Chap1, 2. Statics Review

  • 4대 힘 : Tension & Compression (Axial Loading) / Bending (중립축에 90도) / Shear (단변적 평행방향 힘, 각도 변화) / Twist
  • 외력에 대한 deformation field 구하기 :
    • Governing Equation \(\triangledown σ + \overrightarrow{b} = \rho \frac{\partial v}{\partial t}\),
    • Stress-strain equation \(σ=D\varepsilon\),
    • Displacement - strain equation \(\varepsilon = \frac{\partial u}{\partial x}\)

Statics Review → Equilibrium

Analysis of load acting on physical system
  • Do not experience acceleration
  • Newton's Law
    1. An object tends to either remain at rest or continue to move at constant velocity
    2. \(\sum\overrightarrow{F}=m\overrightarrow{a}\): resultant force로 표현가능
    3. 작용 & 반작용 : 같은 크기 반대방향 힘, 작용점이 같아야 함
      • 반작용 : 움직이지 않는 부분은 같은 크기 반대방향 힘이 걸림 → 가장 안 움직이는 점을 반력 작용점으로.
  • 6개의 힘 요소 : translation & moment ⇒ \(F_x, F_y, F_z, M_x, M_y, M_z\)
  • Rigid body : An object which does not go through deformation(But internal force exists)
  • Deformable body : An object which can be deformed (Shape Change)
  • displacement : Even the same deformation does not have the same effect (different shape)
  • strain
    • normal : 힘의 방향이 가해지는 면에 수직, element의 각도 변화 없음, \(\varepsilon = \frac{δ u}{L_0}\)
    • shear : 힘의 방향이 가해지는 면에 평행, \(γ = \frac{dx}{L}=\tanθ\approx\theta\)
  • 외력 : contact 필요 / Body force : contact 필요 없음

iamge

\[x:Σ F=(σ_{xx}+\triangle σ_{xx})\triangle y\triangle z -σ_{xx}\triangle y\triangle z\\+(σ_{yx}+\triangle σ_{yx})\triangle x\triangle z -σ_{yx}\triangle x\triangle z\\+(σ_{zx}+\triangle σ_{zx})\triangle x\triangle y -σ_{zx}\triangle x\triangle y+B_x=M\overrightarrow{a_x}\\⇒ \div dV\\ ⇒x:\frac{\triangle σ_{xx}}{\triangle x}+\frac{\triangle σ_{yx}}{\triangle y}+\frac{\triangle σ_{zx}}{\triangle z} + b_x = \rho \overrightarrow{a_x} \\⇒ y:\frac{\triangle σ_{xy}}{\triangle x}+\frac{\triangle σ_{yy}}{\triangle y}+\frac{\triangle σ_{zy}}{\triangle z} + b_y = \rho \overrightarrow{a_y}\\ ⇒ z:\frac{\triangle σ_{xz}}{\triangle x}+\frac{\triangle σ_{yz}}{\triangle y}+\frac{\triangle σ_{zz}}{\triangle z} + b_z = \rho \overrightarrow{a_z} \\ b=0 \because no\ weight \\ \rho\overrightarrow{a}=0\because static\]