Solved: The nonuniform loading causes a normal strain in

An air-filled rubber ball has a diameter of 6 in. If the air pressure within the ball is increased until the diameter becomes 7 in., determine the average normal strain in the rubber.

A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5 in., determine the average normal strain in the strip.

If the load P on the beam causes the end C to be 2 displaced 10 mm downward, determine the normal strain in wires CE and BD.

The force applied at the handle of the rigid lever causes the lever to rotate clockwise about the pin B through an angle of \(2^{\circ}\). Determine the average normal strain in each wire. The wires are unstretched when the lever is in the horizontal position.

The rectangular plate is subjected to the deformation shown by the dashed line. Determine the average shear strain \(\gamma_{x y}\) in the plate.

The square deforms into the position shown by the dashed lines. Determine the shear strain at each of its corners, A, B, C, and D, relative to the x, y axes. Side D? B? remains horizontal.

The pin-connected rigid rods AB and BC are inclined at \(\theta=30^{\circ}\) when they are unloaded. When the force P is applied \(\theta\) becomes \(30.2^{\circ}\). Determine the average normal strain in wire AC.

The wire AB is unstretched when \(\theta=45^{\circ}\). If a load is applied to the bar AC, which causes u to become \(47^{\circ}\), determine the normal strain in the wire.

If a horizontal load applied to the bar AC causes point A to be displaced to the right by an amount \(\Delta L\), determine the normal strain in the wire AB. Originally, \(\theta=45^{\circ}\).

Determine the shear strain \(\gamma_{x y}\) at corners A and B if the plastic distorts as shown by the dashed lines.

Determine the shear strain \(\gamma_{x y}\) at corners D and C if the plastic distorts as shown by the dashed lines.

The material distorts into the dashed position shown. Determine the average normal strains \(\epsilon_{x}, \epsilon_{y}\) and the shear strain \(\gamma_{x y}\) at A, and the average normal strain along line BE.

The material distorts into the dashed position shown. Determine the average normal strains along the diagonals AD and CF.

Part of a control linkage for an airplane consists of a rigid member CB and a flexible cable AB. If a force is applied to the end B of the member and causes it to rotate by \(\theta=0.5^{\circ}\), determine the normal strain in the cable. Originally the cable is unstretched.

Part of a control linkage for an airplane consists of a rigid member CB and a flexible cable AB. If a force is applied to the end B of the member and causes a normal strain in the cable of 0.004 mm>mm, determine the displacement of point B. Originally the cable is unstretched.

The nylon cord has an original length L and is tied to a bolt at A and a roller at B. If a force P is applied to the roller, determine the normal strain in the cord when the roller is at C, and at D. If the cord is originally unstrained when it is at C, determine the normal strain \(\epsilon_{D}^{\prime}\) when the roller moves to D. Show that if the displacements \(\Delta_{C} \text { and } \Delta_{D}\)are small, then \(\epsilon_{D}^{\prime}=\epsilon_{D}-\epsilon_{C}\).

A thin wire, lying along the x axis, is strained such that each point on the wire is displaced \(\Delta x=k x^{2}\) along the x axis. If k is constant, what is the normal strain at any point P along the wire?

Determine the shear strain \(\gamma_{x y}\) at corners A and B if the plate distorts as shown by the dashed lines.

Determine the shear strain \(\gamma_{x y}\) at corners D and C if the plate distorts as shown by the dashed lines.

Determine the average normal strain that occurs along the diagonals AC and DB.

The corners of the square plate are given the displacements indicated. Determine the average normal strains \(\epsilon_{x} \text { and } \epsilon_{y}\) along the x and y axes.

The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5 mm. Determine the shear strain, \(\gamma_{x y}\), at A.

The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5 mm. Determine the average normal strain \(\epsilon_{x}\) along the x axis.

The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5 mm. Determine the average normal strain \(\epsilon_{x^{\prime}}\) along the x? axis.

The polysulfone block is glued at its top and bottom to the rigid plates. If a tangential force, applied to the top plate, causes the material to deform so that its sides are described by the equation \(y=3.56 x^{1 / 4}\), determine the shear strain at the corners A and B.

The corners of the square plate are given the displacements indicated. Determine the shear strain at A relative to axes that are directed along AB and AD, and the shear strain at B relative to axes that are directed along BC and BA.

The corners of the square plate are given the displacements indicated. Determine the average normal strains along side AB and diagonals AC and BD.

The block is deformed into the position shown by the dashed lines. Determine the average normal strain along line AB.

The rectangular plate is deformed into the shape shown by the dashed lines. Determine the average normal strain along diagonal AC, and the average shear strain at corner A relative to the x, y axes.

The rectangular plate is deformed into the shape shown by the dashed lines. Determine the average normal strain along diagonal BD, and the average shear strain at corner B relative to the x, y axes.

The nonuniform loading causes a normal strain in the shaft that can be expressed as \(\epsilon_{x}=k \sin \left(\frac{\pi}{L} x\right)\) , where k is a constant. Determine the displacement of the center C and the average normal strain in the entire rod.

The rectangular plate undergoes a deformation shown by the dashed lines. Determine the shear strain \(\gamma_{x y}\) and \(\gamma_{x^{\prime} y^{\prime}}\) at point A.

The fiber AB has a length L and orientation u. If its ends A and B undergo very small displacements \(u_{A} \text { and } v_{B}\) respectively, determine the normal strain in the fiber when it is in position A? B?.

If the normal strain is defined in reference to the final length \(\Delta s^{\prime}\), that is, \(\epsilon^{\prime}=\lim _{\Delta s^{\prime} \rightarrow 0}\left(\frac{\Delta s^{\prime}-\Delta s}{\Delta s^{\prime}}\right)\) 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, \(\epsilon-\epsilon^{\prime}=\epsilon \epsilon^{\prime}\).