Strength of Materials

ASSIGNMENT 1

Table of contents

Ultimate Strength....................................................................2

Yield Strength..........................................................................3

Modulus of Elasticity...............................................................4

Modulus of Rigidity..................................................................5

Moment of inertia:...................................................................6

Section Modulus......................................................................7

SAYED ASADULLAHUNISEL, FACULTY OF ENGINEERING, CIVIL DIVISION

Chap

ter:

Ulti

mat

e St

reng

th

1

ASSIGNMENT 1

Ultimate Strength

Ultimate strength is the maximum stress that a material can withstand while being stretched or pulled before failing or breaking.

Some materials will break sharply without experiencing plastic deformation, in what is called a brittle failure. Others, which are more ductile, including most metals, will experience some plastic deformation and possibly necking before fracture. As shown in [Figure 1], [Figure 2]

Figure 1 Brittle Material Figure 2 Ductile Material

Graph [Figure 3] clearly show us the differences between this two materials.

So in conclusion the strength of a material just before it breaks, is called Ultimate Strength.


Figure 3 [Stress vs Strain]

Ultimate Strength

Chap

ter:

Ulti

mat

e St

reng

th

2

ASSIGNMENT 1

Yield Strength

A yield strength or yield point of a material is defined as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible. As shown in [Figure 4]

We can conclude that yield point of a material is the point where we stretch, and release, it will came back to the original shape.

The formula for Yield strength can be written as;

Yield Strength= Yield Load∈NewtonCorss−Sectional Area∈mm2


Figure 4 [Yield Point]

Chap

ter:

Ulti

mat

e St

reng

th

3

ASSIGNMENT 1

Modulus of Elasticity

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=σε[MPA ]

where ¿ 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.


Chap

ter:

Ulti

mat

e St

reng

th

4

ASSIGNMENT 1

Modulus of Rigidity

The modulus of rigidity or shear modulus are one of several quantities for measuring the stiffness of 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 materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain.

μ=Ʈγ=

FA∆ xl

=FlA∆ x

Where;

Ʈ= FA

=¿ shear stress;

is the force which acts is the area on which the force acts


Figure 5 Figure 6

Chap

ter:

Ulti

mat

e St

reng

th

5

ASSIGNMENT 1

in engineering,γ=∆ xl

=θ=¿ shear strain. Elsewhere,

is the transverse displacement is the initial length

Shear modulus' derived SI unit is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch (ksi).

Moment of inertia:

Moment of inertia is;

It is a measure of an object’s resistance to changes to its rotation. Also defined as the capacity of a cross-section to resist bending. It must be specified with respect to a chosen axis of rotation. It is usually quantified in m4 or kgm2

If we want to find moment of inertia of an object (2d), we need to find out there centroid.

o The centroid, or center of gravity, of any object is the point within that object from which the force of gravity appears to act.

o An object will remain at rest if it is balanced on any point along a vertical line passing through its center of gravity.

o The centroid of a 2D surface is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape.

o If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry.

o The moment of inertia (MI) of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis.

o That means the Moment of Inertia Iz = Ix+Iy

Standard table of moment of inertia for some geometric shapes [Figure 7];


Figure 7

Chap

ter:

Ulti

mat

e St

reng

th

6

ASSIGNMENT 1

Section Modulus

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. And it depends on the shape of the member. There are two types of section modulus, Elastic section modulus & plastic section modulus.

The elastic section modulus is defined as S=Iy

, where I is moment of inertia and y is the

distance from the neutral axis to any given fiber. [Figure 8]

SAYED ASADULLAHUNISEL, FACULTY OF ENGINEERING, CIVIL DIVISIONFigure 7 ; Elastic section modulus of some member shapes

Chap

ter:

Ulti

mat

e St

reng

th

7

ASSIGNMENT 1

The plastic section modulus is used for materials where (irreversible) plastic behavior is dominant. The plastic section modulus depends on the location of the plastic neutral axis (PNA). The PNA is defined as the axis that spits the cross section such that the compression fore from the area in compression equals the tension force from the area in tension. So for section with constant yielding stress, the area above and below the PNA will be equal, but forcomposite sections, this is not necessarily the case.

The plastic section modulus is then the sum of the areas of the cross section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA. Z=A c yc+AT yT. [Figure 8]


Figure 8 ; Plastic section modulus of some member shapes

Chap

ter:

Ulti

mat

e St

reng

th

8

Strength of Materials

Engineering

Transcript of Strength of Materials