Università degli Studi di Bari

Transcription

1 Università degli Studi di Bari mechanics 2 prof. ing. Livio Quagliarella 2/83 3/83 4/83

2 5/83 6/83 Rigid body static balance disaggregation principle D A B C E 7/83 8/83

3 disaggregation principle : external load disaggregation principle : external load A D A A D C 9/83 10/83 disaggregation principle : internal load disaggregation principle : gravitational load 11/83 12/83

4 Rigid body static balance disaggregation principle: ree body diagram disaggregation principle: ree body diagram each part of that body mantains its static balance if 13/83 14/83 disaggregation principle: ree body diagram disaggregation principle: ree body diagram 15/83 16/83

5 Eccentric load General case C 2 C 3 line of action C R C 4 C 1 C 4 C = d R = C = M cm 17/83 18/83 General case statics cardinal equations APPLICABLE ONLY TO: isostatic systems C 4 NOT POINT RELATED R P Necessary conditions for an object to settle into equilibrium R = C = P M equilibrium = 0 CM M = 0 translation rotation 19/83 20/83

6 statically equivalent systems Two force systems are said statically equivalent when they produce the same effect stress strain distribution biological response 21/83 22/83 task The body stress strain distribution in A The body is a system modelled as: point A D E Kinetics study rigid body Static study B C deformable body Stress strain description E Body mass 23/83 24/83

7 Center of Mass (COM) or Gravity (COG) It is an imaginary point where there is intersection of all 3 cardinal plane. Imaginary point where all the mass of the body or system is concentrated Point where the body s mass is equally distributed center of mass of a rigid body. In a homogeneous body each mass infinitesimal element dm can be associated with a vector. the result of such a set of vectors is applied in a point called "center of mass". 25/83 26/83 center of mass of a rigid body. COM belongs to an element of symmetry or to their intersection COG Normally COM COG A 27/83 28/83

8 Moment of inertia Refered to a point (polar) or to a straight line (axial) Scalar quantity P 1 P 2 m 2 Moment of inertia m P r 2 I = mr m 1 r 1 r 2 r 3 m3 3 i= 1 m i r i 2 P 3 s 29/83 30/83 Moment of inertia Moment of inertia a I = dm i r i 2 2a 2a a 31/83 32/83

9 Moment of inertia Moment of inertia m 1 = m 3 I 1 < I 3 33/83 Moment of inertia deformable body I = dm i r i 2 < L 0 L 1 A B A greater couple is required to accelerate B than it is for A 35/83 36/83

10 statically equivalent systems 37/83 perfect elastic body perfect elastic body L 0 L 1 X Instantly! L 0 L 1 r = kx ΔL = L 0 L 1 40/83

11 perfect elastic body perfect elastic body? Linear strain L 0 L 1 Δl ε = l ΔL = L 0 L 1 X Longitudinal stress σ 41/83 42/83 Angular strain Tangential stress Δφ γ = φ τ perfect elastic body ENGINEERING STRESS Tensile stress, σ: Shear stress, τ: t t Area, A Area, A s Angular strain Tangential stress Δφ γ = φ τ t σ= t A o original area before loading Stress has units: N/m 2 τ= s A o s t 43/83 4

12 Tensile strain: ENGINEERING STRAIN Lateral strain: δ/2 ε= δ L o w o L o ε L = δ L w o δ/2 Shear strain: θ/2 δ L /2 δ L /2 stress Normal stress σ π/2 - θ γ = tan θ Strain is always dimensionless. Tangential stress τ ρ π/2 θ/2 8 46/83 Superposition principle Superposition principle the superposition principle states that, for all linear systems, the net response at a given place and time caused by two or more loads is the sum of the responses which would have been caused by each load individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). 47/83 48/83

13 Superposition principle Types of load De Saint Venant beam theory 49/83 50/83 Types of load Types of load 51/83 52/83

14 Types of load Axial load N Bending My, Mz Shear Ty, Tz Torsion Mx De Saint Venant s hypothesis: monodimensional beam straight line small deformation (0,1%) y My Types of load Axial load N Bending My, Mz Shear Ty, Tz Torsion Mx x y z Mz Tz Ty N x z Mx 53/83 54/83 traction Normal load σ = A0 compression 55/83 56/83

15 Normal load Normal load σ = = N A 0 m 2 = Pa STRESS ε = L L 0 L 0 [ ] A0 B 0 L0 A B L STRAIN σ ε = E 57/83 MODULUS O ELASTICITY OR YOUNG S MODULUS 58/83 ε t = B B 0 B 0 υ = ε t ε Poisson's ratio Poisson Effect 59/83 60/83

16 Bending M M b y My σ = J 61/83 62/83 Bending Bending curvature 1 r = M EJ LINEAR STRESS DISTRIBUTION 63/83 64/83

17 Shear Shear Physical example when boards glued together, shear stress is developed at surfaces which prevents slippage. 65/83 Shear Shear Internal shear force creates shear deformation, strain and shear stress! Note: due to nature of shear stress get transverse and longitudinal strain. Notice deformation: key point, deformation not uniform!!

18 first moment of inertia 6.2 Shear Stress ormula: Shear first moment of inertia, is area times distance to an axis. It is a measure of the distribution of the area of a shape in relation to an axis. τ = Moment of inertia of entire cross section (in 4 ) TQ It Internal Shear irst Moment of area (in 3 ) at point of interest Thickness of cross-section at point of interest (in) Shear Torsion 71/83 72/83

19 Torsion Torsion Torsion angle θ = M t GJ p where E G = 2(1+ υ) 73/83 Jp = polar moment of inertia τ = MtR J p 74/83 G = torsional elasticity modulus Jp = polar moment of inertia τ = MtR J p Type of load point load bending Normal load Bending Torsion Shear σ = A0 Μy σ = J ΜtR τ = Jp ΤS i τ = Jb i 75/83 76/83

20 Superposition principle the superposition principle states that, for all linear systems, the net response at a given place and time caused by two or more loads is the sum of the responses which would have been caused by each load individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). = + +s +s +s + = s s s 77/83 78/83 Load types hollow section Μy σ = J ΜtR τ = Jp 79/83 80/83

21 resistant section tie-beam - axis direction 81/83 82/83 tie-beam - axis direction equal strength structure 83/83 84/83

22 Structure evolution Structure evolution Cullmann Cullmann bending bending 85/83 86/83 Structure evolution Cullmann 87/83 88/83

23 Normal load Bending ASSE LONGITUDINALE 89/83 90/83 torsion shear 91/83 92/83