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TALAT Lecture 2301

                        Design of Members

                                 Torsion


    Example 8.2 : Torsion constants for hollow cross
                        section

                                   3 pages

                               Advanced Level

    prepared by Torsten Höglund, Royal Institute of Technology, Stockholm




Date of Issue: 1999
 EAA - European Aluminium Association

TALAT 2301 – Example 8.2                1
Example 8.2. Torsion constants for hollow cross section
a) Polygon
                                                                                                            b
Half width           b     50 . mm                            Half width of                     a                        a = 20.711 mm
                                                              flat parts
Thickness            s     6 . mm                                                                       1       2


Nodes no.,                0                     b                           a                       s
co-ordinates,
                          1                     a                           b                       s
thickness
                          2                     a                           b                       s
                          3                     b                           a                       s
                 i        4            y        b                 z         a               t       s
                          5                     a                           b                       s
                          6                     a                           b                       s
                          7                     b                           a                       s
                          8                     b                           a                       s                          50

                                                                                                                                                              0
Nodes                i     1 .. rows ( y )          1
                                                                                                                    z                                                                     0
                                                                                                                                0
Area of cross                                                                                                       mm
                                ti .
section                                                      2                          2
                     dAi                   yi   yi 1                  zi     zi 1
elements

                                                                                                                               50
Cross                         rows ( y )        1
section                                                                                                                                 50                    0                       50
area                 A                              dAi                                                                                                       y
                                   i =1                                                                                                                      mm



First                          rows ( y )           1
                                                                         dAi                                                                                 Sy
moment               Sy                                 zi        zi 1 .                                                                         z gc
of area.                                                                  2                                                                                   A
                                       i =1
Gravity centre                                                                                                                                   z gc = 1.704 . 10
                                                                                                                                                                                 15
                                                                                                                                                                                      mm


Second                         rows ( y )       1
                                                                                                        dAi
                                                                                            zi . zi 1 .                                                           A . z gc
moment                                                            2                 2                                                                                            2
                     Iy                                  zi                zi 1                                                                  Iy         Iy
of area                                                                                                  3
                                       i =1
                                                                                                                                                 I y = 2.627 . 10 mm
                                                                                                                                                                         6            4



                                rows ( y )          1
Area within
                                                        0.5 . yi           yi 1 . zi                                                             A v = 8.284 . 10 mm
                                                                                                                                                                         3            2
mid-line             Av                                                                         zi 1
                                                                                                                         Check:
                                       i =1
                                                                                                                         ( 2 .b )       2 .( b     a ) = 8.284 . 10 mm
                                                                                                                                    2                   2                        3            2

Sum of l/t                     rows ( y )           1
                                                                                                                                                            4 .A v
                                                                                2                       2                                                            2
                                                             yi       yi 1              zi      zi 1
Torsion              Dn                                                                                             Dn = 55.228                  Iv
                                                                                ti                                                                           Dn
constant                               i =1


                                                                                                                                                 I v = 4.971 . 10 mm
                                                                                                                                                                         6            4

Torsion
                               2 . A v . min ( t )                                                                                               W v = 9.941 . 10 mm
                                                                                                                                                                             4            3
resistance           Wv                                                         min ( t ) = 6 mm


TALAT 2301 – Example 8.2                                                                    2
TALAT 2301 – Example 8.2   3
b) Symmetric hollow extrusion
                    Half width              b           50 . mm
                    Indent                  c           10 . mm
                    Half width of a                     20 . mm
                    flat parts
                    Thickness               s           6 . mm



Nodes no.,
co-ordinates,            0                          b                        a               s
thickness
                         1                      b        c                   a               s
                         2                          a                   b        c           s
                         3                          a                        b               s
                         4                          a                        b               s
                         5                          a                   b        c           s
                         6                      b       c                    a               s
                         7                          b                        a               s
                i        8          y               b             z          a          t    s
                         9                      b       c                    a               s
                         10                         a                    b         c         s
                         11                         a                        b               s
                         12                         a                        b               s
                         13                         a                    b         c         s              50

                         14                     b        c                   a               s                               0

                         15                         b                        a               s
                                                                                                       z                                          0
                         16                         b                        a               s               0
                                                                                                       mm




                                                                                                            50
                                                                                                                 50          0                50
Nodes               i    1 .. rows ( y )        1                                                                            y
                                                                                                                            mm

                               rows ( y )           1
Area within
                                                        0.5 . yi        yi 1 . zi                                     A v = 7.2 . 10 mm
                                                                                                                                 3        2
mid-line            Av                                                                      zi 1
                                     i =1

                               rows ( y )       1                            2                     2
                                                             yi       yi 1             zi   zi 1
Sum of l/t          Dn                                                                                                Dn = 58.856
                                                                              ti
                                    i =1

                              4 .A v
                                       2
Torsion
                                                                                                                      I v = 3.523 . 10 mm
                                                                                                                                     6        4
constant            Iv
                               Dn


                               2 . A v . min ( t )                                                                    W v = 8.64 . 10 mm
                                                                                                                                     4        3
Torsion             Wv                                                       min ( t ) = 6 mm
resistance


TALAT 2301 – Example 8.2                                                                4

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TALAT Lecture 2301: Design of Members Example 8.2: Torsion constants for hollow cross section

  • 1. TALAT Lecture 2301 Design of Members Torsion Example 8.2 : Torsion constants for hollow cross section 3 pages Advanced Level prepared by Torsten Höglund, Royal Institute of Technology, Stockholm Date of Issue: 1999  EAA - European Aluminium Association TALAT 2301 – Example 8.2 1
  • 2. Example 8.2. Torsion constants for hollow cross section a) Polygon b Half width b 50 . mm Half width of a a = 20.711 mm flat parts Thickness s 6 . mm 1 2 Nodes no., 0 b a s co-ordinates, 1 a b s thickness 2 a b s 3 b a s i 4 y b z a t s 5 a b s 6 a b s 7 b a s 8 b a s 50 0 Nodes i 1 .. rows ( y ) 1 z 0 0 Area of cross mm ti . section 2 2 dAi yi yi 1 zi zi 1 elements 50 Cross rows ( y ) 1 section 50 0 50 area A dAi y i =1 mm First rows ( y ) 1 dAi Sy moment Sy zi zi 1 . z gc of area. 2 A i =1 Gravity centre z gc = 1.704 . 10 15 mm Second rows ( y ) 1 dAi zi . zi 1 . A . z gc moment 2 2 2 Iy zi zi 1 Iy Iy of area 3 i =1 I y = 2.627 . 10 mm 6 4 rows ( y ) 1 Area within 0.5 . yi yi 1 . zi A v = 8.284 . 10 mm 3 2 mid-line Av zi 1 Check: i =1 ( 2 .b ) 2 .( b a ) = 8.284 . 10 mm 2 2 3 2 Sum of l/t rows ( y ) 1 4 .A v 2 2 2 yi yi 1 zi zi 1 Torsion Dn Dn = 55.228 Iv ti Dn constant i =1 I v = 4.971 . 10 mm 6 4 Torsion 2 . A v . min ( t ) W v = 9.941 . 10 mm 4 3 resistance Wv min ( t ) = 6 mm TALAT 2301 – Example 8.2 2
  • 3. TALAT 2301 – Example 8.2 3
  • 4. b) Symmetric hollow extrusion Half width b 50 . mm Indent c 10 . mm Half width of a 20 . mm flat parts Thickness s 6 . mm Nodes no., co-ordinates, 0 b a s thickness 1 b c a s 2 a b c s 3 a b s 4 a b s 5 a b c s 6 b c a s 7 b a s i 8 y b z a t s 9 b c a s 10 a b c s 11 a b s 12 a b s 13 a b c s 50 14 b c a s 0 15 b a s z 0 16 b a s 0 mm 50 50 0 50 Nodes i 1 .. rows ( y ) 1 y mm rows ( y ) 1 Area within 0.5 . yi yi 1 . zi A v = 7.2 . 10 mm 3 2 mid-line Av zi 1 i =1 rows ( y ) 1 2 2 yi yi 1 zi zi 1 Sum of l/t Dn Dn = 58.856 ti i =1 4 .A v 2 Torsion I v = 3.523 . 10 mm 6 4 constant Iv Dn 2 . A v . min ( t ) W v = 8.64 . 10 mm 4 3 Torsion Wv min ( t ) = 6 mm resistance TALAT 2301 – Example 8.2 4