUT X                                                        |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|              654.5       4180.       0.157     Cl.F4.2        4      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom L-T-B Cap             : Mnx    =  6980.3     kip-in     Eq.F4-2        |
|  Mom. Distr. factor        : CbX    =  1.0000                Custom         |
|  Limiting Unbraced Length  : LpX    =  53.119     in         Eq.F4-7        |
|  Limiting Unbraced Length  : LrX    =  235.38     in         Eq.F4-8        |
|  Aw Ratio                  : AwX    = 0.99518                Eq.F4-12       |
|  Nom. Compr. Flange Stress : FLX    =  35.000     ksi        Eq.F4-6a       |
|  Compr. Flange Yield Mom.  : MycX   =  5999.4     kip-in     Eq.F4-4        |
|  Web Plastification Factor : RpcX   =  1.1817                Eq.F4-9a       |
|-----------------------------------------------------------------------------|
| COM FLANGE YIELDING                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -677.6       4245.       0.160     Cl.F4.1        2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom comp. Flange Yld      : Mny    =  7089.4     kip-in     Eq.F4-1        |
|  Compr Flange Yld Mom.     : Myc    =  6777.0     kip-in     Eq.F4-4        |
|  Web Plastification Factor : Rpc    =  1.0461                Eq.F4-9a       |
|-----------------------------------------------------------------------------|
| TEN FLANGE YIELDING                                                         |
|              DEMAND      CAPACITY    RATIO     REFERENCE     L/C    LOC     |
|             -677.6       4245.       0.160     Cl.F4.4        2      0.00   |
|                                                                             |
| Intermediate Results :                                                      |
|  Nom Tens. Flange Yielding : Mny    =  7089.4     kip-in     Eq.F4-15       |
|  Tens. Flange Yield Moment : Myt    =  5999.4     kip-in     Eq.F4-4        |
|  Web Plastification Factor : Rpt    =  1.1817                Eq.F4-16a      |
|-----------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.   18
                       STAAD.PRO CODE CHECKING - AISC 360-16 ASD (V1.2)
                     ****************************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted).
                                                       - Member :     2 Contd.
|-----------------------------------------------------------------------------|
|                         CHECKS FOR AXIAL BEND INTERACTION                   |
|-----------------------------------------------------------------------------|
| COMBINED FORCES CLAUSE H1                                                   |
|                            RATIO      CRITERIA           L/C      LOC       |
|                            0.186      Eq.H1-1b            2        0.00     |
|                                                                             |
| Intermediate Results :                                                      |
|  Axial Capacity            : Pc     =  531.48     kip        Cl.H1.1        |
|  Moment Capacity           : Mcx    =  4245.1     kip-in     Cl.H1.1        |
|  Moment Capacity           : Mcy    =  858.28     kip-in     Cl.H1.1        |
|-----------------------------------------------------------------------------|
 ***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.
   AISC Spec.      STAAD.Pro      Description
  ------------    -----------     -------------
       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: -
   AISC Spec.      STAAD.Pro      Description
  ------------    -----------     -------------
       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