Reinforced Concrete Design-I Design of Axial members By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt.

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 Types: Tied Spiral Composite Combination Steel pipe

Analysis and Design of “Short” Columns Tied 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 An elastic analysis using the transformed section method would be: For concentrated load, P uniform stress over section n = Es / Ec Ac = concrete area As = steel area

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 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 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 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 Initial Behavior up to Nominal Load - Tied and spiral columns. 1.

Behavior, Nominal Capacity and Design under Concentric Axial loads

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 ) ACI 10.3.6.3

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 min. of 3 bars in triangular ties

Behavior, Nominal Capacity and Design under Concentric Axial loads 3. Reinforcement Requirements (Lateral Ties) ACI Code 7.10.5.1 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: (ACI 7.10.5.2) 16 db ( db for longitudinal bars ) 48 db ( db for tie bar ) least lateral dimension of column s s s

Behavior, Nominal Capacity and Design under Concentric Axial loads 3. Reinforcement Requirements (Lateral Ties) Arrangement Vertical spacing: (ACI 7.10.5.3) 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. 1.) 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 “ dia. (3/8” f smooth bar, #3 bar dll or wll wire) clear spacing between spirals 1 in. 3 in. ACI 7.10.4.3

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. 10-5 where

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

Behavior, Nominal Capacity and Design under Concentric Axial loads 4. Design for Concentric Axial Loads (b) General Strength Requirement where, f = 0.65 for tied columns f = 0.7 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.

Example: Design Tied Column for Concentric Axial Load Determine the loading Check the compression or tension in the column

Example: Design Tied Column for Concentric Axial Load For a square column r = 0.80 and f = 0.65 and r = 0.03

Example: Design Tied Column for Concentric Axial Load For a square column, As=rAg= 0.03(15.2 in.)2 =6.93 in2 Use 8 #8 bars Ast = 8(0.79 in2) = 6.32 in2

Example: Design Tied Column for Concentric Axial Load Check P0

Example: Design Tied Column for Concentric Axial Load Use #3 ties compute the spacing < 6 in. No cross-ties needed

Example: Design Tied Column for Concentric Axial Load Stirrup design Use #3 stirrups with 16 in. spacing in the column

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 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

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

Columns in Pure Tension Section is completely cracked (no concrete axial capacity) Uniform Strain

Columns Strength Reduction Factor, f (ACI Code 9.3.2) (a) Axial tension, and axial tension with flexure. f = 0.9 Axial compression and axial compression with flexure. (b) Members with spiral reinforcement confirming to 10.9.3 f = 0.70 Other reinforced members f = 0.65

Columns Except for low values of axial compression, f may be increased as follows: when and reinforcement is symmetric and ds = distance from extreme tension fiber to centroid of tension reinforcement. Then f may be increased linearly to 0.9 as fPn decreases from 0.10fc Ag to zero.

Column

Columns Commentary: Other sections: f may be increased linearly to 0.9 as the strain es increase in the tension steel. fPb

Design for Combined Bending and Axial Load (Short Column) Design - select cross-section and reinforcement to resist axial load and moment.

Design for Combined Bending and Axial Load (Short Column) Column Types 1) Spiral Column - more efficient for e/h < 0.1, but forming and spiral expensive Tied Column - Bars in four faces used when e/h < 0.2 and for biaxial bending 2)

General Procedure The interaction diagram for a column is constructed using a series of values for Pn and Mn. The plot shows the outside envelope of the problem.

General Procedure for Construction of ID Compute P0 and determine maximum Pn in compression Select a “c” value (multiple values) Calculate the stress in the steel components. Calculate the forces in the steel and concrete,Cc, Cs1 and Ts. Determine Pn value. Compute the Mn about the center. Compute moment arm,e = Mn / Pn.

General Procedure for Construction of ID Repeat with series of c values (10) to obtain a series of values. Obtain the maximum tension value. Plot Pn verse Mn. Determine fPn and fMn. Find the maximum compression level. Find the f will vary linearly from 0.65 to 0.9 for the strain values The tension component will be f = 0.9

Example: Axial Load vs. Moment Interaction Diagram Consider an square column (20 in x 20 in.) with 8 #10 (r = 0.0254) and fc = 4 ksi and fy = 60 ksi. Draw the interaction diagram.

Example: Axial Load vs. Moment Interaction Diagram Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi

Example: Axial Load vs. Moment Interaction Diagram Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi [ Point 1 ]

Example: Axial Load vs. Moment Interaction Diagram Determine where the balance point, cb.

Example: Axial Load vs. Moment Interaction Diagram Determine where the balance point, cb. Using similar triangles, where d = 20 in. – 2.5 in. = 17.5 in., one can find cb

Example: Axial Load vs. Moment Interaction Diagram Determine the strain of the steel

Example: Axial Load vs. Moment Interaction Diagram Determine the stress in the steel

Example: Axial Load vs. Moment Interaction Diagram Compute the forces in the column

Example: Axial Load vs. Moment Interaction Diagram Compute the forces in the column

Example: Axial Load vs. Moment Interaction Diagram Compute the moment about the center

Example: Axial Load vs. Moment Interaction Diagram A single point from interaction diagram, (585.6 k, 556.9 k-ft). The eccentricity of the point is defined as [ Point 2 ]

Example: Axial Load vs. Moment Interaction Diagram Now select a series of additional points by selecting values of c. Select c = 17.5 in. Determine the strain of the steel. (c is at the location of the tension steel)

Example: Axial Load vs. Moment Interaction Diagram Compute the forces in the column

Example: Axial Load vs. Moment Interaction Diagram Compute the forces in the column

Example: Axial Load vs. Moment Interaction Diagram Compute the moment about the center

Example: Axial Load vs. Moment Interaction Diagram A single point from interaction diagram, (1314 k, 351.1 k-ft). The eccentricity of the point is defined as [ Point 3 ]

Example: Axial Load vs. Moment Interaction Diagram Select c = 6 in. Determine the strain of the steel, c =6 in.

Example: Axial Load vs. Moment Interaction Diagram Compute the forces in the column

Example: Axial Load vs. Moment Interaction Diagram Compute the forces in the column

Example: Axial Load vs. Moment Interaction Diagram Compute the moment about the center

Example: Axial Load Vs. Moment Interaction Diagram A single point from interaction diagram, (151 k, 471 k-ft). The eccentricity of the point is defined as [ Point 4 ]

Example: Axial Load vs. Moment Interaction Diagram Select point of straight tension. The maximum tension in the column is [ Point 5 ]

Example: Axial Load vs. Moment Interaction Diagram Point c (in) Pn Mn e 1 - 1548 k 0 0 2 20 1515 k 253 k-ft 2 in 3 17.5 1314 k 351 k-ft 3.2 in 4 12.5 841 k 500 k-ft 7.13 in 5 10.36 585 k 556 k-ft 11.42 in 6 8.0 393 k 531 k-ft 16.20 in 7 6.0 151 k 471 k-ft 37.35 in 8 ~4.5 0 k 395 k-ft infinity 9 0 -610 k 0 k-ft

Example: Axial Load vs. Moment Interaction Diagram Use a series of c values to obtain the Pn verses Mn.

Example: Axial Load vs. Moment Interaction Diagram Max. compression Location of the linearly varying f. Cb Max. tension