RC Columns

Civil Engineering Design (1)

Civil Engineering Design (1) Design of Reinforced Concrete Columns

2006/7

Dr. Colin Caprani, Chartered Engineer

1

 Dr. C. Caprani

Civil Engineering Design (1)

Contents 1.

2.

3.

4.

Introduction ................... ............................. .................... ................... ................... .................... .................... ................... ................... .................. ........ 3

1.1

Background.......... Background ................... ................... .................... .................... ................... ................... .................... .................... ................... .............. ..... 3

1.2

Failure Modes ................... ............................. .................... .................... .................... .................... .................... .................... ................... ......... 5

1.3

Design Aspects .................... .............................. .................... .................... .................... .................... .................... .................... ................ ...... 7

1.4

Background Information................... Information............................. .................... .................... ................... ................... .................... .............. .... 8

Short Braced Axially Loaded Columns.................. Columns............................ .................... .................... .................... ............ 10

2.1

Development.......................... Development................ .................... .................... ................... ................... .................... .................... .................... ............. ... 10

2.2

Example ................... ............................. ................... ................... .................... .................... .................... ................... ................... .................. ........ 11

2.3

When loaded by Beams ................... ............................. .................... .................... .................... .................... .................... ............ .. 12

Short Braced Columns Resisting Resisting Axial Load and Moment .................... .......................... ...... 13

3.1

Introduction................... Introduction............................. .................... .................... ................... ................... .................... .................... ................... ............ ... 13

3.2

Case 1: 0.9x ≤ h ................... ............................. .................... .................... ................... ................... .................... .................... ............... ..... 14

3.3

Case 2: 0.9x > h .................... .............................. .................... .................... .................... .................... .................... .................... ............. ... 15

3.4

Combination of Cases........................... Cases..................................... .................... .................... .................... .................... ................. ....... 16

3.5

Description of Interaction Diagram......... Diagram ................... .................... .................... .................... .................... .............. .... 18

3.6

Example ................... ............................. ................... ................... .................... .................... .................... ................... ................... .................. ........ 21

3.7

Design Using Standard Interaction Diagrams .................... .............................. ..................... ................. ...... 30

Biaxial Bending.......... Bending .................... .................... ................... ................... .................... .................... .................... .................... ................... ......... 33

4.1

Introduction................... Introduction............................. .................... .................... ................... ................... .................... .................... ................... ............ ... 33

4.2

Method 1 – Resolution of Moments .................... .............................. ..................... ..................... .................... ............ 34

4.3

Method 2 – 3-D Interaction Diagrams........................... Diagrams..................................... ................... ................... ............. ... 37

4.4

Method 3 – Code Methods .................... .............................. .................... .................... .................... .................... ................ ...... 39

4.5

Example ................... ............................. ................... ................... .................... .................... .................... ................... ................... .................. ........ 41

2

 Dr. C. Caprani

Civil Engineering Design (1)

Contents 1.

2.

3.

4.

Introduction ................... ............................. .................... ................... ................... .................... .................... ................... ................... .................. ........ 3

1.1

Background.......... Background ................... ................... .................... .................... ................... ................... .................... .................... ................... .............. ..... 3

1.2

Failure Modes ................... ............................. .................... .................... .................... .................... .................... .................... ................... ......... 5

1.3

Design Aspects .................... .............................. .................... .................... .................... .................... .................... .................... ................ ...... 7

1.4

Background Information................... Information............................. .................... .................... ................... ................... .................... .............. .... 8

Short Braced Axially Loaded Columns.................. Columns............................ .................... .................... .................... ............ 10

2.1

Development.......................... Development................ .................... .................... ................... ................... .................... .................... .................... ............. ... 10

2.2

Example ................... ............................. ................... ................... .................... .................... .................... ................... ................... .................. ........ 11

2.3

When loaded by Beams ................... ............................. .................... .................... .................... .................... .................... ............ .. 12

Short Braced Columns Resisting Resisting Axial Load and Moment .................... .......................... ...... 13

3.1

Introduction................... Introduction............................. .................... .................... ................... ................... .................... .................... ................... ............ ... 13

3.2

Case 1: 0.9x ≤ h ................... ............................. .................... .................... ................... ................... .................... .................... ............... ..... 14

3.3

Case 2: 0.9x > h .................... .............................. .................... .................... .................... .................... .................... .................... ............. ... 15

3.4

Combination of Cases........................... Cases..................................... .................... .................... .................... .................... ................. ....... 16

3.5

Description of Interaction Diagram......... Diagram ................... .................... .................... .................... .................... .............. .... 18

3.6

Example ................... ............................. ................... ................... .................... .................... .................... ................... ................... .................. ........ 21

3.7

Design Using Standard Interaction Diagrams .................... .............................. ..................... ................. ...... 30

Biaxial Bending.......... Bending .................... .................... ................... ................... .................... .................... .................... .................... ................... ......... 33

4.1

Introduction................... Introduction............................. .................... .................... ................... ................... .................... .................... ................... ............ ... 33

4.2

Method 1 – Resolution of Moments .................... .............................. ..................... ..................... .................... ............ 34

4.3

Method 2 – 3-D Interaction Diagrams........................... Diagrams..................................... ................... ................... ............. ... 37

4.4

Method 3 – Code Methods .................... .............................. .................... .................... .................... .................... ................ ...... 39

4.5

Example ................... ............................. ................... ................... .................... .................... .................... ................... ................... .................. ........ 41

2

 Dr. C. Caprani

Civil Engineering Design (1)

1. Introduction 1.1

Background 

The two main parameters governing column design are:

• Bracing: if the column can sway additional moments are generated through the P − δ  effect. This does not affect braced columns

• Slenderness ratio: The effective length divided by the lateral dimension of the column. Low values indicate a crushing failure, while high values denote  buckling.

Bracing

The two kinds of bracing are:

• Braced: Lateral loads on the structure are resisted by elements other than the columns in the structure; e.g. lift cores, shear walls. Hence these columns mainly take axial load; however, bending moments may result in the columns due to unsymmetrical arrangements of loads.

• Unbraced: Lateral loads are resisted by the bending action of the columns. The axial loads are also taken by the columns.

Slenderness ratio

The slenderness ratio is the ratio of the effective length le to the lateral dimension of  the column in that direction. We have two directions on plan, and we can have two ratios:

• Long side of column: slenderness ratio = lex h ; • Short side of column: slenderness ratio = ley b .

3

 Dr. C. Caprani

Civil Engineering Design (1)

Y 

   h

 X

X 

Y  b 

Standard Column

The effective length is calculated from a table in BS8110 and depends on the endconditions of the column, for each axis:

4

 Dr. C. Caprani

Civil Engineering Design (1)

1.2

Failure Modes

We define:

•  N uz as the crushing load of a perfectly axially loaded column; •  N crit  as Euler’s buckling load for the column. The following graph is typical and shows how for different slenderness ratios, different forms of failure are possible:

5

 Dr. C. Caprani

Civil Engineering Design (1)

From the figure:

• For l h ≤ 15 the crushing capacity is much lower than the buckling capacity and so the column crushes.

• For l h > 32 the buckling capacity is less than the crushing capacity and so the column buckles.

• For  l h values in between, the failure mode is not clear and depends on imperfections in the column and the way the load is applied.

Therefore, BS8110 defines:

• Short columns: l h ≤ 15 for a braced column and l h ≤ 10 for an unbraced column. These ratios must be met for both axes.

• Slender columns: any column not meeting the criteria for short columns.

As this module is short, we will only consider short, braced columns.

6

 Dr. C. Caprani

Civil Engineering Design (1)

1.3

Design Aspects

At SLS, the axial compression tends to close up the cracks, which is beneficial. Hence crack widths do not normally need to be checked in columns (crack widths are checked in beams and slabs in water retaining structures and bridges, but don't need to be for columns). Further, deflections do not normally need to be checked in columns.

It is ULS conditions that govern the design of columns, i.e., their strength.

Areas of reinforcement are given for reference:

2

Cross Sectional areas of groups of bars (mm ) Bar Size (mm)

6

8

1 2 3   s   r   a 4    b    f   o 5   r   e 6    b   m 7   u    N 8 9 10 Circumference

28 57 85 113 141 170 198 226 254 283 18.8

50 101 151 201 251 302 352 402 452 503 25.1

10 79 157 236 314 393 471 550 628 707 785 31.4

12 113 226 339 452 565 679 792 905 1018 1131 37.7

7

16 201 402 603 804 1005 1206 1407 1608 1810 2011 50.3

20 314 628 942 1257 1571 1885 2199 2513 2827 3142 62.8

25 491 982 1473 1963 2454 2945 3436 3927 4418 4909 78.5

32 804 1608 2413 3217 4021 4825 5630 6434 7238 8042 100.5

40 1257 2513 3770 5027 6283 7540 8796 10053 11310 12566 125.7

 Dr. C. Caprani