This document discusses key concepts related to the strength of materials, including: 1. Ultimate strength is the maximum stress a material can withstand before failure, while yield strength is the stress at which a material begins to deform plastically. 2. Modulus of elasticity, or Young's modulus, measures a material's resistance to elastic deformation and is defined as the ratio of stress to strain based on Hooke's law. 3. Modulus of rigidity, or shear modulus, describes a material's response to shear stress and is the ratio of shear stress to shear strain. 4. Other concepts covered include moment of inertia, which measures an object's resistance to changes in rotation, and section modulus

1. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
Table of contents
Ultimate Strength............................................................................2
Yield Strength.................................................................................3
Modulus of Elasticity .......................................................................4
Modulus of Rigidity .........................................................................5
Moment of inertia:...........................................................................6
Section Modulus .............................................................................7

2. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
Ultimate strength isthe maximumstressthata material can withstandwhile beingstretchedorpulled
before failingorbreaking.
Some materialswill breaksharplywithoutexperiencing plasticdeformation,inwhatiscalledabrittle
failure.Others,whichare more ductile,includingmostmetals,will experience some plasticdeformation
and possibly neckingbeforefracture. Asshownin [Figure 1], [Figure 2]
Figure 1 Brittle Material Figure 2 Ductile Material
Graph [Figure 3] clearlyshowusthe differencesbetweenthistwomaterials.
So in conclusion the strength of a material just before it breaks, is called Ultimate Strength.
Figure 3 [Stress vs Strain]
Ultimate Strength

3. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
A yieldstrengthor yieldpoint of a material isdefinedasthe stressat whicha material beginsto deform
plastically.Priortothe yieldpointthe material will deform elastically andwill returntoitsoriginal shape
whenthe appliedstressisremoved.Once the yieldpointispassed,some fractionof the deformation
will be permanentandnon-reversible. Asshownin [Figure 4]
We can conclude thatyieldpointof amaterial isthe pointwhere we stretch,andrelease,itwill came
back to the original shape.
The formulafor Yieldstrengthcanbe writtenas;
𝑌𝑖𝑒𝑙𝑑 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ =
𝑌𝑖𝑒𝑙𝑑 𝐿𝑜𝑎𝑑 𝑖𝑛 𝑁𝑒𝑤𝑡𝑜𝑛
𝐶𝑜𝑟𝑠𝑠−𝑆𝑒𝑐𝑡𝑖 𝑜 𝑛𝑎𝑙 𝐴𝑟𝑒𝑎 𝑖𝑛 𝑚𝑚2
Figure 4 [Yield Point]

4. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
An elastic modulus, or modulus of elasticity, is a number that measures an object or
substance's resistance to being deformed elastically (i.e., non-permanently) when a force is
applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in
the elastic deformation region. A stiffer material will have a higher elastic modulus.
Mechanical deformation puts energy into a material. The energy is stored elastically or
dissipated plastically. The way a material stores this energy is summarized in stress-strain
curves. Stress is defined as force per unit area and strain as elongation or contraction per unit
length when a material deforms elastically, the amount of deformation likewise depends on the
size of the material, but the strain for a given stress is always the same and the two are related
by Hooke´s Law (stress is directly proportional to strain):
σ = 𝐸 × ε
Where;
σ is stress [ MPa ]
E modulus of elasticity [MPa]
ε strain [unitless or %]
From the Hook’s law the modulus of elasticity is defined as the ratio of the stress to the strain:
𝐸 =
σ
ε
[𝑀𝑃𝐴]
where [𝛔 stress] is the restoring force caused by the deformation divided by the area to which
the force is applied and [ε strain] is the ratio of the change in some length parameter caused by
the deformation to the original value of the length parameter. If stress is measured in pascals,
then since strain is a dimensionless quantity, the units of E will be pascals as well.

5. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
The modulusof rigidityorshearmodulusare one of several quantitiesformeasuringthe stiffnessof
materials.
The shear modulus describes the material's response to shear stress (like cutting it with dull
scissors).
The shear modulus is concerned with the deformation of a solid when it experiences a force
parallel to one of its surfaces while its opposite face experiences an opposing force (such as
friction). In the case of an object that's shaped like a rectangular prism, it will deform into a
parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals
exhibit differing material response to stress or strain when tested in different directions. In this
case one may need to use the full tensor-expression of the elastic constants, rather than a
single scalar value. As shown in [Figure 5 , Figure 6].
In materialsscience,shearmodulusor modulusof rigidity,denotedby G,or sometimes Sorμ, is
definedasthe ratioof shearstress to the shearstrain.
𝜇 =
Ʈ
𝛾
=
𝐹
𝐴⁄
∆𝑥
𝑙⁄
=
𝐹𝑙
𝐴∆𝑥
Where;
Ʈ = 𝐹
𝐴⁄ = shear stress;
is the force which acts
is the area on which the force acts
in engineering, 𝛾 = ∆𝑥
𝑙⁄ = 𝜃 = shear strain. Elsewhere,
is the transverse displacement
is the initial length
Shearmodulus'derived SIunitisthe pascal (Pa),althoughitisusuallyexpressedin gigapascals(GPa) or
inthousands of poundspersquare inch (ksi).
Figure 5 Figure 6

6. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
Momentof inertiais;
 It isa measure of an object’sresistancetochangestoits rotation.
 Alsodefinedasthe capacityof a cross-sectiontoresistbending.
 It mustbe specifiedwithrespecttoa chosenaxisof rotation.
 It isusuallyquantifiedinm4or kgm2
If we wantto findmomentof inertiaof anobject(2d),we needtofindout there centroid.
o The centroid,or centerof gravity,of any objectisthe pointwithinthatobjectfromwhichthe
force of gravityappearsto act.
o An objectwill remainatrestif itis balancedonanypointalonga vertical line passingthroughits
centerof gravity.
o The centroidof a 2D surface isa pointthat correspondstothe centerof gravityof a verythin
homogeneousplateof the same areaand shape.
o If the area (orsectionor body) hasone line of symmetry,the centroidwill liesomewhere along
the line of symmetry.
o The momentof inertia(MI) of a plane areaabout an axisnormal tothe plane isequal tothe sum
of the momentsof inertiaaboutanytwomutuallyperpendicularaxeslyinginthe plane and
passingthroughthe givenaxis.
o That meansthe Momentof Inertia Iz = Ix+Iy
Standardtable of momentof inertia forsome geometricshapes [Figure 7];
Figure 7

7. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
Sectionmodulusisageometricpropertyfora given cross-sectionusedinthe designof beamsorflexural
members.Anditdependsonthe shape of the member.There are twotypesof sectionmodulus,Elastic
sectionmodulus&plasticsectionmodulus.
 The elasticsection modulusisdefinedas 𝑆 = 𝐼
𝑦⁄ ,where I ismomentof inertiaandy is the
distance fromthe neutral axistoany givenfiber. [Figure 8]
Figure 7 ; Elastic section modulus of some member shapes

8. ASSIGNMENT 1 STRENGTH OF MATERIAL
SAYED ASADULLAH UNISEL, FACULTYOF ENGINEERING,CIVILDIVISION
 The plasticsectionmodulusisusedformaterialswhere(irreversible) plasticbehavioris
dominant. The plasticsectionmodulusdependsonthe locationof the plasticneutral axis(PNA).The
PNA isdefinedasthe axisthatspitsthe cross sectionsuchthat the compressionfore fromthe areain
compressionequalsthe tensionforce fromthe area intension.Soforsectionwithconstantyielding
stress,the area above andbelowthe PNA will be equal,butforcompositesections,thisisnotnecessarily
the case.
The plasticsectionmodulusisthenthe sumof the areas of the crosssectionon eachside of the PNA
(whichmayor may not be equal) multipliedbythe distance fromthe local centroidsof the twoareasto
the PNA. 𝑍 = 𝐴 𝑐 𝑦𝑐 + 𝐴 𝑇 𝑦 𝑇. [Figure 8]
Figure 8 ; Plastic section modulus of some member shapes