MECHANICS OF MATERIALS
(ID4020)
Coordinator : Dr. N. Sathiparan
Credits : 03
Offered by : Department of Civil Engineering
Prerequisite/s : None
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
CIVIL ENGINEERING
• Civil Engineering is a
composite of many specific
disciplines.
– Structural
– Transportation
– Geotechnical
– Hydrology
– Environmental
– Construction Management
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
STRUCTURAL ENGINEERING
• Structural engineers build
structures such as:
– Tall buildings & towers,
– Bridges,
– Dams,
– Retaining walls, & foundations, &.
– Stadiums.
• Design using different
construction materials such as:
steel, concrete, masonry, timber,
or other composite materials
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Why do we study Mechanics of Materials?
• Anyone concerned with the strength and physical
performance of natural/man-made structures
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Course Contents
1Lecture; 2Tutorial; 3Lab/ Fieldwork; 4Assignment
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Method of Assessment
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Equilibrium of a Deformable Body
• External Loads. A body is subjected to only two types of external
loads; namely, surface forces or body forces
• Surface forces are caused by
the direct contact of one
body with the surface of
another
• A body force is developed
when one body exerts a
force on another body
without direct physical
contact between the bodies
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Equilibrium of a Deformable Body
• Support Reactions: The surface forces that develop at
the supports or points of contact between bodies are
called reactions
• If the support prevents translation in a given direction,
then a force must be developed on the member in that
direction. Likewise, if rotation is prevented, a couple
moment must be exerted on the member.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Equilibrium of a Deformable Body
• Most common supports
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
MECHANICS OF MATERIALS
• A branch of mechanics that studies the internal effects of stress
and strain in a solid body that is subjected to an external loading.
• Study the behaviour of solid bodies subjected to various types of
loading
• Study of the body’s stability
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
MECHANICS OF MATERIALS
• Stress is associated with the strength of the material from which
the body is made, while strain is a measure of the deformation of
the body.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Forces
• For a body to prevent from translating or having accelerated
motion along a straight or curved path then ΣF= 0
• For a body not to rotate, ΣM= 0
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Review of Statics
Example 1
• The structure is designed to
support a 30 kN load
• The structure consists of a
boom and rod joined by
pins (zero moment
connections) at the
junctions and supports
• Perform a static analysis to
determine the internal
force in each structural
member and the reaction
forces at the supports
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Review of Statics
Example 2
• Determine the resultant internal loadings acting on the cross
section at C of the cantilevered beam shown in Fig. 1–4a.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Review of Statics
Example 2
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Review of Statics
Example 3
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.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Stress
• Expressed as force intensity, that is, as force per unit area.
𝐹𝑜𝑟𝑐𝑒
𝑠𝑡𝑟𝑒𝑠𝑠 =
𝐴𝑟𝑒𝑎
• Stress has two components, one acting perpendicular to the
plane of the area and the other acting parallel to the area.
Normal stress, Shear stress
• International System of units (SI):
Magnitude of stress is N/m2, called Pascal (Pa)
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Normal stress
Force 𝐧𝐨𝐫𝐦𝐚𝐥 (i. e. , perpendicular) to an area
𝑁𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 =
Area on which the force acts
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Normal stress
Axial Loading: Normal Stress
• The resultant of the internal forces for an axially
loaded member is normal to a section cut
perpendicular to the member axis.
• The force intensity on that section is defined as
the normal stress.
F P
lim ave
A0 A A
• The normal stress at a particular point may not be
equal to the average stress but the resultant of the
stress distribution must satisfy
P ave A dF dA
A
• The sign convention for normal stress: A positive
value indicates tensile stress, a negative indicates
compressive stress.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Normal stress
• Prismatic All cross sections are the same
throughout its length
• Homogeneous material has the same physical
and mechanical properties throughout its
• Isotropic material has these same properties in
all directions.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 4:
• Two solid circular rods are welded to a plate at B to form a
single rod, as shown in figure. Consider the 30-kN force at B
to be uniformly distributed around the circumference of the
collar at B and the 10 kN load at C to be applied at the
centroid of the end cross section. Determine the axial stress
in each portion of the rod.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 1 continuation:
Can the structure safely support the 30 kN
load?
• From a statics analysis
FAB = 40 kN (compression)
FBC = 50 kN (tension)
• At any section through member BC, the
internal force is 50 kN with a force intensity
or stress of
dBC = 20 mm P 50 103 N
BC 159 MPa
A 314 10-6 m 2
• From the material properties for steel, the
allowable stress is
all 165 MPa
• Conclusion: the strength of member BC is
adequate
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 1 continuation:
• Design of new structures requires selection of
appropriate materials and component dimensions
to meet performance requirements
• For reasons based on cost, weight, availability,
etc., the choice is made to construct the rod from
aluminum (all= 100 MPa). What is an
appropriate choice for the rod diameter?
P P 50 103 N
all A 500 10 6 m 2
A all 6
100 10 Pa
d2
A
4
d
4A
(
4 500 10 6 m 2 2.52 102 m 25.2 mm
• An aluminum rod 26 mm or more in diameter is
adequate
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Normal strain
𝐸𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 𝛿
𝑆𝑡𝑟𝑎𝑖𝑛 𝜀 = =
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝐿
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 5:
• The bar in figure has a constant width of 35 mm and a thickness of
10 mm. Determine the maximum average normal stress in the bar
when it is subjected to the loading shown.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 6:
• A short post constructed from a hollow circular tube of
aluminum supports a compressive load of 2.6 kN. The inner
and outer diameters of the tube are d1 = 100 mm and d2 =
120 mm, respectively, and its length is 500 mm. The
shortening of the post due to the load is measured as 2 mm.
• Determine the compressive stress and strain in the post.
(Disregard the weight of the post itself)
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Shear stress
Defined as the stress component that acts in the plane of
the sectioned area.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Shear stress
• Forces P and P’ are applied transversely to the
member AB.
• Corresponding internal forces act in the plane
of section C and are called shearing forces.
• The resultant of the internal shear force
distribution is defined as the shear of the section
and is equal to the load P.
• The corresponding average shear stress is,
P
ave
A
• Shear stress distribution varies from zero at the
member surfaces to maximum values that may be
much larger than the average value.
• The shear stress distribution cannot be assumed to
be uniform.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Shear stress
Shearing Stress Examples
Single Shear Double Shear
P F P F
ave ave
A A A 2A
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Shear strain
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Notation for stress
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Internal equilibrium
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Internal equilibrium
• For equilibrium principles:
τxy = τyx, τxz = τzx , τzy= τyz
• All four shear stresses must have equal magnitude
and be directed either toward or away from each other
at opposite edges of the element,
• This is referred to as the complementary property of shear,
and under the conditions, where the material is subjected
to pure shear
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 7:
The 80-kg lamp is supported by two rods AB and BC as shown in
Fig. If AB has a diameter of 10 mm and BC has a diameter of
8 mm, determine the average normal stress in each rod.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 7:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 7:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 8:
When force P is applied to the rigid lever arm ABC in figure, the
arm rotates counterclockwise about pin A through an angle of 0.05°.
Determine the normal strain developed in wire BD.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 9:
Member AC shown in figure is subjected to a vertical force of 3 kN.
Determine the position x of this force so that the average
compressive stress at the smooth support C is equal to the average
tensile stress in the tie rod AB. The rod has a cross-sectional area of
400 mm2 and the contact area at C is 650 mm2.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 9:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 9:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 10:
Determine the average maximum shear stress in the 20-mm-
diameter pin at A and the 30-mm-diameter pin at B that support the
beam in figure.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 10:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 10:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 10:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 10:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 11:
If the wood joint in figure has a width of 150 mm, determine the
average shear stress developed along shear planes a–a and b–b. For
each plane, represent the state of stress on an element of the
material.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 11:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 11:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Stress in Two Force Members
• Axial forces on a two force
member result in only normal
stresses on a plane cut
perpendicular to the member axis.
• Transverse forces on bolts and
pins result in only shear stresses
on the plane perpendicular to bolt
or pin axis.
• Will show that either axial or
transverse forces may produce both
normal and shear stresses with respect
to a plane other than one cut
perpendicular to the member axis.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Stress on an Oblique Plane
• Pass a section through the member forming
an angle q with the normal plane.
• From equilibrium conditions, the
distributed forces (stresses) on the plane
must be equivalent to the force P.
• Resolve P into components normal and
tangential to the oblique section,
F P cosq V P sinq
• The average normal and shear stresses on
the oblique plane are
F P cosq P
cos2 q
Aq A0 A0
cosq
V P sinq P
sinq cosq
Aq A0 A0
cosq
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Maximum Stresses
• Normal and shearing stresses on an oblique
plane
P P
cos 2 q sinq cosq
A0 A0
• The maximum normal stress occurs when the
reference plane is perpendicular to the member
axis,
P
m 0
A0
• The maximum shear stress occurs for a plane at
+ 45o with respect to the axis,
P P
m sin 45 cos 45
A0 2 A0
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Sign convention
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Stress Transformation Relationship
y y1
x1
y
sin q
s
x
cosq
s
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Stress Transformation Relationship
• Forces on element (considering a unit length)
.s ( yy .x xy .y ) cosq ( yx .x xx .y ) sin q
xx . sin 2 q yy . cos 2 q 2 xy sin q . cosq
1 cos 2q 1 cos 2q
xx yy xy sin 2q
2 2
xx yy xx yy
cos 2q xy sin 2q (1)
2 2
.s ( yy .x xy .y ).sin q ( yx .x xx .y ).cosq
xx sin q cosq yy cosq sin q xy (cos2 q sin 2 q )
xx yy
sin 2q xy cos 2q (2)
2
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Properties of stress tensor
1. There are 2 planes at right angles on which shear stress is zero.
2 xy
For 0, tan 2q
xx yy
2q , ( )
(
q 2, 2 2
These planes on which are referred to as “principal planes”.
The normal stresses on these planes are called the “principal
stresses”.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Properties of stress tensor
2. The principal stresses correspond to the maximum and minimum
values of
Maximum value of principal stress Major Principal stress
Minimum value of principal stress Minor Principal stress
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Properties of stress tensor
Maximum or minimum in-plane normal stress
Maximum in-plane shear stress
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 12:
The state of plane stress at a failure point on the shaft is shown on
the element in figure. Represent this stress state in terms of the
principal stresses.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Properties of stress tensor
3. A relation between and acting on any plane
xx yy xx yy
2
(1)2 + (2)2,
[ ]
2 2
xy 2
2 2
( a) 2 2 r
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Mohr Circle of stress
Sign convention
• (Normal stresses) – Tensile stresses (+) ve
• (Shear stresses) – (+)ve if it gives a clockwise moment about a point inside
the element
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Mohr Circle of stress
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Mohr Circle of stress
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 13:
Due to the applied loading, the element at point A on the solid shaft
in Figure is subjected to the state of stress shown. Determine the
principal stresses acting at this point.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 13:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 14:
• The state of plane stress at a point is shown on the element in
Figure. Determine the maximum in-plane shear stress at this point.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 14:
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
Example 15:
• The state of plane stress at a point is shown on the element in
Figure. Represent this state of stress on an element oriented 30°
counterclockwise from the position shown.
H.M.C.C. Somarathna, Department of Civil Engineering, University of Jaffna
