V. SNiP SP16 2017 - Channel section with UDL

Design an channel section subjected to a uniform distributed load per the SP 16.13330.2017 code.

Details

A 5m long, simply supported beam has an American C15X50 section. The beam is subjected to a uniform distributed load of 100 kN/m in the Y direction. The steel used has a modulus of elasticity of 206,000 MPa and a R_yn = 235 MPa. γ_c1 = 1.1, γ_c2 = 1.1

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:

\frac{M}{W_{n, \min} R_{y} γ_{c}} \leq 1

(Eq. 41)

where

M	=	100 (5)² / 8 = 312.5 kN·m
W_n,min	=	W_x = 883 cm³

Thus, the ratio is $\frac{312.5}{883 {(10)}^{- 3} \times 235 \times 1.1} = 1.37 > 1$

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 / W_x = 312.5 (10)³ / 883 = 353.9 MPa
σ_y	=	0 MPa
τ_xy	=	0 MPa

Thus, the ratio is $\frac{0.87}{235 \times 1.1} \sqrt{{(353.9)}^{2}} = 1.19 > 1$

Check for Stability

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

{\bar{λ}}_{u b} = 0.35 + 0.0032 \frac{b}{t} + (0.76 - 0.02 \frac{b}{t}) \frac{b}{h}

(Eq. 73)

{\bar{λ}}_{u b} = 0.35 + 0.0032 \frac{94.49}{16.51} + (0.76 - 0.02 \frac{94.49}{16.51}) 0.2593 = 0.536

(Eq. 73)

{\bar{λ}}_{b} = \frac{l_{e f}}{b} \sqrt{\frac{R_{y}}{E}} = 1.787 > {\bar{λ}}_{u b}

So, the stability of the beam is not ensured per Cl. 8.4.4.b. Therefore a check per Cl 8.4.1 needs to be made.

Check for Lateral-Torsional Buckling

Need to satisfy the following equation:

\frac{M_{x}}{ϕ_{b} W_{c x} R_{y} γ_{c}} \leq 1

(Eq. 69)

α = 1.54 \frac{I_{t}}{I_{y}} {(\frac{l_{e f}}{h})}^{2} = 63.80

(Eq. G.4)

Since α > 40, from Table G.1:

ψ = 3.15 + 0.04 α - 2.7 {(10)}^{- 5} α^{2} = 5.592

ϕ_{1} = ψ \frac{I_{y}}{I_{x}} {(\frac{h}{l_{e f}})}^{2} \frac{E}{R_{y}} = 0.775

(Eq. G.4)

ϕ_{b} = 0.7 ϕ_{1} = 0.7 (0.775) = 0.543

Thus, the ratio is $\frac{312.5}{0.543 \times 883 {(10)}^{- 3} \times 235 \times 1.1} = 2.52 > 1$

Check for Deflection

f_{\max} = \frac{5}{384} \frac{q l^{4}}{E I_{x}} = \frac{5}{384} \frac{100 \times 5^{4}}{206 {(10)}^{6} 16, 816 {(10)}^{- 8}} = 0.0235 m

The maximum member deflection is limited to l / 200 = 0.025 m

Thus, the ratio is 0.0235 / 0.025 = 0.94

Results

Result Type	Reference	STAAD.Pro	Difference
Ratio of Flexure (Eq. 41)	1.37	1.37	none
Ratio of Shear (Eq. 42)	0	0	none
Ratio of LTB (Eq. 69)	2.52	2.52	none
Ratio of Combined Shear & Flexure (Eq. 44)	1.19	1.19	none
Deflection (m)	0.0235	0.02348	negligible
Deflection Ratio	0.94	0.94	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 - Channel section with UDL.std is typically installed with the program.

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 01-Sep-20
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 5 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 AMERICAN
1 TABLE ST C15X50
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 -100
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE RUSSIAN
CB 1 ALL
PY 235000 ALL
GAMC2 1.1 ALL
GAMC1 1.1 ALL
TB 1 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  [      C15X50       FAIL     SP cl.8.2.1(41)    1.37         1
                          0.000E+00      3.125E+02    0.000E+00   2.500E+00
      1  [      C15X50       PASS     SP cl.8.2.1(42)    0.00         1
                          0.000E+00      3.125E+02    0.000E+00   2.500E+00
*     1  [      C15X50       FAIL     SP cl.8.2.1(44)    1.19         1
                          0.000E+00      3.125E+02    0.000E+00   2.500E+00
*     1  [      C15X50        FAIL      SP cl.8.4.1      2.52         1
                          0.000E+00      3.125E+02    0.000E+00   2.500E+00
      1  [      C15X50        PASS         DISPL         0.94         1
                          0.000E+00      3.125E+02    0.000E+00   2.500E+00
   MATERIAL DATA
      Steel                         = User                 
      Modulus of elasticity         = 206.E+06 kPa
      Design Strength (Ry)          = 235.E+03 kPa
   SECTION PROPERTIES (units - m, m^2, m^3, m^4)
      Member Length                 = 5.00E+00
      Gross Area                    = 9.48E-03
      Net Area                      = 9.48E-03
                                         x-axis      y-axis
      Moment of inertia (I)         :   168.E-06    458.E-08
      Section modulus (W)           :   883.E-06    617.E-07
      First moment of area (S)      :   561.E-06    667.E-07
      Radius of gyration (i)        :   133.E-03    220.E-04
      Effective Length              :   5.00E+00    5.00E+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                       :   312.5E+00    0.000E+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.(41)  M/(Wn,min*Ry*GammaC)= 312.5E+00/( 8.83E-04* 235.0E+03* 1.10E+00= 1.37E+00>1 
      F.(44)  0.87/(Ry*GammaC)*SQRT(SIGMx^2+3*TAUxy^2)=
              0.87/( 235.0E+03* 1.10E+00)*SQRT(-354.0E+03^2+3* 0.000E+00^2)=
              1.19E+00>1 
              TAUxy/(Rs*GammaC)= 0.000E+00/( 136.3E+03* 1.10E+00)= 0.00E+00=&lt;1
      F.(69)  Mx/(FIb*Wcx*Ry*GammaC)=
              312.5E+00/( 5.43E-01* 8.83E-04* 235.0E+03* 1.10E+00)= 2.52E+00>1 
          LIMIT SPAN/DEFLECTION (DFF) =    200.00   (DEFLECTION LIMIT=      0.025 M)
          SPAN/DEFLECTION = 213.0E+00 (DEFLECTION=  2.348E-02M)
          LOAD=    1     RATIO=    0.939     LOCATION=    2.500