# V. GB500017-2017 H Section Subject to Bending

Verify the strength, stability, and deflection of an H section subject to bending per GB50017-2017.

## Reference

MOHURD. 2017. GB 50017-2017 Standard for design of steel structures . Beijing, China: Ministry of Housing and Urban-Rural Development

## Problem

The section is an HN500x200 with a length of 2.5 m. The structure is a two-story frame. Member #40 assigned with an H-section (HN500x200) is designed per GB 50017-2017. The section has an unbraced length of 1.0 m about the x direction and 2.5 m about the y direction.

Material Properties

The material is Q235 type steel.

• Design strength in tension, compression, and flexure: fy = 215 MPa
• Design strength in shear: fv = 125 MPa

Section Properties

• Section depth, h = 500 mm
• Section width, b = 200 mm
• Flange thickness, tf = 16 mm
• Web thickness, tw = 10 mm
• Cross-sectional area, A = 11,225 mm2
• Moment of inertia about x, Ix = 468,110,000 mm4
• Moment of inertia about y, Iy = 21,380,000 mm4
• Area moment about x, Sx = 1,872,400 mm3
• Area moment about y, Sx = 213,800 mm3

## Calculations

Slenderness Ratio

The effective length is:

loy = ly = 2,500 mm

According to the formula (c.0.1-2) in Appendix C, the slenderness ratio is:

$λ y = l 0 y i y = 2,500 43.6 = 57.3$

The steel type is Q235. According to note 1 in table 3.5.1 of Standard for design of steel structures and table 1 of Standard for design of steel structures. Commentary 2.2, the steel grade correction coefficient is εk = 1.

Overall Stability Coefficient

According to the formula (c.0.5-1) in Appendix C of Standard for design of steel structures, the overall stability coefficient of the beam determined by the bending around the strong axis is calculated as:

$ϕ b x = ϕ b = 1.07 - λ y 2 44,000 ε k 2 = 1.07 - ( 57.3 ) 2 44,000 = 0.995$

Plastic Development Coefficient

According to clause 6.1.2 of Standard for design of steel structures, and the width thickness ratio of cross-section plate is grade S3,

yx = 1.05

yy = 1.20

Check Web Thickness to Height Ratio

The calculated height of the web does not include the chamfering arc segment,

h0 = 500 mm - 2× (16 mm) = 468 mm

$h 0 t f = 468 10 = 46.8$

According to table 3.5.1 of Standard for design of steel structures, and the width thickness ratio of cross-section plate is grade S3, so the limit value of height thickness ratio is:

$93 ε k = 93$

Ratio: $46.8 93 = 0.50$

Check Flange Thickness to Width Ratio

The overhanging width of flange does not include chamfering arc segment,

$b 0 t f = 95 16 = 5.94$

According to table 3.5.1 of Standard for design of steel structures, and the width thickness ratio of cross-section plate is grade S3, so the limit value of width thickness ratio is:

$13 ε k = 13$
Ratio: $5.94 13 = 0.46$

Check Overall Stability

The controlling load condition for the overall stability is combination 59. F : 1.35DL+0.84WF:

Mx = -111.2 kN·m

My = 2.68 kN·m

 $M x ϕ b x W x f + M y γ y W y f$ (Cl. 6.2.3)
$-111.2 ( 10 ) 6 0.995 × 1,872,400 × 215 + 2.67 ( 10 ) 6 1.2 × 213,800 × 215 = 0.33$

Deflection

From the STAAD analysis, the maximum displacement of the beam is 5.8 mm. The total length of the physical beam member is 10 m.

According to table B.1.1 of Appendix B of Standard for design of steel structures, the allowable deflection ratio is 1/400.

$1 / 1,730 1 / 400 = 0.23$

Equivalent Stress in Beam

The controlling load condition for equivalent stress is combination 56. F : 1.35DL+0.98LL+0.84WL:

M = -111.2 kN·m

V = 43.3 kN

According to clause 6.1.5 of Standard for design of steel structures and taking the top of web as the calculation point, the distance from calculated point to neutral axis of beam:

The normal stress:

According to clause 6.1.3 of Standard for design of steel structures and taking the top of web as the calculation point, calculate the area moment of the above rough section to the neutral axis at the shear stress:

S = 774,400 mm3

Shear stress:

According to clause 6.1.5 of Standard for design of steel structures, the local compressive stress σc = 0，so, β1 = 1.1. The resultant stress is:

$σ2+σc2-σσc+3τ2$

The allowable stress:

β1f = 1.1 × 215 N/mm2 = 237 N/mm2

$σ 2 + σ c 2 - σ σ c + 3 τ 2 < β 1 f$

Ratio: $σ 2 + σ c 2 - σ σ c + 3 τ 2 β 1 f = 57.0 237 = 0.24$

Shear Strength

The controlling load condition for shear is combination 10. F : 1.20DL+1.40LL+0.84WL:

Vy = 49.9 kN

Vz = 540.3 kN

Take the neutral axis as the calculation point of shear stress, calculate the area moment:

Sy = 1,048,000 mm3

According to clause 6.1.3 of Standard for design of steel structures, shear stress

τ > fv = 125 N/mm2

Therefore, the ratio is:

$τ f v = 11.18 125 = 0.09$

Bending Strength

The controlling load condition for bending is combination 59. F : 1.35DL+0.84WF:

Mx = -111.2 kN·m

My = 2.68 kN·m

According to Clause 6.1.1 of Standard for design of steel structures,

Ratio: $M x γ x W n x + M y γ y W n y f = 67.0 215 = 0 . 31$

## Comparison

Table 1. Comparison of results
Result Type Reference STAAD.Pro Difference Comment
Web Slenderness 0.50 0.50 none
Flange Slenderness 0.46 0.46 none
Overall Stability 0.33 0.33 none
Beam Deflection 0.23 0.23 none
Equivalent Stress 0.24 0.24 none
Shear Strength 0.09 0.09 none
Bending Strength 0.31 0.31 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\\Verification Models\09 Steel Design\China\GB500017-2017 H-section with Bending.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 03-Aug-18
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 4.5 0; 3 0 9 0; 5 10 0 0; 6 10 4.5 0; 7 10 9 0; 17 0 0 6;
18 0 4.5 6; 19 0 9 6; 20 10 0 6; 21 10 4.5 6; 22 10 9 6; 23 2.5 4.5 6;
24 5 4.5 6; 25 7.5 4.5 6; 26 2.5 4.5 0; 27 5 4.5 0; 28 7.5 4.5 0; 29 2.5 9 0;
30 5 9 0; 31 7.5 9 0; 32 2.5 9 6; 33 5 9 6; 34 7.5 9 6;
MEMBER INCIDENCES
1 1 2; 2 2 3; 4 2 26; 5 3 29; 7 5 6; 8 6 7; 23 3 19; 24 6 21; 25 7 22;
30 17 18; 31 18 19; 32 18 23; 33 19 32; 34 20 21; 35 21 22; 36 23 24; 37 24 25;
38 25 21; 39 26 27; 40 27 28; 41 28 6; 42 29 30; 43 30 31; 44 31 7; 45 32 33;
46 33 34; 47 34 22; 48 23 26; 49 24 27; 50 25 28; 51 32 29; 52 33 30; 53 34 31;
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 CHINESE
1 2 7 8 30 31 34 35 TABLE ST HW400X408
4 5 32 33 36 TO 47 TABLE ST HN500X200
23 TO 25 48 TO 53 TABLE ST HN300X150
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 5 17 20 FIXED
*{ TYPE AUTO, DON'T MODIFY FOLLOWING DATA
TYPE 2
INT 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 HEIG 0.9 1.8 2.7 3.6 -
4.5 5.4 6.3 7.2 8.1 9
*{ END MAIN TYPE
*{ END INCLINE TYPE
SELFWEIGHT Y -1
23 TO 25 48 TO 53 UNI GY -12.5
23 TO 25 48 TO 53 UMOM GY -10
*{ THE FIRST AUTO BUILD WIND LOAD
*{ 自动定义的风荷载的类型号
*{ TYPE NO : 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
*{ 建筑结构的总高度
*{ STRUCTURE HIGH : 9
*{ 是否考虑风振系数的影响
*{ IS BETAZ : 0
*{ 建筑结构基本自振周期
*{ BASE PERIOD : 0.5
*{ 基本风压值
*{ BASE PRESS : 0.35
*{ 场地土粗造度类别
*{ SOIL TYPE : B
*{ 四个风向的迎风面宽度
*{ AWEATHER WIDTH : 6 6 10 10
*{ 四个风向的迎风面宽度
*{ AWEATHER WIDTH : 6 6 10 10
*{ 结构类型（空间或平面）
*{ STRUCTURE TYPE 2
*{ 风荷载体型系数，四个风向，四个面
*{ LEFTWIND MIUS : 0.8 -0.5 -0.6 -0.6
*{ RIGHTWIND MIUS : -0.5 0.8 -0.6 -0.6
*{ FRONTWIND MIUS : -0.6 -0.6 0.8 -0.5
*{ BACKWIND MIUS : -0.6 -0.6 -0.5 0.8
*{ 计算风压强度/高度曲线的方式
*{ INTENSITY FLAG : 1
*{ 用户指定需要计算风压强度的点数
*{ INTENSITY NUMBER : 10
*{ 等距离计算风压强度时的距离值
*{ INTENSITY ISOMETRY : 5
*{ 按照层高计算各点风压强度时的层高度
*{ INTENSITY FLOOR : 4.5
WIND LOAD X 0.8 TYPE 2
WIND LOAD -X 0.5 TYPE 2
WIND LOAD -Z 0.6 TYPE 2
WIND LOAD -Z -0.6 TYPE 2
WIND LOAD X -0.5 TYPE 2
WIND LOAD -X -0.8 TYPE 2
WIND LOAD -Z 0.6 TYPE 2
WIND LOAD -Z -0.6 TYPE 2
WIND LOAD -X -0.6 TYPE 2
WIND LOAD -X 0.6 TYPE 2
WIND LOAD -Z -0.8 TYPE 2
WIND LOAD Z -0.5 TYPE 2
WIND LOAD -X -0.6 TYPE 2
WIND LOAD -X 0.6 TYPE 2
WIND LOAD -Z 0.5 TYPE 2
WIND LOAD Z 0.8 TYPE 2
LOAD COMB 7 F : 1.20DL
1 1.2
LOAD COMB 8 F : 1.20DL+1.40LL
1 1.2 2 1.4
LOAD COMB 9 F : 1.20DL+0.84WL
1 1.2 3 0.84
LOAD COMB 10 F : 1.20DL+1.40LL+0.84WL
1 1.2 2 1.4 3 0.84
LOAD COMB 11 F : 1.20DL+0.84WR
1 1.2 4 0.84
LOAD COMB 12 F : 1.20DL+1.40LL+0.84WR
1 1.2 2 1.4 4 0.84
LOAD COMB 13 F : 1.20DL+0.84WF
1 1.2 5 0.84
LOAD COMB 14 F : 1.20DL+1.40LL+0.84WF
1 1.2 2 1.4 5 0.84
LOAD COMB 15 F : 1.20DL+0.84WB
1 1.2 6 0.84
LOAD COMB 16 F : 1.20DL+1.40LL+0.84WB
1 1.2 2 1.4 6 0.84
LOAD COMB 17 F : 1.00DL
1 1.0
LOAD COMB 18 F : 1.00DL+1.40LL
1 1.0 2 1.4
LOAD COMB 19 F : 1.00DL+0.84WL
1 1.0 3 0.84
LOAD COMB 20 F : 1.00DL+1.40LL+0.84WL
1 1.0 2 1.4 3 0.84
LOAD COMB 21 F : 1.00DL+0.84WR
1 1.0 4 0.84
LOAD COMB 22 F : 1.00DL+1.40LL+0.84WR
1 1.0 2 1.4 4 0.84
LOAD COMB 23 F : 1.00DL+0.84WF
1 1.0 5 0.84
LOAD COMB 24 F : 1.00DL+1.40LL+0.84WF
1 1.0 2 1.4 5 0.84
LOAD COMB 25 F : 1.00DL+0.84WB
1 1.0 6 0.84
LOAD COMB 26 F : 1.00DL+1.40LL+0.84WB
1 1.0 2 1.4 6 0.84
LOAD COMB 27 F : 1.20DL+0.98LL
1 1.2 2 0.98
LOAD COMB 28 F : 1.20DL+0.98LL+0.84WL
1 1.2 2 0.98 3 0.84
LOAD COMB 29 F : 1.20DL+0.98LL+0.84WR
1 1.2 2 0.98 4 0.84
LOAD COMB 30 F : 1.20DL+0.98LL+0.84WF
1 1.2 2 0.98 5 0.84
LOAD COMB 31 F : 1.20DL+0.98LL+0.84WB
1 1.2 2 0.98 6 0.84
LOAD COMB 32 F : 1.00DL+0.98LL
1 1.0 2 0.98
LOAD COMB 33 F : 1.00DL+0.98LL+0.84WL
1 1.0 2 0.98 3 0.84
LOAD COMB 34 F : 1.00DL+0.98LL+0.84WR
1 1.0 2 0.98 4 0.84
LOAD COMB 35 F : 1.00DL+0.98LL+0.84WF
1 1.0 2 0.98 5 0.84
LOAD COMB 36 F : 1.00DL+0.98LL+0.84WB
1 1.0 2 0.98 6 0.84
LOAD COMB 37 F : 1.20DL+1.40WL
1 1.2 3 1.4
LOAD COMB 38 F : 1.20DL+0.98LL+1.40WL
1 1.2 2 0.98 3 1.4
LOAD COMB 39 F : 1.20DL+1.40WR
1 1.2 4 1.4
LOAD COMB 40 F : 1.20DL+0.98LL+1.40WR
1 1.2 2 0.98 4 1.4
LOAD COMB 41 F : 1.20DL+1.40WF
1 1.2 5 1.4
LOAD COMB 42 F : 1.20DL+0.98LL+1.40WF
1 1.2 2 0.98 5 1.4
LOAD COMB 43 F : 1.20DL+1.40WB
1 1.2 6 1.4
LOAD COMB 44 F : 1.20DL+0.98LL+1.40WB
1 1.2 2 0.98 6 1.4
LOAD COMB 45 F : 1.00DL+1.40WL
1 1.0 3 1.4
LOAD COMB 46 F : 1.00DL+0.98LL+1.40WL
1 1.0 2 0.98 3 1.4
LOAD COMB 47 F : 1.00DL+1.40WR
1 1.0 4 1.4
LOAD COMB 48 F : 1.00DL+0.98LL+1.40WR
1 1.0 2 0.98 4 1.4
LOAD COMB 49 F : 1.00DL+1.40WF
1 1.0 5 1.4
LOAD COMB 50 F : 1.00DL+0.98LL+1.40WF
1 1.0 2 0.98 5 1.4
LOAD COMB 51 F : 1.00DL+1.40WB
1 1.0 6 1.4
LOAD COMB 52 F : 1.00DL+0.98LL+1.40WB
1 1.0 2 0.98 6 1.4
LOAD COMB 53 F : 1.35DL
1 1.35
LOAD COMB 54 F : 1.35DL+0.98LL
1 1.35 2 0.98
LOAD COMB 55 F : 1.35DL+0.84WL
1 1.35 3 0.84
LOAD COMB 56 F : 1.35DL+0.98LL+0.84WL
1 1.35 2 0.98 3 0.84
LOAD COMB 57 F : 1.35DL+0.84WR
1 1.35 4 0.84
LOAD COMB 58 F : 1.35DL+0.98LL+0.84WR
1 1.35 2 0.98 4 0.84
LOAD COMB 59 F : 1.35DL+0.84WF
1 1.35 5 0.84
LOAD COMB 60 F : 1.35DL+0.98LL+0.84WF
1 1.35 2 0.98 5 0.84
LOAD COMB 61 F : 1.35DL+0.84WB
1 1.35 6 0.84
LOAD COMB 62 F : 1.35DL+0.98LL+0.84WB
1 1.35 2 0.98 6 0.84
LOAD COMB 63 D : 1.00DL
1 1.0
LOAD COMB 64 D : 1.00DL+1.00LL
1 1.0 2 1.0
LOAD COMB 65 D : 1.00DL+0.60WL
1 1.0 3 0.6
LOAD COMB 66 D : 1.00DL+1.00LL+0.60WL
1 1.0 2 1.0 3 0.6
LOAD COMB 67 D : 1.00DL+0.60WR
1 1.0 4 0.6
LOAD COMB 68 D : 1.00DL+1.00LL+0.60WR
1 1.0 2 1.0 4 0.6
LOAD COMB 69 D : 1.00DL+0.60WF
1 1.0 5 0.6
LOAD COMB 70 D : 1.00DL+1.00LL+0.60WF
1 1.0 2 1.0 5 0.6
LOAD COMB 71 D : 1.00DL+0.60WB
1 1.0 6 0.6
LOAD COMB 72 D : 1.00DL+1.00LL+0.60WB
1 1.0 2 1.0 6 0.6
LOAD COMB 73 D : 1.00DL+0.70LL
1 1.0 2 0.7
LOAD COMB 74 D : 1.00DL+0.70LL+0.60WL
1 1.0 2 0.7 3 0.6
LOAD COMB 75 D : 1.00DL+0.70LL+0.60WR
1 1.0 2 0.7 4 0.6
LOAD COMB 76 D : 1.00DL+0.70LL+0.60WF
1 1.0 2 0.7 5 0.6
LOAD COMB 77 D : 1.00DL+0.70LL+0.60WB
1 1.0 2 0.7 6 0.6
LOAD COMB 78 D : 1.00DL+1.00WL
1 1.0 3 1.0
LOAD COMB 79 D : 1.00DL+0.70LL+1.00WL
1 1.0 2 0.7 3 1.0
LOAD COMB 80 D : 1.00DL+1.00WR
1 1.0 4 1.0
LOAD COMB 81 D : 1.00DL+0.70LL+1.00WR
1 1.0 2 0.7 4 1.0
LOAD COMB 82 D : 1.00DL+1.00WF
1 1.0 5 1.0
LOAD COMB 83 D : 1.00DL+0.70LL+1.00WF
1 1.0 2 0.7 5 1.0
LOAD COMB 84 D : 1.00DL+1.00WB
1 1.0 6 1.0
LOAD COMB 85 D : 1.00DL+0.70LL+1.00WB
1 1.0 2 0.7 6 1.0
PERFORM ANALYSIS
FINISH

Chinese steel design parameters (.gsp file):

[version=2207]
*{ The below data is for code check general information, please do not modify it.
[CodeCheck]
BeamBendingStrength=1
BeamShearStrength=1
BeamEquivalentStress=1
BeamOverallStability=1
BeamSlendernessWeb=1
BeamSlendernessFlange=1
TrussStrength=1
TrussStability=1
TrussShearStrength=1
ColumnStrength=1
ColumnStabilityMzMy=1
ColumnStabilityMyMz=1
PressedTrussSlenderness=1
TensionTrussSlenderness=1
ColumnSlendernessFlange=1
ColumnSlendernessWeb=1
BeamDeflection=1
SelectAll=0
GroupOptimize=0
FastOptimize=0
Iteration=0
SecondaryMembers=
SectCollectionOrder=0
[CheckOptionAngle]
PrimaryAxis=60.000000
SecondaryAxis=60.000000
ExtendLine=10.000000
*{ The above data is for code check general information, please do not modify it.

[GROUP=1]
Name(Parameter Name)=MAINBEAM
Type(Member Type)=1
Principle(Principle Rules)=0
SteelNo()=Q235
Fatigue(Fatigue Calculation)=0
Optimization(Perform optimized design)=0
MaxFailure(Failure Ratio)=1
MinTooSafe(Safety Ratio)=0.3
BeamBendingStrength()=1
BeamShearStrength()=1
BeamEquivalentStress()=1
BeamOverallStability()=1
BeamSlendernessFlange(b/t on beam)=1
BeamSlendernessWeb(h0/tw on beam)=1
TrussStrength(Axial Force Strength)=1
SecondaryMoment(Secondary Moment of Truss)=0
TrussStability(Solid-web Axial Compression Stability)=1
TrussShearStrength(Axial Shear Strength)=1
PressedTrussSlenderness(Pressed Member Slenderness)=1
TensionTrussSlenderness(Tension Member Slenderness)=1
ColumnStrength(Column Member Strength)=1
ColumnStabilityMzMy(Column Stability In-plane)=1
ColumnStabilityMyMz(Column Stability Out-plane)=1
ColumnSlendernessFlange(b/t on column)=1
ColumnSlendernessWeb(h0/tw on column)=1
CheckItemAPPENDIX_B11(Beam Deflection)=1
lmdc(Slenderness Limit of Compression Member)=0
lmdt(Slenderness Limit of Tension Member)=0
Lmd831(Slenderness of Seismic Column)=0
Lmd841(Slenderness of Seismic Brace)=0
Lmd9213(Slenderness of Seismic Single-story Plant)=0
LmdH28(Slenderness of Seismic Multi-story Plant)=0
rz(Plastic Development Factor in Major Axis)=0
ry(Plastic Development Factor in Minor Axis)=0
gamaSharp(Plastic Development Factor of sharp side)=0
betamz(the equivalent moment factor in Major Axis plane)=0
betamy(the equivalent moment factor in Minor Axis plane)=0
betatz(the equivalent moment factor out Major Axis plane)=0
betaty(the equivalent moment factor out Minor Axis plane)=0
DFF(Deflection Limit of Beam)=400
DJ1(Start Node Number in Major Axis)=0
DJ2(End Node Number in Major Axis)=0
Horizontal(Check for Deflection in Minor Axis)=0
Cantilever(Cantilever Member)=0
fabz(Overall Stability Factor in Major Axis of Bending Member)=0
faby(Overall Stability Factor in Minor Axis of Bending Member)=0
StressFeature(Select the Stress Feature to calulate stability factor of beam)=1
faz(Overall Stability Factor in Major Axis of Axial Compression Member=0
fay(Overall Stability Factor in Minor Axis of Axial Compression Member)=0
lz(Unbraced Length in Major Axis)=0
ly(Unbraced Length in Minor Axis)=0
miuz(Effective Length Factor for Column in Major Axis)=0
miuy(Effective Length Factor for Column in Minor Axis)=0
Lateral(Member in Frame Without Sidesway or not)=0
APZ(Gyration Radius Calculation as Z-Axis Parallel Leg)=0
rFlange(Limit Ratio of Width to Thickness for Flange)=0
rWeb(Limit Ratio of High to Thickness for Web)=0
BucklingStrength(Axis forced member bulking strength)=0
ZSectType(Section Type in Z-Axis)=0
YSectType(Section Type in Y-Axis)=0
HSectWebInTrussPlane(Web of H in Truss Plane)=0
rAn(Net Factor of Section Area)=1
rWnz(Net Factor of Resistance Moment in Z-Axis)=1
rWny(Net Factor of Resistance Moment in Y-Axis)=1
CapReduce(Seismic Reduction Factor of Load-bearing Capacity for Brace)=1
AngleReduce(Angle Strength Reduce)=0
LAglConSta(Connect Type of unequal single angle)=0
LAngleStrength(Reduction Factor of Angle Strength)=0
LAngleStability(Reduction Factor of Angle Stability)=0
rTrussSectReduce(Effective Factor of Axial Force Section)=1
Members(Member Number)=40