CHAPTER OBJECTIVES. To define the concept of normal strain To define the concept of shear strain To determine the normal and shear strain in engineering applications. CHAPTER OUTLINE. Deformation Strain. 2.1 DEFORMATION. Deformation Occurs when a force is applied to a body
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CHAPTER OBJECTIVES • To define the concept of normal strain • To define the concept of shear strain • To determine the normal and shear strain in engineering applications
CHAPTER OUTLINE • Deformation • Strain
2.1 DEFORMATION Deformation • Occurs when a force is applied to a body • Can be highly visible or practically unnoticeable • Can also occur when temperature of a body is changed • Is not uniform throughout a body’s volume, thus change in geometry of any line segment within body may vary along its length
2.1 DEFORMATION Assumptions to simplify study of deformation • Assume lines to be very short and located in neighborhood of a point, and • Take into account the orientation of the line segment at the point
Deformed body Undeformed body 2.2 STRAIN Normal strain Defined as the elongation or contraction of a line segment per unit of length Consider line AB in figure above After deformation, Δs changes to Δs’
Δs’ − Δs avg = Δs Deformed body Undeformed body lim B→A along n Δs − Δs’ = Δs 2.2 STRAIN • Defining average normal strain using avg (e = epsilon) As Δs → 0, Δs’ → 0
Δs’ ≈(1 + )Δs 2.2 STRAIN • If normal strain is known, use the equation to obtain approx. final length of a short line segment in direction of n after deformation. Hence, when is positive, initial line will elongate, if is negative, the line contracts
2.2 STRAIN Units • Normal strain is a dimensionless quantity, as it’s a ratio of two lengths • But common practice to state it in terms of meters/meter (m/m) • is small for most engineering applications, so it is normally expressed as micrometers per meter (μm/m) where 1 μm = 10−6m • Also expressed as a percentage, e.g.,0.001 m/m = 0.1 %
2.2 STRAIN Shear strain • Defined as the change in angle that occurs between two line segments that were originally perpendicular to one another • This angle is denoted by γ(gamma) and measured in radians (rad).
Undeformed body Deformed body 2.2 STRAIN • Consider line segments AB and AC originating from same point A in a body, and directed along the perpendicular n and t axes After deformation, lines become curves, such that angle between them at A is θ’
n t 2 ’ lim B→A along n C→A along t − γnt = Undeformed body Deformed body 2.2 STRAIN Shear strain • Hence, shear strain at point A associated with n andt axes is Shear strain positive if q’ < /2, and Shear strain negative if q’ > /2.
Undeformed point 2.2 STRAIN Cartesian strain components Using above definitions of normal and shear strain, we show how to describe the deformation of the body A point in the un-deformed body is regarded as a small element having dimensions of Δx, Δy and Δz
Undeformed point Approx. lengths of sides of parallelepiped are (1+z)Δz (1+x)Δx (1+y)Δy Deformed point 2.2 STRAIN • Since element is very small, deformed shape of element is a parallelepiped (1 + x)Δx (1 + y)Δy (1 + z)Δz
(1+z)Δz 2 2 2 (1+x)Δx − γxy − γyz − γxz (1+y)Δy Deformed point 2.2 STRAIN Approx. angles between the sides are Normal strains, e, cause a change in its volume Shear strains, g, cause a change in its shape To summarize, state of strain at a point requires specifying 3 normal strains of x,y,zand 3 shear strains of γxy,γyz,γxz
2.2 STRAIN Small strain analysis • Most engineering design involves applications for which only small deformations are allowed • We’ll assume that deformations that take place within a body are almost infinitesimal, so normal strains occurring within material are very small compared to 1, i.e., << 1. • This assumption is widely applied in practical engineering problems, and is referred to as small strain analysis • E.g., it can be used to approximate sinθ= θ, cosθ = θ and tanθ = θ, provided θ is very small
EXAMPLE The rigid beam is supported by a pin at A and wires BD and CE. If the load P on the beam is displaced 10 mm downward, determine the normal strain developed in each wire.
dBD dCE A 10 mm B C dBD = 4.28 mm dCE = 7.14 mm EXAMPLE (CONT.) Free-body diagram dBD = change in length of BD dCE = change in length of CE
EXAMPLE (CONT.) Normal strain developed in each wire
CHAPTER REVIEW • Loads cause bodies to deform, thus points in the body will undergo displacements or changes in position • Normal strain is a measure of elongation or contraction of small line segment in the body • Shear strain is a measure of the change in angle that occurs between two small line segments that are originally perpendicular to each other • State of strain at a point is described by six strain components: • Three normal strains: x, y, z • Three shear strains: γxy, γxz, γyz • These components depend upon the orientation of the line segments and their location in the body
CHAPTER REVIEW • Strain is a geometrical quantity measured by experimental techniques. Stress in body is then determined from material property relations • Most engineering materials undergo small deformations, so normal strain << 1. This assumption of “small strain analysis” allows us to simplify calculations for normal strain, since first-order approximations can be made about their size