Strength of Materials

Prepared by, Jenifer S

STRESS

• Resistance force offered by the material

per unit area

• Applied force or system of forces that

tends to deform a body

• Stress distribution may or may not be

uniform, depending on the nature of the

loading conditionSTRAIN

• When the external force applied in the

material it tends to go some deformation

• Difference between the change in

dimension of the object to the original

dimension of the object

Normal Stress(𝜎)

Result of load applied perpendicular to a member

Shear Stress (𝜏)

Results when a load is applied parallel to an area

Normal Strain (ε)

Ratio between change in dimension to original

dimension

Shear strain (γ)

Change in angle

States of Stress:

• Simply a general state of stress at a point involves all the

normal stress components, together with all the shear stress

components and it remains under equilibrium

• Six components to specify the state of stress at a point

(i.e) 𝜎x , 𝜎y , 𝜎z , 𝜏xy , 𝜏yz , 𝜏zx

States of Strain:

• Homogenous strain always deforms circles and spheres into

strain ellipses and strain ellipsoids, respectively. The strain

ellipsoid completely defines the State of strain

• εx , εy , εz , γxy , γyz , γzx

Stress transformation equation,

[𝜎’] = [q] [𝜎] [q]ᵀ

Strain transformation

ε’ = RAR¯¹ε

Where,

R – Reuter’s matrix

Mohr’s Circle

• 2-D graphical representation of the transformation

law for the Cauchy stress tensor

• Used to determine graphically the stress components

acting on a rotated coordinate system

• Locus of points that represent the state of stress on

individual planes at all their orientations, where the

axes represent the principal axes of the stress

element

• Can be applied to any symmetric 2x2 tensor matrix,

including the strain and moment of inertia tensors.

• s

Derivation of Mohr's circle parametric equations - Equilibrium of forces

Derivation of Mohr's circle parametric equations - Tensor transformation

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