1. ES2501: Statics/Unit 20-1: Internal Forces in Beams
Introduction:
Beam: Slender structural member subjected to lateral loading
In contrast to:
bar:
Subjected to axial loading
shaft:
Internal forces in a beam:
Subjected to torque
Three types of internal forces in a beam:
N ---- Normal force (to prevent relative axial
motion at C of two parts)
V ---- Shear force (to prevent relative lateral
sliding at C of two parts)
M ---- Bending moment (to prevent relative
rotation at C of two parts)
Cross-section

2. ES2501: Statics/Unit 20-2: Internal Forces in Beams
Introduction:
Sign conventions:
Conventional “+” direction
Nature of Internal forces:
Distributed shear
stresses
Distributed normal
stresses
Internal forces are in fact normal and shear stresses distributed over a cross-section area. N, V, and M are simplified models as resultant force in the axial direction, the resultant force in the lateral direction, and the resultant moment about the so-called neutral axism respectively.

3. ES2501: Statics/Unit 20-3: Internal Forces in Beams
Example 1: Find the internal forces of a beam, subjected to a point load,
at a specified location
Find internal forces on cross-sections
at C and D
Support Reactions:
Internal forces at C:
Note: these are reactions that do
not exactly the conventional signs
of internal forces

4. ES2501: Statics/Unit 20-4: Internal Forces in Beams
Example 1 (con’d):
Find internal forces on cross-sections at C and D
Note: the problem can also been equivalently
solved by equilibriun of the BD part
The same results but
more convenient
Internal forces at D:
Note: The normal force is zero in general for the case where a load
acts perpendicularly to the axis of a straight beam

5. ES2501: Statics/Unit 20-5: Internal Forces in Beams
Example 2: Find internal forces on cross-sections at C and D
Distributed load
Support Reactions:
Internal forces at C:

6. ES2501: Statics/Unit 20-6: Internal Forces in Beams
Example 2 (con’d): Find internal forces on cross-sections at C and D
Distributed load
Statically equivalent load
Internal forces at D:
Or, equivalently by equilibrium of BD:
The same results

7. ES2501: Statics/Unit 20-7: Internal Forces in Beams
Comments:
N-diagram
V-diagram
M-diagram
Find internal forces at any position
Graphic
representation
Finding internal forces in a beam is the first step of a complete beam analysis
Both need M(x) first

8. ES2501: Statics/Unit 20-8: Internal Forces in Beams
Comments:
Superposition principle
Problem II:
by distributed load only
Problem I:
by P only
By Beam analysis or results in Handbook
Reactions
Internal forces
Superposition
Total solution

9. ES2501: Statics/Unit 20-9: Internal Forces in Beams
Comments:
Statiacally inderminate problems:
# of static equilibrium equations =3
# of unknowns =6
Properties of symmetry may help:
By symmetry.