202.121.48.120b70c3b42-49ac-47d2-88a2-9867279132b

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					§2-5     拉(压)杆内的应变能

 §2-5   Strain Energy in the Axially
              Loaded Bar
    应变能(Strain Energy):
    伴随着弹性变形的增减而改
    变的能量
       V

       V  W       2-8

          功能定理




F
                F
                          F
                                             F

                            F

                                                        F
          l                 l
          l1
                                         O               l
    1     1
 W  FL  FN L              FN L
    2     2              L                     l
                              EA
     1           FN2 L
 V  FN L                    2-9
     2           2 EA
                                                1        2-10
应变能密度(Strain Energy                               FL
                                           V   2      1
                                      v             
Density): 单位体积内的应变                         V      AL   2
          图示的杆系是由两根圆截面钢杆铰接而成。已知
例题         α=300,杆长L=2m,杆的直径d=25mm,材料的
2.12       弹性模量E=2.1×105MPa,设在结点A处悬挂一重
           物F=100kN,试求结点A的位移δA。
                            X 0               FNAC sin   FNAB sin   0
                                                FNAC  FNAB 
                                                                   F
 B     1         2    C     Y  0           FNAC cos   FNAB cos   F  0
                                                                2 cos 
       FNAB FNAC
           α α              LAB  LAC        
                                                 FNAC L
                                                        
                                                            FL
                                                  EA      2EA cos 
             A
                                                   LAC 
                                                                 FL
       A                                A  AA
                                                   cos     2EA cos 2 
            F                     A          1.3mm 100 103  2
                                         
                     LAC                                         
                                      LAB   2  2.1105 106         252 10 6  cos 300
                                                                  4


                             A
 小结

REVIEW
Stiffness (Robert Hooke, 1648)

 Robert Hooke was the first to experiment
 with and define the stiffness of materials.
 He suspended various masses from
 springs, and measured the extension.




                  The force is proportional to
                  the extension of the spring.

    m
             u
Material Properties (Thomas Young, 1810)

                    Thomas Young helped to develop the
                    theory of how materials deform elastically.
                    In particular, he defined an important
                    material constant, “Young’s Modulus”.

             P                                           P

                                   L
                      Internal Normal Force
Let Axial Stress,  
                                                      (in Units of N/m2
                            C.S. Area                 or Pascals (Pa))

                        Change in Length
Let Axial Strain,                                (Dimensionless)
                         Original Length
Material Properties (Cont.) (Simon Poisson, 1825)
 Poisson made important observations and
 theories about lateral deflections of materials.

 When a bar is placed in tension, lateral
 contractions accompany the extension.

              Initial
              Shape
  Final
  Shape                                        P    A real math nut.




     P

				
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