Stress: Mathematical problems & solutions

Problem 1: Simple Stress Calculation

Problem: A rod with a cross-sectional area of $50{\text{mm}}^2$ is subjected to a tensile force of $1000\text{N}$. Calculate the stress in the rod. Solution: $𝜎=\frac{𝐹}{𝐴}=\frac{1000\text{N}}{50\times 10^{-6}{\text{m}}^2}=20\times 10^6\text{Pa}=20\text{MPa}$ The stress in the rod is $20\text{MPa}$.

Problem 2: Compressive Stress

Problem: A column with a cross-sectional area of $0.01{\text{m}}^2$ is subjected to a compressive force of $5000\text{N}$. Calculate the compressive stress. Solution: $𝜎=\frac{𝐹}{𝐴}=\frac{5000\text{N}}{0.01{\text{m}}^2}=500,000\text{Pa}=500\text{kPa}$ The compressive stress is $500\text{kPa}$.

Problem 3: Shear Stress

Problem: A bolt with a cross-sectional area of $25{\text{mm}}^2$ is subjected to a shear force of $200\text{N}$. Calculate the shear stress. Solution: $𝜏=\frac{𝐹}{𝐴}=\frac{200\text{N}}{25\times 10^{-6}{\text{m}}^2}=8\times 10^6\text{Pa}=8\text{MPa}$ The shear stress is $8\text{MPa}$.

Problem 4: Tensile Stress in a Wire

Problem: A steel wire with a diameter of $2\text{mm}$ is subjected to a tensile force of $1500\text{N}$. Calculate the tensile stress in the wire. Solution: $𝐴=𝜋{\left(\frac{𝑑}{2}\right)}^2=𝜋{\left(\frac{2\times 10^{-3}\text{m}}{2}\right)}^2=3.14\times 10^{-6}{\text{m}}^2$ $𝜎=\frac{𝐹}{𝐴}=\frac{1500\text{N}}{3.14\times 10^{-6}{\text{m}}^2}=477.7\times 10^6\text{Pa}=477.7\text{MPa}$ The tensile stress in the wire is $477.7\text{MPa}$.

Problem 5: Compressive Stress in a Concrete Cylinder

Problem: A concrete cylinder with a diameter of $0.1\text{m}$ and height of $0.2\text{m}$ is subjected to a compressive force of $10,000\text{N}$. Calculate the compressive stress. Solution: $𝐴=𝜋{\left(\frac{𝑑}{2}\right)}^2=𝜋{\left(\frac{0.1\text{m}}{2}\right)}^2=0.00785{\text{m}}^2$ $𝜎=\frac{𝐹}{𝐴}=\frac{10,000\text{N}}{0.00785{\text{m}}^2}=1,273,885\text{Pa}=1.27\text{MPa}$ The compressive stress is $1.27\text{MPa}$.

Problem 6: Bearing Stress

Problem: A pin with a diameter of $10\text{mm}$ is subjected to a load of $500\text{N}$. Calculate the bearing stress. Solution: $𝐴=𝑑\cdot 𝑡\text{ (assuming t is the thickness of the material in contact)}$ Assuming $𝑡=5\text{mm}$: $𝐴=10\text{mm}\times 5\text{mm}=50{\text{mm}}^2=50\times 10^{-6}{\text{m}}^2$ ${𝜎}_{𝑏}=\frac{𝐹}{𝐴}=\frac{500\text{N}}{50\times 10^{-6}{\text{m}}^2}=10\times 10^6\text{Pa}=10\text{MPa}$ The bearing stress is $10\text{MPa}$.

Problem 7: Thermal Stress

Problem: A steel rod with a length of $2\text{m}$ and a cross-sectional area of $100{\text{mm}}^2$ is subjected to a temperature increase of $50^{\circ }\text{C}$. Calculate the thermal stress if the coefficient of thermal expansion is $12\times 10^{-6}{\text{C}}^{-1}$ and the Young's modulus is $210\text{GPa}$. Solution: $\mathrm{\Delta }𝐿=𝛼𝐿\mathrm{\Delta }𝑇$ $\mathrm{\Delta }𝐿=12\times 10^{-6}{\text{C}}^{-1}\times 2\text{m}\times 50\text{C}=0.0012\text{m}$ $𝜎=\frac{𝐸\mathrm{\Delta }𝐿}{𝐿}=\frac{210\times 10^9\text{Pa}\times 0.0012\text{m}}{2\text{m}}=126\times 10^6\text{Pa}=126\text{MPa}$ The thermal stress is $126\text{MPa}$.

Problem 8: Bending Stress

Problem: A beam with a rectangular cross-section of width $50\text{mm}$ and height $100\text{mm}$ is subjected to a bending moment of $500\text{Nm}$. Calculate the maximum bending stress. Solution: $𝐼=\frac{1}{12}𝑏{ℎ}^3=\frac{1}{12}(0.05\text{m})(0.1\text{m}{)}^3=4.17\times 10^{-7}{\text{m}}^4$ $𝜎=\frac{𝑀𝑐}{𝐼}=\frac{500\text{Nm}\times 0.05\text{m}}{4.17\times 10^{-7}{\text{m}}^4}=60\times 10^6\text{Pa}=60\text{MPa}$ The maximum bending stress is $60\text{MPa}$.

Problem 9: Stress in a Tapered Bar

Problem: A tapered bar with a small diameter of $10\text{mm}$ and a large diameter of $20\text{mm}$ is subjected to an axial force of $1000\text{N}$. Calculate the average stress in the bar. Solution: $𝐴=\frac{𝜋}{4}({𝑑}_1^2+{𝑑}_2^2)=\frac{𝜋}{4}((0.01\text{m}{)}^2+(0.02\text{m}{)}^2)=\frac{𝜋}{4}(0.0001{\text{m}}^2+0.0004{\text{m}}^2)=0.00039{\text{m}}^2$ $𝜎=\frac{𝐹}{𝐴}=\frac{1000\text{N}}{0.00039{\text{m}}^2}=2.56\times 10^6\text{Pa}=2.56\text{MPa}$ The average stress is $2.56\text{MPa}$.

Problem 10: Stress in a Composite Bar

Problem: A composite bar made of steel and aluminum is subjected to an axial force of $20,000\text{N}$. The cross-sectional area of the steel part is $200{\text{mm}}^2$ and of the aluminum part is $300{\text{mm}}^2$. Calculate the stress in each material. Solution: Steel: ${𝜎}_{𝑠𝑡𝑒𝑒𝑙}=\frac{{𝐹}_{𝑠𝑡𝑒𝑒𝑙}}{{𝐴}_{𝑠𝑡𝑒𝑒𝑙}}=\frac{20,000\text{N}\times \frac{200{\text{mm}}^2}{500{\text{mm}}^2}}{200{\text{mm}}^2}=\frac{8,000\text{N}}{200{\text{mm}}^2}=40\text{MPa}$ Aluminum: ${𝜎}_{𝑎𝑙𝑢𝑚𝑖𝑛𝑢𝑚}=\frac{{𝐹}_{𝑎𝑙𝑢𝑚𝑖𝑛𝑢𝑚}}{{𝐴}_{𝑎𝑙𝑢𝑚𝑖𝑛𝑢𝑚}}=\frac{20,000\text{N}\times \frac{300{\text{mm}}^2}{500{\text{mm}}^2}}{300{\text{mm}}^2}=\frac{12,000\text{N}}{300{\text{mm}}^2}=40\text{MPa}$ The stress in both steel and aluminum is $40\text{MPa}$.

Problem 11: Stress Concentration

Problem: A plate with a hole in the center is subjected to a tensile force. The nominal stress in the plate is $100\text{MPa}$. If the stress concentration factor is $3$, calculate the maximum stress around the hole. Solution: ${𝜎}_{𝑚𝑎𝑥}={𝐾}_{𝑡}{𝜎}_{𝑛𝑜𝑚𝑖𝑛𝑎𝑙}=3\times 100\text{MPa}=300\text{MPa}$ The maximum stress around the hole is $300\text{MPa}$.

Problem 12: Stress in a Thin-walled Cylinder

Problem: A thin-walled cylindrical tank with a radius of $0.5\text{m}$ and a wall thickness of $0.01\text{m}$ is subjected to an internal pressure of $2\text{MPa}$. Calculate the hoop stress. Solution: ${𝜎}_{ℎ𝑜𝑜𝑝}=\frac{𝑝𝑟}{𝑡}=\frac{2\times 10^6\text{Pa}\times 0.5\text{m}}{0.01\text{m}}=100\times 10^6\text{Pa}=100\text{MPa}$ The hoop stress is $100\text{MPa}$.

Problem 13: Von Mises Stress

Problem: A material is subjected to principal stresses of ${𝜎}_1=80\text{MPa}$, ${𝜎}_2=40\text{MPa}$, and ${𝜎}_3=0\text{MPa}$. Calculate the von Mises stress. Solution: ${𝜎}_{𝑣𝑚}=\sqrt{\frac{1}{2}[({𝜎}_1-{𝜎}_2{)}^2+({𝜎}_2-{𝜎}_3{)}^2+({𝜎}_3-{𝜎}_1{)}^2]}$ ${𝜎}_{𝑣𝑚}=\sqrt{\frac{1}{2}[(80-40{)}^2+(40-0{)}^2+(0-80{)}^2]}$ ${𝜎}_{𝑣𝑚}=\sqrt{\frac{1}{2}[1600+1600+6400]}$ ${𝜎}_{𝑣𝑚}=\sqrt{4800}$ ${𝜎}_{𝑣𝑚}=69.3\text{MPa}$ The von Mises stress is $69.3\text{MPa}$.

Problem 14: Torsional Stress

Problem: A solid circular shaft with a diameter of $50\text{mm}$ is subjected to a torque of $200\text{Nm}$. Calculate the shear stress. Solution: $𝐽=\frac{𝜋{𝑑}^4}{32}=\frac{𝜋(0.05\text{m}{)}^4}{32}=3.07\times 10^{-7}{\text{m}}^4$ $𝜏=\frac{𝑇𝑐}{𝐽}=\frac{200\text{Nm}\times 0.025\text{m}}{3.07\times 10^{-7}{\text{m}}^4}=16.3\times 10^6\text{Pa}=16.3\text{MPa}$ The shear stress is $16.3\text{MPa}$.

Problem 15: Bending Stress in a Beam

Problem: A beam with a rectangular cross-section of width $100\text{mm}$ and height $200\text{mm}$ is subjected to a bending moment of $1000\text{Nm}$. Calculate the maximum bending stress. Solution: $𝐼=\frac{1}{12}𝑏{ℎ}^3=\frac{1}{12}(0.1\text{m})(0.2\text{m}{)}^3=6.67\times 10^{-6}{\text{m}}^4$ $𝜎=\frac{𝑀𝑐}{𝐼}=\frac{1000\text{Nm}\times 0.1\text{m}}{6.67\times 10^{-6}{\text{m}}^4}=15\times 10^6\text{Pa}=15\text{MPa}$ The maximum bending stress is $15\text{MPa}$.

Problem 16: Stress in a Composite Beam

Problem: A composite beam made of two different materials is subjected to a bending moment. The moduli of elasticity are ${𝐸}_1=200\text{GPa}$ and ${𝐸}_2=100\text{GPa}$, and the respective areas are ${𝐴}_1=100{\text{mm}}^2$ and ${𝐴}_2=200{\text{mm}}^2$. Calculate the transformed section modulus. Solution: $𝑛=\frac{{𝐸}_1}{{𝐸}_2}=\frac{200\text{GPa}}{100\text{GPa}}=2$ ${𝐴}_2^{\mathrm{\prime }}=𝑛{𝐴}_2=2\times 200{\text{mm}}^2=400{\text{mm}}^2$ ${𝐴}_{𝑡𝑜𝑡𝑎𝑙}={𝐴}_1+{𝐴}_2^{\mathrm{\prime }}=100{\text{mm}}^2+400{\text{mm}}^2=500{\text{mm}}^2$ The transformed section modulus is $500{\text{mm}}^2$.

Problem 17: Stress in a Plate with a Circular Hole

Problem: A plate with a circular hole of radius $5\text{mm}$ is subjected to a tensile force. The nominal stress is $120\text{MPa}$. If the stress concentration factor is $3$, calculate the maximum stress around the hole. Solution: ${𝜎}_{𝑚𝑎𝑥}={𝐾}_{𝑡}{𝜎}_{𝑛𝑜𝑚𝑖𝑛𝑎𝑙}=3\times 120\text{MPa}=360\text{MPa}$ The maximum stress around the hole is $360\text{MPa}$.

Problem 18: Hydrostatic Stress

Problem: A solid sphere is subjected to an external pressure of $5\text{MPa}$. Calculate the hydrostatic stress. Solution: ${𝜎}_{ℎ}=-𝑝=-5\text{MPa}$ The hydrostatic stress is $-5\text{MPa}$.

Problem 19: Principal Stresses

Problem: A material is subjected to stresses ${𝜎}_{𝑥}=50\text{MPa}$, ${𝜎}_{𝑦}=30\text{MPa}$, and ${𝜏}_{𝑥𝑦}=20\text{MPa}$. Calculate the principal stresses. Solution: ${𝜎}_1,{𝜎}_2=\frac{{𝜎}_{𝑥}+{𝜎}_{𝑦}}{2}\pm \sqrt{{\left(\frac{{𝜎}_{𝑥}-{𝜎}_{𝑦}}{2}\right)}^2+{𝜏}_{𝑥𝑦}^2}$ ${𝜎}_1,{𝜎}_2=\frac{50+30}{2}\pm \sqrt{{\left(\frac{50-30}{2}\right)}^2+20^2}$ ${𝜎}_1,{𝜎}_2=40\pm \sqrt{100+400}$ ${𝜎}_1,{𝜎}_2=40\pm \sqrt{500}$ ${𝜎}_1,{𝜎}_2=40\pm 22.4$