1/81/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars INTERNAL FORCES IN BARS

2/82/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars Definitions L H B Bar – a body for which L»H,B Bar axis - locus of gravitational centres of bar sections cutting its surface Prismatic bar – when generator of bar surface is parallel to the bar axis Straight bar – when bar axis is a straight line Bar axis

3/83/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars Assumptions Bar axis represents the whole body and loading is applied not to the bar surface but the bar axis Set of bar and loading will be considered as the plane one if forces acts in plane of the bar. P q M. M

4/84/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars Agreements Reduction centre O is located on the bar axis by vector r 0 Internal forces are determined on the planes perpendicular to the bar axis (vector n is parallel to the axis) Vector n is an outward normal vector n n O x y z r0r0

5/85/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars Components of internal forces resultants S wx, S wy, S wz and M wx, M wy, M wz are called cross-sectional forces In 3D vectors of internal forces resultants have three components each S w { S wx, S wy, S wz } M w { M wx, M wy, M wz } x y z S wz SnySny S wx SwSw M wz M wx M wy MwMw

6/86/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars S w = S w (r O, n) M w = M w (r O, n) S w = S w (r O ) M w = M w (r O ) Vector n is known if we know the shape of bar axis. n. n. n. n. n Thus, resultants of internal forces for known bar structure are function of only one vector r 0 Resultants of internal forces are vector functions of two vectors r o and n

7/87/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars In 2D number of cross-sectional forces is reduced, because loading and bars axes are in the same plane ( x, z ): S w { S x, 0, S z } M w { 0, M y, 0 } x y z P q. M M SxSx SzSz MyMy We will use following notations and names for these components: S x =N - axial forces S z =Q - shear force M y = M - bending moment

8/88/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars Special cases of internal forces reductions are called: TENSION – when internal forces reduce to the sum vector only, which is parallel to the bar axis SHEAR – when internal forces reduce to the sum vector only, which is perpendicular to the bar axis BENDING – when internal forces reduce to the moment vector only, which is perpendicular to the bar axis TORSION – when internal forces reduce to the moment vector only, which is parallel to the bar axis M MsMs Q N

9/89/8 M.Chrzanowski: Strength of Materials SM1-02: Statics 1: Internal forces in bars  stop