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If the normal strain is defined in reference to the final length Δs′, that is,
Instead of in reference to the original length, Eq.2–2, show that the difference in these strains is represented as a second-order term, namely, ε − ε′ = εε′.
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Chapter 2 Solutions
Mechanics of Materials
- The state of strain at the point on the pin leaf has components of ϵx=200(10−6)ϵx=200(10−6) , ϵy=180(10−6)ϵy=180(10−6) , and γxy=−300(10−6)γxy=−300(10−6) . (Figure 1) -Use the strain transformation equations and determine the normal strain in the xx direction on an element oriented at an angle of θ=−55∘θ=−55∘ clockwise from the original position. -Determine the shear strain along the xy plain Determine the normal strain in the y direction.arrow_forwardThe deviatoric principal strains at a point in a loaded member are ₁, 2, 3. Show that ₁ + ₂ + 3 = 0 (ej)² + (e₂)² + (ez)² = [(e₁ = €₂)² + (€2 − €3)² + (€3 − €₁)²] -arrow_forwardThe state of strain in a plane element is ex =-200 x 10-6, Ey = 0, and yxy = 75 × 10-6 , as shown below. Determine the equivalent state of strain which represents (a) the principal strains (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element. y Yxy 2 dy Yxy FExdx dxarrow_forward
- For the strain rate shown below, determine the principal strains [εxx ε = Ey Ex Exy Eyy Exy Exs Eyz W ti [-1.1 0.5 01 0.5 2.4 0 0 0 0 X 10-4arrow_forwardFor some element, the displacement is obtained as u = [1 + 3x +4y +5xy]x10^(-3) mm. The normal strain along x-axis at point (1.2, 0.8) m located inside the element isarrow_forwardThe state of strain in a plane element is €x = -200 x 10-6 , Ey = 100 × 10-6 , and Yxy = 75 x 10-6 , as shown below. Determine the equivalent state of strain which represents (a) the principal strains (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element. y Eydy Yxy 2 dy Yxy FExdx 2 dxarrow_forward
- The block is deformed into the position shown by the dashed lines. Determine the average normal strain along with line AB.arrow_forwardThe strain components ɛ, Ey, and yxy are given for a point in a body subjected to plane strain. Using Mohr's circle, determine the principal strains, the maximum in-plane shear strain, and the absolute maximum shear strain at the point. Show the angle 0p, the principal strain deformations, and the maximum in-plane shear strain distortion in a sketch. Ex = 0 µE, ɛ, = 320 µɛ, Vxy = 240 µrad. Enter the angle suen that -45° s 0,s+45°.arrow_forwardDetermine the total strain (mm/mm) of a 2.70-m bar with a diameter of 21 mm subjected to a tensile force of 63 kN at a temperature increase of 77 C°. Consider the a=37.8 µm/mC° and E = 114 GPa.arrow_forward
- On the free surface of an extruded acrylic polymer [E = 3.5 GPa; v = 0.32] sheet, three strain gages arranged as shown record the following strains at a point: ε--420 με, ε, -+600 με, ε, -+490 με. Determine (a) the strain components E, Ɛy, and Yxy at the point. (b) the maximum in-plane shear strain Ymax- (c) the stresses O, Oy, and Ty at the point. y 45° 45° a Use the strain transformation equations to determine the strain components Ex, Ey, and yxy at the point. Answer: Ex = με Ey = i με Yxy = pradarrow_forwardA 45° strain rosette was placed on the surface of a critical point on an engineering part. The following were measured: Ea = 400 μ C ли 45° mm mm 45° ли Gauge a was aligned with the x-axis. a. Determine Ex, Ey, Yxy b. Using Mohr's Circle, find the principal strains and the maximum shear strain at that point, and find the orientation of the principal planes from the given x-y axes. y ли & = 450 μ ஆ b a mm X mm & c = 500 μ y+ ос mm mm eb 10₂ Xarrow_forward5 decimal places Determine the total strain (mm/mm) of a 2.64-m bar with a diameter of 21 mm subjected to a tensile force of 71 kN at a temperature increase of 49 C°. Consider the α=27.3 µm/mC° and E = 122 GPa.arrow_forward
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