Lecture 13: Beams: Internal shear-stress by Dr R. Kromanis (Roland) r.kromanis@utwente.nl Horst Room Z230 (now) 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 1 Contents • Lab sessions this week. Any questions? • Recap on calculating moment of inertia for non-standard sections • Shear stress 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 2 Laboratory reports: Truss and bending stress in beams • Read the brief carefully • Attendance is compulsory © TecQuipment 1/26/2025 © TecQuipment SM1-Lecture 13: Beams: Internal shear-stress 3 Calculating the moment of inertia for singly symmetric beams 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 4 Moment of Inertia for a non-standard section 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 5 Moment of Inertia for a non-standard section 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 6 Moment of Inertia for a non-standard section πΌπ₯π₯ = 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress πΌππ + π΄β2 7 Shear-stress in beams 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 8 Longitudinal shear stress 12_02b 12_02a 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 9 Transverse shear stress 12_03 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 10 Shear-stress -> transverse shear stress and longitudinal shear stress οThe transverse shear stress (on the cross section) exists together with the complementary shear stress in the longitudinal direction (on the longitudinal section). οThe transverse and longitudinal shear stress are complementary and numerically equal. 12_01 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 11 Geometry of the top section plane 12_04c 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 12 The shear formula Shear stress (π) formula ππ π = ππΏ = π π = πΌπ‘ π is an internal shear force at a point along the beam π = π¦ ′ π΄′ where π¦ ′ is the distance from NA to the centroid of π΄′ , which is the area of the top (or bottom) part of the cross section, above (or below) the section plain where π is being calculated πΌ is the moment of inertia of beam x-section about NA π‘ is the thickness of the section where π is being calculated 12_04c 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 13 Calculating Q 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 14 Example 12.1 from Hibbeler's book The given beam is made from two timber boards. Determine the maximum shear stress in the glue necessary to hold the boards together along the seam where they are joined. 12_10a-EX01 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 15 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 16 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 17 Example 12.3 from Hibbeler's book A steel wide-flange beam (or H section, see right) is subjected to a shear of π = 80 kN, draw the shear-stress distribution acting over the beam’s cross section. ππ π= πΌπ‘ π = π¦ ′ π΄′ 12_12a-EX03 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 18 Shear-stress distribution 12_12b-EX03 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 19 Relate to: stress distribution 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 20 Homework • From Hibbeler’s book: F12-1 to F12-5. 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 21 This Photo by Unknown Author is licensed under CC BY-SA Dr R. Kromanis (Roland) r.kromanis@utwente.nl Horst Room Z230 Solving questions correctly QUESTION / PROBLEM 1. How to solve it? Schematic presentation – model, e.g., model of a system, free body diagram 2. Present the data; what is given? What are the assumptions 3. Explain the questioned and provide equations 4. Provide solutions; derive them in a clear and sequential order 5. Give the result(-s)/answer(-s) to the question, problem 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 23 Sources • R. C. Hibbler, SI conversion by Kai Beng Yap, 2019, Statics and mechanics of materials, Fifth edition in SI units. 1/26/2025 SM1-Lecture 13: Beams: Internal shear-stress 24
