CHAPTER 5: ANALYSIS AND DESIGN OF BEAMS FOR BENDING
5.1 Introduction
•
Beams - structural members supporting loads at various points along the member
•
Transverse loadings of beams are classified as concentrated loads or distributed loads
•
Applied loads result in internal forces consisting of a shear force (from the shear stress
distribution) and a bending couple (from the normal stress distribution).
•
For example, a simply supported beam AB carrying two concentrated load and a
uniformly distributed load, Figure (a).
•
To determine the internal forces in a section through point C we first draw the freebody diagram of the entire beam to obtain the reactions at the supports, Figure (b).
•
Passing a section through C, we then draw the free body diagram of AC, Figure (c), from
which we determine the shear force V and the bending couple M.
(a) Transversely-loaded beam
(b) Free-body diagram to find support
reactions
(c) Free-body diagram to find internal forces at C.

Since the distribution of the normal stresses in a given section depends only upon the
value of the bending moment M in that section and the geometry of the section, the
elastic flexure formulas can be used to determine the maximum stress,
Normal stress is often the critical design criteria requires determination of the location
and magnitude of largest bending moment:
x  
My
I
m 
Mc M

I
S
Where,
I = moment of inertia of the cross section with respect to a centroidal axis perpendicular
to the plane of the couple
y = the distance from the neutral surface
c = the maximum value of the distance
1
S = 6 𝑏ℎ2 where b is the width and h is the depth of the cross section

Classification of Beam Supports
Steps in designing a beam for bending
Assume that E,G,sall, and tall for the material selected
are known
1. Construct the diagrams corresponding to the
specified loading conditions for the beam and define
|V|max and |M|max .
2. Assume that the design of the beam is
controlled by the normal stress at +,-c in the section
and determine Smin .
3. From available tables, select beams with S>Smin
CONSIDERATIONS:
-SMALL WEIGHT PER UNIT LENGTH
-SMALL DISPLACEMENT (SHOW IN THE LATTER CHAPTER
Shear and Bending Moment Diagram
Complete Beam Analysis Example
• For the beam loaded and supported as shown, determine the max tensile
and compressive stresses and where they occur.
1. Determine the location of the centroid.
• The area of the cross section (A) = 18.75 in2
2. Determine the area moment of inertia with respect to the centroidal axis (as
shown in figure on the lower left) and use transfer formula to calculate I of the
section.
3. Begin the analysis by defining reactions at A and B using FBD, one writes the
equations of equilibrium.
4. Next, construct the shear diagram
6. Next, we compute stresses. At x=3.5 ft, moment is negative so the top is in
tension while the bottom is under compression.
7. At x=7 ft, moment is positive so the top is under compression while the bottom
is in tension.
EXAMPLE OF REAL LIFE APPLICATION
Skateboard
Discuss the shear forces and bending
moment set up in the skateboard when
you stand on it sideways balanced on
your heels. Calculate the maximum
normal stress when;
Weight of a person = 50 kg, Length of
skateboard = 65 cm, Thickness = 4 cm,
Width = 18 cm, Distance x = a = 20 cm