4
1 Elasticity
Civil engineering applications involve important contributions to stress and deflection analysis of structures, such as beams, plates, shells and rods. Further applications include in geomechanics involving the stresses in materials such as soil, rock,
concrete and asphalt.
Mechanical engineering includes application of elasticity in many problems such
as in the analysis and design of machine elements. These applications include analysis
of general stresses distribution in solids, contact stresses, thermal stresses, fatigue
and fracture mechanics.
Aerospace and aeronautical engineering uses elasticity in the analysis of stress,
fracture and fatigue analysis in aerostructures.
Materials engineering applies elasticity in the determination of stress fields in
crystalline solids, s around dislocations and in materials with microstructure.
Although, elasticity, mechanics of materials and structural mechanics are the three
branches of solid mechanics, they differ from one to other both in the objects and
methods of analysis.
Mechanics of materials deals essentially with the stresses and displacements of
a structural or machine element in the shape of a bar, straight or curved, which is
subjected to tension, compression, shear, bending or torsion. Structural mechanics,
on the basis of mechanics of materials, deals with the stresses and displacements of
a structure in the form of a bar system, such as a truss or a rigid frame. As to the
structural elements that are not in form of a bar, such as blocks, plates, shells, dams
and foundations, they are analysed only in the theory of elasticity. Moreover, in order
to analyse a bar element thoroughly and very precisely, it is necessary to apply the
theory of elasticity.
Although bar-shaped elements are studied both in mechanics of materials and
in theory of elasticity, the methods of analysis used here are not entirely the same.
When the element is studied in mechanics of materials, some assumptions are usually
made on the strain condition or the stress distribution. These assumptions simplify
the mathematical derivation to a certain extent, but many a times inevitably reduce
the degree of accuracy of the results obtained. However, in elasticity, the study of barshaped element usually does not need those assumptions. Thus, the results obtained
by the application of elasticity theory are more accurate and may be used to check
the appropriate results obtained in mechanics of materials.
While analysing the problems of bending of straight beam under transverse loads
by the mechanics of materials, it is usual to assume that a plane section before bending
of the beam remains plane even after the bending. This assumption leads to the linear
distribution of bending stresses. In the theory of elasticity, however one can solve
the problem without this assumption and prove that the stress distribution will be far
from linear variation as shown in the next sections.
Further, while analysing for the distribution of stresses in a tension member with
a hole, it is assumed in mechanics of materials that the tensile stresses are uniformly
distributed across the net section of the member, whereas the exact analysis in the
theory of elasticity shows that the stresses are by no means uniform, but are concentrated near the hole; the maximum stress at the edge of the hole is far greater than
the average stress across the net section.
1 Elasticity
Civil engineering applications involve important contributions to stress and deflection analysis of structures, such as beams, plates, shells and rods. Further applications include in geomechanics involving the stresses in materials such as soil, rock,
concrete and asphalt.
Mechanical engineering includes application of elasticity in many problems such
as in the analysis and design of machine elements. These applications include analysis
of general stresses distribution in solids, contact stresses, thermal stresses, fatigue
and fracture mechanics.
Aerospace and aeronautical engineering uses elasticity in the analysis of stress,
fracture and fatigue analysis in aerostructures.
Materials engineering applies elasticity in the determination of stress fields in
crystalline solids, s around dislocations and in materials with microstructure.
Although, elasticity, mechanics of materials and structural mechanics are the three
branches of solid mechanics, they differ from one to other both in the objects and
methods of analysis.
Mechanics of materials deals essentially with the stresses and displacements of
a structural or machine element in the shape of a bar, straight or curved, which is
subjected to tension, compression, shear, bending or torsion. Structural mechanics,
on the basis of mechanics of materials, deals with the stresses and displacements of
a structure in the form of a bar system, such as a truss or a rigid frame. As to the
structural elements that are not in form of a bar, such as blocks, plates, shells, dams
and foundations, they are analysed only in the theory of elasticity. Moreover, in order
to analyse a bar element thoroughly and very precisely, it is necessary to apply the
theory of elasticity.
Although bar-shaped elements are studied both in mechanics of materials and
in theory of elasticity, the methods of analysis used here are not entirely the same.
When the element is studied in mechanics of materials, some assumptions are usually
made on the strain condition or the stress distribution. These assumptions simplify
the mathematical derivation to a certain extent, but many a times inevitably reduce
the degree of accuracy of the results obtained. However, in elasticity, the study of barshaped element usually does not need those assumptions. Thus, the results obtained
by the application of elasticity theory are more accurate and may be used to check
the appropriate results obtained in mechanics of materials.
While analysing the problems of bending of straight beam under transverse loads
by the mechanics of materials, it is usual to assume that a plane section before bending
of the beam remains plane even after the bending. This assumption leads to the linear
distribution of bending stresses. In the theory of elasticity, however one can solve
the problem without this assumption and prove that the stress distribution will be far
from linear variation as shown in the next sections.
Further, while analysing for the distribution of stresses in a tension member with
a hole, it is assumed in mechanics of materials that the tensile stresses are uniformly
distributed across the net section of the member, whereas the exact analysis in the
theory of elasticity shows that the stresses are by no means uniform, but are concentrated near the hole; the maximum stress at the edge of the hole is far greater than
the average stress across the net section.