SHEAR & NON-UNIFORM BENDING

1. SHEAR&NON-UNIFORM BENDING

2. y Qy x z y y x Qz x Qz My z z Mx = 0, My = 0, Mz = 0 Formal definition: the case when set of internal forces reduces solely to the shear force vector perpendicular to the bar axis Qx=N= 0, Qy ≠ 0, Qz = 0 Mx = 0, My ≠ 0, Mz = 0 or Qx= N=0, Qy= 0, Qz ≠ 0 Qx=N=0, Qy= 0, Qz ≠ 0 Shear is associated with bending ! NON-UNOFRM BENDING PURE SHEAR

3. M Q Pure shear Reactions Mu0 Q=const ?

4. P P t t t t A A P P A A Pure shear Mean shear stress 4At = P t = P/4A

5. Z Z X X Non-uniform bending h l Bernoulli hypothesis of plane cross-sections does not hold!! For h/l<<1 distorsion is small and we will use the formula for normal stress derived from this assumption :

6. Z Z X X txz Z txz= txz(z,y) X ? txz txz Qz = P = Non-uniform bending P P tzx „point” image tzx

7. Z A* z D,F B,C B,D C,F y b (z) x dx D F b (z) B C Non-uniform bending A* Prismatic bar!

8. Kinematic Boundary Conditions in A and B: Formula holds for prismatic bars only! For B:S*(zmax)=0 since A*=A For A:S*(zmax)=0 since A*=0 Non-uniform bending and is given in main principal axes of cross-section inertia. Distribution along z-axis A* z A Also: zmax z y A zmin B

9. z Parabola 2o z tmax= 3Q/2bh h/2 2h/3 y y h/2 h/3 b z z b tmax= ? Parabola 2o tb tc c h/2 h/2 tmax y y h/2 h/2 b b Non-uniform bending Distribution along z-axis; special cases b(z) = b =const|z b(z) – linear function of z Parabola 3o b(z) – step-wise change tb/tc=c/b

10. z tzx txz sx sx sx txz x tzx 2 z s2 a2 s1 x s1 a1 1 s2 Non-uniform bending Stress distribution in beams – trajectories of main principal stresses For z=0:

11. s2=t z 2 s1=t a2=45o x a1=45o s2=t s1=t 2 1 2 z z s2 a2=90º a2 s1 s1= sx x 1 s1 s1 x a1 a1=0º 1 s2 s2=0 Non-uniform bending l/2 Principal stress trajectories

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