V. SNiP SP16 2017 - CLASS 2 UPT I section

Check the capacity of a user-provided table I section subject to biaxial moment per the SP 16.13330.2017 code.

Details

A 6m, simply supported beam uses a welded, built-up I section. The section is defined as Class 2 (elasto-plastic). The flanges are 300 mm × 23 mm; the total section depth is 490 mm; the web thickness is 12 mm. The steel used has a modulus of elasticity of 206,000 MPa and a R_yn = 235 MPa. γ_m = 1.025. γ_c = 1.0

The member is subjected to uniformly distributed loads of 150 kN/m in the Y direction and 30 kN/m in the Z direction.

Validation

R_y = R_yn/ γ_m = 229.3 MPa

R_s = 0.58×R_y/ γ_m = 133.0 MPa

Section Properties

I_{x} = \frac{12 {(490 - 2 \times 23)}^{3}}{12} + 2 [\frac{300 {(23)}^{3}}{12} + 300 (23) {(\frac{490}{2} - \frac{23}{2})}^{2}] = 84, 054 {cm}^{4}

I_{y} = \frac{(490 - 2 \times 23) 12^{3}}{12} + 2 [\frac{23 {(300)}^{3}}{12}] = 10, 356 {cm}^{4}

W_{x} = \frac{84, 054 (2)}{49} = 3, 431 {cm}^{3}

W_{x} = \frac{10, 356 (2)}{30} = 690.4 {cm}^{3}

Check for Flexure

Ratio of area of the flange to the area of the web:

\frac{A_{f}}{A_{w}} = \frac{300 \times 23}{(490 - 2 \times 23) 12} = 1.295

Need to satisfy the following equation from Cl. 8.2.3:

\frac{M_{x}}{c_{x} β W_{x n, \min} R_{y} γ_{c}} + \frac{M_{y}}{c_{y} β W_{y n, \min} R_{y} γ_{c}} \leq 1

(Eq. 51)

where

M_x	=	150 (6)² / 8 = 675 kN·m
M_y	=	30 (6)² / 8 = 135 kN·m
β	=	1 per Eq. 52, as Q_x = Q_y = 0, and therefore τ_x = τ_y = 0
c_x	=	1.0611 (from Table E.1)
c_y	=	1.47 > 1.15, thus use 1.15

Thus, the ratio is $\frac{675}{1.061 \times 1 \times 3, 431 {\times (10)}^{- 3} \times 229.3 \times 1} + \frac{135}{1.15 \times 1 \times 690.4 {\times (10)}^{- 3} \times 229.3 \times 1} = 1.55 > 1$

Check for Stability

ϕ_{x} = \frac{M_{x}}{W_{c} \times γ_{c}} = \frac{675}{3, 431 \times {(10)}^{- 3} \times 1} = 196.7 MPa

{\bar{λ}}_{b} = \frac{l_{e f}}{b} \sqrt{\frac{R_{y}}{E}} = \frac{6}{0.3} \sqrt{\frac{229.3}{206, 000}} = 0.667

{\bar{λ}}_{u b} = [0.35 + 0.0032 \frac{b}{t} + (0.76 - 0.02 \frac{b}{t}) \frac{b}{h}] \times δ \sqrt{\frac{R_{y}}{ϕ_{x}}}

(Eq. 76)

Per Cl. 8.4.6, the value ${\bar{λ}}_{u b}$ is multiplied by δ.

where

δ	=	$1 - 0.6 (c_{1 x} - 1) / (c_{x} - 1)$
c_1x	=	is the larger of $\frac{M_{x}}{W_{x n} R_{y} γ_{c}} = \frac{675}{3, 431 \times {(10)}^{- 3} 229.3 \times 1} = 0.858, or β c_{x} = 1.061$ , so =1.061

δ = 1 - 0.6 \frac{1}{1} = 0.4

{\bar{λ}}_{u b} = [0.35 + 0.0032 \frac{300}{23} + (0.76 - 0.02 \frac{300}{23}) \frac{300}{490 - 23}] \times 0.4 \sqrt{\frac{229.3}{196.7}} = 0.308 < {\bar{λ}}_{b}

(Eq. 73)

So, the stability of the beam is not ensured per Cl. 8.4.4.b.

Check for Deflection

f_{y} = \frac{5}{384} \frac{q l^{4}}{E I_{x}} = \frac{5}{384} \frac{150 \times 6^{4}}{206 {(10)}^{6} 84, 054 {(10)}^{- 8}} = 0.0237 m

f_{z} = \frac{5}{384} \frac{q l^{4}}{E I_{x}} = \frac{5}{384} \frac{30 \times 6^{4}}{206 {(10)}^{6} 10, 356 {(10)}^{- 8}} = 0.0146 m

f_{\max} = \sqrt{{(0.0237)}^{2} + {(0.0146)}^{2}} = 0.0279 m

The maximum member deflection is limited to l / 200 = 0.03 m

Thus, the ratio is 0.0279 / 0.03 = 0.93

Results

Result Type	Reference	STAAD.Pro	Difference
Ratio for flexure (Eq. 51)	1.55	1.55	none
Deflection (m)	0.0279	0.02785	negligible
Deflection ratio	0.93	0.93	none

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Russia\SNiP SP16 2017 - CLASS 2 UPT I section with biaxial moment.std is typically installed with the program.

STAAD SPACE
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 6 0 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
TABLE 1
UNIT METER KN
WIDE FLANGE
I_500
0.019128 0.49 0.012 0.3 0.023 0.000840544 0.000103564 2.68914e-06 -
0.00588 0.0092
END
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
TYPE STEEL
STRENGTH FY 253200 FU 407800 RY 1.5 RT 1.2
END DEFINE MATERIAL
****************************************
MEMBER PROPERTY
1 UPTABLE 1 I_500
****************************************
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 -150
1 UNI GZ 30
PERFORM ANALYSIS
********************************
PARAMETER 1
CODE RUSSIAN
TRACK 2 ALL
DFF 200 ALL
TB 2 ALL
STP 2 ALL
SGR 9 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - (SP 16.13330.2017)   V1.0
                       ********************************************
   ALL UNITS ARE - KN METRE
   ========================================================================
   MEMBER     CROSS          RESULT/   CRITICAL COND/    RATIO/    LOADING/
              SECTION NO.      N             Mx            My      LOCATION
   ========================================================================
*     1  B I    I_500         FAIL      SP cl.8.2.3      1.55         1
                          0.000E+00      6.750E+02    1.350E+02   3.000E+00
      1  B I    I_500         PASS         DISPL         0.93         1
                          0.000E+00      6.750E+02    1.350E+02   3.000E+00
   MATERIAL DATA
      Steel                         = C255       SP16.13330
      Modulus of elasticity         = 206.E+06 kPa
      Design Strength (Ry)          = 229.E+03 kPa
   SECTION PROPERTIES (units - m, m^2, m^3, m^4)
      Member Length                 = 6.00E+00
      Gross Area                    = 1.91E-02
      Net Area                      = 1.91E-02
                                         x-axis      y-axis
      Moment of inertia (I)         :   841.E-06    104.E-06
      Section modulus (W)           :   343.E-05    690.E-06
      First moment of area (S)      :   191.E-05    525.E-06
      Radius of gyration (i)        :   210.E-03    736.E-04
      Effective Length              :   6.00E+00    6.00E+00
      Slenderness                   :   0.00E+00    0.00E+00
   DESIGN DATA (units -kN,m) SP16.13330.2017
      Axial force                   :   0.000E+00
                                         x-axis      y-axis
      Moments                       :   675.0E+00    135.0E+00
      Shear force                   :   0.000E+00    0.000E+00
      Bi-moment                     :   0.000E+00 Value of Bi-moment not being entered!!!
      Stress-strain state checked as:   Class    2
   CRITICAL CONDITIONS FOR EACH CLAUSE CHECK
      F.(51)  Mx/(Cx*beta*Wxn,min*Ry*GammaC)+My/(Cy*Wyn,min*Ry*GammaC)=
               675.0E+00/( 1.06E+00* 1.00E+00* 3.43E-03* 229.3E+03* 1.00E+00)+
               135.0E+00/( 1.15E+00* 6.90E-04* 229.3E+03* 1.00E+00)=
               1.55E+00>1 
              TAUx=Qy/Aw= 0.000E+00/ 532.8E-05= 0.00E+00 &lt;= 0,9*RS= 119.7E+03
              TAUy=Qx/Af= 0.000E+00/ 138.0E-04= 0.00E+00 &lt;= 0,5*RS= 664.9E+02
      LAMBDA_b=(Lef/b)*SQRT(Ry/E)=
              ( 600.0E-02/( 3.000E-01))*SQRT( 229.3E+03/ 206.0E+06)= 6.672E-01
      SIGMA_x=Mx/(Wc*GammaC)= 675.0E+00/( 343.1E-05* 100.0E-02)= 1.967E+05 kPa
      LAMBDA_ub=(0.35+0.0032*b/t+(0.76-0.02*b/t)*b/h)*delta*SQRT(Ry/SIGMA_x)=
              =(0.35+0.0032* 1.304E+01+(0.76-0.02* 1.304E+01)* 6.424E-01)* 4.000E-01* 1.079E+00
              = 3.076E-01&lt; LAMBDA_b= 6.672E-01
      ***WARNING: Stability of the beam is not ensured according to cl. 8.4.4 b) 
          LIMIT SPAN/DEFLECTION (DFF) =    200.00   (DEFLECTION LIMIT=      0.030 M)
          SPAN/DEFLECTION = 215.4E+00 (DEFLECTION=  2.785E-02M)
          LOAD=    1     RATIO=    0.928     LOCATION=    3.000