--- title: "Resultant stress on planes EF and GE, principal stresses" subject: "Strength of Materials" chapter: 4 key: "semester_3/strength_of_materials:resultant-stress-plane-ef-ge" source: https://www.bctengineer.com/q/semester_3/strength_of_materials/resultant-stress-plane-ef-ge site: BCT Engineering --- # Resultant stress on planes EF and GE, principal stresses Strength of Materials · Chapter 4: Stress and Strain Analysis - Asked in: 2080 Bhadra, 8 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/resultant-stress-plane-ef-ge - Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-4 ## Question At a point in a material under stress, the intensity of resultant stress on a plane EF is $50\ \text{MN/m}^2$ (tensile) inclined at $30^\circ$ to the normal to that plane. The stress on another plane at right angles to this plane is $30\ \text{MN/m}^2$. Determine: (i) The resultant stress on the plane GE. (ii) The principal stresses and the principle planes. (iii) Maximum shear stress and its plane. [Figure: a corner of the material with plane GE horizontal (stress of $30\ \text{MN/m}^2$ acting upward on it) and plane EF vertical (resultant stress 50 acting at $30^\circ$ to the normal to EF).] ## Short answer Resolve 50 MN/m² on EF: σn = 50 cos30° = 43.30 MN/m², τ = 25 MN/m². With σ on GE = 30 MN/m², τxy = 25. Resultant on GE = √(30²+25²) = 39.05 MN/m² at 39.8° to normal. Principal stresses = 62.52 and 10.78 MN/m²; planes at 37.55° to EF normal. τmax = 25.87 MN/m². ## Answer **Assumption:** the 30 MN/m² on plane GE is the normal (tensile) stress on it, as in the figure. Shear stresses on two perpendicular planes are equal in magnitude, so the shear on GE equals the shear on EF. #### Components on plane EF The 50 MN/m² resultant is at $30^\circ$ to the normal of EF: $$ \sigma_{EF} = 50\cos 30^\circ = 43.30\ \text{MN/m}^2, \qquad \tau_{EF} = 50\sin 30^\circ = 25.00\ \text{MN/m}^2 $$ On the perpendicular plane GE: $\sigma_{GE} = 30$ and $\tau_{GE} = 25.00$ MN/m². So $\sigma_x = 43.30$, $\sigma_y = 30$, $\tau_{xy} = 25.00$ (all in MN/m²). #### (i) Resultant stress on plane GE $$ \sigma_R = \sqrt{30^2 + 25.00^2} = 39.05\ \text{MN/m}^2, \qquad \tan\alpha = \frac{25.00}{30} \Rightarrow \alpha = 39.81^\circ\ \text{to the normal of GE} $$ #### (ii) Principal stresses and planes $$ \sigma_{1,2} = \frac{43.30+30}{2} \pm \sqrt{\left(\frac{43.30-30}{2}\right)^2 + 25.00^2} = 36.65 \pm 25.87 $$ $$ \sigma_1 = 62.52\ \text{MN/m}^2, \qquad \sigma_2 = 10.78\ \text{MN/m}^2 $$ $$ \tan 2\theta_p = \frac{2(25)}{43.30-30} \Rightarrow 2\theta_p = 75.10^\circ,\ \theta_p = 37.55^\circ $$ $\sigma_1$ acts on the plane whose normal is at $37.55^\circ$ to the normal of EF; $\sigma_2$ acts on the plane at $127.55^\circ$ (both planes are at right angles). #### (iii) Maximum shear stress $$ \tau_{max} = \frac{\sigma_1-\sigma_2}{2} = 25.87\ \text{MN/m}^2 $$ It acts on planes at $45^\circ$ to the principal planes, i.e. at $82.55^\circ$ and $172.55^\circ$ from the normal of EF, with a normal stress of $36.65$ MN/m² on them. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the 50 MN/m^2 resultant on EF resolved? A. Normal 50 sin30 = 25 and shear 50 cos30 = 43.30 B. Normal 50 and shear 30 C. Normal 50 cos30 = 43.30 and shear 50 sin30 = 25.00 D. Normal 43.30 and shear 30 Answer: C. The angle is measured to the normal, so normal uses cos and shear uses sin. (Key point: Components on EF) 2. What is the shear stress on plane GE? A. 25 MN/m^2, equal to shear on EF (complementary shear) B. 0 C. 30 MN/m^2 D. 43.30 MN/m^2 Answer: A. Shear stresses on perpendicular planes are equal in magnitude. (Key point: Shear on GE) 3. What is the resultant stress on GE? A. 50 MN/m^2 B. 55 MN/m^2 C. 39.05 MN/m^2 D. 25 MN/m^2 Answer: C. sqrt(30^2 + 25^2) = 39.05. (Key point: Resultant on GE) 4. What are sigma_1 and sigma_2? A. 62.52 and 10.78 MN/m^2 B. 73.30 and 0 C. 43.30 and 30 D. 36.65 and 25.87 Answer: A. Centre is 36.65 and radius 25.87, so 36.65 +/- 25.87. (Key point: Principal stresses) ## Related in Strength of Materials - [Normal/shear stress on 25 degree plane, principal stresses](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ainclined-plane-30-25deg-mohr-verify) — Strength of Materials, chapter 4, asked 3× - [Plane stress: principal planes and max shear](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aplane-stress-15-30-60-principal) — Strength of Materials, chapter 4, asked 2× - [Plane stress: principal, max shear, normal](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aplane-stress-80-60-50-mpa) — Strength of Materials, chapter 4, asked 2× - [Principal planes, stresses on plane AC](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-planes-max-shear-120-180) — Strength of Materials, chapter 4 - [Stresses on 20 degree plane; principal and max shear](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ainclined-plane-20deg-50-40-80) — Strength of Materials, chapter 4, asked 4× - [Normal and resultant stress on 60 degree plane](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Anormal-resultant-stress-60deg-plane-100) — Strength of Materials, chapter 4 ## Related in other subjects - [Stresses on inclined plane and principal stresses](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Ainclined-plane-stresses-principal-stresses) — Strength of Materials, chapter 6 - [Plane stress: principal stresses and oblique plane](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Aplane-stress-principal-numerical) — Mechanics of Solids, chapter 2 - [Principal stresses in a shaft under combined loads](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Ashaft-combined-bending-torsion-stresses) — Strength of Materials, chapter 6 - [Principal stresses from a 3D stress tensor](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Athree-d-stress-invariants-numerical) — Mechanics of Solids, chapter 2