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Design of
Compression
Members
Design of Compression Members Design of Steel Structures to EC3
Page(2) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
2.1 Classification of cross sections
Classifying cross-sections may mainly depend on four critical factors:
1- Width to thickness (c/t) ratio.
2- Support condition.
3- Yield strength of the material.
4- Stress distribution across the width of the plate element.
There are basically four different cross-section classes and they are defined
as:
1- Class 1 cross-sections: are those which can form a plastic hinge with the rotation
capacity required from plastic analysis without reduction of the resistance.
2- Class 2 cross-sections: are those which can develop their plastic moment
resistance, but have limited rotation capacity because of local buckling.
3- Class3 cross-sections: are those in which the stress in the extreme compression
fiber of the steel member assuming an elastic distribution of stresses can reach the
yield strength, but local buckling is liable to prevent development of the plastic
moment resistance.
4- Class 4 cross-sections: are those in which local buckling will occur before the
attainment of yield stress in one or more parts of the cross section.
Design of Compression Members Design of Steel Structures to EC3
Page(3) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Design of Compression Members Design of Steel Structures to EC3
Page(4) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Design of Compression Members Design of Steel Structures to EC3
Page(5) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Design of Compression Members Design of Steel Structures to EC3
Page(6) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
2.2 Support Conditions
Note:
1- If buckling occurs, it will take place in a plane perpendicular to the crossponding
principal axis of inertia.
2- The slenderness to be used, ฮป, is the larger of those calculated in the y and z
directions, that is ฮป = max (ฮปy, ฮปz). For example, it is possible to make reference to
the compression member in the below Figure, restrained in different ways in the xโ
y and xโz planes, where the x-axis is the one along the length of the member. The
effective length in the xโy plane has to be taken as L/4 (i.e. LZ = 2.25 m), whereas
in the xโz plane it is 0.7L(Ly = 6.3 m).
Design of Compression Members Design of Steel Structures to EC3
Page(7) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
2.3 Design for compression
1- Plastic Resistance:
The design values of the compression force ๐๐ธ๐ at each cross-section shall satisfy:
๐๐ธ๐
๐๐,๐ ๐โค 1.0
The design resistance of the cross-section for uniform compression ๐๐,๐ ๐ should be
determined as follows:
๐๐,๐ ๐ = ๐ด ๐๐ฆ ๐พ๐0โ for class 1,2 or 3 cross-sections
๐๐,๐ ๐ = ๐ด๐๐๐ ๐๐ฆ ๐พ๐0โ For class 4 cross-sections
2- Buckling Resistance:
A compression member should be verified against buckling as follows:
๐๐ธ๐
๐๐,๐ ๐โค 1.0
The design buckling resistance of a compression member should be taken as:
๐๐,๐ ๐ = ฯ ๐ด ๐๐ฆ ๐พ๐1โ for class 1,2 or 3 cross-sections
๐๐,๐ ๐ = ฯ ๐ด๐๐๐ ๐๐ฆ ๐พ๐1โ for class 4 cross-sections
Where ฯ is the reduction factor for the relevant buckling mode.
Design of Compression Members Design of Steel Structures to EC3
Page(8) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
2.4 Buckling curves
For axial compression in members the value of ฯ for the appropriate non-dimensional
slenderness ๏ฟฝ๏ฟฝ should be determined from the relevant buckling curve according to:
ฯ =1
โ + โโ 2 โ ๏ฟฝ๏ฟฝ2 , ๐๐ข๐ก ฯ โค 1.0
where โ = 0.5[1 + ๐ผ(๏ฟฝ๏ฟฝ โ 0.2) + ๏ฟฝ๏ฟฝ2]
๏ฟฝ๏ฟฝ = โ๐ด ๐๐ฆ ๐๐๐โ =๐ฟ๐๐
๐
1
๐1 for class 1,2 or 3 cross-sections
๏ฟฝ๏ฟฝ = โ๐ด๐๐๐ ๐๐ฆ
๐๐๐=
๐ฟ๐๐
๐
โ๐ด๐๐๐ /๐ด
๐1 for class 4 cross-sections
ฮฑ is an imperfection factor.
๐๐๐ is the elastic critical force for the relevant buckling mode based on the gross
cross-sectional properties.
The imperfection factor ฮฑ corresponding to the appropriate buckling curve should be
obtained from Table 6.1 and Table 6.2.
Design of Compression Members Design of Steel Structures to EC3
Page(9) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Design of Compression Members Design of Steel Structures to EC3
Page(10) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Values of the reduction factor ฯ for the appropriate non-dimensional slenderness ๏ฟฝ๏ฟฝ may be
obtained from Figure 6.4.
For slenderness ๏ฟฝ๏ฟฝ โค 0.2 or for ๐๐ธ๐
๐๐๐โค 0.04 the buckling effects may be ignored and only cross
sectional checks apply.
Design of Compression Members Design of Steel Structures to EC3
Page(11) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
2.5 Solved Problems
Problem (1)
Design a lighting column subjected to an axial compression force 30 KN, using a CHS
(circular hollow section) cross section in S 275 steel, according to EC3-1-1. The
column is fixed at the base and free from the other end. With length of 3 m.
Solution:
Preliminary design โ Assuming class 1, 2 or 3 cross sections, yields:
๐๐ธ๐ = 30 ๐พ๐ โค ๐๐,๐ ๐ = ๐ด๐๐ฆ ๐พ๐0โ = ๐ด ร 275 ร 103/1.0
โ ๐ด โฅ 1.09 ร 10โ4๐2 = 1.09 ๐๐2
Use ๐ช๐ฏ๐บ ๐๐. ๐ ร ๐. ๐ section with A=2.38 cm2, d/t=8.41, I = 1.7 cm4, i= 0.846 cm.
Classification of the section
d/t=8.41,๐ = โ235/275 = 0.92 โ๐
๐ก< 50๐2 โ 8.41 < 50(0.92)2 โ 8.41 <
42.32 โโ ๐ถ๐๐๐ ๐ 1
Buckling lengths โ According to the support conditions, the buckling lengths are equal
in both planes, given by:
Buckling in the plane of the structure - ๐ฟ๐ธ = 2 ร 3 = 6 ๐
Determination of the slenderness coefficients
๐1 = ๐โ210ร109
275ร106 = 86.81
๐ =๐ฟ
๐=
6ร102
0.846= 709.21
๏ฟฝ๏ฟฝ =๐
๐1=
709.21
86.81= 8.16 > 1 โ ๐ฟ๐๐๐ ๐๐๐๐ข๐๐
Calculation of the reduction factor ๐
๐๐ข๐๐ฃ๐ ๐ โ ๐ผ = 0.21
๐ฅ =1
โ + โโ 2 โ ๏ฟฝ๏ฟฝ2 , ๐๐ข๐ก ๐ฅ โค 1.0
โ = 0.5[1 + ๐ผ(๏ฟฝ๏ฟฝ โ 0.2) + ๏ฟฝ๏ฟฝ2]
โ = 0.5[1 + 0.21 ร (8.16 โ 0.2) + 8.162] =34.62
๐ฅ =1
34.62 + โ34.622 โ 8.162= 0.0146
Design of Compression Members Design of Steel Structures to EC3
Page(12) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
-Safety verification
๐๐,๐ ๐ = ๐ฅ๐ด๐๐ฆ ๐พ๐1โ = 0.0146 ร 1.09 ร 10โ4 ร 275 ร 103 1.0โ = 0.4376 ๐พ๐
As NEd = 30 kN > Nb,Rd = 0.4376 kN Safety is not verified
Try heavier section such ๐ช๐ฏ๐บ ๐๐๐. ๐ ร ๐๐. ๐ section with A=81.1 cm2, d/t=17.5,
I = 4350 cm4, i= 7.32 cm.
Classification of the section
d/t=17.5,๐ = โ235/275 = 0.92 โ๐
๐ก< 50๐2 โ 17.5 < 50(0.92)2 โ 8.41 <
42.32 โโ ๐ถ๐๐๐ ๐ 1
Buckling lengths โ According to the support conditions, the buckling lengths are equal
in both planes, given by:
Buckling in the plane of the structure - ๐ฟ๐ธ = 2 ร 3 = 6 ๐
Determination of the slenderness coefficients
๐1 = ๐โ210ร109
275ร106 = 86.81
๐ =๐ฟ
๐=
6ร102
7.32= 81.96
๏ฟฝ๏ฟฝ =๐
๐1=
81.96
86.81= 0.944 < 1 โ ๐ โ๐๐๐ก ๐๐๐๐ข๐๐
Calculation of the reduction factor ๐
๐๐ข๐๐ฃ๐ ๐ โ ๐ผ = 0.21
โ = 0.5[1 + 0.21 ร (0.944 โ 0.2) + 0.9442] =1.02
๐ฅ =1
1.02 + โ1.022 โ 0.9442= 0.711
-Safety verification
๐๐,๐ ๐ = ๐ฅ๐ด๐๐ฆ ๐พ๐1โ = 0.711 ร 81.1 ร 10โ4 ร 275 ร 103 1.0โ = 1585.70 ๐พ๐
As NEd = 30 kN < Nb,Rd = 1585.70 kN Safety is verified
Design of Compression Members Design of Steel Structures to EC3
Page(13) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Problem (2)
Check a column subjected to an axial compression force 6000 KN, using a UC 254 ร
254 ร 167 (universal column) cross section in S 355 steel, according to EC3-1-1. The
column is supported as shown in the figure. With length of 5 m.
buckling in XโY plan buckling in XโZ plane
Solution:
Section with A=213 cm2, (c/t) flange=3.48, (c/t) web=10.4, Iy-y = 30000 cm4, Iz-z = 9870
cm4, iy-y = 11.9 cm, iz-z = 6.81 cm
Classification of the section
Flange: (c/t) =3.48, ๐ = โ235
355= 0.81 โ
๐
๐ก< 9๐ โ 3.48 < 9 โ 0.81 โ
3.48 < 7.29 โโ ๐ถ๐๐๐ ๐ 1
Web: (c/t) =10.4, ๐ = โ235
355= 0.81 โ
๐
๐ก< 33๐ โ 10.4 < 33 โ 0.81 โ
10.4 < 26.73 โโ ๐ถ๐๐๐ ๐ 1
โด ๐โ๐ ๐๐๐๐ ๐ ๐ ๐๐๐ก๐๐๐ ๐๐ ๐ถ๐๐๐ ๐ 1
๐๐,๐ ๐ = ๐ด๐๐ฆ ๐พ๐0โ = 213 ร 10โ4 ร 355 ร103
1.0= 7561.5 > 6000 โ ๐๐๐๐
Buckling lengths โ According to the support conditions, the buckling lengths are equal
in both planes, given by:
Buckling in the plane of the structure (plane x-z) - ๐ฟ๐ธ๐ฆ = 1.0 ร 5.0 = 5.0 ๐
Buckling in the plane of the structure (plane x-y) - ๐ฟ๐ธ๐ง = 1.0 ร 3.0 = 3.0 ๐
Design of Compression Members Design of Steel Structures to EC3
Page(14) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Determination of the slenderness coefficients
๐1 = ๐โ210ร109
355ร106 = 76.4
๐๐ฆ =๐ฟ๐ธ๐ฆ
๐๐ฆ=
5ร102
11.9= 42.01, ๐๐ฆ
=๐๐ฆ
๐1=
42.01
76.4= 0.55 < 1 โ ๐โ๐๐๐ก ๐๐๐๐ข๐๐
๐๐ง =๐ฟ๐ธ๐ง
๐๐ง=
3ร102
6.81= 44.05, ๐๐ง
=๐๐ง
๐1=
44.05
76.4= 0.57 < 1 โ ๐โ๐๐๐ก ๐๐๐๐ข๐๐
Calculation of the reduction factor ๐
h/b = 289.1/265.2 = 1.09 < 1.2, tf=31.7 mm < 100 mm
๐๐๐๐๐๐๐ ๐๐๐๐ข๐๐ ๐ง โ ๐๐ข๐๐ฃ๐ ๐ โ ๐ผ = 0.49
โ = 0.5[1 + 0.49 ร (0.57 โ 0.2) + 0.572] = 0.75
๐ฅ =1
0.75 + โ0.752 โ 0.572= 0.808
-Safety verification
๐๐,๐ ๐ = ๐ฅ๐ด๐๐ฆ ๐พ๐1โ = 0.808 ร 213 ร 10โ4 ร 355 ร 103 1.0โ = 6110.58 ๐พ๐
As NEd = 6000 kN < Nb,Rd = 6110.58 kN Safety is verified.
Design of Compression Members Design of Steel Structures to EC3
Page(15) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Problem (3)
Check a column subjected to an axial compression force 2500 KN, using a UB 533 ร
210 ร 82 (universal beam) cross section in S 275 steel, according to EC3-1-1. The
column is supported as shown in the figure. With length of 6 m.
Solution:
Section with A=105 cm2, (c/t) flange=6.58, (c/t) web=49.6, Iy-y = 47500 cm4, Iz-z = 2010
cm4, iy-y = 21.3 cm, iz-z = 4.38 cm
Classification of the section
Flange: (c/t) =6.58, ๐ = โ235
275= 0.92 โ
๐
๐ก< 9๐ โ 6.58 < 9 โ 0.92 โ
6.58 < 8.28 โโ ๐ถ๐๐๐ ๐ 1
Web: (c/t) =49.6, ๐ = โ235
275= 0.92 โ
๐
๐ก< 42๐ โ 49.6 < 42 โ 0.92 โ
49.6 < 38.64 โโ ๐ถ๐๐๐ ๐ 4
From bluebook ๐ด๐๐๐ = 96.4 ๐๐2
๐๐,๐ ๐ = ๐ด๐๐๐ ๐๐ฆ ๐พ๐0โ = 96.4 ร 10โ4 ร 275 ร103
1.0= 2651 > 2500 โ ๐๐๐๐
Buckling lengths โ According to the support conditions, the buckling lengths are equal
in both planes, given by:
Buckling in the plane of the structure (plane x-z) - ๐ฟ๐ธ๐ฆ = 1.0 ร 6.0 = 6.0 ๐
Buckling in the plane of the structure (plane x-y) - ๐ฟ๐ธ๐ง = 1.0 ร 2.0 = 2.0 ๐
Design of Compression Members Design of Steel Structures to EC3
Page(16) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Determination of the slenderness coefficients
๐1 = ๐โ210ร109
275ร106 = 86.81
๐๐ฆ =
๐ฟ๐๐
๐๐ฆ
โ๐ด๐๐๐ /๐ด
๐1 =
6ร100รโ96.4/105
21.3ร86.81= 0.31 < 1 โ ๐โ๐๐๐ก ๐๐๐๐ข๐๐
๐๐ง =
๐ฟ๐๐
๐๐ง
โ๐ด๐๐๐ /๐ด
๐1 =
2ร100รโ96.4/105
4.38ร86.81= 0.50 < 1 โ ๐โ๐๐๐ก ๐๐๐๐ข๐๐
Calculation of the reduction factor ๐
h/b = 528.3/208.8 = 2.53 > 1.2, tf=13.2 mm < 40 mm
๐๐๐๐๐๐๐ ๐๐๐๐ข๐๐ ๐ง โ ๐๐ข๐๐ฃ๐ ๐ โ ๐ผ = 0.34
โ = 0.5[1 + 0.34 ร (0.50 โ 0.2) + 0.502] = 0.676
๐ฅ =1
0.676 + โ0.6762 โ 0.502= 0.88
-Safety verification
๐๐,๐ ๐ = ๐ฅ ๐ด๐๐๐ ๐๐ฆ ๐พ๐1โ = 0.88 ร 96.4 ร 10โ4 ร 275 ร 103 1.0โ = 2332.88 ๐พ๐
As NEd = 2500 kN > Nb,Rd = 2332.88 kN Safety is not verified.
Design of Compression Members Design of Steel Structures to EC3
Page(17) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
Problem (4)
The following truss design the upper cord members compressed members, considering
the same type of cross section, that is: Square hollow sections (SHS), with welded
connections between the members of the structure.
Solution:
Based on the axial force diagrams represented in Figure 3.53, the most
compressed chord member is under an axial force of 742.6 kN and it is
simultaneously one of the longest members, with L = 3.00 m; For the definition of the
buckling lengths of the members, it is assumed that all the nodes of the truss are braced
in the direction perpendicular to the plane of the structure.
Preliminary design โ Assuming class 1, 2 or 3 cross sections, yields:
Upper cord:
๐๐ธ๐ = 742.6 ๐พ๐ โค ๐๐,๐ ๐ = ๐ด๐๐ฆ ๐พ๐0โ = ๐ด ร 275 ร 103/1.0
โ ๐ด โฅ 27 ร 10โ4๐2 = 27 ๐๐2
Use ๐บ๐ฏ๐บ ๐๐๐ ร ๐๐๐ ร ๐ for upper cord with A=35.5 cm2, I=738cm4, i=4.56cm
Classification of the section
Upper cord: c/t=12,๐ = โ235/275 = 0.92 โ๐
๐ก< 33๐ โ 12 < 33(0.92) โ 12 <
30.4 โโ ๐ถ๐๐๐ ๐ 1
Determination of the slenderness coefficients
๐1 = ๐โ210ร109
275ร106= 86.81 , LE=3.0 m
๐ =๐ฟ๐ธ
๐=
3ร102
4.56= 65.78 ๏ฟฝ๏ฟฝ =
๐
๐1=
65.78
86.81= 0.757 < 1 โ ๐โ๐๐๐ก ๐๐๐๐ข๐๐
Calculation of the reduction factor ๐
๐๐ข๐๐ฃ๐ ๐ โ ๐ผ = 0.21
Design of Compression Members Design of Steel Structures to EC3
Page(18) Dr.Mamoun Alqedra Eng.Mohammed AbuRahma Eng. Haya Baker
โ = 0.5[1 + 0.21 ร (0.757 โ 0.2) + 0.7572] =0.84
๐ฅ =1
0.84 + โ0.842 โ 0.7572= 0.83
-Safety verification
๐๐,๐ ๐ = ๐ฅ๐ด๐๐ฆ ๐พ๐1โ = 0.83 ร 35.5 ร 10โ4 ร 275 ร 103 1.0โ = 810.28 ๐พ๐
As NEd = 742.6 kN < Nb,Rd = 810.28 kN Safety is verified