 # V. SNiP SP16 2017 - Interaction check of a column

Design a column per the SP 16.13330.2017 code.

## Details

A 6.78m tall, simply supported column has a European HE600B section. The column is subject to a 310 kN axial load along with a uniformly distributed load of 50 kN/m in the local Y axis. The steel used has a modulus of elasticity of 206,000 MPa and a Ry = 239 MPa. γc = 1, γm = 1.05

## Validation

Ry = Ryn/ γm = 223.8 MPa

Rs = 0.58×Ry/ γm = 129.8 MPa

Shear force at support:

Q = qx × L / 2 = 171.8 kN

Mx = qx × L2/ 8 = 295.0 kN·m

Bending moment:

Mx = qx × L2 / 8 = 171.8 (6.78)2 / 8 = 295.0 kN·m

Design for Strength (Cl. 9.1.1)

$σ = N A n = 310 270 ( 10 ) − 1 = 11.48 < 0.1 R y = 22.38$

Ryn ≤ 440 N/mm2

τ = 0; i.e., > 0.5×Rs

So, as per Cl. 9.1.1, F.105 should not be checked. Rather F.106 needs to be checked.

$m e f = η × m$
where
 m = e×A / Wc = 4.51 (e = M/N = 0.9516)
$λ x = k x l r x = 1.0 × 6.87 0.2517 = 27.3$
$λ y = k y l r y = 1.0 × 6.87 0.0708 = 97.0$
$λ ¯ x = λ x R y E = 27.3 223.8 206 , 000 = 0.900$
$λ ¯ y = λ y R y E = 97.0 223.8 206 , 000 = 3.198$

Therefore, $λ ¯ = min ⁡ ( λ x ¯ , λ y ¯ ) = 0.900$

$η = ( 1.90 − 0.1 m ) − 0.02 ( 6 − m ) λ ¯ = ( 1.90 − 0.1 × 4.51 ) − 0.02 ( 6 − 4.51 ) 0.900 = 1.422$

So, $m e f = 1.422 × 4.51 = 6.41 < 20$ [As per F.(110)]

$A f / A w = 90 / 90 = 1$
 $N A n ± M x y I x n ± M y x I y n ± B ω I ω n R y γ c ≤ 1$ (F.(106) )
where
 x = 150 mm y = 300 mm Bω = 0

So, the ratio is $11.48 + 295.0 ( 300 ) 171 , 000 × 10 − 2 223.8 × 1 = 0.28 < 1$

Design for Stability (Cl. 9.2.2)

From Table E.3, depending on conditional slenderness and reduced relative eccentricity:

ϕe = 0.2144

 $Nϕe×A×Ry×γc=3100.2144×270(10)−1×223.8×1=0.24<1$ (F.(109) )

Design for Stability (Cl. 9.2.4)

Calculate the stability of eccentrically compressed elements of constant cross-section, out-of-plane bending moment in the plan of maximum stiffness (Ix > Iy), coinciding with the plane of symmetry:

 $Nc×ϕy×A×Ry×γc≤1$ (F.(111) )
where
 ϕy = $0.5(δ−δ2−39.48λ¯2λ¯2$ $λ¯$ = conditional slenderness $=max⁡(λx¯,λy¯)=3.198$ δ = $9.87(1−α+βλ¯)+λ¯2$, per Eq. 9

From Table 7, α = 0.03 and β = 0.06.

$δ=9.87[1−0.03+0.06(3.198)]+(3.198)2=21.70$
$ϕy=0.521.70−(21.70)2−39.48×(3.198)2(3.198)2=0.660$
 $c=β1+α×mx$ (Cl. 9.2.5)
where
 α = 0.65 + 0.05×mx = 0.875 $λy¯$ = 3.14 δc = $9.87[1−0.03+0.06(3.14)]+(3.14)2=21.29$ ϕc = $0.521.29−(21.29)2−39.48×(3.14)2(3.14)2=0.674$ β = $ϕcϕy=0.6740.661=1.009$

Therefore, $c=1.0091+0.875×4.51=0.204<1$

So, the ratio is $3100.204×0.66×270(10)−1×223.9×1=0.38<1$

Calculate Cmax Per Annex E.1

 $cmax⁡=21+δβ+(1+δβ)2+16μ(α−exh)2$ (F.(E.1) )
where
 δ = 4×ρ / μ ρ = $Ix+IyAh2+α2$ μ = $8ω+0.156×Itλy2Ah2$ ω = It = $k3Σbiti3=1.293[2(30 cm)(3 cm)3+(60 cm−2×3 cm)(1.55 cm)3]=783.1 cm4$ (k = 1.29) ex = M / N = 295 / 310 = 0.952 B = $1+2(β/ρ)(ex/hc)=1$

As per Table E.6, α = 0, β = 0

$μ=8(0.249)+0.156×(783.1)(97.0)2270×(60)2=3.17$
$ρ=171,000+13,530270×602+0=0.190$
$δ=4×0.193.17=0.24$
$cmax⁡=21+0.24(0)+(1+0.24×0)2+163.17(0−0.9520.6)2=0.43$

## Results

Ratio per Cl. 9.1.1 0.28 0.28 none
Ratio per Cl. 9.2.2 0.24 0.24 none
Ratio per Cl. 9.2.4 0.38 0.38 none
mef 6.41 6.41
mx 4.51 4.51 none
C 0.204 0.204 none
Cmax 0.43 0.40 negligible
Φy 0.66 0.66 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Russia\SNiP SP16 2017 - Interaction check of a column.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 02-Sep-20
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 6.87 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+008
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-005
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 HE600B
****************************************
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FY MX MZ
****************************************
2 FY -310
1 UNI GX 50
********************************
PERFORM ANALYSIS
*********************************
PARAMETER 1
CODE RUSSIAN
TB 1 ALL
ENSGR 1 ALL
GAMM 2 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH


                       STAAD.PRO CODE CHECKING - (SP 16.13330.2017)   V1.0
********************************************
ALL UNITS ARE - KN METRE
========================================================================
SECTION NO.      N             Mx            My      LOCATION
========================================================================
1  I      HE600B        PASS      SP cl.9.1.1      0.28         1
3.100E+02 C    2.950E+02    0.000E+00   3.435E+00
1  I      HE600B        PASS      SP cl.9.2.2      0.24         1
3.100E+02 C    2.950E+02    0.000E+00   3.435E+00
1  I      HE600B        PASS      SP cl.9.2.4      0.38         1
3.100E+02 C    2.950E+02    0.000E+00   3.435E+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                 = 6.87E+00
Gross Area                    = 2.70E-02
Net Area                      = 2.70E-02
x-axis      y-axis
Moment of inertia (I)         :   171.E-05    135.E-06
Section modulus (W)           :   570.E-05    902.E-06
First moment of area (S)      :   321.E-05    696.E-06
Radius of gyration (i)        :   252.E-03    708.E-04
Effective Length              :   6.87E+00    6.87E+00
Slenderness                   :   273.E-01    970.E-01
DESIGN DATA (units -kN,m) SP16.13330.2017
Axial force                   :   310.0E+00
x-axis      y-axis
Moments                       :   295.0E+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.(106) (N/A+Mx*y/Ix+My*x/Iy+B*w/Iw)/(Ry*GammaC)=
( 310.0E+00/ 2.7E-02+ 295.0E+00* 3.00E-01/ 1.71E-03+ 0.000E+00* 1.50E-01/
1.35E-04+ 0.000E+00* 2.49E-01/ 1.10E-05)/( 223.8E+03* 1.00E+00)
= 2.83E-01=&lt;1
cl.9.2.2  m_ef=eta*mx= 1.42E+00* 4.51E+00= 6.41E+00
F.(109)  N/(FIe*A*Ry*GammaC)= 310.0E+00/( 2.14E-01* 2.70E-02* 223.8E+03* 1.00E+00)
= 2.39E-01=&lt;1
F.(112)  c=beta/(1+alfa*mx)= 1.01E+00/(1+8.75E-01* 4.51E+00)= 2.04E-01
c_max= 4.00E-01
F.(111)  N/(c*FIy*A*Ry*GammaC)
= 0.31E+03/( 0.20E+00* 0.66E+00* 0.27E-01* 223.8E+03* 1.00E+00)
= 3.80E-01=&lt;1