CENG 212 – Mechanics of Materials
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Welcome to the Course!
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
1
CENG 212 – Mechanics of Materials
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
CHAPTER 1:
Average Normal and Shear Stresses
(Based on Dr. Semih Erhan’s Material)
Objectives:
• Review important principles of statics.
• Principles to determine internal resultant loadings in a
body.
• Normal and shear stress.
• Applications of analysis and design of members subjected
to an axial load or direct shear
For UoB Students: It is the pre-requisite of :
• CENG 311 – Structural Analysis 1
• CENG 341 – Soil Mechanics
Dr. Mahmoud Jahjouh
2
Mechanics
University of Bahrain
College of Eng.
Civil Eng. Dept.
Mechanics of Materials: is a branch of mechanics that studies the internal
effects of stress and strain in a solid body that is subjected to an external
loading.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Important to Understand after this Chapter:
• External Loads:
(Applied Loads on a body)
• Internal Loads:
(Internal response to applied loads)
• Equilibrium of deformable bodies: (External = Internal/Action = Reaction)
• Stresses:
(Force per Area)
• Strains:
(dL/Original L).
Those terms are going to be explained further.
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
3
External Loads
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
External Loads: A body is subjected to two types of external loads; surface
forces or body forces.
• Surface Force: Applied from outside contact.
• Body Force:
Resulting from mass and movement.
• Surface Forces:
Distributed over Area
Distributed over Line
Concentrated
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
4
External Loads
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
External Loads: A body is subjected to two types of external loads; surface
forces or body forces.
• Surface Force: Applied from outside contact.
• Body Force:
Resulting from mass and movement.
• Body Force :
• Own Weight (Gravitational Field)
• Electro Magnetic Force (Electro Magnetic Field)
• Inertia Loads.
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
5
Internal Loads
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Internal Loads: Loads that are generated in response to the applied
surface loads and body loads.
• Internal Forces:
• Axial Forces: Force Perpendicular to the plane of the section.
• Shear Forces: Force Parallel to the plane of the section
• Internal Moments:
• Bending Moments: Moment in the plane of the section.
• Torsion: Moment perpendicular to the plane of the section.
Book:
Mechanics of Materials
Hibbeler 9thed.
3D General
3D Beam
2D Beam
Dr. Mahmoud Jahjouh
6
Equilibrium
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Equilibrium: Sum of Forces and Moments should be zero.
For equilibrium (External), forces and moments are balanced
F = 0
M = 0
General procedure for solution is based on two step:
i. Draw a free-body diagram to account for all forces acting on the body
ii. Apply the equilibrium equations to achieve equilibrium state
For equilibrium (Internal Loads)
•
Draw free body diagram
•
Calculate support reaction
•
Choose segment to analyze
•
Indicate the internal forces (N, V, M) on the segment in positive
directions as follows and apply equilibrium.
Dr. Mahmoud Jahjouh
7
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
Reminder to support Reactions:
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
8
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: Determine resultant loadings acting on cross section at C of
the cantilever beam.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
9
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: Determine resultant loadings acting on cross section at C of
the cantilever beam.
Intensity (w) of loading at C
Course:
CENG 212 - MoM
w 270
=
 w = 180 N / m
6
9
1
F = ( 180 )( 6 ) = 540 N
2
F acts 1/3 of length 6 m from C
= 2 m from C.
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
+
∑ Fx = 0;
− Nc = 0
Nc = 0
+
∑ Fy = 0;
Vc − 540 N = 0
Vc = 540 N
+
∑ Mc = 0;
−Mc − 540 N (2 m) = 0
Mc = −1080 N·m
Dr. Mahmoud Jahjouh
10
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: The 500-kg engine is suspended from the crane boom in
Figure. Determine the resultant internal loadings acting on the cross
section of the boom at point E.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
11
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
FCD
Course:
CENG 212 - MoM
FCD
C
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Ax
B
Ay
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
12
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
3
+  M A = −500  9.81 (3m) + FCD  (2m) = 0
5
FCD = 12262.5 N
+  Fx = Ax −
4
FCD = 0
5
Ax = 9810 N
3
+   Fy = − Ay + FCD − 500  9.81 = 0
5
Ay = 2452.5 N
+ Fx = 9810 + N E = 0  N E = −9810 N
 + Fy = −2452.5 − VE = 0  VE = −2452.5N
+ M = 2452.5  (1m) + M E = 0  M E = −2452.5 N .m
Dr. Mahmoud Jahjouh
13
Statics Review
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: Determine the resultant internal loadings acting on the cross
section at G of the beam shown in Figure. Each joint is pin connected.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
N
m
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
N/m
m
m
m
Dr. Mahmoud Jahjouh
14
Statics Review
+ M E = 900  (4m) + 1500  (10m) − FBC  (3m) = 0
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
FBC= 6200 N
 + Fy = −1500 − 900 + Ey = 0
Ey = 2400 N
+ Fx = − Ex + FBC = 0
Ex = 6200 N
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
15
Statics Review
4
FBA = 0
5
FBA= 7750 N
+  Fx = 6200 −
University of Bahrain
College of Eng.
Civil Eng. Dept.
3
 +  Fy = FBD −  FBA = 0
5
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
FBD= 4650 N
3
+  M E = 1500  (2m) −  7750  (2m) + M G = 0
5
MG = 6300 N.m
3
 +  Fy = −1500 +  7750 − VG = 0
5
3
+  Fx =  7750 + NG = 0 VG = 3150 N.m
5
NG = -6200 N.m
Dr. Mahmoud Jahjouh
16
Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Stress: The force per unit area => The intensity of force.
Stress (SIMPLIFIED)/AVERAGE: Average Stress is Force divided by Area.
Loads alone: Do not give a full picture as of the effect of loads on the
structural components.
100kN
100kN
Is more dangerous, because it has a smaller section
100kN
100kN
Is less dangerous, because it has a larger section
So, stress, which is the force per unit area, is a better measurement.
A simplified version is: Stress = Force / Area
A more accurate version is:
σz =
lim ΔFz
ΔA →0
Dr. Mahmoud Jahjouh
ΔA
17
Axial Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
Axial Stress: Stresses induced by axial forces, which are forces
Perpendicular to the section.
_
Average Stress
Actual Stress
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
𝜎𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
𝑃
𝐴
σz =
lim ΔFz
ΔA →0
Dr. Mahmoud Jahjouh
ΔA
18
Axial Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Normal Stress: The intensity of the force acting normal to (ΔA) is
defined as the normal stress, σ (sigma).
_
σz =
lim ΔFz
ΔA →0
ΔA
Tensile stress: Normal force “pulls” or “stretches” the area
element ΔA. (We use POSITIVE SIGN for that)
Compressive stress: Normal force “pushes” or “compresses” area
element ΔA. (We use NEGATIVE SIGN for that)
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
Tension
Compression
19
Shear Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
Shear Stress: Stresses induced by shear forces, which are forces Parallel to
the section.
_
Average Stress
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
𝜏𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
𝑃
𝐴
Dr. Mahmoud Jahjouh
20
Shear Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
Shear Stress: Stresses induced by shear forces, which are forces Parallel to
the section.
_
Actual Stress
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
τzx =
τzy =
lim ΔFx
ΔA →0
ΔA
lim ΔFy
ΔA →0
ΔA
Shear Stress: The intensity of force acting tangent to ΔA is called
the shear stress, τ (tau).
Dr. Mahmoud Jahjouh
21
General Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
• General state of stress: Figure shows the state of stress acting
around a chosen point in a body. This state of stress is then
characterized by three components acting on each face of the
element.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
•Units (SI system)
•Unit of stress is pascal (1 Pa = 1 N/m2)
•kPa = 103 N/m2 (kilo-pascal)
•MPa = 106 N/m2 (mega-pascal)
•GPa = 109 N/m2 (giga-pascal)
Dr. Mahmoud Jahjouh
22
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Average normal stress
• From equilibrium: we need to add all the “mini forces” from
those little areas (dA)
+ FRz = ∑ Fz
∫ dF = ∫A σ dA
P = σA
P
σ=
A
Book:
Mechanics of Materials
Hibbeler 9thed.
σ = average normal stress at any point on cross sectional area
P = internal resultant normal force
Dr. Mahmoud Jahjouh
A = x-sectional area of the bar
23
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Maximum average normal stress
If internal force P and x-sectional area A are constant along the bar,
σ = P/A
constant along the bar.
• If the bar is subjected to several external loads along its axis and
there is a change in x-sectional area we have to find the section
where the P/A is maximum.
• For such cases, first draw an axial or normal force diagram along
bar’s length. Then determine the maximum average normal stress
from the plot.
• P is positive (+) (Tension) P is negative (-) (Compression)
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
24
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: Determine max. average normal stress in bar when
subjected to loading shown if the bar width = 35 mm, thickness = 10
mm
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
25
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Solution:
Internal Forces:
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Normal force diagram:
Book:
Mechanics of Materials
Hibbeler 9thed.
Largest loading area is
BC, where PBC = 30 kN
Dr. Mahmoud Jahjouh
26
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Solution:
Maximum Average normal stress:
Note (It is NOT always where the maximum force is!)
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
σBC =
PBC
A
=
30(103) N
(35 mm)(10 mm)
= 85.7 MPa
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
27
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Example: Determine average compressive stress acting at points A and
B if the specific unit weight γst = 80 kN/m3
200 mm
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
800 mm
100 mm
200 mm
200 mm
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
28
Average Normal Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Solution:
Free Body Diagram:
Course:
CENG 212 - MoM
Weight of segment AB determined from
Wst = γstVst
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
800 mm
+   Fz = 0  P − Wst = 0
P − ( 80kN / m3 )( 0.8m )( 0.2m )2 = 0
P = 8.042 kN
P 8.042 103 N
= =
= 0.064 MPa
2
2
A  200 mm
Note that stress distribution is uniform in the case of
axially loaded member. Therefore, normal stresses at point
of A and B are equal.
Dr. Mahmoud Jahjouh
29
Average Shear Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
• Shear stress is the stress component that act in the plane of the
sectioned area as shown in figure.
σz
Course:
CENG 212 - MoM
τzx
τzy
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
• Consider the effect of applying a force F acting to the bar as shown
below:
• If the supports are considered rigid, the bar will deform and fail
along the planes identified by AB and CD
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
Free body diagram
30
Average Shear Stresses
The average shear stress over each section is defined by:
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
V
τavg =
A
τavg = Average shear stress at section.
V = Internal shear force at section determined
from equations of equilibrium
A = Area of section
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
The distribution of average shear stress acting over the sectioned is
shown in Figure.
(Uniform distribution)
Book:
Mechanics of Materials
Hibbeler 9thed.
This is example of simple or direct shear. The shear is caused by the
direct action of applied load F. This type of shear occurs in
connections that use, bolts, pins, welded material
Dr. Mahmoud Jahjouh
31
Average Shear Stresses
Single, double
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
32
Average Shear Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Single shear
Steel and wood joints shown below are examples of single-shear
connections, also known as lap joints.
For equilibrium, V = F.
The average shear stress can be determined on colored section in (d).
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
33
Average Shear Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Double shear
The joints shown below are examples of double-shear connections,
often called double lap joints.
For equilibrium, V = F/2.
Apply average shear stress equation to determine average shear stress
acting on colored section in (d).
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
34
Average Shear Stresses
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Procedure for analysis
Internal shear
1.
Section member at the point where the τavg is to be determined
2.
Draw free-body diagram
3.
Calculate the internal shear force V
Average shear stress
1.
Determine sectioned area A
2.
Compute average shear stress τavg = V/A
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
35
Average Shear Stresses
Single Shear
University of Bahrain
College of Eng.
Civil Eng. Dept.
12.5 kN
12.5 kN
VB 12.5 103 N
B =
=
= 17.7 MPa
 2
AB
30 mm2
4
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Pin diameter
=30 mm
VB=12.5 kN
Double Shear
VA=10.68 kN
21.36 kN
Pin diameter
=20 mm
VA=10.68 kN
VA 10.68 103 N
A =
=
= 34MPa
 2
AA
2
20 mm
4
Dr. Mahmoud Jahjouh
36
Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: Determine average normal stress and average shear stress
acting along (a) section planes a-a, and (b) section plane b-b. (depth
and thickness = 40 mm)
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
37
Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Solution:
Section a-a:
Internal loading: Based on free-body diagram, Resultant loading of
axial force, P = 800 N
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Fa-a=800 N
P=800 N
Average stress:
P
=
σ=
A
800 N
(40 mm)(40 mm)
= 0.5 MPa = 500 kPa
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
38
Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Solution:
Section b-b:
Internal loading:
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
❖
+
F
+
F
x
y
Or directly using x’, y’ axes:
= −800 + N sin 60o + V cos 60o = 0
+ Fx = N − 800 cos 30o = 0
= V sin 60o − N cos 60o = 0
+ Fy = V − 800 sin 30o = 0
N = 692.8N
V = 400 N
Dr. Mahmoud Jahjouh
39
Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Solution:
Section b-b:
Cross section area:
Course:
CENG 212 - MoM
40 mm
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
x = 40 / sin 60o
60o
x
A = ( 40mm )  ( 40mm / sin 60o )
40 mm
Average normal stress:
=
N
692.8 N
=
= 0.375MPa
A ( 40mm )( 40mm / sin 60o )
Average shear stress:
=
V
400 N
=
= 0.217 MPa
A ( 40mm )( 40mm / sin 60o )
Stress distribution
Dr. Mahmoud Jahjouh
40
Allowable Stress
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
When designing a structural member or mechanical element, the
stress in this member or element must be restricted to safe level. This
level or value of stress is allowable stress.
One method of specifying the allowable stress for a member is to use
a number called the factor of safety. The factor of safety (F.S.) is a
ratio of the failure stress (σfail) to the allowable load (σallow).
F .S . =
F .S . =
 fail
 allow
 fail
 allow
allow =
allow =
 fail
F .S.
 fail
F .S .
Here (σfail) is found from experimental testing of the material, and the
factor of safety (F.S.) is selected based on experience. The factor of
safety must be greater than 1 in order to avoid the potential for
Dr. Mahmoud Jahjouh
41
failure.
Allowable Stress
Factor of Safety: Why???? How is it determined?
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
•
•
•
•
•
•
•
•
Material Uncertainty
Loading Uncertainty
Analysis Uncertainty
# Loading Cycles
Type of failure (with warning or without).
Importance of the member.
Risk to life and property
Influence on the function of the structure
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
42
Simple Design Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
By making simplifying assumptions regarding the behavior of the
material, the following equations can often be used to analyze or
design a simple connection or mechanical element.
P
=
A
avg
V
=
A
Cross-sectional area of member subjected to normal force
A=
P
allow
Dr. Mahmoud Jahjouh
43
Simple Design Examples
Cross-sectional area of a connecter subjected to shear
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
44
Simple Design Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Required area to resist shear caused by axial load
The embedded length l of this rod in concrete can be determined
using the allowable shear stress of the bonding glue.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
45
Simple Design Examples
University of Bahrain
College of Eng.
Civil Eng. Dept.
Required area to resist bearing:
Bearing stress is normal stress produced by the compression of one
surface against another.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
The area of the column base plate B is determined from the allowable
bearing stress for the concrete
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
46
Example
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Procedure for analysis
1. Draw internal force diagrams.
2. Determine the section over which the critical forces (maximum
forces) is acting.
3. Then use the following formulas to calculate the required area to
sustain these loads.
A=
P
A=
allow
V
allow
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
47
Example
University of Bahrain
College of Eng.
Civil Eng. Dept.
Example: Determine the maximum magnitude P of the load the beam
will support if the average shear stress in each pin is not to allowed to
exceed 60 MPa. All pins are subjected to double shear as shown, and
each has a diameter of 18 mm.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Dr. Mahmoud Jahjouh
48
Example
Free Body Diagram
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
FBC
Ax
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Ay
+ M A = FBC  sin 30o  (6) − P  (2) − P  (4) = 0
FBC = 2 P
+
F
= Ax − 2  P  cos30o = 0
x
Ax = 1.732 P
Book:
Mechanics of Materials
Hibbeler 9thed.
+
F
y
= Ay − P − P + FBC  sin 30o = 0
Dr. Mahmoud Jahjouh
Ay = P
49
Example
Solution:
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
❖Note that the shear forces acting on the pins at B and C are:
VB = VC = FBC = 2P
❖The force acting on pin A is:
VA =
Ax 2 + Ay 2 =
P 2 + 1.732 P = 2 P
❖All pins are subjected to same force and double shear.
2P
V=
=P
2
❖Then P is calculated as:
V
P
allow =  60Mpa =
 P = 15268 N = 15.3kN

A
( 18mm )2
4
Dr. Mahmoud Jahjouh
50
Example
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Example: The two members pinned together at B. If the pins have an
allowable shear stress of τallow=90 MPa, and allowable tensile stress of
rod CB is (σt)allow=115 MPa. Determine to nearest mm the smallest
diameter of pins A and B and the diameter of rod CB necessary to
support the load.
A
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
C
5
6 kN
4
A
B
2m
3
B
1m
Dr. Mahmoud Jahjouh
51
Example
Free Body Diagram
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
6 kN
5
Ax
FBC 4
2m
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
3
1m
3
+  M A = FBC   (3) − 6  (2) = 0
5
FBC = 6.67 kN
Ay
+
 Fx = Ax − FBC 
4
=0
5
Ax = −5.32 kN
Book:
Mechanics of Materials
Hibbeler 9thed.
+
3
F
=
A
−
6
+
F

=0
 y y
BC
5
Ay = 2 kN
Dr. Mahmoud Jahjouh
52
Example
Solution:
University of Bahrain
College of Eng.
Civil Eng. Dept.
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Pin at A:
Double Shear
VA = 5.322 + 22 = 5.68 kN
VA / 2 2.84  103 N
AA =
=
= 31.56 mm 2
allow
90 Mpa
 d A2
= 31.56 mm 2  d A = 6.3 mm
4
Pin at B:
VB = 6.67 kN
Single Shear
VA
6.67  103 N
AB =
=
= 74.1 mm 2
allow
90 Mpa
 d B 2
= 74.1 mm 2  d B = 9.7 mm
4
Dr. Mahmoud Jahjouh
53
Example
Diameters of pins:
University of Bahrain
College of Eng.
Civil Eng. Dept.
❖
dA = 7 mm
Course:
CENG 212 - MoM
Instructor:
Dr. Mahmoud Jahjouh
Table of Contents:
• Title
• Mechanics
• Statics Review
• Stress
• Average N. Stress
• Average Sh. Stress
• Examples
• Allowable Stress
• Design Example
Book:
Mechanics of Materials
Hibbeler 9thed.
Choose a size larger to nearest millimeter.
dB = 10 mm
Diameter of rod:
PBC
6.67  103 N
A=
=
= 58 mm 2
 allow
115Mpa
❖
 d BC 2
= 58 mm 2  d BC = 8.59 mm
4
Choose a size larger to nearest millimeter.
dBC = 9 mm
Dr. Mahmoud Jahjouh
54