V. AISC N690 1984 Pipe Section With SLC

Verify the design of a pipe section used for a beam member per the AISC N690 1984 code.

References

Gere, J. M. and S. P. Timoshenko. 1984. Mechanics of Materials, 2nd Edition. Boston, MA: PWS Publishers. p.238.

Details

The 10 ft long span is a propped cantilever which is free to move axially at the fixed end. The beam is subject to the following four different load cases:

a 10 kip axial tension force,
a 10 kip axial compression force,
a uniformly distributed load of 0.3 kip/ft in the Global Y and Z directions, the member selfweight (acts in Global Y), and the tension load from case 1
a uniformly distributed load of 0.3 kip/ft in the Global Y and Z directions, the member selfweight (acts in Global Y), and the compression load from case 2

The member is Pipe 5-Sched. 40 (standard) pipe section with Grade A36 steel. The self-weight is 14.6 lbs/ft.

Validation

Slenderness Ratio

The slenderness ratio is the same about either local z or y axes: KL/r = 120 in / 1.888 in = 63.56 < 300, OK

Axial Tension

From section Q1.5.1.1, the allowable tensile stress:

F_{t} = {}_{min}{| \begin{matrix} 0.6 F_{y} = 0.6 \times 36 = 21.6 ksi \\ 0.5 F_{u} = 0.5 \times 58 = 29 ksi \end{matrix}}

The actual tensile stress from load case 1:

f_{t} = \frac{P}{A} = \frac{10}{4.01} = 2.49 ksi < F_{t}

Stress ratio = 2.49 / 21.6 = 0.115, hence OK.

Axial Compression

Allowable compressive stress per Cl Q1.9.2.3:

\frac{OD}{t} = \frac{5.56}{0.24} = 23.17 < \frac{3,300}{F_{y}} = \frac{3,300}{36} = 91.67

C_{c} = \sqrt{\frac{2 π^{2} E}{F_{y}}} = 126.1

Since Kl/r < C_c, per eq. Q1.5-1, the allowable axial compressive stress:

F_{a} = \frac{1 - \frac{{(\frac{k l}{r})}^{2}}{2 \times C_{c}^{2}}}{\frac{5}{3} + 3 \times \frac{\frac{k l}{r}}{8 \times C_{c}} - \frac{{(\frac{k l}{r})}^{3}}{8 \times C_{c}^{3}}} F_{y} = 17.08 ksi

The actual compressive stress from load case 2:

f_{a} = \frac{P}{A} = \frac{10}{4.01} = 2.49 ksi < F_{a}

Stress ratio = 2.49 / 17.08 = 0.146, hence OK.

Bending

Section is compact per Cl Q1.5.1.4.1: $\frac{O D}{t} = 23.17 < \frac{3,300}{F_{y}} = 91.67$

Allowable bending stress about both axes: F_b = 0.66×F_y = 23.76 ksi

Moment due to load case 3 or 4 on a propped cantilever (ref. AISC Steel Construction Manual):

M_{z} = \frac{w \times l^{2}}{8} = \frac{0.314 \times 10^{2}}{8} = 3.93 ft·k = 47.2 in·k

M_{y} = \frac{w \times l^{2}}{8} = \frac{0.3 \times 10^{2}}{8} = 3.75 ft·k = 45 in·k

Actual bending stress about the major axis: $f_{bz} = \frac{M_{z}}{I_{z}} (\frac{O D}{2}) = \frac{47.2}{14.3} (\frac{5.56}{2}) = 9.17 ksi$

Actual bending stress about the minor axis:

f_{by} = \frac{M_{z}}{I_{z}} (\frac{O D}{2}) = \frac{45.0}{14.3} (\frac{5.56}{2}) = 8.75 ksi

Stress ratio: 9.17 / 23.76 = 0.386, hence OK.

Combined Bending and Axial Tension

Interaction per Cl Q1.6.2 in load case 3 (tension + bending):

\frac{f_{a}}{SFT \times 0.6 F_{y}} + \frac{f_{b y}}{SMY \times F_{b y}} + \frac{f_{b z}}{SMZ \times F_{b z}} = \frac{2.49}{1.3 \times 21.6} + \frac{8.75}{1.4 \times 23.76} + \frac{9.17}{1.4 \times 23.76} = 0.627

(Eq. Q1.6-1b)

Combined Bending and Axial Compression

Interaction per Cl Q1.6 in load case 4 (compression + bending):

\frac{f_{a}}{F_{a}} = 0.145 < 0.15

(Eq. Q1.6-2)

\frac{f_{a}}{SFC \times F_{a}} + \frac{f_{b y}}{SMY \times F_{b y}} + \frac{f_{b z}}{SMZ \times F_{b z}} = \frac{2.49}{1.3 \times 17.08} + \frac{8.75}{1.4 \times 23.76} + \frac{9.17}{1.4 \times 23.76} = 0.651

Shear

Actual shear (at supports):

f_{v,y} = \frac{5 w \times l}{8} = \frac{5 \times 0.315 \times 10}{8} = 1.97 kip

f_{v,z} = \frac{5 w \times l}{8} = \frac{5 \times 0.3 \times 10}{8} = 1.88 kip

Shear area (calculated as per Gere & Timoshenko reference):

A_{z} = \frac{\frac{π}{4} ({r_{2}}^{4} - {r_{1}}^{4}) \times 2 (r_{2} - r_{1})}{\frac{2}{3} ({r_{2}}^{3} - {r_{2}}^{3})}

where

$r_{1}$	=	the inner radius of the pipe = 5.56 / 2 - 0.24 = 2.54 in
$r_{2}$	=	the outer radius of the pipe = 5.56 / 2 = 2.78 in

Thus, the shear area along both axes = 2.008 in².

F_{vy} = \frac{F_{v,y}}{A_{y}} = \frac{1.97}{2.008} = 0.98 ksi

F_{vz} = \frac{F_{v,z}}{A_{y}} = \frac{1.88}{2.008} = 0.94 ksi

The allowable shear stress:

F_{v} = 0.4 F_{y} = 14.4 ksi

Stress ratio in Y (minor axis): 0.98/14.4 = 0.068, hence OK.

Stress ratio in Z (major axis): 0.94/14.4 = 0.065, hence OK.

Results

Table 1. Comparison of results
Parameter	Hand Calculation	STAAD.Pro	Difference
Slenderness ratio	63.56	63.55	negligible
Actual axial tensile stress (ksi)	2.49	2.49	none
Allowable axial tensile stress	21.6	21.6	none
Tensile stress ratio (LC1)	0.115	0.115	none
Actual axial compressive stress (ksi)	2.49	2.49	none
Allowable axial compressive stress	17.08	17.1	negligible
Compressive stress ratio (LC2)	0.146	0.146	none
Allowable bending stress (ksi)	23.76	23.8	negligible
Actual bending stress (ksi)	9.17	9.15	negligible
Combined axial tension and bending stress ratio (LC3)	0.627	0.625	negligible
Combined axial compression and bending stress ratio (LC4)	0.651	0.649	negligible
Allowable shear stress (ksi)	14.4	14.4	none
Actual shear stress (ksi)	0.98	0.97	negligible
Shear stress ratio in major axis	0.065	0.064	negligible
Shear stress ratio in minor axis	0.068	0.067	negligible

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\US\AISC\AISC N690 1984 Pipe Section With SLC.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 05-Apr-19
END JOB INFORMATION
INPUT WIDTH 79
SET SHEAR
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 10 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 4.176e+006
POISSON 0.3
DENSITY 0.489024
ALPHA 1.2e-005
DAMP 0.03
END DEFINE MATERIAL
UNIT INCHES KIP
MEMBER PROPERTY AMERICAN
1 TABLE ST PIPS50
UNIT FEET KIP
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX
LOAD 1 TENSION ONLY
JOINT LOAD
2 FX 10
LOAD 2 COMPRESSION ONLY
JOINT LOAD
2 FX -10
LOAD 3 TENSION+BENDING
SELFWEIGHT Y -1
MEMBER LOAD
1 UNI GY -0.3
JOINT LOAD
2 FX 10
MEMBER LOAD
1 UNI GZ 0.3
LOAD 4 COMPRESSION+BENDING
SELFWEIGHT Y -1
MEMBER LOAD
1 UNI GY -0.3
JOINT LOAD
2 FX -10
MEMBER LOAD
1 UNI GZ 0.3
PERFORM ANALYSIS PRINT LOAD DATA
PRINT MEMBER PROPERTIES ALL
UNIT INCHES KIP
PARAMETER 1
CODE AISC N690 1984
CB 1 ALL
CMY 1 ALL
CMZ 1 ALL
FYLD 36 ALL
FU 58 ALL
KY 1 ALL
KZ 1 ALL
MAIN 200 ALL
NSF 1 ALL
RATIO 1 ALL
SMZ 1.4 ALL
SMY 1.4 ALL
SFT 1.3 ALL
SFC 1.3 ALL
TMAIN 300 ALL
TRACK 2 ALL
UNB 30 ALL
UNT 30 ALL
CT 0.75 ALL
CHECK CODE ALL
FINISH

STAAD Output

      STAAD SPACE                                              -- PAGE NO.    6
                       STAAD.PRO CODE CHECKING - (  AISC N690 1984)   v1.0
                       ********************************************
 ALL UNITS ARE - KIP  INCH (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
      1  ST   PIPS50                   (AISC SECTIONS)
                           PASS      Q1.6-Eqn 2        0.651         4
                       10.00 C         45.00          47.04      120.00
 |-----------------------------------------------------------------------------|
 | SLENDERNESS CHECK:    ACTUAL RATIO:  63.55 ALLOWABLE RATIO: 200.00          |
 | ALLOWABLE STRESSES:   (UNIT - KIP  INCH)                                    |
 | AXIAL: 1.71E+01  FCZ: 2.38E+01  FCY: 2.38E+01  FTZ: 2.38E+01  FTY: 2.38E+01 |
 | SHEAR: 1.44E+01                                                             |
 | ACTUAL STRESSES: (UNIT - KIP  INCH)                                         |
 | AXIAL: 2.49E+00  FBZ: 9.15E+00  FBY: 8.75E+00 SHEAR: 9.72E-01               |
 |-----------------------------------------------------------------------------|
 | SECTION PROPERTIES: (UNIT - INCH)                                           |
 | AXX:   4.01  AYY:   2.02  AZZ:   2.02  RZZ:  1.89  RYY:  1.89               |
 | SZZ:       5.14  SYY:       5.14                                            |
 |-----------------------------------------------------------------------------|
 | PARAMETER: (UNIT - KIP  INCH)                                               |
 | KL/R-Z: 63.55   KL/R-Y: 63.55   UNL:   30.0   CMZ:  1.00   CMY:  1.00       |
 | CB: 1.00   FYLD:    36.00   FU:    58.00   NET SECTION FACTOR: 1.00         |
 | CT:  0.75   STEEL TYPE:  0.0                                                |
 |-----------------------------------------------------------------------------|
 | CRITICAL LOADS FOR EACH CLAUSE CHECK (UNITS KIP -INCH)                      |
 | CLAUSE       RATIO  LOAD       FX        VY        VZ        MZ        MY   |
 | TENSION      0.115    1    1.00E+01       -         -         -         -   |
 | COMPRESSION  0.146    2    1.00E+01       -         -         -         -   |
 | COMP&BEND    0.651    4    1.00E+01       -         -    4.70E+01 4.50E+01  |
 | TEN&BEND     0.627    3    1.00E+01       -         -    4.70E+01 4.50E+01  |
 | SHEAR-Y      0.067    3        -     1.96E+00       -         -         -   |
 | SHEAR-Z      0.065    3        -          -    1.88E+00       -         -   |
 |-----------------------------------------------------------------------------|