1 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM WHAT STRENGTH OF MATERIALS IS IT ABOUT?

2 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM Why You Don’t Fall Through the Floor

3 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM Linguistics Соротивле матеиаов Résistance des matériaux Festigkeitlehre Hållfasthetslära 材料力学 Wytrzymałość materiałówWytrzymać: co? Ile? Jak długo? Opór materiałów? Nauka o sile materiałów? Nauka o spójności materiałów? No, to już zupełna „chińszczyzna”! Moc materiałów?Strength of materials SM is about the resistance of materials (and structures) against external environmental actions (forces, deformations, temperatures etc.) which may lead to the loss of load bearing capacity

4 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM SM (of a deformable body) Theoretical mechanics (of a rigid body) EXPERIMENTS PhysicsMathematics HYPOTHESES Theory of elasticity Theory of plasticity Material Science Differential calculus Matrix algebra Calculus of variations Numerical methods Origin of SM

5 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM Idealisation of: Material Loadings Structure geometry Continuous matter distribution (material continuum ) Continuous mass distribution ρ (x) Intact, unstressed initial state of a material Permanent versus movable Constant versus variable in time (static versus dynamic) Bulk structures (H ~ L~B) Surface structures (H«L~B) Bar structures (L»H~B) Modelling scheme

6 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM B L External surface forces Point force P [N] Line distributed force q [N/m] Point moment M [Nm] Surface distributed force p [N/m 2 ] H b External volume forces M [Nm] P [N] p [N/m 2 ] l l q [N/m] (gravitational forces, inertia forces, electromagnetic forces etc.) X [N/m3] Displacements u (u,v,w) [m] (e.g. supports, forced shift of structural members) Mechanical loadings X [N/m3] u=0, v=0v=0 + ≡ u v w

7 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM A body (structure) under external loadings changes its shape (material points of this body are subjected to the displacement) This change in material points position influences forces of interaction and results in creation of internal forces If a body (structure) is in equilibrium – each point of this body is also in mechanical equilibrium i.e. resultant of forces and moments is equal to zero. Internal forces Fundamental observations

8 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM A P1P1 P3P3 PnPn P2P2 PiPi wiwi w1w1 w2w2 w3w3 A body in equilibrium Coulomb particle interaction assumed (convergent set of internal forces) { w i } – convergent, infinite, zero valued set of internal forces Internal forces { w i }, i =1,2 … ∞

9 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM n n wiwi w2w2 w1w1 w3w3 A PiPi A P1P1 P3P3 PnPn P2P2 I II A P1P1 PnPn I w w w A P3P3 P2P2 PiPi w n r w= f(r,n) Internal forces n - outward normal vector n r – point position vector n ∞ ∞

10 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM n A P1P1 PnPn I A P3P3 P2P2 PiPi II n {wI}{wI} {ZI}{ZI} {w II } {Z II } { Z I } + { w I } ≡ {0} { Z II } + { w II } ≡ {0} { Z } = { Z I } + { Z II } ≡ {0} { w I } + { w II } ≡ {0} { Z I } ≡ - { w I }{ Z II } ≡ - { w II } {w II } ≡ {Z I } { w I } ≡{ Z II } Body in equilibrium Internal forces ∞ ∞

11 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM n A P1P1 PnPn I A P3P3 P2P2 PiPi II {wI}{wI} {ZI}{ZI} {w II } {Z II } n { w I } ≡ { Z II }{ w II } ≡ { Z I } The set of internal forces in part I is equal to the set of external forcces acting on II The set of internal forces in part II is equal to the set of external forcces acting on I Internal forces

12 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM M wII S wII SwISwI M wI P3P3 PiPi O n O P1P1 PnPn P2P2 n {wI}{wI} {w II } { w II } ≡ { Z I }{ w I } ≡ { Z II } S wII ≡ S zI M wII ≡ M zI S wI ≡ - S wII M wI ≡ - M wII S wI ≡ S zII M wI ≡ M zII S zII M zII M zI S zI Cross-sectional forces O is assumed to be the reduction point of internal and external forces ∞ ∞

13 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM P3P3 PiPi O n n S w ≡ S w (r O, n) M wI ≡ M z (r O, n) rOrO P1P1 PnPn P2P2 Cross-sectional forces The components of the resultants of internal forces reduced to the point O will be called cross-sectional forces

14 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM The immediate goal of SM is to evaluate internal forces These forces will define the conditions of material cohesion and its deformation As the first step the components of the sum and moment of cross-sectional forces will be evaluated as a function of chosen reduction point O, and cross-section plane n In what follows we will limit ourselves to bar structures, as the simplest approximation of 3D bodies (structures). Cross-sectional forces

15 /14 M.Chrzanowski: Strength of Materials SM1-01: About SM  stop