Civil Engineering Materials – CIVE 2110 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, 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.

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.

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.

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.

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.

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.

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

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

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

BEAM BENDING THEORY

BEAM BENDING THEORY

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

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

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

Internal Moment must resist External Moment. 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: Σ Internal Forces must = ZERO Neutral Axis Compressive Stress Tensile Stress

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

Internal Moment must resist External Moment. 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

Internal Moment must resist External Moment. 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

Flexural Bending Stress Equation: For Stress in the Direction of the 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

Beam Bending 2nd Moment of Area Calculation

A Rectangular Cross Section

PARALLEL AXIS THEOREM FOR 2nd MOMENTS OF AREA 2nd MOMENTS OF COMPOSITE AREAS B & J 8th,9.6, 9.7 Z Z Y2

PARALLEL AXIS THEOREM FOR 2nd MOMENTS OF AREA

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

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

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

BEAM END CONDITIONS Pin VL=RLY

BEAM END CONDITIONS Roller Pin VL=RLY