Stress (Mechanics Of Materials)

Question

Anu Bhardwaj · Accepted Answer

Step 1:- To solve this problem, we need to calculate the bending moment at critical locations along the beam and then determine the maximum tensile and compressive bending stresses. Explanation:- 𝑊 1 W 1 ​ : Distributed load of 1600 lb/ft 𝑊 2 W 2 ​ : Concentrated load of 4000 lb 𝑅 1 R 1 ​ : Reaction at support 1 𝑅 2 R 2 ​ : Reaction at support 2 𝐿 L : Total length of the beam Given: 𝐿 = 3 ft + 6 ft + 6 ft = 15 ft L = 3 ft + 6 ft + 6 ft = 15 ft 𝑑 1 = 7 in d 1 ​ = 7 in (distance from the bottom to the centroid of the T-section) 𝑑 2 = 2 in d 2 ​ = 2 in (thickness of the T-section) 𝐼 UA = 84 in 4 I NA ​ = 84 in 4 (moment of inertia of the T-section) Step 2:- First, let's calculate the reactions 𝑅 1 R 1 ​ and 𝑅 2 R 2 ​ : 𝑅 1 = 1 2 × ( 𝑊 1 × 3 ft ) + 𝑊 1 × 6 ft + 𝑊 2 = 1 2 × ( 1600 lb/ft × 3 ft ) + 1600 lb/ft × 6 ft + 4000 lb = 15400 lb R 1 ​ = 2 1 ​ × ( W 1 ​ × 3 ft ) + W 1 ​ × 6 ft + W 2 ​ = 2 1 ​ × ( 1600 lb/ft × 3 ft ) + 1600 lb/ft × 6 ft + 4000 lb = 15400 lb 𝑅 2 = 1 2 × ( 𝑊 1 × 3 ft ) + 𝑊 1 × 6 ft + 𝑊 2 − 𝑅 1 = 15400 lb R 2 ​ = 2 1 ​ × ( W 1 ​ × 3 ft ) + W 1 ​ × 6 ft + W 2 ​ − R 1 ​ = 15400 lb Next, let's calculate the bending moments at critical locations. At the midspan (point where the concentrated load is applied), the bending moment: 𝑀 mid = 4000 lb × 6 ft 2 = 12000 lb-ft M mid ​ = 2 4000 lb × 6 ft ​ = 12000 lb-ft At support 1 or 2, the bending moment: 𝑀 support = 15400 lb × 6 ft 2 = 46200 lb-ft M support ​ = 2 15400 lb × 6 ft ​ = 46200 lb-ft Now, let's calculate the maximum tensile and compressive bending stresses. The maximum tensile stress ( 𝜎 max σ max ​ ) occurs at the bottom of the beam: 𝜎 max = 𝑀 support × 𝑑 1 𝐼 UA σ max ​ = I NA ​ M support ​ × d 1 ​ ​ 𝜎 max = 46200 lb-ft × 7 in 84 in 4 σ max ​ = 84 in 4 46200 lb-ft × 7 in ​ 𝜎 max ≈ 3842.86 psi σ max ​ =3850 psi The maximum compressive stress occurs at the top of the beam: 𝜎 min = 𝑀 support × 𝑑 2 𝐼 UA σ min ​ = I NA ​ M support ​ × d 2 ​ ​ 𝜎 min = 46200 lb-ft × 2 in 84 in 4 σ min ​ = 84 in 4 46200 lb-ft × 2 in ​ 𝜎 min ≈ 1100 psi σ min ​ =1100 psi Explanation:- the maximum tensile bending stress developed in the beam with the inverted T-section is approximately 3842.86 psi 3850 psi , and the maximum compressive bending stress developed is approximately 1100 psi 1100 psi . Final Answer:- the maximum tensile bending stress developed in the beam with the inverted T-section is approximately 3842.86 psi 3850 psi , and the maximum compressive bending stress developed is approximately 1100 psi 1100 psi .

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Anu Bhardwaj

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