📘 mechanics of materials

1. **Problem Statement:** A cube-shaped solid is subjected to normal stresses: 30 MPa downward on the top face (along $x_3$), and in-plane stresses of 30 MPa and 10 MPa on the side

1. **Problem Statement:** A rod consists of two cylindrical portions AB (steel) and BC (brass), fixed at both ends. Given the temperature rise $\Delta T = 50 ^\circ C$, find the co

1. **Problem statement:** Given principal stresses $\sigma_{p1} = 7.88$ ksi and $\sigma_{p2} = 30.49$ ksi, and $b=2.15$ with $\theta_p = \frac{1}{2} \tan^{-1}(b) = 32.5^\circ$, use

1. **Problem Statement:** Calculate the net change in length of a prismatic bar of diameter 10 mm subjected to axial forces along three segments of lengths 3 m, 2 m, and 1 m respec

1. The problem asks where the value 175 comes from in the calculation of the first moment $Q_b$. 2. The first moment $Q_b$ is calculated as $$Q_b = 2(150)(60 \times 100) + (175)(30

1. **مقدمة للمشكلة:** نحن ندرس دائرة مور التي تمثل حالة الإجهاد في نقطة معينة داخل مادة تحت تأثير إجهادات نظامية وقص.

1. **مقدمة للمشكلة:** نحن ندرس حالة إجهاد في نقطة داخل مادة، حيث لدينا إجهادات طبيعية $\sigma_x$ و $\sigma_y$ وإجهاد قص $\tau_{xy}$. الهدف هو إيجاد الإجهادات الرئيسية $\sigma_{max}

1. **Problem Statement:** Determine the reactions at supports A and B for an A-36 steel rod of diameter 50 mm, initially at temperature $T_1=80^\circ C$, cooled to $T_2=20^\circ C$

1. **Problem Statement:** Determine the normal stress in cables AB, CD, EF and cable A, and the extension of point E for a rigid bar ACE supported by three steel cables AB, CD, EF.

1. **Problem Statement:** Calculate the maximum shear stress developed in the steel shaft in regions CF and BC.

1. **Problem Statement:** Calculate the maximum shear stresses $\tau_{max}$ at points A and B on the cross section of a cantilevered round peg subjected to a force $F=50000$ N and

1. **Problem Statement:** Determine the internal forces and average normal stresses in the aluminium and steel rods subjected to a 50 kN load at point B.

1. **Problem statement:** A wire DE made of A-36 steel with diameter 5 mm supports a beam with a man of mass 80 kg sitting at seat C. We need to find the elongation of wire DE due

1. **Stating the problem:** We need to determine the magnitude of the average normal stress developed in rods AB, CD, and EF. Rods AB and CD are made of C83400 copper, and rod EF i

1. **State the problem:** We have a bar with cross-sectional area $A = 1385$ mm$^2$ under a compressive load $P = 60000$ N (since 60 kN = 60000 N). The bar is cut at an angle $B =

1. **State the problem:** Calculate the direct axial stress at point A on the top of a rectangular beam with breadth $b=0.03$ m, depth $d=0.21$ m, subjected to an axial compressive

1. **Problem Statement:** Calculate the maximum shear stress in the hollow shaft segment CD subjected to a torque $T_C = 28$ kNm, with inner radius $R_i = 22$ mm and outer radius $

1. **Problem Statement:** Determine the maximum normal stress in bar ABCD with cross-sectional area $600\ \text{mm}^2$ under applied loads 25 kN at B, 20 kN at C, and 30 kN at D.

1. **Problem 01:** Calculate average shear stress in two 20 mm diameter bolts in single shear carrying 150 kN. Formula: Average shear stress $\tau = \frac{F}{A}$ where $F$ is force

1. **Problem Statement:** Determine the minimum width $b$ of a beam subjected to a distributed load of 2000 N/m over 4 m total length, a point load of 5000 N at 1 m from the left s

1. **Problem Statement:** A steel I-section has flange dimensions 200 mm by 12 mm and web dimensions 376 mm by 8 mm. It is subjected to a pure torque. The maximum shear stress allo