V. SNiP SP16 2017 - CLASS 2 Rolled I Section with Bi-Moment

Verify a rolled I section subjected to uniform distributed loads per the SP 16.13330.2017 code.

Details

A 6m long, simply supported beam has a UPT HE500A section. The beam is subjected to a uniform distributed load of 132.2 kN/m in the Y direction and a bi-moment of 10 kN·. 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

Section Properties

D = 490 mm

B = 300 mm

t_f = 23 mm

t_w = 12 mm

I_x = 84,054 cm⁴

I_y = 10,356 cm⁴

W_x = 3,431 cm³

W_y = 690.4 cm³

Validation

R_y = R_yn/ γ_m = 229.3 MPa

R_s = 0.58×R_y/ γ_m = 133.0 MPa

Check for Flexure

As this is a Class 2 section (elasto-plastic), it must satisfy equation 53 from Cl. 8.2.3:

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

(Eq. 53)

where

M_x	=	132.2 (6)² / 8 = 594.9 kN·m
β	=	1 per Eq. 52 (Cl. 8.2.3), as Q_x = Q_y = 0, and therefore τ_x = τ_y = 0
c_x	=	1.063 (from Table E.1); for A_f / A_w = 69 / 53.28 = 1.295
c_w	=	2.915 (from Table 10a)

Thus, the ratio is $\frac{594.9 {(10)}^{3}}{1.063 (3,431) (229.3) (1.0)} + \frac{10 {(10)}^{3}}{2.915 (690.4) (229.3) (1.0)} = 0.711 + 0.022 = 0.733$

Check for Stability

Check per Cl. 8.4.4. From Table 11 of SP 16.13330-2017:

σ_{x} = \frac{M_{x}}{W_{c} \times γ_{c}} = \frac{594.9}{3,431 \times {(10)}^{-3} \times 1} = 173.4 MPa

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

Per Cl. 8.4.6, this value is multiplied by δ as indicated above, where:

δ = 1 - 0.6 \frac{(c_{1x} - 1)}{(c_{x} - 1)}

(Eq. 76)

where

c_1x

max | \begin{matrix} M_{x} / (W_{xn} \times R_{y} \times γ_{c}) = \frac{594.9 {(10)}^{3}}{3,431 (229.3) (1.0)} = 0.756 \\ β c_{x} = 1.0 \times 1.063 = 1.063 \end{matrix}

Therefore, since c_1x = c_x, δ = 1 - 0.6 = 0.4

{\overline{λ}}_{u b} = [0.35 + 0.0032 \frac{300}{23} + (0.76 - 0.02 \frac{300}{23}) \frac{300}{490}] 0.4 \sqrt{\frac{229.3}{173.4}} = 0.328

{\overline{λ}}_{b} = \frac{l_{e f}}{b} \sqrt{\frac{R_{y}}{E}} \frac{6,000}{300} \sqrt{\frac{229.3}{206,000}} = 0.667 > {\bar{λ}}_{u b}

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

Results

Check for Deflection

f_{y} = \frac{5}{384} \frac{q l^{4}}{E I_{x}} = \frac{5}{384} \frac{132 \times 6^{4}}{206 {(10)}^{6} 84,054 {(10)}^{- 8}} = 0.0129 m

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

Thus, the ratio is 0.0129 / 0.03 = 0.429

Result Type	Reference	STAAD.Pro	Difference
Ratio of Flexure (Eq. 53)	0.733	0.73	negligible
C_x	1.063	1.06	negligible
C_w	2.915	2.92	negligible
σ_x (MPa)	173.4	173.4	none
${\overline{λ}}_{b}$	0.667	0.6672	negligible
${\overline{λ}}_{ub}$	0.328	0.3277	negligible
Deflection (m)	0.0129	0.01287	negligible
Deflection Ratio	0.429	0.429	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 Rolled I Section with Bi-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
UTP-HE-500-A
0.019128 0.49 0.012 0.3 0.023 0.000840544 0.000103564 2.68914e-06 -
0.00588 0.0092 0.3 0.023
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 RUSSIAN
1 UPTABLE 1 UTP-HE-500-A
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -132.2
PERFORM ANALYSIS
PARAMETER 1
CODE RUSSIAN
BMT 10 ALL
GAMM 1 ALL
SGR 9 ALL
GAMC2 1 ALL
GAMC1 1 ALL
GAMM 1 ALL
STP 2 ALL
TB 2 ALL
DFF 200 ALL
TRACK 2 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    UTP-HE-500-A  PASS      SP cl.8.2.3      0.73         1
                          0.000E+00      5.949E+02    0.000E+00   3.000E+00
      1  B I    UTP-HE-500-A  PASS         DISPL         0.43         1
                          0.000E+00      5.949E+02    0.000E+00   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
      Sectorial moment of inertia (Iw)   :    564.E-08 [m^6]
      Sectorial section modulus (Ww)     :    226.E-07 [m^4]
      Sectorial area (coordinate) (w)    :    250.E-03 [m^2]
   DESIGN DATA (units -kN,m) SP16.13330.2017
      Axial force                   :   0.000E+00
                                         x-axis      y-axis
      Moments                       :   594.9E+00    0.000E+00
      Shear force                   :   0.000E+00    0.000E+00
      Bi-moment                     :   100.0E-01 [kNm^2]
      Stress-strain state checked as: Class    2
   CRITICAL CONDITIONS FOR EACH CLAUSE CHECK
      F.(53)  Mx/(Cx*beta*Wxn,min*Ry*GammaC)+B/(Cw*Wyn,min*Ry*GammaC)=
               594.9E+00/( 1.06E+00* 1.00E+00* 3.43E-03* 229.3E+03* 1.00E+00)+
               100.0E-01/( 2.92E+00* 6.90E-04* 229.3E+03* 1.00E+00)=
               7.33E-01=&lt;1
              TAUx=Qy/Aw= 0.000E+00/ 532.8E-05= 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)= 594.9E+00/( 343.1E-05* 100.0E-02)= 1.734E+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* 1.150E+00
              = 3.277E-01&amp;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 = 466.0E+00 (DEFLECTION=  1.287E-02M)
          LOAD=    1     RATIO=    0.429     LOCATION=    3.000