A beam has a bending moment of 4.5 kN-m applied to a section with a hollow circular cross-section of external diameter 3.1 cm and internal diameter 2.1 cm . The modulus of elasticity for the material is 210 x 109 N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.6 cm from the neutral axis Solution: - The bending stress at the point 0.6 cm from the neutral axis  is

Find the shear force and bending moment at points C and D. 60 lb/ft Answer: VD= 637 lb (up) Mp= = 3185 lb ft (ccw) Vc = 559 lb (up) Mc = 3033 lb ft (cw) -10 ft- -15 ft- 40 lb/ft B B ID -5 ft- -10 ft- 690 Ib 13 12 A

consider a beam with two 60-inch-long rods. The rods have a cross section area of 0.50 in 2 . Find the axial stress in each rod, draw the shear and moment diagrams for the beam, and find the maximum stress in the beam.

Calculate the shear force V (in kN) and bending moment M (in kN - m) at a cross section located just right of the 4 kN load on the cantilever beam AB shown in the figure. (Use the deformation sign convention.) | 4.0 kN 1.4 kN/m 3.9 m 3.9 m 7.8 m V = 10.92 v kN M - -85.2 kN m

Calculate the support reaction moment at the clamped part of the cantilever beam and find the maximum bending stress.   Note that the beam has a rectanglular cross-section with height of ℎh and thickness b. The 2nd moment of area of this beam is Iz.

. A composite beam, made of aluminum (Al) and magnesium (Mg), has the cross section shown below. The moduli of elasticity for aluminum and magnesium are EAI = 70 GPa and EMg = 45 GPa, respectively. If the allowable flexural stresses (in tension and com- pression) are All 135 MPa for the aluminum and Mg 120 MPa for the magnesium, determine the maximum allowable bending moment for the beam. all - 0.2 m 0.2 m Al 0.2 m Mg C

A beam has a bending moment of 3.5 kN-m applied to a section with a hollow circular cross-section of external diameter 3 cm and internal diameter 2.3 cm . The modulus of elasticity for the material is 210 x 109 N/m2. Calculate the radius of curvature and maximum bending stress. Also, calculate the stress at the point at 0.6 cm from the neutral axis (i) The moment of inertia in (mm^4)=  ii) The radius of curvature in (mm) = (iii) The maximum bending stress  in (N/mm^2)= iv) The bending stress at the point 0.6 cm from the neutral axis  in (N/mm^2)=

A simply supported reinforced concrete beam carries a uniformly distributed load of 12 kN/m on a span of 4 m. The beam has a rectangular cross section 250 mm wide by 450 mm deep, with three 20 mm diameter steel bars placed 60 mm from the bottom of the beam, as shown below. The moduli of elasticity for the concrete and steel are 15 GPa and 200 GPa, respectively. Determine the average tensile stress in the steel and the maximum compressive stress in the concrete at the section of maximum bending moment. 450 mm 250 mm 60 mm

A composite beam is constructed using a steel plate(0.5 in.x 6 in.) with two wood beams (3in. X 6 in.) on either side. The wood and steel are securely fastened to act as a single beam. The beam is subjected to a positive bending moment Mz=60 kip-in. Calculate the maximum tensile and compressive stresses in the wood and steel if Ew= 1500 ksi and Es=29,500 ksi.

4. The box beam is made of an elastic-perfectly plastic material for which the yield 250 MPa. Draw the residual stress distribution in the beam after the elastic-plastic moment, Mep = 548.5 kNm, is applied and then released. It can be shown that under the applied moment, the flanges (260 mm × 30 mm) yield, while the webs (2 x 200mm x 30mm) show linear elastic behavior. The moment of inertia and plastic section modulus of the section are given below: I = 247.48 x 106 mm4; Z, = 2394000 mm³ stress oy = y 30 mm 30 mm 200 mm 200 mm 30 mm 30 mm