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					Institute of Engineering   Mechanics of Materials
                                 410211
Dr.Tanongsak Bisarnsin
Email: tanong@ccs.sut.ac.th
Phone: 224227
Office Hours:
Monday      13.30-15.30

                              2
Mechanics I
                                 • Structural analysis

                                        ΣF = 0;

                   P                  ΣMA = 0;
              C
                                   • Internal forces
                                        M
               A   Ax        B                    V
                                            N
              Ay        By                                 B

                                                       By
                                   • Section properties of member BC




                                                       A, I, x, y
                                                                       3
Mechanics of Materials I

    • Stresses             • Strain
     M
                           P                   P
           V                          L
    N                                          ∆L
                   B

                  By                      ∆L
                                 ε=
                                          L
        σn = N
            A

        τv = VQ
            It

        σb = My
            I

                                                    4
• Stresses
M                          σn
      V           N
N
              B

             By
                      V    τv




                          σbending
                  M




                                     5
                σ (MPa)
                                         σu = 390 MPa
               400
               350
                                                         σfail = 295 MPa
               300
               250
               200
σy = 225 MPa                             σpl = 205 MPa
               150
               100                    200
                                 E=           = 200x103 MPa
                50                    0.001
                 0                                          ε (mm/mm)
                 0.00     0.10    0.20      0.30   0.40      Upper scale
               0.0000   0.0010 0.0020 0.0030 0.0040           Lower scale




                                                                            6
•Beam

                        P

        A                                        B
                        C        D


                        P
   Ax   A                    NC
                                       MC
                       C
            Ay              VC

                       VC
                                 D ND
                 MC                         MD
                      NC C
                                  VD
                        P
   Ax   A                          ND
                                            MD
                             D
            Ay                    VD
                                                     7
D ND
           MD
 VD




 D          ND   MD

      VD
                      Cross-section of D




                                           8
                                                      P
Axial                                          σN =
                                                      A
                                   1

               N     NA            2


                                   3


                                                          Cross-section of D
Normal Force              Normal stress



                               1          σN



                               2          σN



                               3          σN

                   State of stress at 1,2 and 3                                9
Bending Stress
                                             (σb)top
                                                                    My
                                     1                   (σ b ) =
                 M                                   y               I
                       NA            2      O
                                     3

                                                (σb)bottom
Bending Moment            Bending stress



                                 1         (σb)top


                                 2



                                 3         (σb)bottom

                     State of stress at 1,2 and 3                        10
Transverse Shear
                                   1

                     NA                             (τV)max
                                   2


                                   3
          V                                              VQ   A
                                              (τ V ) =
Shear Force             Shear stress                     IT




                               1



                               2       τmax


                               3

                   State of stress at 1,2 and 3                   11
 Combine Stress                         σV = 20 kPa
                                    1


                 Normal stress      2
                                    2


                                    3

                                                (σb)top = 50 kPa
                                   1

             Bending stress        2            O
                                   3

                                                    (σb)bottom = 50 kPa
                                   1

                  Shear stress                    (τV)max = 15 kPa
                                   2


                                   3


                                                         (σb)top = 30 kPa
                       1

Combine stress                   (τV)max = 15 kPa             (σb)NA = 20 kPa
                       2


                       3

                                                            (σb)bottom = 70 kPa   12
                                               (σb)top = 30 kPa
1

        N   M             (τV)max = 15 kPa           (σb)NA = 20 kPa
2


3
    V
                                                    (σb)bottom = 70 kPa



                            1         σ = 30 kPa


                            2          σ = 20 kPa
                                  τmax= 15 kPa

                            3         σ = 70 kPa

                State of stress at 1,2 and 3                              13
• Shear diagram, bending moment diagram and deflected curve of a beam
                                 Po
                 Mo                        wo

                      A
                                   D       B                 C
                               L       L                 L
                               2       2                 2
                          RA                     RB


              V                                      +
                               +
           diagram
                                       -


              M M
                   0
           diagram                 +
                                                 -

          Deflected
           Curve
                                               Point of inflection      14
     15 kN•m          20 kN                          2 kN/m
                                                                        15 kN•m                     37.5 kN•m
            A
                              D              B                   C
                   3m                 3m                  3m                       A            D-
                RA = 7.5 kN                            RB = 24.5 kN      7.5 kN         3m       7.5 kN

                7.5                              6
  V (kN)                                                               37.5 kN•m 2 kN/m
                      +                                   +                                      18.5 kN

                          -12.5          -
                                                                                   D+           B-     9 kN•m
                                                      -18.5              12.5 kN        3m
                                  37.5
M (kN•m) 15
                                  +
                                                      -                9 kN•m          2 kN/m

                                                          -9
Deflected
                                                                                   B+           C
 Curve
                                                                           6 kN         3m
                                                 Point of inflection                                        15
• Shaft
Torsion Shear Stress
                                      Tρ
                               τT =
              T                        J          (τT)max

                                           NA     O     ρ



                                                   (τT)max

                               1

                           2       3




                       State of stress at 1, 2 and 3         16
Combine Torsion and Shear Stress

         1
                      T                                              (τT)max = 25 kPa

                              NA                 (τV)max = 15 kPa    O           ρ
     2           V

                                                                             (τT)max = 25 kPa
                                                 VQ                         Tρ
                                      (τ V ) =                      τT =
                                                 IT                          J
         3




             1       25 kPa                        2     15 + 25 = 40 kPa            3    25 kPa



                                   State of stress at 1, 2 and 3                                   17
•Connection
Simple Shear Stress
  F1
                                           F1



                              F1
                F1

                          d

                              F1

                      F´1 = F1


                                  F1   τaverage = F1 = F1
                                                  A   π d2
                       τaverage                        4
                                                             18
                        V   V
           τaverage =     =
                        A   π d2
                            4




τaverage




                                   19
Stress Transformation


                                 σ

                            θ


                                 τ


                        θ   σθ




                            τθ
                                     20
Deflection of Beams




                                                 v(x) = f (M, EI)

                                       Deflection curve



Column Buckling
                          Pcr = f (EI, A, L)




                      L




                                                                    21