V. SNiP SP16 2017 - I section with biaxial moment

Design an I section subjected to uniform distributed loads in both major in minor axes per the SP 16.13330.2017 code.

Details

A 6m long, simply supported beam has a European HE500A section. The beam is subjected to a uniform distributed load of 132.2 kN/m in the Y direction and 30 kN/m in the Z direction. The steel used has a modulus of elasticity of 206,000 MPa and a R_yn = 235 MPa. γ_m = 1.05, γ_c = 1

Validation

R_y = R_yn/ γ_m = 223.8 MPa

R_s = 0.58×R_y/ γ_m = 129.8 MPa

Check for Flexure

Need to satisfy the following equation from Cl. 8.2.1:

\frac{M_{x}}{I_{x n} R_{y} γ_{c}} y \pm \frac{M_{y}}{I_{y n} R_{y} γ_{c}} s \pm \frac{B ω}{I_{ω n} R_{y} γ_{c}} \leq 1

(Eq. 43)

where

M_x	=	132.2 (6)² / 8 = 594.9 kN·m
M_y	=	30 (6)² / 8 = 135 kN·m
B_x	=	0

Thus, the ratio is $\frac{M_{x}}{I_{x n} R_{y} γ_{c}} y \pm \frac{M_{y}}{I_{y n} R_{y} γ_{c}} s \pm \frac{B ω}{I_{ω n} R_{y} γ_{c}} = 1.62 > 1$

Check for Shear

Need to satisfy the following equation:

\frac{Q S}{I t_{w} R_{s} γ_{c}} \leq 1

(Eq. 42)

where

0 kN

Thus, the ratio is 0.0 < 1

Check for Combined Flexure & Shear

Need to satisfy the following equation:

\frac{0.87}{R_{y} γ_{c}} \sqrt{{σ_{x}}^{2} - σ_{x} σ_{y} + {σ_{y}}^{2} + 3 {τ_{x y}}^{2}} \leq 1

(Eq. 44)

where

σ_x	=	M_x / W_x = 594.9 (10)³ / 3,550 = 167.6 MPa
σ_y	=	M_y / W_y = 135 (10)³ / 691.1 = 195.3 MPa
τ_xy	=	0 MPa

Thus, the ratio is $\frac{0.87}{223.8 \times 1.} \sqrt{{(167.6)}^{2} - 167.6 \times 195.3 + {(195.3)}^{2}} = 0.71 < 1$

Check for Stability

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

ϕ_{x} = \frac{M_{x}}{W_{c} \times γ_{c}} = \frac{594.9}{3,550 \times {(10)}^{-3} \times 1} = 167.6 MPa

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

{\bar{λ}}_{u b} = [0.35 + 0.0032 \frac{300}{23} + (0.76 - 0.02 \frac{300}{23}) \frac{300}{490}] \sqrt{\frac{235}{167.6}} = 0.823

{\bar{λ}}_{b} = \frac{l_{e f}}{b} \sqrt{\frac{R_{y}}{E}} \frac{6}{0.3} \sqrt{\frac{235}{206,000}} = 0.659 < {\bar{λ}}_{u b}

So, the stability of the beam is ensured per Cl. 8.4.4.b. Check per Cl 8.4.1 is not required.

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} 86,970 {(10)}^{- 8}} = 0.0124 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}} = 0.0237 m

f_{\max} = \sqrt{{(0.0124)}^{2} + {(0.0237)}^{2}} = 0.0267 m

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

Thus, the ratio is 0.0267 / 0.03 = 0.89

Result Type	Reference	STAAD.Pro	Difference
Ratio of Flexure (Eq. 43)	1.62	1.62	none
Ratio of Shear (Eq. 42)	0	0	none
Ratio of Combined Shear & Flexure (Eq. 44)	0.71	0.71	none
Deflection (m)	0.0267	0.02675	negligible
Deflection Ratio	0.89	0.89	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 - 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;
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 EUROPEAN
1 TABLE ST HE500A
****************************************
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
*2 FIXED BUT FX MZ
2 FIXED BUT FX MY MZ
****************************************
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -132.2
1 UNI GZ 30
********************************
PERFORM ANALYSIS
***********************
PARAMETER 1
CODE RUSSIAN
ENSGR 1 ALL
GAMM 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  I      HE500A       FAIL     SP cl.8.2.1(43)    1.62         1
                          0.000E+00      5.949E+02    1.350E+02   3.000E+00
      1  I      HE500A       PASS     SP cl.8.2.1(42)    0.00         1
                          0.000E+00      5.949E+02    1.350E+02   3.000E+00
      1  I      HE500A       PASS     SP cl.8.2.1(44)    0.71         1
                          0.000E+00      5.949E+02    1.350E+02   3.000E+00
      1  I      HE500A        PASS         DISPL         0.89         1
                          0.000E+00      5.949E+02    1.350E+02   3.000E+00
   MATERIAL DATA
      Steel                         = S235       EN10025-2 
      Modulus of elasticity         = 206.E+06 kPa
      Design Strength (Ry)          = 224.E+03 kPa
   SECTION PROPERTIES (units - m, m^2, m^3, m^4)
      Member Length                 = 6.00E+00
      Gross Area                    = 1.98E-02
      Net Area                      = 1.98E-02
                                         x-axis      y-axis
      Moment of inertia (I)         :   870.E-06    104.E-06
      Section modulus (W)           :   355.E-05    691.E-06
      First moment of area (S)      :   197.E-05    530.E-06
      Radius of gyration (i)        :   210.E-03    724.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                       :   594.9E+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    1
   CRITICAL CONDITIONS FOR EACH CLAUSE CHECK
      F.(43)  (Mx*y)/(Ixn*Ry*GammaC)+(My*x)/(Iyn*Ry*GammaC)+
              +(B*w)/(Iwn*Ry*GammaC)=
              ( 594.9E+00* 2.45E-01)/( 8.70E-04* 223.8E+03* 1.00E+00)+
              ( 135.0E+00* 1.50E-01/( 1.04E-04* 223.8E+03* 1.00E+00)+
              ( 0.000E+00* 2.50E-01)/( 5.64E-06* 223.8E+03* 1.00E+00)=
              = 1.62E+00>1 
      F.(44)  0.87/(Ry*GammaC)*SQRT(SIGMx^2-SIGMx*SIGMy+SIGMy^2+3*TAUxy^2)=
              0.87/( 223.8E+03* 1.00E+00)*SQRT(-167.6E+03^2--167.6E+03
              *-195.3E+03+-195.3E+03^2+3* 0.000E+00^2)= 7.11E-01=&lt;1
              TAUxy/(Rs*GammaC)= 0.000E+00/( 129.8E+03* 1.00E+00)= 0.00E+00=&lt;1
      LAMBDA_b=(Lef/b)*SQRT(Ry/E)=
              ( 600.0E-02/( 3.000E-01))*SQRT( 223.8E+03/ 206.0E+06)= 6.592E-01
      SIGMA_x=Mx/(Wc*GammaC)= 594.9E+00/( 355.0E-05* 100.0E-02)= 1.676E+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)* 1.000E+00* 1.156E+00
              = 8.232E-01>=LAMBDA_b= 6.592E-01
      Stability of the beam is ensured according to cl. 8.4.4 b)
      Check according to cl. 8.4.1 is not required
          LIMIT SPAN/DEFLECTION (DFF) =    200.00   (DEFLECTION LIMIT=      0.030 M)
          SPAN/DEFLECTION = 224.3E+00 (DEFLECTION=  2.675E-02M)
          LOAD=    1     RATIO=    0.892     LOCATION=    3.000