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Analysis-Design-of-Tension-Member

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 CHAPTER
                    ANALYSIS & DESIGN OF
  04                TENSION MEMBER




4.1     INTRODUCTION
        Tension member is a structural steel member subjected to axial tensile load. Tension member occur
        usually at roof trusses, truss bridge, suspension cable, stayed cable, lateral bracing, threaded road etc.
        Tension member is connected with bolt or rivet, welded using gusset plate at the end of member.


        This chapter describes the analysis and design procedure of tension member such as tension member
        with bolt end connection, tension member with weld end connection, threaded road usually used in roof
        and tension member with pin connection.




4.2     NET AREA OF TENSION MEMBER
        4.2.1      GENERAL
        If tension member is connected with bolt fastener its gross section is reduced because of the holes of
        bolt. Based on the theory of elasticity there is tensile stress concentration in the holes area, the
        stress becomes 3 times of the average stress on the net area.



             T                                             T    T                                   T




                                      FIGURE 4.1   STRESS DISTRIBUTION ON NET AREA


        4.2.2      BOLT HOLES
        The most usually method to cut the bolt holes are by punch the holes 1.6 mm (1/16 inch) larger than
        the bolt diameter.


        The diameter of bolt holes then becomes :


                                 dh = db + 2(1.6 ) = db + 3.2                          [4.1]


        where :
        dh           = diameter of holes
        db           = diameter of bolt




                                                                                                             4-1
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        4.2.3      UN-STAGGERED HOLES




             T                                                    T




                               FIGURE 4.2     NET AREA WITH UN-STAGGERED HOLES


        A net area of tension member with un-staggered holes is :


                                       A n = A g − ∑ (dht )                                    [4.2]


        where :
        An           = net area
        Ag           = gross area
        t            = thickness of hole (usually same as plate thickness)
        dh           = diameter of holes


        4.2.4      STAGGERED HOLES
        If the staggered holes are used then there is more than one potential failure. The critical section is in
        the section has the minimum net area.


        The net length of failure section is :


                                                          s2
                                      Ln = L − ∑ dh + ∑                                        [4.3]
                                                          4g


        where :
        Ln           = net length perpendicular to tensile load
        L            = gross length perpendicular to tensile load
        dh           = diameter of holes
        s            = bolt spacing parallel to tensile load          (longitudinal spacing)
        g            = bolt spacing perpendicular to tensile load (transverse spacing)
         ∑ dh        = number of holes in a failure section




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                                 FIGURE 4.3    NET AREA WITH STAGGERED HOLES


        For example see the figure above, the net length of section AB and AC is :


                                       Ln AB = L AB − dh

                                                             s2                        [4.4]
                                  Ln AC = L AB − 2(dh ) +
                                                             4g


                    s2
        The term       is used as correction length between transverse length perpendicular to tensile load and
                    4g

        inclined length. The plus sign indicates that the inclined length is greater than the transverse length.


        The net area then becomes :


                                         ⎛              s2 ⎞
                                   A n = ⎜ L − ∑ dh + ∑    ⎟t                          [4.5]
                                         ⎜              4g ⎟
                                         ⎝                 ⎠

        where :
        An           = net area
        t            = plate thickness


        or the net area can be written as :


                                                             s2
                                  A n = A g − ∑ (dht ) + ∑      t                      [4.6]
                                                             4g




                                                                                                             4-3
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        4.2.5       ANGLE, CHANNEL AND FLANGE SECTION
        If angle, channel or flange section is used as tension member the calculation of net area is also use the
        same equation. The difference is in the determination of transverse spacing g.
        TABLE 4.1    TRANSVERSE SPACING OF ANGLE, CHANNEL & FLANGE SECTION
                                                     TRANSVERSE
            TYPE                 SECTION
                                                        SPACING

                                                     ⎛     t⎞ ⎛       t⎞
                                                 g = ⎜ ga − ⎟ + ⎜ gb − ⎟
                                                     ⎝     2⎠ ⎝       2⎠

                                                 g = ga + gb − t
           ANGLE




                                                      g = ga + gb − t




          CHANNEL




                                                        gb ⎛      t ⎞
                                                   g=     + ⎜ ga − f ⎟
                                                        2 ⎝        2⎠


           FLANGE




        4.2.6       EFFECTIVE NET AREA
        Effective net area is used to account the effect of non-uniformity of tensile load acts at the member
        connection (member end). The non-uniformity of tensile load occurs when the tension member is not
        connected at all sides. The simple example of this condition is if the angle section is only
        connected at one leg, the tensile stress distribution is concentrated at the leg being connected.
        Lengthening the connected region will reduce this effect.




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                                         FIGURE 4.4    DETERMINATION OF X


        The effective net area is computed using the reduction factor U as follows :


                                                 x
                                        U = 1−     p 0.90                                      [4.7]
                                                 L
        where :
        U            = reduction factor
        x            = distance from center of connected member to plane of connection
        L            = length of connection parallel to load


        L is the distance from center to center of end bolt in the direction of loading for bolt end connection
        and the distance of weld to end of connection.
        x also can defined as eccentricity of the loading in the connection.
        Determination of x is shown in the figure above.
        The effective net area then becomes :


        TABLE 4.2    EFFECTIVE NET AREA FOR BOLT & WELD CONNECTION
                                                                   WELD
            BOLD CONNECTION
                                                               CONNECTION

                                      Load transmitted by longitudinal weld,
                                                                                            A e = UA g
                                      or by combination of longitudinal & transverse weld

                                      Load transmitted by transverse weld                   A e = A con
                                      Load transmitted by longitudinal weld along both
                A e = UAn             side of plate

                                                                                            A e = UA g
                                      Lw ≥ w                                   U = 1.00

                                      2w > Lw ≥ 1.5w                           U = 0.87

                                      1.5w > Lw ≥ w                            U = 0.75




                                                                                                                    4-5
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                                             FIGURE 4.5   LENGTH OF WELDS


        where :
        Ae           = effective net area
        An           = net area
        Ag           = gross area
        Acon         = area of directly connected element
        U            = reduction factor
        Lw           = length of pair of welds
        w            = distance between longitudinal welds




        The length of connection for bolt end connection and weld connection is shown in the figure below :




                                      FIGURE 4.6   LENGTH OF CONNECTION




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        Approximately the reduction factor can be determined based on the AISC Commentary, as follows :




        TABLE 4.3      APPROXIMATE REDUCTION FACTOR U
                                                                   BOLD         WELD
                             CATEGORIES
                                                                 CONNECTION   CONNECTION

                             bf 2
          WF section with      >
                             d    3                                            U = 0.90
                                                                  U = 0.90
          Connected at the flanges,
          (Minimum three bolts per line in the load direction)
          All shapes including built up shape
                                                                  U = 0.85     U = 0.85
          (Minimum three bolts per line in the load direction)
          All shapes
                                                                  U = 0.75        –
          (Minimum two bolts per line in the load direction)




4.3     ANALYSIS OF TENSION MEMBER – BOLT END CONNECTION
        4.3.1       GENERAL
        Tension member with bolt end connection is a tension member connected by bolt at the end of
        member. The bolt holes reduces the section area at the connection and this area is a critical section of
        failure.


        The strength of tension member with bolt end connection based on the LRFD is the minimum of three
        categories below, as follows :
             Yielding of Gross Section, yielding of tension member over the member length away from the
             connection.
             Fracture of Effective Net Section, fracture of tension member in the connection region.
             Block Shear Rupture, tearing out the connection these is combination of tension and shear
             failure.


        4.3.2       YIELDING OF GROSS SECTION
        The nominal strength of tension member based on the yielding of gross section or section area
        without holes is :


                                          Tn = Fy A g                                     [4.8]


        where :
        Tn             = nominal strength of tension member
        Fy             = steel yield strength
        Ag             = gross section area


        The design strength of tension member then becomes :


                                                 (
                                      φT Tn = φT Fy A g   )
                                                                                          [4.9]
                                          φT = 0.90




                                                                                                               4-7
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        where :
        φT           = strength reduction factor of tension member




        Yielding of gross section occurs over the member length away from the connection.
        4.3.3      FRACTURE OF EFFECTIVE NET SECTION
        The nominal strength of tension member based on the fracture of effective net section or section
        area with holes is :


                                          Tn = Fu A e                                [4.10]


        where :
        Tn           = nominal strength of tension member
        Fu           = steel tensile strength
        Ae           = effective net section area


        The design strength of tension member then becomes :


                                      φT Tn = φT (Fu A e )
                                                                                     [4.11]
                                          φT = 0.75


        where :
        φT           = strength reduction factor of tension member


        Fracture of effective net area occurs at the connection through the holes. When the member yield at
        gross section, the connection may deform with large elongation and reach the strain hardening
        region with its tensile strength.


        The strength reduction factor is lower than when yielding occurs because when the large elongation
        condition in net section is more dangerous.


        4.3.4      BLOCK SHEAR RUPTURE
        The block shear rupture (BSR) occurs when the bolt end connection is tearing out. The block shear
        strength of the section is provided by the tensile yield strength and shear rupture strength.


        Combination of shear and tension tearing is uncommon in tension member design, it is useful when we
        design the bolt end connection to determine the minimum distance of end bolt to edge of gusset
        plate.


        The block shear rupture controls for short connection.




        The following figure shows the block shear rupture of bolt end connection.




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                         FIGURE 4.7    BLOCK SHEAR RUPTURE – BOLT END CONNECTION


        LRFD provides two modes of block shear failure, as follows :
              Fracture of Tensile Plane followed by Yielding of Shear Plane
              Fracture of Shear Plane followed by Yielding of Tensile Plane


        TABLE 4.4    BLOCK SHEAR STRENGTH – BOLT END CONNECTION
                                       SHEAR YIELDING                 SHEAR FRACTURE
              CONDITION
                                      TENSION FRACTURE                TENSION YIELDING

          Fu A nt ≥ 0.6Fu Anv           (           )
                                  Rn = 0.6Fy A gv + (Fu A nt )                  –

          0.6Fu A nv ≥ Fu A nt                  –                                       (
                                                                   Rn = (0.6Fu A nv ) + Fy A gt   )

                                                                 φT = 0.75

        where :
        Rn           = block shear strength
        Fu           = steel tensile strength
        Fy           = steel yield strength
        Agt          = gross section area at tension plane
        Agv          = gross section area at shear plane
        Ant          = net section area at tension plane
        Anv          = net section area at shear plane


        The fracture is occurs after yielding has taken place, so the equation used has the greater ratio of
        fracture strength to yield strength.
4.4     ANALYSIS OF TENSION MEMBER – WELD END CONNECTION
        4.4.1       GENERAL
        Tension member with weld end connection is a tension member connected by longitudinal and
        transverse welds at the end of member. Because there are no bolt holes then the section area is not
        reduced so the gross section is used to determine the tensile strength.


        The strength of tension member with weld end connection based on the LRFD is the minimum of three
        categories below, as follows :




                                                                                                                           4-9
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              Yielding of Gross Section, yielding of tension member over the member length away from the
              connection.
              Fracture of Effective Section, fracture of tension member in the connection region.
              Block Shear Rupture, tearing out the connection these is combination of tension and shear
              failure.


         4.4.2     YIELDING OF GROSS SECTION
         The nominal strength of tension member based on the yielding of gross section or section area
         without holes is :


                                         Tn = Fy A g                                [4.12]


         where :
         Tn          = nominal strength of tension member
         Fy          = steel yield strength
         Ag          = gross section area


         The design strength of tension member then becomes :


                                                  (
                                      φT Tn = φT Fy A g   )
                                                                                    [4.13]
                                          φT = 0.90


         where :
         φT          = strength reduction factor of tension member


         Yielding of gross section occurs over the member length away from the connection.
         4.4.3     FRACTURE OF EFFECTIVE SECTION
         The nominal strength of tension member based on the fracture of effective section or section area
         with holes is :


                                          Tn = Fu A e                               [4.14]


         where :
         Tn          = nominal strength of tension member
         Fu          = steel tensile strength
         Ae          = effective net section area


         The design strength of tension member then becomes :


                                      φT Tn = φT (Fu A e )
                                                                                    [4.15]
                                          φT = 0.75


         where :
         φT          = strength reduction factor of tension member




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        Fracture of effective area occurs at the end of connection. When the member yield at gross section,
        the connection may deform with large elongation and reach the strain hardening region with its
        tensile strength.


        The strength reduction factor is lower than when yielding occurs because when the large elongation
        condition in net section is more dangerous.


        4.4.4       BLOCK SHEAR RUPTURE
        As in bolt end connection, LRFD provides two modes of block shear failure, as follows :
              Fracture of Tensile Plane followed by Yielding of Shear Plane
              Fracture of Shear Plane followed by Yielding of Tensile Plane


        The following figure shows the block shear rupture of weld end connection.




                         FIGURE 4.8    BLOCK SHEAR RUPTURE – WELD END CONNECTION


        TABLE 4.5    BLOCK SHEAR STRENGTH – WELD END CONNECTION
                                       SHEAR YIELDING                  SHEAR FRACTURE
              CONDITION
                                      TENSION FRACTURE                 TENSION YIELDING

          Fu A gt ≥ 0.6Fu A gv          (         ) (
                                  Rn = 0.6Fy A gv + Fu A gt   )                 –

          0.6Fu A gv ≥ Fu A gt                –                          (          ) (
                                                                    Rn = 0.6Fu A gv + Fy A gt   )

                                                                  φT = 0.75

        where :
        Rn           = block shear strength
        Fu           = steel tensile strength
        Fy           = steel yield strength
        Agt          = gross section area at tension plane
        Agv          = gross section area at shear plane


        The fracture is occurs after yielding has taken place, so the equation used has the greater ratio of
        fracture strength to yield strength.



                                                                                                                       4 - 11
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4.5      ANALYSIS OF TENSION MEMBER – THREADED ROD
         4.5.1       GENERAL
         Threaded rod is a tension member commonly used in the roof structure to stiffened the purlin.
         Threaded rod is a round section with thread at the end of the member, the thread reduces the gross
         section area.


         4.5.2       TENSILE STRENGTH
         The nominal strength of threaded road is :


                                      Tn = 0.75 A g (0.75Fu )
                                                                                  [4.16]
                                            φT = 0.75


         where :
         Tn           = nominal strength tension member
         Ag           = gross section area of rod
         0.75Ag       = effective section area of rod (stress area)
         Fu           = steel tensile strength


         The equation above also can be used to design the tension cable.




                                      FIGURE 4.9   TRIBUTARY AREA OF SAG ROD




         As previously mentioned that rod commonly used in roof truss its called sag rod. The sag rod is
         assumed resist the entire load below the rod. As shown in the figure above the sag rod (thick line)
         is designed due to load based on the shaded tributary area.
4.6      ANALYSIS OF TENSION MEMBER – PIN CONNECTED MEMBER
         4.6.1       GENERAL
         Pin connected member is tension member that connected using massive steel pin at the connection,
         this pin is placed through the holes. Pin connection is a moment – free connection. Design of pin
         connected member is based on the gross section.


         The strength of pin connected member based on the LRFD is the minimum of four categories below, as
         follows :



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             Yielding of Gross Section, yielding of tension member over the member length away from the
             connection.
             Fracture of Effective Net Section, fracture of tension member in the connection region.
             Shear on Effective Section, tearing out the connection these is a shear failure.
             Bearing, bearing strength of the gusset plate.


        4.6.2      YIELDING OF GROSS SECTION
        The nominal strength of pin connected member based on the yielding of gross section or section
        area without holes is :


                                            Tn = Fy A g                              [4.17]


        where :
        Tn           = nominal strength of tension member
        Fy           = steel yield strength
        Ag           = gross section area


        The design strength of tension member then becomes :


                                                     (
                                        φT Tn = φT Fy A g    )
                                                                                     [4.18]
                                            φT = 0.90


        where :
        φT           = strength reduction factor of tension member
        Yielding of gross section occurs over the member length away from the connection.
        4.6.3      FRACTURE OF EFFECTIVE NET SECTION
        The nominal strength of pin connected member based on the fracture of net section or section
        area with holes is :


                                         Tn = (2beff t )Fu
                                                                                     [4.19]
                                      b eff = 2t + 0.63 ≤ b


        where :
        Tn           = nominal strength of tension member
        Fu           = steel tensile strength
        t            = thickness of connected member


        The design strength of pin connected member then becomes :


                                      φT Tn = φT [(2beff t )Fu ]
                                                                                     [4.20]
                                            φT = 0.75


        where :
        φT           = strength reduction factor of tension member




                                                                                                         4 - 13
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                              FIGURE 4.10       FRACTURE OF EFFECTIVE NET SECTION


         4.6.4     SHEAR OF EFFECTIVE SECTION
         The nominal strength of pin connected member based on the shear of effective section is :


                                         Tn = (2tLh )0.6Fu                          [4.21]
         where :
         Tn          = nominal strength of tension member
         Fu          = steel tensile strength
         t           = thickness of connected member
         Lh          = distance from edge of member to center of hole


         The design strength of pin connected member then becomes :


                                      φT Tn = φT [(2tLh )0.6Fu ]
                                                                                    [4.22]
                                             φT = 0.75


         where :
         φT          = strength reduction factor of tension member




                                      FIGURE 4.11    SHEAR OF EFFECTIVE SECTION




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        4.6.5      BEARING STRENGTH
        The bearing strength is based on the strength of connected part due to bearing load. If the strength
        of pin is greater than strength of connected part, the connected member will be deformed.




                                        FIGURE 4.12   BEARING STRENGTH


        The nominal strength of pin connected member based on the bearing strength is :


                                         Tn = 1.8Fy dht                              [4.23]


        where :
        Tn            = nominal strength of tension member
        Fy            = steel yield strength
        t             = thickness of connected member
        dh            = diameter of hole


        The design strength of pin connected member then becomes :


                                                (
                                      φTTn = φT 1.8Fy dht   )
                                                                                     [4.24]
                                          φT = 0.75


        where :
        φT            = strength reduction factor of tension member




4.7     DESIGN OF TENSION MEMBER
        4.7.1      GENERAL
        Design is a state of the art (SOA) rather than a science. In a design process a structural engineer
        combine the ability of analysis, engineering judgment, experience, construction method, economic
        design etc.


        The design procedure is similar as analysis, we try to find the required steel section for a tension
        member. Design cannot be done if engineer do not know the basic concept of analysis.
        There are two consideration when a tension member is designed, as follows :
             Strength, the tension member must adequate to resist the ultimate axial tensile load.
             Stiffness, the tension member must not fail due to serviceability requirements.




                                                                                                          4 - 15
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         4.7.2          STRENGTH CONSIDERATION
         The basic equation of tension member design is :


                                              φT Tn ≥ Tu                                          [4.25]
         where :
         Tn                = nominal strength of tension member
         Tu                = ultimate axial tensile load
         φT                = strength reduction factor of tension member


         TABLE 4.6         DESIGN OF TENSION MEMBER – STRENGTH CONSIDERATION
                                      BOLT END                  WELD END                 THREADED
             CONDITION
                                    CONNECTION                 CONNECTION                   ROD

                                         Tu    Tu                    Tu    Tu
             Yielding of         Ag ≥       =                Ag ≥       =                     –
                                        φTFy 0.90Fy                 φTFy 0.90Fy
                   Ag

                                                                                                Tu
                                                                                  Ag ≥
                                         Tu    Tu                    Tu    Tu            φT [0.75(0.75Fu )]
             Fracture of         Ae ≥       =                Ae ≥       =
                                        φTFu 0.75Fu                 φTFu 0.75Fu                  Tu
                A e / Ag                                                          Ag ≥
                                                                                         0.75[0.75(0.75Fu )]



         After a preliminary section area is known, we can choose a steel section and then check the
         strength if the basic equation of tension member is not achieved try another section.


         For preliminary design the maximum net section area is can be taken as :
                                            A n ≥ 0.85 A g                                        [4.26]


         The maximum net section area reduced by bolt holes is not permitted less than 85% of gross section
         area.
         4.7.3          STIFFNESS CONSIDERATION
         The stability effect is not major in the tension member but it is still necessary to limit the length of
         tension member to prevent the following items, as follows :
               Erection Purpose, if the tension member too flexible it may be deform during erection.
               Vibrate, to prevent the tension member is vibrating.
               Sag, if the tension member is too long it may be sag excessively due to its self weight.


         To control the stiffness of tension member the slenderness ratio is used. This control is used to
         provide enough bending strength during fabrication, shipping and erection.


         The maximum slenderness ratio of tension member is :


                                               L
                                                 ≤ 300                                            [4.27]
                                               r
         where :
         L                 = length of tension member
         r                 = minimum radius of gyration



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        The minimum radius of gyration of principal axis is defined as :


                                                    Iy
                                           ry =                                        [4.28]
                                                    A


        where :
        Iy           = minimum moment of inertia
        A            = gross section area


        The following consideration above is cannot applied to design a tension rod.
        For tension rod to the minimum of rod diameter for stiffness consideration is :


                                                   5
                                        dr min =     inch
                                                   8                                   [4.29]
                                        dr min   = 16mm




4.8     STEP – BY – STEP PROCEDURES
        The followings are step-by-step procedure can be used as a guide for design of tension member with
        bolt end connection, as follows :
             Determine the ultimate axial tensile load Tu from elastic structural analysis.
             Choose the steel section, calculate the gross section area Ag and effective net area Ae.
             Calculate the nominal and design tensile strength based on the yielding of Ag and fracture of Ae.


                                      NOMINAL
              CONDITION                                      φ
                                      STRENGTH
              Yielding of
                                      Tn = Fy A g           0.90
                    Ag
              Fracture of
                                      Tn = Fu A e           0.75
                    Ae


             Calculate the block shear rupture at the end of connection.
             Check for the stiffness consideration, this is the maximum slenderness ratio.


                                                             L
                                                               ≤ 300
                                                             r


             Repeat the design process until the basic equation is achieved.


                                                            φT Tn ≥ Tu


        Appendix B shows the nominal strength and design strength of tension member for many of
        steel profile usually used for tension member such as angle section, C-channel section and U-
        channel section.




                                                                                                            4 - 17

				
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