Beam – Columns

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Beam–Columns

Pang Sze Dai Associate Professor B.Eng., NUS, 2001; M.Eng., NUS, 2002; Ph.D. Northwestern University, 2005

Associate Head (Academic) Department of Civil & Environmental Engineering Email : [email protected] Tel: 65162799 Room: E1 07-11 1

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

What About Members Subject to the Following Forces?? Biaxial Moments and Shear

Major Axis Moment, Shear and Axial Load

Minor Axis Moment, Shear and Axial Load

NEd

NEd My1,Ed

My1,Ed

My1,Ed

Mz1,Ed

Mz1,Ed

Vz1,Ed

Mz2,Ed

Mz1,Ed

My2,Ed

Vy2,Ed Vz2,Ed

Vy1,Ed

Vz1,Ed

My2,Ed

My2,Ed

NEd

Vy1,Ed

Vy1,Ed

Vz1,Ed

Biaxial Moments, Shear and Axial Load

Vy2,Ed

Vy2,Ed NEd Vz2,Ed

Mz2,Ed

NEd

Vz2,Ed

Mz2,Ed

NEd

Members subject to such forces are also known as BEAM-COLUMNS 2

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Identify the slabs, beams, tension/compression members, and beam-columns A E

B

D What is the Load Path?

C 3

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Design Flow Chart for Beam–Columns Resistance Check Determine design forces VEd, NEd, My,Ed, Mz,Ed NO Shear check VEd/Vpl,Rd ≤ 1 YES Low Shear Use yield strength fy

Select section

Restrained Beams Slide 7

High Shear Beam-Columns Slide 11

Reduce yield strength or thickness of shear area

4

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Moment and Axial Force Resistances in the Presence of Shear Force N EN 1993-1-1 (Clause 6.2.10)

When the design value of the shear force VEd exceed 50% of the design plastic shear resistance Vpl,Rd, the yield strength fy in the shear area should be reduced by (1 – ). Alternatively the plate thickness of the relevant part of the cross section may be reduced.

My Mz

Vy

Vz

f yr  (1   ) f y 2

 2VEd    1 where     V pl , Rd   Av ( f y / 3 ) V pl , Rd 

M0

5

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Moment and Axial Force Resistances in the Presence of Shear Force fy

When shear force is high,

Option 1 Reduce yield strength of shear area

hw

(1-) fy (1-) fy

tw fy

hw tw

fy

Welded I, H & box sections, load parallel to web

Option 2 Reduce thickness of shear area

hw

fy fy

(1-) tw fy

6

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Design Flow Chart for Beam–Columns Resistance Check Determine design forces VEd, NEd, My,Ed, Mz,Ed NO Shear check VEd/Vpl,Rd ≤ 1 YES Low Shear Use yield strength fy

Select section

Restrained Beams Slide 7

Classify section

High Shear Beam-Columns Slide 15

Class 1 & 2

Reduce yield strength or thickness of shear area

Class 3

Class 4

Cross-Section Capacity Check 



 M y , Ed   M z , Ed      1  M N , y , Rd   M N , z , Rd 

M y , Ed

M z , Ed

N Ed   1 N Rd M y , Rd M z , Rd

N Ed  N Rd

M y , Ed  N Ed eN y M y , Rd



M z , Ed  N Ed eN z M z , Rd

NO 1

How does the Axial Force affect the Bending Moment Resistance?

7

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Moment Capacity in the Presence of Axial Load First, consider a rectangular section under a tension load Nt,Ed.

Nt

 t , Ed  f y

Nt,Ed

Refer to Appendix I – Plastic Moment Capacity in the Presence of Axial Force.ppt in the IVLE

8

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Moment Capacity in the Presence of Axial Load Next, consider the gradual increase of the bending moment My,Ed in the presence of the constant axial load Nt,Ed.

Nt My

 t , Ed   fy Elastic moment capacity is reached

Nt,Ed Plastic moment capacity is reached

My,Ed  c , Ed   f yy

Refer to Appendix I – Plastic Moment Capacity in the Presence of Axial Force.ppt in the IVLE

9

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Moment Capacity in the Presence of Axial Load The plastic moment capacity is reached when the entire cross section has yielded (in either tension or compression)

Nt My

 t , Ed  f y

Nt,Ed

centroid

My,Ed  c , Ed  f y

               

The force resultant force the green have The resultant of the redofregion haveregion net zero axial zero moment centroid. resistance because theyresistance cancel outabout each the other but they It only contributes to about the axial resistance. create a moment resistance centroid of section.

Refer to Appendix I – Plastic Moment Capacity in the Presence of Axial Force.ppt in the IVLE

10

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Moment Capacity in the Presence of Axial Load Comparison of Moment Capacity in the Absence and Presence of Axial Load

My

 t , Ed  f y

centroid

My,Ed  c , Ed  f y

Nt My

 t , Ed  f y

Nt,Ed

centroid

My,Ed

How do we compute the reduced moment capacity in the presence of axial force?

 c , Ed  f y

Refer to Appendix I – Plastic Moment Capacity in the Presence of Axial Force.ppt in the IVLE

11

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Cross-Section Capacity Check for Class 1&2 Cross-Sections

N

EN 1993-1-1 (Clause 6.2.9.1)

Check at critical locations, i.e. moments & axial force are the largest 



Valid for  M y , Ed   M z , Ed       1 Class 1 & Class 2  M N , y , Rd   M N , z , Rd  sections ONLY Type of Section





I and H

2

5n ≥ 1

circular hollow

2

2

My Mz

rectangular hollow 1.66/(1 – 1.13n2) ≤ 6 others

1.0

1.0

Af y N Ed where n  and N pl , Rd  N pl , Rd M0

How can we determine the moment resistance, MN,y,Rd & MN,z,Rd in the presence of axial load?

12

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Doubly symmetrical I– and H– sections Reduced moment resistance about y–y axis M N , y , Rd

0.5hwt w f y  for n  0.25 & N Ed   M pl , y , Rd M0  M pl , y , Rd  1  n   M pl , y , Rd otherwise   1  0.5a 

n

Reduced moment resistance about z–z axis M N , z , Rd

 M pl , z , Rd     n  a 2   M pl , z , Rd 1      a 1      

for N Ed  for N Ed 

hwt w f y

What does this mean?

where a 

( A  2bt f ) A

 0.5

N Ed N pl , Rd

M0 hwt w f y

M0

where a 

( A  2bt f ) A

 0.5

13

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Rectangular hollow & welded box sections of uniform thickness Reduced moment resistance about y–y axis M N , y , Rd

 1 n    M pl , y , Rd  M pl , y , Rd   1  0.5aw 

A  2bt  a   0.5 for hollow sections  w A where  A  2bt f  aw   0.5 for welded box sections A 

Reduced moment resistance about z–z axis M N , z , Rd

 1 n  M pl , z , Rd   1  0.5a f 

   M pl , z , Rd  

n

N Ed N pl , Rd

A  2ht  a   0.5 for hollow sections  f A where  A  2ht w a f   0.5 for welded box sections A 

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CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Cross-Section Capacity Check for Class 3 Cross-Sections  t , Ed  f y

EN 1993-1-1 (Clause 6.2.1(7))

Check at critical locations:

Nt,Ed

N Ed M y , Ed M z , Ed   1 N Rd M y , Rd M z , Rd

My,Ed  c , Ed  f y

where N Rd 

 t , Ed  f y

Af y

M0

, M y , Rd 

Wel , y f y

M0

, M z , Rd 

Wel , z f y

Elastic moment capacity Plastic moment capacity is reached is reached

M0

Refer to Appendix II – Elastic Moment Capacity in the Presence of Axial Force.ppt in the IVLE

This check can also be used as a conservative approximation for Class 1 & 2 cross-sections where NRd, My,Rd and Mz,Rd are defined as follows: N Rd 

Af y

M0

, M y , Rd 

W pl , y f y

M0

, M z , Rd 

W pl , z f y

M0 15

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Capacity Check for Class 4 Cross-Sections (for info)

EN 1993-1-1 (Clause 6.2.9.3)

Check at critical locations:

N Ed M y , Ed  N Ed e N y M z , Ed  N Ed e N z   1 N Rd M y , Rd M z , Rd where N Rd 

Aeff f y

M0

, M y , Rd 

Weff , y f y

M0

, M z , Rd 

Weff , z f y

M0

eN = shift in the relevant centroidal axis when the effective cross section is subjected to compression only. top flange is slender when under compression

centroidal axis of gross section

ineffective segment

eN y

centroidal axis of effective section

16

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Perform section capacity check on a steel column using 203x203x60 UC that is subjected to a compressive load of N = 380kN, linearly varying major & minor axis moments with the end moments being My1 = 82.0kNm, My2 = –41.0kNm & Mz1 = 35.4kNm, Mz2 = 17.7kNm. The steel grade is S275 and the effective length is Lcr,y = Lcr,z = L = 3m.

N My1

Mz1 L

My2 Mz2

N

My,1= 82.0kNm My,2= –41.0kNm Mz,1= 35.4kNm

Mz,2= 17.7kNm

Identify the critical point(s) to perform cross-section capacity check 17

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Determine design forces VEd, NEd, My,Ed, Mz,Ed NO Shear check VEd/Vpl,Rd ≤ 1 YES Low Shear Use yield strength fy

Select section

Restrained Beams Slide 7

High Shear Beam-Columns Slide 15

Reduce yield strength or thickness of shear area

18

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check N My Vy Vz

Mz L

Design Forces NEd = 380 kN My,Ed = 82.0 kNm Mz,Ed = 35.4 kNm Vz,Ed = (82.0 – (–41.0))/3.0 = 61.5kN Vy,Ed = (35.4 – 17.7)/3.0 = 5.9kN

Design Strength

Vz

My,1

Mz,1

Vy

My Vz Vz My,2

Vy Mz,2

tw = 9.4mm, tf = 14.2mm. Maximum thickness = 14.2mm < 16mm (EN 10025-2) For S275 steel, fy = 275N/mm2

Vy Mz N

My,1= 82.0kNm My,2= –41.0kNm Mz,1= 35.4kNm

Mz,2= 17.7kNm 19

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Shear Check

Restrained Beams Slide 8

Vz,pl,Rd = 0.577fy [A – 2btf + (tw+2r)tf ] = 0.577*275*[7640 – 2*205.8*14.2 + (9.4+2*10.2)*14.2]*10-3 = 352kN Vz,Ed = 61.5kN < Vz,pl,Rd = 352kN Shear Check OK! Vz,Ed = 61.5kN < 0.5Vz,pl,Rd = 176kN Low Shear  No change in fy Vy,pl,Rd = 0.577fy (2btf ) = 0.577*275*[2*205.8*14.2]*10-3 = 927kN Vy,Ed = 5.9kN < Vy,pl,Rd = 927kN Shear Check OK! Vy,Ed = 5.9kN < 0.5Vy,pl,Rd = 464kN Low Shear  No change in fy

r

N

tf

My

tw

Vy

b

Av  A  2bt f  (t w  2r )t f

V pl , Rd 

Av ( f y / 3 )

Vz

Mz

M0 r

L tf

tw b Av  2bt f

My Vz

Vy Mz N

20

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Determine design forces VEd, NEd, My,Ed, Mz,Ed NO Shear check VEd/Vpl,Rd ≤ 1 YES Low Shear Use yield strength fy

Select section

Restrained Beams Slide 7

High Shear Beam-Columns Slide 15

Classify section Class 1 & 2

Reduce yield strength or thickness of shear area

Class 3

Class 4

Cross-Section Capacity Check 



 M y , Ed   M z , Ed      1  M N , y , Rd   M N , z , Rd 

N Ed M y , Ed M z , Ed   1 N Rd M y , Rd M z , Rd

N Ed M y , Ed  N Ed eN y M z , Ed  N Ed eN z   1 N Rd M y , Rd M z , Rd

NO

21

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Section Classification Classification of flange cf / tf = 6.20 ≤ 9 = 9*0.92 = 8.28  Flange is Class 1. Part subject to Class compression

1

Stress distribution (compression +ve) Tip in compression

c / t  9

c/t 

 = (235/fy)0.5 = 0.92

Part subject to bending and compression Stress distribution Tip in (compression +ve) tension

9

c/t 



Stress distribution (compression +ve)

9

 

Classification of web Assume the web is in pure compression (worst case scenario resulting in smaller limits for cw/tw). cw / tw = 17.1 ≤ 33 = 33*0.92 = 30.4  Web is Class 1. Class

1

Part subject to Part subject to bending compression

c / t  72

c / t  33

Part subject to bending and compression

when   0.5 : c / t  when   0.5 : c / t  

396  13  1 36



Stress distribution (compression +ve)

Try to classify the section by treating the web as “Part subjected to bending & compression”



 M y , Ed   M z , Ed  Section is Class 1  Use      1 for Section Capacity Check M M  N , y , Rd   N , z , Rd 

22

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Moment Resistance in the Presence of Axial Load First check if axial force will affect the moment resistance

N pl , Rd 

n

Af y

M0

(76.4 * 102 )(275) * 10 3   2100kN 1.0

N Ed 380   0.181 N pl , Rd 2100

hwt w f y

M0

M0

M N , y , Rd

0.5hwt w f y  for n  0.25 & N Ed   M pl , y , Rd M0  M pl , y , Rd  1  n   M pl , y , Rd otherwise   1  0.5a 

Reduced moment resistance about z–z axis

(209.6 - 2 * 14.2) * 9.4 * 275 * 10-3   468kN 1.0

0.5hwt w f y

Reduced moment resistance about y–y axis

M N , z , Rd

 M pl , z , Rd   2    M pl , z , Rd 1   n  a      1  a   

for N Ed  for N Ed 

hwt w f y

M0 hwt w f y

M0

 234kN

Since N Ed  380kN 

Since N Ed  380kN 

0.5hwt w f y

M0 hwt w f y

M0

 1 n  , M N , y , Rd  M pl , y , Rd    1  0.5a 

, M N , z , Rd  M pl , z , Rd 23

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check Now we determine the moment resistance

a

( A  2bt f )

M pl,y,Rd

(7640  2 * 205.8 * 14.2)   0.235  0.5 A 7640 W pl,y f y (656 * 103 ) * 275 * 10 6    180 kNm 1.0 γM 0

Reduced moment resistance about y–y axis  1 n  M N , y , Rd  M pl , y , Rd    M pl , y , Rd  1  0.5a  where a 

( A  2bt f ) A

 0.5

 1  0.181   1 n   180   167 kNm M N,y,Rd  M pl,y,Rd     1  0.5a  1  0.5 * 0.235 

M N,z,Rd  M pl,z,Rd 

W pl,z f y γM 0

(305 * 103 ) * 275 * 10 6   84 kNm 1.0

Reduced moment resistance about z–z axis M N , z , Rd  M pl , z , Rd

Interaction Check for Bi-Axial Bending   5n  5 * 0.181  0.905

Type of Section





 Use  = 1

I and H

2

5n ≥ 1

circular hollow

2

2

α

β

 M y,Ed   M z,Ed   82.0   35.4     0.663  1          167   84   M N,y,Rd   M N,z,Rd  2

The cross-section resistance is adequate.

1

rectangular hollow others

1.66/(1 – 1.13n2) ≤ 6 1.0

1.0 24

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check

Use of Design Tables

N Ed 380   0.181 n N pl , Rd 2100 Consider n = 0.2 (more conservative)

  5n  5 * 0.181  0.905  Use  = 1 α

β

2 1  M y,Ed   M z,Ed  82.0 35.4       0.671  1          164   84   M N,y,Rd   M N,z,Rd 

The cross-section resistance is adequate.

Design Forces NEd = 380 kN My,Ed = 82.0 kNm Mz,Ed = 35.4 kNm Type of Section





I and H

2

5n ≥ 1

circular hollow

2

2

rectangular hollow others

1.66/(1 – 1.13n2) ≤ 6 1.0

1.0

25

CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Example BC-1: Cross-section capacity check

If the conservative check is used,

N My

N Ed M y , Ed M z , Ed 380 82.0 35.4       1.06  1 N Rd M y , Rd M z , Rd 2100 180 84

For the same section, it fails using the conservative cross-section capacity check:

Vy Vz

Mz L

N Ed M y , Ed M z , Ed   1 N Rd M y , Rd M z , Rd

but pass the more exact cross-section capacity check: 



 M y , Ed   M z , Ed      1  M N , y , Rd   M N , z , Rd 

My Vz

Vy Mz N

Will the answer be different if N is a tension force of 380kN?

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CE3166 Structural Steel Design and System

J Y R Liew & S D Pang

Review Questions • When do you use beam–column check? • How do you treat the sign for the forces in cross-section capacity check? • For Class 1 or 2 sections, you have the option of using the more exact crosssection capacity check involving reduced moment resistance, or the more conservative check involving a direct superposition of the effects of axial loads and bending moments. How do you know which check to use? 



 M y , Ed   M z , Ed  •     1  M N , y , Rd   M N , z , Rd 

vs

N Ed M y , Ed M z , Ed   1 N Rd M y , Rd M z , Rd

Challenge Question • Determine the reduced moment resistance of a circular hollow section (CHS).

27