WHAT STRENGTH OF MATERIALS IS IT ABOUT?

1. WHATSTRENGTH OF MATERIALSIS IT ABOUT?

2. Why You Don’t Fall Through the Floor

3. Linguistics Strength of materials Moc materiałów? 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 Wytrzymałość materiałów Wytrzymać: co? Ile? Jak długo? Соротивлематеиаов Opór materiałów? Résistance des matériaux Opór materiałów? Festigkeitlehre Nauka o sile materiałów? Hållfasthetslära Nauka o spójności materiałów? 材料力学 No, to już zupełna „chińszczyzna”!

4. Physics Mathematics Theoretical mechanics(of a rigid body) Origin of SM • Differential calculus • Matrix algebra • Calculus of variations • Numerical methods • Theory of elasticity • Theory of plasticity • Material Science SM (of a deformable body) HYPOTHESES EXPERIMENTS

5. Modelling scheme Idealisation of: Material • Continuous matter distribution (material continuum) • Continuous mass distribution ρ(x) • Intact, unstressed initial state of a material Loadings • Permanent versus movable • Constant versus variable in time (static versus dynamic) Structure geometry • Bulk structures (H ~ L~B) • Surface structures (H«L~B) • Bar structures (L»H~B)

6. P[N] l q[N/m] p[N/m2] l B L v u w Mechanical loadings External surface forces Surface distributed force p[N/m2] Line distributed force q[N/m] Point force P [N] b Point moment M [Nm] H External volume forces M[Nm] (gravitational forces, inertia forces, electromagnetic forces etc.) X [N/m3] X [N/m3] Displacementsu(u,v,w)[m](e.g. supports, forced shift of structural members) u=0, v=0 v=0 ≡ +

7. Internal forces Fundamental observations • 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.

8. w2 w1 wi w3 Internal forces {wi}, i=1,2 …∞ P2 P3 P1 A Coulomb particle interaction assumed(convergent set of internal forces) Pn Pi A body in equilibrium {wi} – convergent, infinite, zero valued set of internal forces

9. n n n w2 P2 A P3 P1 A w1 w wi w w w3 Pn Pi P2 P3 P1 w A A II I Pn Pi ∞ Internal forces ∞ r II I r – point position vector n n - outward normal vector n w= f(r,n)

10. n {ZII} {wI} {wII} P2 {ZI} P3 P1 A A II I Pn Pi Internal forces n ∞ ∞ {Z} = {ZI} + {ZII} ≡ {0} {ZI} + {wI} ≡ {0} Body in equilibrium {ZII} + {wII} ≡ {0} {wI} + {wII} ≡ {0} {ZI} ≡ - {wI} {ZII} ≡ - {wII} {wII}≡{ZI} {wI} ≡{ZII}

11. n {ZII} n {wI} {wII} P2 {ZI} P3 P1 A A II I Pn Pi Internal forces {wI} ≡ {ZII} {wII} ≡ {ZI} 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

12. MzI n P3 P2 MwII SzII SwI O P1 SzI O Pi SwII MwI MzII n Pn {wI} {wII} Ois assumed to be the reduction point of internal and external forces Cross-sectional forces ∞ ∞ {wI} ≡ {ZII} {wII} ≡ {ZI} SwI≡ SzII MwI≡ MzII SwII ≡SzI MwII≡MzI SwI≡ - SwII MwI≡ - MwII

13. n P3 P2 P1 O Pi rO n Pn Cross-sectional forces The components of the resultants of internal forces reduced to the point O will be called cross-sectional forces Sw≡Sw(rO, n) MwI≡Mz(rO, n)

14. Cross-sectional forces • 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 planen • In what follows we will limit ourselves to bar structures, as the simplest approximation of 3D bodies (structures).

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