A beam of the cross section shown in Figure Q3 is made of a steel that is assumed to be elastic- perfectectly plastic material with E = 200 GPa and σy = 240 MPa. Determine: i. The shape factor of the cross section ii. The bending moment at which the plastic zones at the top and bottom of the bar are 30 mm thick. 15 mm 30 mm 15 mm 30 mm 30 mm 30 mm

A plastic-lined steel pipe has the cross-sectional shape shown in the figure. The steel pipe has an outer diameter d1= 100 mm and an inner diameter d2= 94 mm. The plastic liner has an inner diameter d1= 82 mm. The modulus of elasticity of the steel is 75 times the modulus of the plastic. Determine the allowable bending moment Mallowif the allowable stress in the steel is 35 M Pa and in the plastic is 600 kPa. If pipe and liner diameters remain unchanged, what new value of allowable stress for the steel pipe will result in the steel pipe and plastic liner reaching their allowable stress values under the same maximum moment (i.e., a balanced design)? What is the new maximum moment?

The cross section of a beam made of thin strips of aluminum separated by a lightweight plastic is shown in the figure. The beam has width b = 3.0 in., the aluminum strips have thickness t = 0.1 in., and the plastic segments have heights d = 1.2 in. and 3d = 3.6 in. The total height of the beam is h = 6.4 in. The moduli of elasticity for the aluminum and plastic are EM= 11 X 106 psi and Ep= 440 X 10* psi, respectively. Determine the maximum stresses trAiand pin the aluminum and plastic, respectively, due to a bending moment of 6,0 kip-in.

The Z-section of Example D-7 is subjected to M = 5 kN · m, as shown. Determine the orientation of the neutral axis and calculate the maximum tensile stress c1and maximum compressive stress ocin the beam. Use the following numerical data: height; = 200 mm, width ft = 90 mm, constant thickness a = 15 mm, and B = 19.2e. Use = 32.6 × 106 mm4 and I2= 2.4 × 10e mm4 from Example D-7

A flat brass bar has length L, constant thickness t, and a rectangular cross section whose width varies linearly between b2at the fixed support to b1at the free end (see figure). Assume that the taper of the bar is small. The bar has modulus of elasticity E. Calculate the displacements ??Band ??cif P = 200 kN, L = 2 m, t = 20 mm, b, = 100 mm, b, = 115 mm, and E = 96 GPa.

5. A 4-point bending test is performed on a beam of length L = 1 m. A load of P = 2000 N is applied across two loading points a distance a = 0.4 m from the sides of the beam. If the beam has a radius of r = 1 cm and a yield strength of σ = 250 MPa, does the bar yield? If it's an elastic-perfectly plastic material, does it fail all the way through the thickness of the bar? What would the yield strength of the material need to be if we wanted a factor of safety of 1.5?

An elastomeric bearing pad is used to support a bridge girder as shown. The bearing pad provides flexibility in the horizontal direction to accommodate thermal expansion and earthquake motions. The shear modulus of the bearing pad is G = 1.50 MPa. The beam must not displace horizontally more than 25 mm when a load of P = 45 kN is applied as shown. Determine the maximum thickness t that can be used for the bearing pad. Assume a = 300 mm and b = 420 mm. Answer: tmax = i mm

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)=

The figure below shows the cross-section of an axisymmetric composite beam that comprises steel (Young's modulus 270 GPa) and aluminum (Young's modulus 90 GPa) sections that are bonded together. The steel section is of wall thickness 15 mm and the aluminum section is of wall thickness 10mm. The steel section comprises 4 axisymmetric holes of 5 mm diameter as shown. Given that the beam is bent by a couple moment of 1200 Nm, determine the maximum stress in steel and aluminum. 4 holes of diameter 5 mm. 12 mm steel aluminum

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