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Sample Quiz: Strength of Materials Chapter 5

Stresses in Beams — Strength of Materials (Andrew Pytel)

10 questions · 23 points · answer key included

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Multiple Choice

  1. In floor framing design, if floor joists are spaced a distance a apart on centers and each joist has length L, what is the distributed load intensity w0 that each joist supports when the floor load per unit area is p? (2 PTS)
    1. w0 = pa
    2. w0 = pL
    3. w0 = p/aL
    4. w0 = p(a + L)
  2. The flexure formula σ = My/I indicates that bending stress varies directly with which of the following? (2 PTS)
    1. The distance y from the neutral axis and inversely with the moment of inertia I
    2. The moment of inertia I only
    3. The square of the distance from the neutral axis
    4. The bending moment M only
  3. In the derivation of the flexure formula, the neutral surface is defined as the plane where fibers remain unchanged in length. Which of the following statements correctly describes the neutral surface? (2 PTS)
    1. It moves depending on the magnitude of the applied load
    2. It contains the centroids of all transverse sections and carries no stress
    3. It is located at the outer fibers of the beam
    4. It is where maximum tensile stress occurs
  4. For an unsymmetrical beam section where the allowable tensile stress is σt and the allowable compressive stress is σc, what is the ideal relationship between the distances yt and yc from the neutral axis to the tensile and compressive fibers, respectively? (2 PTS)
    1. yt + yc = σt + σc
    2. yt = yc
    3. yt/yc = σt/σc
    4. yt/yc = σc/σt
  5. In the worked example for a rectangular beam section, the resisting moment Mr is formed by a couple consisting of equal compressive and tensile forces. If the maximum stress is σ and the section has width b and height h, what is the magnitude of each force (C or T)? (2 PTS)
    1. σ · bh/4
    2. σ · b · h
    3. 1/2σ · b · h/2
    4. 1/2σ · b · h

True or False

  1. The neutral axis of a beam always coincides with the centroidal axis of the cross section. (1 PTS)
  2. In a solid beam, horizontal shear resistance is developed at layers parallel to the neutral axis to prevent layers from sliding past one another during bending. (1 PTS)
  3. The modulus of rupture is a true stress value that represents the actual stress distribution in a beam loaded beyond its proportional limit. (1 PTS)

Problem Solving

  1. A timber floor joist in a residential building is 50 mm wide by 200 mm high and simply supported on a 4 m span. The floor is designed to carry a distributed load of 5 kN/m2. If the joists are spaced 0.30 m apart on centers, determine (a) the distributed load intensity w0 acting on each joist, and (b) the maximum bending moment in the joist. (c) Using the flexure formula with c = h/2 = 100 mm, calculate the maximum flexural stress in the joist. (5 PTS)
  2. A cast-iron beam with an inverted T cross section carries a uniformly distributed load on a simple span. The section has a total height of 140 mm, with the neutral axis located 105 mm from the top (in the web). The moment of inertia about the neutral axis is I = 50 × 10-6 m4. The allowable stresses are σt = 40 MPa (tension in lower fibers) and σc = 100 MPa (compression in upper fibers). At a critical section, the bending moment is M = 0.50 kN·m. Determine (a) the maximum compressive stress at this section, (b) the maximum tensile stress at this section, and (c) whether the section is safe under these stresses. (5 PTS)

Answer key

Multiple Choice

  1. 1.A — According to the floor framing concept, the total load W = p · aL is divided by the joist length L to give the distributed load intensity w0 = pa.
  2. 2.A — The flexure formula explicitly shows that stress is proportional to both the bending moment M and the distance y from the neutral axis, and inversely proportional to the moment of inertia I.
  3. 3.B — The neutral surface is the plane of fibers that experience no change in length during bending; it contains the centroids of all transverse sections and therefore carries no stress.
  4. 4.C — For unsymmetrical beams, flexure stresses vary directly with distance from the neutral axis, so to reach allowable stresses simultaneously, the ratio of distances must equal the ratio of allowable stresses: yt/yc = σt/σc.
  5. 5.C — In the rectangular section analysis, the average stress in the linear distribution is one-half the maximum stress, and this acts over the area of half the section, giving T = C = (1/2σ)(bh/2).

True or False

  1. 1.True — According to the derivation of the flexure formula, the neutral surface (and thus the neutral axis) contains the centroids of all transverse sections.
  2. 2.True — The analysis of flexure action demonstrates that unbalanced horizontal forces exist at layers throughout the beam depth, requiring shear resistance to prevent layer slippage, as illustrated by the playing card analogy.
  3. 3.False — The modulus of rupture is a fictitious stress obtained by applying the flexure formula to a beam loaded to rupture; it is not a true stress because the proportional limit has been exceeded and Hooke's law no longer applies.

Problem Solving

  1. 1.(a) w0 = pa = 5 × 0.30 = 1.5 kN/m. (b) For a simply supported beam with uniform load, Mmax = (w0 L2)/8 = (1.5 × 42)/8 = 3.0 kN·m. (c) First, calculate I = bh3/12 = (0.050 × 0.2003)/12 = 33.33 × 10-6 m4. Then, σmax = Mc/I = (3.0 × 103 × 0.100)/(33.33 × 10-6) = 9.0 MPa. — This problem applies the floor framing concept to find the load per joist, then uses standard beam theory and the flexure formula to determine maximum stress, integrating multiple concepts from the chapter.
  2. 2.(a) Distance from NA to top (compression fibers): yc = 0.105 m. σc = Myc/I = (0.50 × 103 × 0.105)/(50 × 10-6) = 1.05 MPa. (b) Distance from NA to bottom (tension fibers): yt = 0.140 - 0.105 = 0.035 m. σt = Myt/I = (0.50 × 103 × 0.035)/(50 × 10-6) = 0.35 MPa. (c) Both stresses are well below their allowable limits (1.05 < 100 MPa and 0.35 < 40 MPa), so the section is safe. — This problem requires applying the flexure formula to an unsymmetrical section with different allowable stresses in tension and compression, demonstrating the design considerations for cast-iron beams.