Strength of Materials Exercise Collection

Conical Rod and Compressive Force

A conical rod with a circular cross-section and length $l=250\mathrm{mm}$ is loaded, as shown in Figure 1, by a compressive force $F=10\mathrm{kN}$ along the axis of the rod. The diameter at $x=0$ is twice as large as the diameter at $x=l$ with $2d_0=150\mathrm{mm}$ and $d_0=75\mathrm{mm}$, respectively.

Suspended Cone

An icicle is hanging from a gutter. The icicle has the shape of a circular cone with a length $l=10\mathrm{cm}$, a diameter $d_0=5\mathrm{cm}$ at the suspension point, and a cross-sectional area $A_0$. The ice has a density of $\varrho =0,91\mathrm{g}/{\mathrm{cm}}^3$.

A Loaded Homogeneous Bar with Constant Thickness and Linearly Variable Width

A homogeneous bar with constant thickness $d=20\mathrm{mm}$ and linearly varying width is subjected to a compressive force $F=1\mathrm{kN}$.

Given: $b=15\mathrm{mm}$, $h=80\mathrm{mm}$, $\varrho =7,85\mathrm{g}/{\mathrm{cm}}^3$

Normal and Shear Stress at an Arbitrary Section Angle

A clamped beam with a square cross-section (side length $d=20\mathrm{mm}$) is subjected to a tensile force $F=10\mathrm{kN}$ along the beam axis.

Determine the average normal stress and the average shear stress...

Plane Stress Transformation, x,y-Coordinates

In a sheet, the stresses ${\sigma }_x=-250\mathrm{MPa}$, ${\sigma }_y=80\mathrm{MPa}$ and ${\tau }_{xy}=50\mathrm{MPa}$ are given.

What normal and shear stresses occur at a section angle of $\phi =30°$?

Plane Stress Transformation, x,z-Coordinates

In a sheet, the stresses ${\sigma }_x=-250\mathrm{MPa}$, ${\sigma }_z=80\mathrm{MPa}$ and ${\tau }_{xz}=50\mathrm{MPa}$ are given.

What normal and shear stresses occur at a section angle of $\phi =30°$?

Determine the Magnitude and Direction of the Principal Stresses in the x-y Coordinate System

In a sheet, the stresses ${\sigma }_x=20\mathrm{MPa}$, ${\sigma }_y=30\mathrm{MPa}$ and ${\tau }_{xy}=10\mathrm{MPa}$ are given.

Determine the magnitude and direction of the principal stresses.

Determine the Magnitude and Direction of the Principal Stresses in the x-z Coordinate System

In a sheet, the stresses ${\sigma }_x=20\mathrm{MPa}$, ${\sigma }_z=30\mathrm{MPa}$ and ${\tau }_{xz}=10\mathrm{MPa}$ are given.

Determine the magnitude and direction of the principal stresses.

Determine the Area Moment of Inertia

For the depicted quarter-circle area, the following area moments of inertia are to be determined with respect to the illustrated $y$, $z$-coordinate system:

Determine the Area Moment of Inertia

For the rectangle area depicted, the following area moments of inertia are to be determined with respect to the illustrated $y$, $z$-coordinate system:

Determine the Area Moment of Inertia

For the triangle area depicted, the following area moments of inertia are to be determined with respect to the illustrated $y$, $z$-coordinate system:

Determine the Area Moment of Inertia

For the circular area depicted, the following area moments of inertia are to be determined with respect to the illustrated $y$, $z$-coordinate system:

Determine Area Moment of Inertia for a Coordinate System Parallel to the Principal Axis System

For the depicted rectangle, the following area moments of inertia are to be determined with respect to the illustrated $\bar{y}$, $\bar{z}$-coordinate system:

The area moments of inertia with respect to the principal axes are given as follows:

Determine Area Moment of Inertia for a Coordinate System Parallel to the Principal Axis System

The depicted rectangular cross-section has an area of $A=72{\mathrm{cm}}^2$.

The axial area moment of inertia with respect to the ${\bar{y}}_1$-axis ($a_1=5\mathrm{cm}$) is known and is $I_{{\bar{y}}_1}=2664{\mathrm{cm}}^4$.

Calculate the axial area moment of inertia $I_{{\bar{y}}_2}$ with respect to the ${\bar{y}}_2$-axis ($a_2=2\mathrm{cm}$).