1. Civil Engineering Materials – CIVE 2110
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
Compressive Strain
For material that is:
Homogeneous
Isotropic
Linear-Elastic
We can conclude for
STRESS, σ
Tensile Strain
Compressive Stress
Tensile Stress

16. 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

17. 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

18. 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

19. 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

20. 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

21. Beam Bending 2nd Moment of Area Calculation

22. A Rectangular Cross Section

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

25. PARALLEL AXIS THEOREM FOR 2nd MOMENTS OF AREA

27. 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=-

29. Steps for V and BM diagrams
1.Draw FBD 2.Obtain reactions: SM support) to obtain reaction at right; SM 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

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

31. BEAM END CONDITIONS
Pin
VL=RLY

32. BEAM END CONDITIONS
Roller Pin
VL=RLY