Civil Engineering - Strength of Material- quick revision

Quick Revision

Strength of Material Civil engineering

Maximum principal stress theory (Rankine’s theory)

According to this theory, permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.

For design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.

• σ1,2 ≤ σy for no failure
• σ1,2 ≤ σ/FOS for design
• Note: For no shear failure τ ≤ 0.57 σy

Graphical representation

For brittle material, which do not fail by yielding but fail by brittle fracture, this theory gives satisfactory result.

The graph is always square even for different values of σ1 and σ2.

Maximum principal strain theory (ST. Venant’s theory)

According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.

• ϵ1,2≤σy/E1 For no failure in uni – axial loading.
• σ1/E−μσ2/E−μσ3/E≤σy/E For no failure in tri – axial loading.
• σ1−μσ2−μσ3≤(σy/FOS)For design, Here, ϵ = Principal strain
• σ1, σ2 and σ3 = Principal stresses   

Graphical Representation

This story over estimate the elastic strength of ductile material.

Maximum shear stress theory (Guest & Tresca’s Theory)

According to this theory, failure of specimen subjected to any combination of load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

• τmax  ≤ σy/2 For no failure
• σ1−σ2 ≤ (σy/FOS)) For design

σ1 and σ2 are maximum and minimum principal stress respectively.

Here, τmax = Maximum shear stress

σy = permissible stress

This theory gives satisfactory result for ductile material.

Maximum strain energy theory (Haigh’s theory)

According to this theory, a body complex stress fails when the total strain energy at elastic limit in simple tension.

Graphical Representation.

• {σ1^2+σ2^2+σ3^2−2μ(σ1σ2+σ2σ3+σ3σ1)}≤σy^2 for no failure
• {σ1^2+σ2^2+σ3^−2μ(σ1σ2+σ2σ3+σ3σ1)}≤(σy/FOS)^2for design

This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory.

It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension / compression test.

• 1/2[(σ1−σ2)^2+(σ2−σ3)^2+(σ3−σ1)^2]≤σy^2 for no failure
• 1/2[(σ1−σ2)^2+(σ2−σ3)^2+(σ3−σ1)^2]≤(σy/FOS)^2 For design

It gives very good result in ductile material.  

It cannot be applied for material under hydrostatic pressure.

All theories will give same results if loading is uniaxial​

The Strain Energy of Distortion or Distortion energy per unit volume is given by:

Distortion energy per unit volume = (Total Strain energy) - (Energy of dilation)

Total Strain energy is given by

U=12E{σ21+σ22+σ23−2μ(σ1σ2+σ2σ3+σ3σ1)}

Energy of dilation:

Uv=(1−2μ)6E×(σ1+σ2+σ3)23

Distortion energy per unit volume = (Total Strain energy) - (Energy of dilation) = Ud 

1 + μ6E[(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2

In simple tension test:

(1 + μ6E)2σ2ym

Maximum distortion energy theory (Von mises theory)

According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy per unit volume reaches the limiting distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple tension test.

von misses stress under triaxial condition is given by:

σvm=12{(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2}−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√

Now if we compare von misses stress and distortion energy per unit volume equation then,

Ud =1 + μ3E×σ2vm$\frac{1+\mu }{3E}\times {\sigma }_{vm}^2$

Ud ∝ σ2vm

σvm ∝  Ud−−√

The von Mises stress at a point in a body subjected to forces is proportional to the square root of the distortional strain energy per unit volume.

A material may fail if

According to theories of failure, a material may fail for any of the following conditions and it depends upon the inherent properties of the material.

a) Maximum principal stress theory (Rankine’s Theory):

If the maximum principal stress (σ1) exceeds direct stress (σ).

b) Maximum principal strain theory (St. Venant’s Theory):

If the maximum principal strain (ε1) exceeds maximum strain (σ/E).

c) Maximum shear stress theory (Guest & Tresca’s Theory):

If maximum shear stress (τmax) exceeds half of direct stress (σ/2).

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Theories of failure	Other Name	Shape
Maximum Principal Stress Theory	RANKINE’S THEORY	Square
Maximum Principal Strain Theory	St. VENANT’S THEORY	Rhombus
Total Strain Energy Theory	HAIGH’S THEORY	Ellipse
Maximum Shear Stress Theory	GUEST AND TRESCA’S THEORY	Hexagon
Maximum Distortion Energy Theory	VON MISES AND HENCKY’S THEORY	Ellipse

Following are the assumptions made in the theory of Simple Bending:

• The material of the beam is homogenous and isotropic.
• The beam is initially straight, and all the longitudinal fibres bend in circular arcs with a common centre of curvature.
• Members have symmetric cross-sections and are subjected to bending in the plane of symmetry.
• The beam is subjected to pure bending and the effect of shear is neglected.
• Plane sections through a beam, taken normal to the axis of the beam remain plane after the beam is subjected to bending.
• The radius of curvature is large as compared to the dimensions of the beam.
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To obtain beams of uniform strength the sections of the beam may be varied by 

• keeping the width constant throughout and varying the depth, 
• keeping the depth constant throughout the length and varying the width, 
• by varying the width and depth in a suitable way and 
• a circular beam of uniform strength can be made by varying diameter in such a way that M/Z is a constant. 

Equation of pure bending:

Pure bending or bending is that in which bending moment M is constant along the length i.e. dM/dx=0, or shear force is zero.

Its empirical relationship is given by –

M/I=σ/y=E/R

where, M = Bending moment, I = MOI of cross-section about the neutral axis (NA), E = Young’s Modulus of Elasticity, σ = Bending stress at a distance y from NA 

• When a beam is suitably designed such that the extreme fibres are loaded to the maximum permissible stress σ max by varying the Cross-section it will be known as a beam of uniform strength.

Section Modulus (Z):

• The ratio of Moment of Inertia I of beam cross-section about NA to the distance of extreme fibre ymax from the neutral axis is known as section modulus.
• It also represents the strength of the section. It is given by

• Z=I/ymax
• σ=M/Z
• For pure bending, when M = Constant
•   σ=M/Z = Constant

Different assumptions made in torsion theory are as follows:

• Shaft must be straight and should have uniform cross-section.
• The shear stress induced in shaft should not exceed the elastic limit.
• Twist along the shaft is uniform.
• Twisting has no effect on circularity of shaft.

Deflection 

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Elasticity

• It is the property of a material to regain its original shape after deformation when the external forces are removed.

Plasticity

• It is the property of a material that retains the deformation produced under load permanently. 
• Thus, it is a property of material which allows it to deform without fracture

Ductility

• The property of the material that allows it to be drawn into wires or elongated before failure is known as ductility.

Malleability 

• The property of a material to deform under compression.
• The metals having malleable property can be rolled or beaten into sheets.
• An example is aluminium foil.

Toughness

• The ability of the material to withstand stress (resist fracture due to high impact loads) without fracture is known as toughness.
• It is defined as the ability to absorb energy in the plastic state.