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					Design of Concrete                                University of
   Structure II          ‫بسم هللا الرحمن الرحيم‬    Palestine




                     Lecture # 3

                           Instructor:
                     Eng. Mazen Alshorafa
Design of Concrete                                                    University of
   Structure II                                                        Palestine
                                       Columns

     According to ACI Code a structural element with a ratio of
     height-to       least   lateral   dimension     exceeding    three   used
     primarily to support compressive loads is defined as column.

                                           P
                                       h
                                           b
                             h                   l

                                  b




       Instructor:
                                                                 Page 1
       Eng. Mazen Alshorafa
Design of Concrete                                                   University of
   Structure II                                                       Palestine
                                   Columns

     Columns are vertical compression members of a structural frame
     intended to support the load-carrying beams. They transmit loads
     from the upper floors to the lower levels and then to the soil through
     the foundations.

                                                             Loads
      Column


                                                                         Column


                                           Beam
      Sec A                        Sec A




                                                      Sec A-A
                     Main beam




       Instructor:
                                                            Page 2
       Eng. Mazen Alshorafa
Design of Concrete                                              University of
   Structure II                                                  Palestine
                                     Columns

     Usually columns carry bending moment as well, about one or both
     axes of the cross section, and the bending action may produce tensile
     forces over a part of the cross section




     The main reinforcement in columns is
     longitudinal, parallel to the direction of
     the load and consists of bars arranged
      in a square, rectangular, or circular
       Instructor:
                                                           Page 3
       Eng. Mazen Alshorafa
Design of Concrete                                                        University of
   Structure II                                                            Palestine
                                  Types of Columns

     1- Form and arrangement of reinforcement

     Columns are divided into three types

     1- Tied Columns

     It is a column in which the longitudinal reinforcement bars are tied together

     with separate smaller diameter transverse bars (ties) spaced         at   some

     interval along the column height. (Figure a)

     2- Spirally-Reinforced Columns

     It is a column in which the longitudinal bars are arranged in a circle

     surrounded by a closely spaced continuous spiral. (Figure b)

     3- Composite Columns

     It is a column made of structural steel shapes or pipes surrounded by or filled

     by concrete with or without longitudinal reinforcement. (Figure c)

       Instructor:
                                                                    Page 4
       Eng. Mazen Alshorafa
Design of Concrete                                    University of
   Structure II                                        Palestine
                              Types of Columns




       Instructor:
                                                 Page 5
       Eng. Mazen Alshorafa
Design of Concrete                                                    University of
   Structure II                                                        Palestine
                               Types of Columns

     2- Length of the column in relation to its lateral dimensions.

     Columns may be divided into two categories

     1- Short Columns, for which the strength is governed by the
     strength of the materials and the geometry of the cross section
      2- Slender columns, for which the strength may be significantly
     reduced by lateral deflections.

     3- Position of the load on the cross-section

     Columns can be classified as

     1-Concentrically loaded columns, are subjected to axial force only
     2-Eccentrically loaded columns, are subjected to moment in addition
     to the axial force.


       Instructor:
                                                              Page 6
       Eng. Mazen Alshorafa
Design of Concrete                                                      University of
   Structure II                                                          Palestine
                                        Columns

     Behavior of Tied and Spirally-Reinforced Columns




                              Failure of a tied column   Failure of a spiral column

                                           Deformation

       Instructor:
                                                                  Page 7
       Eng. Mazen Alshorafa
Design of Concrete                                            University of
   Structure II                                                Palestine
                                   Columns

     Factored Loads and Strength Reduction Factors

     Factored Loads

     For gravity loads only,
                              Pu = 1.2 PD+1.6 PL
     For dead, live and wind loads,
                          Pu = 1.2 PD+1.0 PL+1.6 PW
     For dead and wind loads,
               Pu = 0.9 PD + 1.6 PW    or Pu = 1.2 PD + 0.8 PW
     For dead, live and earthquake loads,
                          Pu = 1.2 PD+1.0 PL+1.0 PE
     For dead and earthquake loads,
                              Pu = 0.9 PD + 1.0 PE

       Instructor:
                                                         Page 8
       Eng. Mazen Alshorafa
Design of Concrete                                                University of
   Structure II                                                    Palestine
                                     Columns

     Strength Reduction Factors

     ACI Code specifies Φ values or strength reduction factors for most
     situations as in the following table
         Strength condition                                       Φ
         Tension-controlled sections (εt ≥ 0.005)                0.90
         Compression-controlled sections (εt ≤ 0.002)
             Members with spiral reinforcement                  0.70
             Other reinforced members                           0.65




       Instructor:
                                                            Page 9
       Eng. Mazen Alshorafa
Design of Concrete                                                          University of
   Structure II                                                              Palestine
                               Sway and Nonsway Frames

     1- Nonsway Frames (braced)
     It is a structural frames whose joints are restrained against lateral
     displacement by attachment to rigid elements or bracing
                 Columns           Shear wall    Columns                Brace X




                       Beams
                                                           Beams




                                                            P
     According to ACI Code                                          M
      a column in a structure is nonsway if
                                                             ∆
                                                      lc
          Secondary moment Pu 
                                   0.05
           Primary momen t   vu lc                                 M
                                                            P
       Instructor:
                                                                   Page 10
       Eng. Mazen Alshorafa
Design of Concrete                                                      University of
   Structure II                                                          Palestine
                                 Sway and Nonsway Frames

     1- Nonsway Frames (braced)
     Moreover, ACI Code assumes a story within a structure is nonsway if:




             Q
                 P   0.05
                        u

                    Vu lc

     Where,
     Q is the stability index which is the ratio of secondary moment due
        to lateral displacement and primary moment,
     ΣPu is the total vertical load in the story,
     Vu     is the story shear in the story under consideration,
     Lc     is length of column measured center-to center of the joints in the
            frame, and
     Δ      is the first-order relative deflection between the top and bottom
            of that story.

          Instructor:
                                                                   Page 11
          Eng. Mazen Alshorafa
Design of Concrete                                                 University of
   Structure II                                                     Palestine
                              Sway and Nonsway Frames

     2- Sway Frames (Unbraced)
     Structural frames, not attached to an effective bracing element, but
     depend on the bending stiffness of the columns and girders to provide
     resistance to lateral displacement are called “sway frames”




       Instructor:
                                                            Page 12
       Eng. Mazen Alshorafa
Design of Concrete                                                University of
   Structure II                                                    Palestine
                                Slenderness effect

     The slenderness of columns is based on their geometry and on their
     lateral bracing. As their slenderness increases, their bending stresses
     increase, and thus buckling may occur.
     Several items involved in the calculation of slenderness ratios, these
     item unsupported column lengths, effective length factors and radii of
     gyration.

     Unsupported lengths (lu)

     It is clear distance between floor slabs, beams, or other members
     capable of providing lateral support as shown in figure.




       Instructor:
                                                            Page 13
       Eng. Mazen Alshorafa
Design of Concrete                                                   University of
   Structure II                                                       Palestine
                                  Slenderness effect
     Effective length factors K

     The effective length is the distance between points of zero moment in
     the column Thus, the effective length factor k, is the ratio of the
     effective length to the original length of column.
     Typical cases illustrating the buckled shape of the column for several
     end conditions and the corresponding length factor K




         Points of
         inflection




       Instructor:
                                                              Page 14
       Eng. Mazen Alshorafa
Design of Concrete                                                 University of
   Structure II                                                     Palestine
                                  Slenderness effect
     Effective length factors K

     For members in a structural frame, the end restraint lies between the
     hinged and fixed conditions. The actual k value can be estimated from
     the Jackson and Moreland alignment charts

     The effective length factor k is a function of the relative stiffness at

     each end of the column. In these charts, k is determined as the

     intersection of a line joining the values of ψ at the two ends of the

     column. The relative stiffness of the beams and columns at each end

     of the column ψ is given by the following equation


                                    
                                       E    c   I c / lc
                                       E    b   I b / lb

       Instructor:
                                                             Page 15
       Eng. Mazen Alshorafa
Design of Concrete                                University of
   Structure II                                    Palestine
                              Braced frame




       Instructor:
                                             Page 16
       Eng. Mazen Alshorafa
Design of Concrete                                  University of
   Structure II                                      Palestine
                              Unbraced frame




       Instructor:
                                               Page 17
       Eng. Mazen Alshorafa
Design of Concrete                                                                   University of
   Structure II                                                                       Palestine
                                       Slenderness effect
     Effective length factors K


                                        
                                           E       c   I c / lc
     where,                                E       b   I b / lb
     lc = length of column center-to-center of the joints
     lb = length of beam center-to-center of the joints
     Ec = modulus of elasticity of column concrete
     Eb = modulus of elasticity of beam concrete
     Ic =moment of inertia of column cross section about an axis
     perpendicular to the plane of buckling being considered.
     Ib    =moment      of   inertia    of   beam       cross      section   about    an   axis
     perpendicular to the plane of buckling being considered.
     Σ indicates a summation of all member stiffness connected to the joint
     and lying in the plane in which buckling of the column is being
     considered

          Instructor:
                                                                             Page 18
          Eng. Mazen Alshorafa
Design of Concrete                                                         University of
   Structure II                                                             Palestine
                                         Slenderness effect
     Effective length factors K

     Consider the two-story frame shown in Figure. To determine the
     effective length factor k for column EF

                                                         C            F             I
          E ( I / h )  Ec ( I EF / h2 )
      E  c DE 1
           Eb ( I BE / l1 )  Eb ( I EH / l2 )     h2

     and                                                 B            E            H

                     Ec ( I EF / h2 )              h1
       F                                               A            D            G
            Eb ( I CF / l1 )  Eb ( I FI / l2 )
                                                                 L1         L2


     For           ψ = ------       and      for        ψ = ------

     ACI Code specifies that for columns in nonsway frames, the effective
     length factor k should be taken as 1.0

       Instructor:
                                                                      Page 19
       Eng. Mazen Alshorafa
Design of Concrete                                                  University of
   Structure II                                                      Palestine
                                  Slenderness effect
     Effective length factors K

     To calculate the ψ values it is necessary to use realistic moments of
     inertia. Usually, the girder will be appreciably cracked on their tensile
     sides, whereas the columns will probably have only a few cracks.


     In the ACI code, it is stated that for determining ψ values for use in
     evaluating K factors, the rigidity for beams = 0.35 Ig and for
     columns= 0.7 Ig as follows
     Where, Ig is the gross moment of inertia




       Instructor:
                                                              Page 20
       Eng. Mazen Alshorafa
Design of Concrete                                                     University of
   Structure II                                                         Palestine
                                  Slenderness effect
     Effective length factors K

     ACI Code provides the following simplified equations for computing
     the effective length factors for nonsway and sway frame members



     For Nonsway frames,
                              k  0.7  0.05  A   B   1.0
     K is the smaller of
                              k  0.85  0.05  min  1.0

     Where, ψA and ψB are the values of ψ at the two ends of the column,
              ψmin is the smaller of the two values.




       Instructor:
                                                                  Page 21
       Eng. Mazen Alshorafa
Design of Concrete                                                 University of
   Structure II                                                     Palestine
                                  Slenderness effect
     Effective length factors K

     For Sway frames,
     a) Restrained at both end

             For ψm > 2.0 ,
                                 20   m
                              k          1  m
                                    20
             For ψm ≥ 2.0 ,   k  0.9 1   m
     Where, ψm is the average of ψ at the two ends of the column



     b) Hinged at one end

                k  2.0  0.3 
     Where, ψ is the values at the restrained end of the column




       Instructor:
                                                           Page 22
       Eng. Mazen Alshorafa
Design of Concrete                                                  University of
   Structure II                                                      Palestine

            The ACI Procedure for Classifying Short and Slender Columns

     According to ACI Code, columns can be classified as short when their
     effective slenderness ratios satisfy the following criteria:

     For Nonsway frames

                       k lu          M
                             34  12 1  40
                        r            M2
     For sway frames

                       k lu
                             22
     Where,             r
     k = effective length factor
     lu = unsupported length of member
     r = radius of gyration, for rectangular cross sections r = 0.30 h, and
           for circular sections, r = 0.25 h
     h    = column dimension in the direction of bending.

         Instructor:
                                                               Page 23
         Eng. Mazen Alshorafa
Design of Concrete                                                 University of
   Structure II                                                     Palestine

          The ACI Procedure for Classifying Short and Slender Columns

     M1 = smaller factored end moment on column, positive if member is
           bent single curvature, negative if bent in double curvature.
     M2 = larger factored end moment on column, always positive.
     [M1/M2] = ratio of moments at two column ends [Range -1 to 1]




                                 M1                        M1
                                    0                        0
                                 M2                        M2




              Single curvature               Double curvature

       Instructor:
                                                             Page 24
       Eng. Mazen Alshorafa
Design of Concrete                                                              University of
   Structure II                                                                  Palestine

      Chart summarizes the process of column design as per the ACI Code

                                       Column Design

                     Non-sway frame                    Non-sway frame



                                           Neglect
                        k lu                            k lu          M
                              22       Slenderness           34  12 1  40
                         r                [ Short ]      r            M2



                                         Moment
                          k lu         magnification           k lu          M
                     22        100                   100          34  12 1 .
                           r             [ long ]               r            M2


                                         Exact P ∆
                       k lu                                     k lu
                             100        analysis                     100
                        r                 [ long ]               r


       Instructor:
                                                                       Page 25
       Eng. Mazen Alshorafa
Design of Concrete                                University of
   Structure II          ‫بسم هللا الرحمن الرحيم‬    Palestine




                     Example # 1

                           Instructor:
                     Eng. Mazen Alshorafa
Design of Concrete                                                                     University of
   Structure II                         ‫بسم هللا الرحمن الرحيم‬                          Palestine


       Example # 1
     The frame shown in Figure consists of members with rectangular cross
     sections, made of the same strength concrete. Considering buckling in
     the plane of the figure.
     Categorize column bc as long or short if the frame is:
     a)Nonsway              270 kN.m             0.6x0.3                     0.6x0.3
     b)Sway




                                                                 0.3x0.35
                                4.0 m




                                                 0.6x0.3                    0.6x0.3

                 400 kN.m




                                                                 0.3x0.35
                                4.5 m




                                                   9.0 m                      7.5 m


       Instructor:
                                                                            Page Ex1-1
       Eng. Mazen Alshorafa
Design of Concrete                                                          University of
   Structure II                     ‫بسم هللا الرحمن الرحيم‬                   Palestine


           Solution
     a- Nonsway
     For a column to be short,
          k lu           M1
                34  12     40
           r             M2
     Lu = 4-0.3-0.3=3.40 m
     k is conservatively taken as 1.0
         k lu    1(3.4)
                          32.38
          r     0.3(0.35)
                   M1            27 
         34  12       34  12        42.1 taken as 40  32.38
                   M2           40 
         i.e., column is classified as being short



       Instructor:
                                                                     Page Ex1-2
       Eng. Mazen Alshorafa
Design of Concrete                                                            University of
   Structure II                             ‫بسم هللا الرحمن الرحيم‬             Palestine


            Solution
     b- Sway
     For a column to be short,                        k lu
                                                            22
                          (0.3)(0.35)3               r
                     0.7                 
      C                 12 (4)                     0.406
                (0.3)(0.6) 
                            3
                                       (0.3)(0.6) 
                                                  3
           0.7                 0.7              
                12 (9)               12 (7.5) 
                (0.3)(0.35) 312            (0.3)(0.4) 3 
           0.7                      0.7  12 (4.5) 
      b            12 (4)                               0.945
                   (0.3)(0.6) 3         (0.3)(0.6)3 
             0.7                   0.7 
                     12 (9)            12 (7.5) 
                                                       

     Using the appropriate alignment chart, k = 1.21, and
      k lu 1.21(3.4)
                        39 .18  22
       r    0.3 (0.35 )
      i.e., column is classified as being slender

        Instructor:
                                                                       Page Ex1-3
        Eng. Mazen Alshorafa
Design of Concrete                                                      University of
   Structure II                                                          Palestine

                              Short axially loaded columns




     For tied reinforced columns

              Pu  0.52 Ag [0.85 fc '  g ( fy  0.85 fc ' )]
     For spirally reinforced columns

               Pu  0.595 Ag [0.85 fc '  g ( fy  0.85 fc ' )]
       Instructor:
                                                                   Page 26
       Eng. Mazen Alshorafa
Design of Concrete                                         University of
   Structure II                                             Palestine
                              Design Considerations

     Maximum and Minimum Reinforcement Ratios




       Instructor:
                                                      Page 27
       Eng. Mazen Alshorafa

				
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