Lecture - Design of Columns CEN 347

Introduction

Types of Column

Types of Column

Types of Column

Strength of short or Slender Columns

Strength of short or Slender Columns

Strength of short or Slender Columns

0.70 0.65

Analysis and Design of “Short” Columns General Information Column: Vertical Structural members Transmits axial compressive loads with or without moment transmit loads from the floor & roof to the foundation

Analysis and Design of “Short” Columns General Information Column:

Analysis and Design of “Short” Columns Tie Columns - 95% of all columns in buildings are tied Tie spacing h (except for seismic) tie support long bars (reduce buckling) ties provide negligible restraint to lateral expose of core

Analysis and Design of “Short” Columns Spiral Columns Pitch = 1.375 in. to 3.375 in. spiral restrains lateral (Poisson’s effect) axial load delays failure (ductile)

Analysis and Design of “Short” Columns Elastic Behavior Concrete creeps and shrinks, therefore we can not calculate the stresses in the steel and concrete due to “acting” loads using an elastic analysis.

Analysis and Design of “Short” Columns Elastic Behavior An elastic analysis using the transformed section method would be: For concentrated load, P uniform stress over section

Analysis and Design of “Short” Columns Elastic Behavior An elastic analysis does not work because creep and shrinkage affect the acting concrete compression strain as follows:

Analysis and Design of “Short” Columns Elastic Behavior The change in concrete strain with respect to time will effect the concrete and steel stresses as follows: Concrete stress Steel stress

Analysis and Design of “Short” Columns Elastic Behavior Therefore, we are not able to calculate the real stresses in the reinforced concrete column under acting loads over time. As a result, an “allowable stress” design procedure using an elastic analysis was found to be unacceptable. Reinforced concrete columns have been designed by a “strength” method since the 1940’s. Note: Creep and shrinkage do not affect the strength of the member.

Behavior, Nominal Capacity and Design under concentric Axial loads 1. Initial Behavior up to Nominal Load - Tied and spiral columns.

Behavior, Nominal Capacity and Design under concentric Axial loads Let Ag = Gross Area = b*h Ast = area of long steel fc =concrete compressive strength fy = steel yield strength Factor due to less than ideal consolidation and curing conditions for column as compared to a cylinder. It is not related to Whitney’s stress block.

Behavior, Nominal Capacity and Design under concentric Axial loads 2. Maximum Nominal Capacity for Design Pn (max) r = Reduction factor to account for accidents/bending r = 0.80 ( tied ) r = 0.85 ( spiral )

Behavior, Nominal Capacity and Design under concentric Axial loads 3. Reinforcement Requirements (Longitudinal Steel Ast) Let - ACI Code 10.9.1 requires

Behavior, Nominal Capacity and Design under concentric Axial loads 3. Reinforcement Requirements (Longitudinal Steel Ast) - Minimum # of Bars ACI Code 10.9.2 min. of 6 bars in circular arrangement w/min. spiral reinforcement. min. of 4 bars in rectangular arrangement

Behavior, Nominal Capacity and Design under concentric Axial loads 3. Reinforcement Requirements (Lateral Ties) ACI Code 7.10.5 size # 3 bar if longitudinal bar # 10 bar # 4 bar if longitudinal bar # 11 bar # 4 bar if longitudinal bars are bundled

Behavior, Nominal Capacity and Design under concentric Axial loads 3. Reinforcement Requirements (Lateral Ties) Vertical spacing: s s s 16 db ( db for longitudinal bars ) 48 db ( db for tie bar ) least lateral dimension of column

Behavior, Nominal Capacity and Design under concentric Axial loads 3. Reinforcement Requirements (Lateral Ties) Vertical spacing: Arrangement, 1.) At least every other longitudinal bar shall have lateral support from the corner of a tie with an included angle 135o. No longitudinal bar shall be more than 6 in. clear on either side from “support” bar. 2.)

Behavior, Nominal Capacity and Design under concentric Axial loads Examples of lateral ties.

Behavior, Nominal Capacity and Design under concentric Axial loads Reinforcement Requirements (Spirals ) ACI Code 7.10.4 - size 3/8 “ f (3/8” f smooth bar, #3 bar dll or wll wire) - clear spacing: 1 in. 3 in.

Behavior, Nominal Capacity and Design under concentric Axial loads Reinforcement Requirements (Spiral) Spiral Reinforcement Ratio, rs

Behavior, Nominal Capacity and Design under concentric Axial loads Reinforcement Requirements (Spiral) ACI Eqn. 10-6 where

Behavior, Nominal Capacity and Design under concentric Axial loads 4. Design for Concentric Axial Loads (a) Load Combination Gravity: Gravity + Wind: and Etc.

Behavior, Nominal Capacity and Design under concentric Axial loads 4. Design for Concentric Axial Loads (b) General Strength Requirement where, f = 0.7 for tied columns f = 0.75 for spiral columns

Behavior, Nominal Capacity and Design under concentric Axial loads 4. Design for Concentric Axial Loads (c) Expression for Design defined:

Behavior, Nominal Capacity and Design under concentric Axial loads

Behavior, Nominal Capacity and Design under concentric Axial loads * when rg is known or assumed: * when Ag is known or assumed:

Example: Design tied Column for concentric Axial Load Pdl = 150 k; Pll =300 k; Pw = 50 k fc =4500 psi fy = 60 ksi Design a square column aim for rg =0.03. Select longitudinal transverse reinforcement.

Behavior under Combined Bending and Axial Loads Usually moment is represented by axial load times eccentricity, i.e.

Behavior under Combined Bending and Axial Loads Resultant Forces action at Centroid ( h/2 in this case ) Moment about geometric center

Behavior under Combined Bending and Axial Loads Interaction Diagram Between Axial Load and Moment ( Failure Envelope ) Concrete crushes before steel yields Steel yields before concrete crushes Note: Any combination of P and M outside the envelope will cause failure.

Behavior under Combined Bending and Axial Loads Axial Load and Moment Interaction Diagram -General

Example: Axial Load vs. Moment Interaction Diagram