Sheet 08: Twist of circle cross-sections. Arkusz przeznaczony do ćwiczeń z przedmiotu „Strength of Materials” na II roku dziennych studiów Wydziału Inżynierii Mechanicznej i Robotyki AGH na kierunku „Mechatronics in English” w roku akademickim 2014/2015. Notes for exercise classes of: „Strength of Materials” Sheet 08: Twist of circle cross-sections 1. Twist of circle cross-sections – definitions, concepts, formulas Theory on twisting - from lectures and books: [1], [2]. The drawings below illustrate concepts and formulas associated with torsion of circular cross-sections. Rysunek 2: The stress distribution in the cross section Rysunek 1: simple twist Rysunek 3: Stress vectors Rysunek 4: Deformation and strain 1.a) STRESSES The stress tensor in twist has the form set out below; in this state, according to Hooke's equations there is generated a spatial strain state: [ 0 τ xy T σ = τ yx 0 τ zx 0 τ xz 0 0 ] [ constitutive relations ⇔ T ε= 0 1 γ 2 yx 1 γ 2 zx 1 γ 2 xy 1 γ 2 xz 0 0 0 0 ] The relationship of the stress function with the cross-sectional force function has the following form: xy ( x , z) = xz ( x , y) = [ M x ( x) ⋅z I x ( x) M x ( x) ⋅y I x( x) [ Nm 4 ⋅m = Pa m Nm ⋅m = Pa 4 m ] ] where Mx(x) is the twist moment in the cross-section with the abscissa x and Ix(x) is the polar moment of inertia in this cross-section. © Copyright: Anna Stręk. Autorem arkusza jest Anna Stręk. Arkusz stanowi przedmiot prawa autorskiego określonego w Ustawie o prawie autorskim i prawach pokrewnych (Dz. U. 1994 r. Nr 24 poz.83 z późn. zmianami). Autor nie wyraża zgody na inne wykorzystywanie arkusza niż podane w jego przeznaczeniu. 1 Sheet 08: Twist of circle cross-sections. Arkusz przeznaczony do ćwiczeń z przedmiotu „Strength of Materials” na II roku dziennych studiów Wydziału Inżynierii Mechanicznej i Robotyki AGH na kierunku „Mechatronics in English” w roku akademickim 2014/2015. If we use the polar coordinate system instead of the Cartesian, the formula for stress is: ( x , ρ) = M x ( x) ⋅ρ . I x ( x) 4 4 πR πD = The formula for the polar moment of a circular cross-section: I x = I 0 = I p = I y + I z = . 2 32 1.b) STRAIN AND DEFORMATION The relationship between the twist moment and the function of the strain has the form: M x ( x) ⋅z G( x)⋅I x ( x) M x ( x) ⋅y γ xz ( x , y) = G ( x)⋅I x ( x) γ xy ( x , z) = where Mx(x) is the twist moment in the cross-section with the abscissa x and Ix(x) is the polar moment of inertia in this cross-section. G(x) is the modulus of rigidity (Kirchhoff's modulus; depends on the material and therefore may vary over the element). The following concepts are related to the deformation and strain for a simple twisting: θ( x) = • unit angle of twist: • current angle of twist: [ ] M x ( x) rad G⋅I x ( x) m α ( x) = ∫ θ( x) dx = ∫ M x ( x) dx [ rad ] G ( x)⋅I x ( x) l M x ( x) dx [ rad ] G ( x)⋅I x ( x) 0 i i M ⋅l α ( x) = ∑ i x i [ rad ] i G ⋅I x α ( x) = ∫ • total angle of twist (for an element of continuous characteristics): • total angle of twist (for an element of step-changing characteristics): 2. Circular cross-sections – statically determinate structures Just to remind: reactions can be determined from the equilibrium equations when the number of these equations corresponds to the number of unknown reactions. In such a case we are talking about the statically determinate systems. For statically determinate systems that operate under simple twisting the following aspects are usually considered: finding the cross-sectional forces, stresses, findinfg unit, current and total angle of twist. According to the strength condition τmax ⩽ k s (ks - allowable shear stress) and the utility condition θmax ⩽ θ allow or max total α ⩽ α allow or α ⩽ α allow one can also design the parameters of a structure: material, dimensions, load. You should solve examples no.: 15.5.1, 15.5.2 from book [2] and tasks no.: 5.1 / p. 63 i 5.3 / p. 65 from book [3]. Caution! It may happen (as in task 5.1) that there will be a need to convert rotation and power of an engine to a torque that they cause. Please find the appropriate formula in the literature – it will be required. 3. Circular cross-sections – statically indeterminate structures In the cases when we do not have a sufficient number of equations to calculate the reactions, one needs to find conditions that will provide additional equations for the extra unknowns. For a simple twist state this is often the case of a bar fixed at both ends. Therefore, the additional condition is the geometric condition: total angle of twist is 0. You should solve the example no.: 15.5.3 from book [2] and tasks no.: 5.10 / p. 69 , 5.12 / p. 70 from book [3]. © Copyright: Anna Stręk. Autorem arkusza jest Anna Stręk. Arkusz stanowi przedmiot prawa autorskiego określonego w Ustawie o prawie autorskim i prawach pokrewnych (Dz. U. 1994 r. Nr 24 poz.83 z późn. zmianami). Autor nie wyraża zgody na inne wykorzystywanie arkusza niż podane w jego przeznaczeniu. 2 Sheet 08: Twist of circle cross-sections. Arkusz przeznaczony do ćwiczeń z przedmiotu „Strength of Materials” na II roku dziennych studiów Wydziału Inżynierii Mechanicznej i Robotyki AGH na kierunku „Mechatronics in English” w roku akademickim 2014/2015. • Ability to solve problems for statically determinate and indeterminate systems in twist (graph of the angle of twist, unit angle of twist, twist moments; graphs of stress in the element and on the face of the cross-section; design conditions, determination of diameters and allowable loads). • Knowledge of formulas and the ability to use them (stress, strain, deformation, the torque of the engine running). • The form of stress and strain matrices in the twist state. Definition of the simple twist. 4. Literatura [1] Piechnik S. "Mechanika techniczna ciała stałego", Wydawnictwo PK, Kraków 2007 [2] Bodnar A. „Wytrzymałość materiałów. Podręcznik dla studentów wyższych szkół technicznych”, wydanie drugie poszerzone i poprawione, Kraków 2004, rozdział 15 [3] Niezgodziński M., Niezgodziński T. "Zadania z wytrzymałości materiałów", Wydawnictwo WNT, Warszawa 2012 © Copyright: Anna Stręk. Autorem arkusza jest Anna Stręk. Arkusz stanowi przedmiot prawa autorskiego określonego w Ustawie o prawie autorskim i prawach pokrewnych (Dz. U. 1994 r. Nr 24 poz.83 z późn. zmianami). Autor nie wyraża zgody na inne wykorzystywanie arkusza niż podane w jego przeznaczeniu. 3
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