机械设计05弯曲剪切和扭转.ppt

1、05,Bending, Shear and Torsion,05 - 2,Shigley, Fig. 4-15,Bending of Straight Beams: Deformed Beams,Deformed Beams,The material is isotropic, homogeneous and linear elastic. The beam is initially straight. Plane cross sections remain plane during bending (i.e., the deformation due to transverse shear

2、is ignored). The beam axes are the principal axes of the section, i.e., Iyz = 0. The beam would fail by bending rather than by crushing, wrinkling (local buckling) or global buckling.,Assump-tions,05 - 3,There is at least one set of coordinate system about which Iyz = 0.,The axis about which Iyz = 0

3、 is called the principal axis.,If the section is symmetric about an axis, this axis is a principal axis.,For most shapes, there are only two mutually perpendicular principal axes.,Straight Beams: Moments of Inertia,Definition,Principal Axis,05 - 4,Straight Beams: Normal Stresses,Normal Stress,Notes,

4、05 - 5,Straight Beams: Normal Stress Distribution,Equation,05 - 6,Shear Stress in Beams: Shear Stress,For equilibrium of beam element,05 - 7,Shear Stress in Beams: First Moment of Area,Shear Stress,First Moment of Area,Note,Shear Flow,b is the width of the cross section at the particular distance y1

5、.,I is the second moment of area of the entire section about N.A.,is the distance from the neutral axis (N.A.) to the centroid (C.G.) of the isolated area,05 - 8,Shear Stress in a Narrow Rectangular Beam,Free Body Diagram,Shear Stress,parabolic,A: Area of the shaded area. A0: Area of the entire cros

6、s section,05 - 9,I Beam: Shear Stress txy,Shear Stress,For Wide Beams,05 - 10,I Beam: Shear Stress txz in the Flange,Shear Stress txz,q in Web,05 - 11,Example,A long beam is made of three planks, nailed together. Knowing that the spacing between nails is 25 mm and that the vertical shear in the beam

7、 is V = 500 N, determine the shear force in each nail.,Problem,05 - 12,Example: Solution Strategy,If the beam were one piece, there would be shear stress (shear flow) between the cap and the web.,Solution Strategy,The nails will carry the force resulted from the shear stress.,05 - 13,Example: Soluti

8、on,I,Shear Flow,Shear Force in Each Nail,Q of the cap,05 - 14,Shear Strain and Stress for Circular Shafts,Angle of Twist,Shear Strain,Shear Stress,Observa- tions,The square grid on the surface become a rhombus grid, indicating a pure shear phenomenon.,05 - 15,Shear Stress Distribution for Circular S

9、hafts,Shear Stress Distribution,Thin Walled Tubes,Solid,Hollow,05 - 16,Thin-Walled Hollow Shafts,Shear stress is uniformly distributed along the thickness and tangent to the cross section median line.,Assumption,Constant Shear Flow, q,Angle of Twist,Shear Stress,05 - 17,Applications: Design of Trans

10、mission Shafts,Designers must select shaft material and cross-section to meet performance specifications without exceeding allowable shearing stress.,1 hp = 33000 ft lbf/min = 550 ft lbf/s = 746 W,Shaft,Relation between Torque, Power and Speed,Unit Conversion,05 - 18,Example,Be aware that two bendin

11、g moment components existing at each point. Cut sections through shafts AB and BC and perform static equilibrium analysis to find torques and bending moments Find the locations of the maximum bending moment and torque. Calculate normal and shear stresses from the bending moments and torque. Calculat

12、e the value and orientation of principal stresses.,Problem,Solution Strategy,05 - 19,1200 lbf,1600 lbf.in,600 lbf,1600 lbf.in,400 lbf,200 lbf,800 lbf,400 lbf,Example: Analysis Model,05 - 20,Example: Internal Forces,Torque (= 1600 lbf-in) is constant in BC,05 - 21,Example: Critical Points,Either Poin

13、t B or Point C will be the critical point.,Point B is the critical point.,Bending Moments in Two Planes,Critical Points,The torque is the same at points B and C,If the cross-section is not circular, we cant use M to determine the critical point.,Note,05 - 22,Example: Bending Moment and Torque at Poi

14、nt B,Net Bending Moment at B,Torque,05 - 23,Example: Stresses Components,05 - 24,Example: Maximum Normal Stress at E,qp1 = 5.49, qp2 = 84.51 ,05 - 25,Example: Maximum Shear Stress at E,s s = 12,445 psi,Maximum Shear Stress.,05 - 26,Example: Maximum Normal Stress at F,05 - 27,Example: Maximum Shear S

15、tress at F,s s = 12,445 psi,Maximum Shear Stress.,05 - 28,Example: Summary,05 - 29,Curved Beams: Assumptions & Neutral Axis,The beam has a plane of symmetry. Plane cross sections remain plane after bending. Youngs Modulus is the same in tension as in compression.,The neutral and centroidal axes are NOT coincident.,Neutral Axis,Assump-tions,05 - 30,Curved Beams in Bending: Normal Stress,Normal Stress,The normal stress is NOT linearly distributed along the r-direction,Distribution,