Advanced Structural Mechanics and Steel Structures

for Transportation Engineers:

Applications in Logistics and Infrastructure

CHAPTER I: INTRODUCTION TO STRENGTH OF

MATERIALS
Course Lecturer: Mrs.Laila HFA,
Eng & Phd Candidate
 Ground rules for this course
1 2
Punctuality: Arrive on time. Put away your cell phones and laptops
Arriving late is a sign of unless they are part of the class
disrespect to the course lecturer and activities.
to your colleagues.

3 4
Private conversations during class are Remember, you are your most valuable
inappropriate, especially when others resource. The success of this course
are speaking. relies on it being a two-way exchange
where everyone engages actively.
 The chapter’s objectives
● The main objective of this chapter is to understand:
 The different types of beams
 The different types of loads
 The internal effects of loads on beams
 The shear force
 The bending moment
 How to sketch shear force and bending moment diagrams.
 01.

Types of beams,
supports and
loads
 BEAMS
● The term beam refers to a slender bar that carries transverse loading;
that is, the applied forces perpendicular to the longitudinal axis of the
beam.
● In a beam, the internal force system consists of a shear force and a
bending moment acting on a cross section of the bar. The internal
forces give rise to two kinds of stresses on a transverse section of a
beam:
1. Normal stress that is caused by the bending moment
2. Shear stress due to the shear force.
 BEAMS
● Normal stress that is caused by the bending moment and shear
stress due to the shear force.
 TYPES OF BEAMS AND SUPPORTS
● Beams are classified according to their supports;
● A beam that has a pin support at one end and a roller
support at the other end is called a simply supported beam.
● A cantilever beam is built into a rigid support at one end, with the
other end being free.
● An overhanging beam is supported by a pin and a roller support,
with one or both ends of the beam extending beyond the supports;
 TYPES OF BEAMS AND SUPPORTS
● The three types of beams are statically determinate because the
support reactions can be found from the equilibrium equations.
● Some beams that are over supported in the sense that each beam
has at least one more reaction than is necessary for support. Such beams are
statically indeterminate. The presence of these redundant supports requires
the use of additional equations obtained by considering the deformation of the
beam.
 TYPES OF LOADINGS
● Concentrated or Point Load: A concentrated load is one, which is considered to act at a point,
although in practical it really be distributed over a small area.

● Uniformly Distributed Load (UDL): A UDL is one which is spread over a beam in such a manner
that rate of loading 'w' is uniform along the length (i.e. each unit length is loaded to the same
rate). The rate of l expressed as w/q N/m run. For solving problems, the total UDL is converted
into a point load, acting at the center of UDL.

● Uniformly Varying Load (UVL): A UVL is one which is spread over a beam in such a manner that
rate of loading varies from point to point along the beam, in which load is zero at one end and
increase uniformly to the other end. Such load is known as triangular load. For solving
problems the total load is equal to the area of the triangle and this total load is assumed to be
acting at the Centroid of the triangle ( at a distance of 2/3 of the total length of the beam from
left end.
02.

Shear force &

Bending Moment
 SHEAR FORCE AND BENDING MOMENT
● Shear Force (S.F.): The algebraic sum of all
the vertical forces at any section of a beam
to the right or left of the section is known
as shear force.

● Bending Moment (B.M.): The algebraic sum

of all the moment of all the forces acting to
the right or left of known as bending
Moment.

● Shear Force (S.F.) and Bending Moment

(B.M.) Diagrams: S.F. diagram is the one
that shows the variation of the shear force
along the length of the beam. And a
bending moment diagram is the one that
shows the variation moment along the
length of the beam.
 SHEAR FORCE AND BENDING MOMENT
● Both shear force and bending moment are
vector quantities requiring a convention of
signs in order that values of opposite sense
may be separated.

● For consistency, it is necessary to adopt sign

conventions for applied loading, shear
forces and bending moments. Assume the
following to be positive:

● External forces that are directed upward,

external couples that are directed clockwise
● Shear forces that tend to rotate beam
element clockwise.
● Bending moments that tend to bend a beam
element concave upward.
 SHEAR FORCE AND BENDING MOMENT
● The shear force is positive if it tends to rotate the beam section clockwise with
respect to a point inside the beam section.

● The bending moment is positive if it tends to bend the beam section concave
facing upward (if it tends to put the top of the beam into compression and the
bottom into tension.
 SHEAR FORCE AND BENDING MOMENT
● The bending moment is considered as Sagging Bending Moment (concave) if it
tends to bend the beam to a curvature having convexity at the bottom. Sagging
Bending Moment is considered as positive bending moment.

● The bending moment is considered as Hogging Bending Moment (convex), if it

tends to bend the beam to a curvature having convexity at the top. Hogging
Bending Moment is considered as Negative Bending Moment
 SHEAR FORCE AND BENDING MOMENT
● In the Bending Moment Diagram, the point at which the BM
changes its sign form positive to negative or negative to positive
is called “Contraflexure”.

● It’s a point where the beam tends to bend in opposite direction.

It is the point at which the curvature of the beam changes.
 SHEAR FORCE AND BENDING MOMENT
 The shear and moment diagrams for a beam can be constructed using the following
procedure:
 Support Reactions.
 Determine all the reactive forces and couple moments acting on the beam, and
resolve all the forces into components acting perpendicular and parallel to the
beam’s axis:
 Shear and Moment Functions.

 Specify separate coordinates x having an origin at the beam’s left end and extending
to regions of
 the beam between concentrated forces and/or couple moments, or where there is no
discontinuity of distributed loading.
 Section the beam at each distance x, and draw the free-body diagram of one of the
segments. Be sure V and M are shown acting in their positive sense, in accordance
with the sign convention.
 SHEAR FORCE AND BENDING MOMENT
 The shear is obtained by summing forces perpendicular to the beam’s axis.
 To eliminate V, the moment is obtained directly by summing moments about the
sectioned end of the segment.
 Shear and Moment Diagrams.
 Plot the shear diagram (V versus x) and the moment diagram (M versus x). If
numerical values of the functions describing V and M are positive, the values are
plotted above the x axis, whereas negative values are plotted below the axis.
 Generally, it is convenient to show the shear and moment diagrams below the free
body diagram of the beam.
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● The relationship between loads, shear forces and bending moments in
beams are useful and helpful when calculating the values and
constructing the diagrams related the shear force and the bending
moment.
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● Distributed load:

For this equation, the rate of change of the shear

force, at any point on the axis of the beam is equal
to the negative of the intensity of the distributed
load at the same point.
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● Distributed load:
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● Distributed load:

This equation shows that the rate of change of the bending moment at any point on
the axis of a beam is equal to the shear force at the same point.
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● Distributed load:
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● Concentrated (Point) load:

The result means that an abrupt change in the

shear force occurs at any point where
concentrated load acts.

The bending moment does not change as we pass through the point of
application of a concentrated load
 SHEAR FORCE, BENDING MOMENT & LOAD
INTENSITY
● Load in the form of a couple:

For equilibrium of forces we obtain that the shear

force does not change at the point of the application
of a couple.
Equilibrium of moments about the left-hand side of the element gives:

This equation shows that the bending moment decreses by M0 as we move

left to right through the point of load application, thus the bending moment
changes abruptly at the point of application of a couple.
 03

Bending stress
in beams
 BENDING STRESS IN BEAMS
 Axial Stress or Bending stress (tension and compression) and the Shear Stress
(horizontal stress) which develop in a loaded beam depend on the values of the
Bending Moments and the Shear Forces in the beam.
 BENDING STRESS IN BEAMS
 The stresses in beams caused by the bending moment are known as bending
stresses or flexural stresses.
 These stresses are normal on beam section and represented by couple of
compressive and tensile stresses according to bending moment direction.
 BENDING STRESS IN BEAMS
 PURE BENDING OR SIMPLE BENDING: when a length of the beam is subjected
to a constant bending moment (shear force = 0 ) or when the beam is subjected
to equal opposite couples acting in the same longitudinal plane.
 BENDING STRESS IN BEAMS
 Bending stress assumptions:

● The member (beam) remains symmetric

● The material is homogeneous and isotropic

● The beam bends uniformly to for a circular arc

● The cross sectional remain planar after

deformation
● The fiber in tension increases in length and the
one in compression decreases in length
● A neutral surface ( axis) is parallel to the upper
and lower surfaces (axes) and for which the
length does not change.
 STRESS AND STRAIN DUE TO BENDING
 The figure shows a part of a beam is subjected to simple bending. Due to the
action of the bending moment.

 The fibers above the neutral axis of the beam are in compression and
decrease in length.
 The fibers under the neutral axis of the beam are in tension and increase in
length.
 The neutral axis doesn’t change in length.
 STRESS AND STRAIN DUE TO BENDING
 Let’s consider a small length dx of a beam subjected to a simple bending (a),
and due to the action of bending, the length dx will be deformed as shown in
figure (b).

 The layers above NN are subjected to compressive stress.

 The layers under NN are subjected to tensile stress.

 The amount by which a layer increases or decreases in length is known as the

theory of simple (pure) bending.
 STRESS AND STRAIN DUE TO BENDING
R = Radius of curvature of neutral layer N’-N’.
θ = Angle subjected at O by A’B’ and C’D’.
y = Distance from the neutral layer.
Original length of the layer = EF = δx = NN = N’N’= R θ
Change (Increase) in length δ of the EF =E’F’ –EF = (R + y) θ – R θ = y θ
Change in length y θ y
Strain in the layer EF = ϵ = = =
original length Rθ R
σ
According to linear elasticity, That is, ϵ = (Hooke’s law)
E
Stress variation from the neutral axis:
y σ E
= σ= y
R E R
E
In general: σ = ± y
R
NEUTRAL AXIS AND MOMENT OF RESISTANCE
Due to pure bending the layers above the NA are subjected to compressive
stresses whereas the layers below the NA are subjected to tensile stresses. Due
to these stresses forces acts on the layers.
E
The stress at a distance y from the neutral axis is given by σ =± y
R
Let dA = Area of layer
E
Force on layer = stress on layer × area of layer = σ × dA = y × d A.
R
E E
Moment of the force about NA =M = ∫ y × y × dA = ∫ y𝟐 ×dA
R R
The expression ∫ y𝟐 ×dA represents the 2nd moment of inertia of the area of
the section about the neutral axis.
E My
Then M = 𝐈  σx =± known also as flexure formula.
R I𝒛
 MAXIMUM STRESSES ON A CROSS SECTION
Since the intensity of the bending stress varies linearly with distance y
from the neutral surface, the maximum bending stress occurs on either the
top or the bottom surface of the beam, depending on which surface is
M y𝒎𝒂𝒙
farther from the neutral surface . Then σ𝒎𝒂𝒙 =
I𝒛

The quantities S𝑡𝑜𝑝 and S𝑏𝑜𝑡 are called the section moduli

M𝒄 M
For Symmetric Cross-Sections c𝒕𝒐𝒑 = c𝒃𝒐𝒕 = 𝒄. Then σ𝒎𝒂𝒙 =±
𝑰𝒛 I𝒛 = ± S
, Where 𝑺 =±
𝒄
 BENDING STRESSES IN BEAMS
Bending stresses in a beam are evaluated in a three step process:
1- Determine the Internal Bending Moment ( Shear force and bending moment
diagrams)
2- Calculate the Geometric properties of the cross section of the beam.
3- Use the Flexure Formula to calculate Bending Stresses and the maximum stresses.
 04

Application
APPLICATION
 Thanks!
See you next
week !
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