1 / 32

Classes #11 & #12

Classes #11 & #12. Civil Engineering Materials – CIVE 2110 Bending Fall 2010 Dr. Gupta Dr. Pickett. ASSUMPTIONS OF BEAM BENDING THEORY. Beam Length is Much Larger Than Beam Width or Depth. so most of the deflection is caused by bending,

martena-barker
Download Presentation

Classes #11 & #12

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Classes #11 & #12 Civil Engineering Materials – CIVE 2110 Bending Fall 2010 Dr. Gupta Dr. Pickett

  2. ASSUMPTIONS OF BEAM BENDING THEORY Beam Length is Much Larger Than Beam Width or Depth. so most of the deflection is caused by bending, very little deflection is caused by shear Beam Deflections are small. Beam is Perfectly Straight, With a Constant Cross Section (beam is prismatic). Beam has a Plane of Symmetry. Resultant of All Loads acts in the Plane of Symmetry. Beam has a Linear Stress-Strain Relationship.

  3. ASSUMPTIONS OF BEAM BENDING THEORY Beam Material is Homogeneous. Beam Material is Isotropic. Beam is Loaded ONLY by a Moment about an axis Perpendicular to the long axis of Symmetry. Thus Moment is CONSTANT across the Length of the Beam. There is NO SHEAR.

  4. ASSUMPTIONS OF BEAM BENDING THEORY Plane Sections Remain Plane. No Warping (no buckling, no rotation about vertical axis). Motion is only in Vertical Plane. Beam Cross Sections originally Perpendicular to Longitudinal Axis Remain Perpendicular.

  5. BEAM BENDING THEORY When a POSITVE moment is applied, TOP of beam is in COMPRESSION BOTTOM of beam is in TENSION. NEUTRAL SURFACE: - plane on which NO change in LENGTH occurs.

  6. BEAM BENDING THEORY When a POSITVE moment is applied, (POSITIVE Bending) TOP of beam is in COMPRESSION BOTTOM of beam is in TENSION. NEUTRAL SURFACE: - plane on which NO change in LENGTH occurs. Cross Sections perpendicular to Longitudinal axis Rotate about the NEUTRAL (Z) axis.

  7. BEAM BENDING THEORY When a POSITVE moment is applied, (POSITIVE Bending) TOP of beam is in COMPRESSION BOTTOM of beam is in TENSION. NEUTRAL SURFACE: - plane on which NO change in LENGTH occurs. Cross Sections perpendicular to Longitudinal axis Rotate about the NEUTRAL (Z) axis.

  8. BEAM BENDING THEORY For M = + Any line segment, Δx : - shortens, if located above Neutral Surface.

  9. BEAM BENDING THEORY For M = + Any line segment, Δx : - does not change length, if located at Neutral Surface.

  10. BEAM BENDING THEORY For M = + Any line segment, Δx : - lengthens, if located below Neutral Surface.

  11. BEAM BENDING THEORY

  12. BEAM BENDING THEORY

  13. Flexural Bending Equation We assumed: Cross Sections remain constant However, do to the Poisson’s Effect; there will be strains in the 2 directions perpendicular to the Longitudinal Axis. Axial Compressive Strain Axial Tensile Strain

  14. BEAM BENDING THEORY For material that is: Homogeneous Isotropic Linear-Elastic We can conclude for STRESS, σ

  15. BEAM BENDING THEORY For material that is: Homogeneous Isotropic Linear-Elastic We can conclude for STRESS, σ Compressive Strain Tensile Strain Compressive Stress Tensile Stress

  16. BEAM BENDING THEORY Internal Moment must resist External Moment. Internal Resisting Moment: Caused by an Internal Force resisting an External force Can find Neutral Axis by balance of Forces: ΣInternal Forces must = ZERO Neutral Axis Compressive Stress Tensile Stress

  17. BEAM BENDING THEORY Can find Neutral Axis by balance of Forces: ΣInternal Forces must = ZERO Neutral Axis = Centroidal Axis 0 =1st Moment of Area about Neutral Axis Neutral Axis Compressive Stress Tensile Stress

  18. BEAM BENDING THEORY Internal Moment must resist External Moment. M = (Lever Arm)x(Internal Force) M = (Lever Arm)x(Stress x Area) Neutral Axis Compressive Stress Tensile Stress

  19. BEAM BENDING THEORY Internal Moment must resist External Moment. M = (Lever Arm)x(Internal Force) M = (Lever Arm)x(Stress x Area) Neutral Axis Compressive Stress Tensile Stress

  20. BEAM BENDING THEORY Flexural Bending Stress Equation: For Stress in the Direction of the Long Axis (X), At any location, Y, above or below the Neutral Axis Compressive Stress Neutral Axis Tensile Stress

  21. Beam Bending2nd Moment of AreaCalculation

  22. A Rectangular Cross Section

  23. PARALLEL AXIS THEOREM FOR 2ndMOMENTS OF AREA2ndMOMENTS OF COMPOSITE AREASB & J 8th,9.6, 9.7 Z Z Y2

  24. PARALLEL AXIS THEOREM FOR 2ndMOMENTS OF AREA

  25. Sign Convention for Diagrams Tension MIntrnl=+ MIntrnl=+ MIntrnl=- MExtrnl=- Tension MIntrnl=+ Compression MIntrnl=- Compression Tension Tension MExtrnl=+ Compression Free End Or Pinned End Fixed End Fixed End V=+ Free End Or Pinned End V=- Tension Tension MIntrnl=- Compression MIntrnl=- MIntrnl=-

  26. Steps for V and BM diagrams 1.Draw FBD 2.Obtain reactions: SM (@left support) to obtain reaction at right; SM (@Right support) to obtain reaction at left; Check SFy = 0 3. Cut a section ; Obtain internal F (or P), V, M at cut section ; SM, SFy, SFx 4. Record, draw internal F (or P), V, M on both sides of cut sections ; - magnitude - units - direction on both sides of cut

  27. BEAM END CONDITIONS VL=RLY Roller Pin - Pin Fixed - Free Fixed - ?

  28. BEAM END CONDITIONS VL=RLY Pin

  29. BEAM END CONDITIONS Roller Pin VL=RLY

More Related