Question 1:
A simply supported beam has a span of 6 meters and carries a uniformly distributed load of 5 kN/m. Calculate the maximum bending moment in the beam.
Solution:
The maximum bending moment occurs at the center of the span and can be calculated using the formula: Mmax = wl^2/8 where w is the uniformly distributed load, and l is the span.
Substituting the values given in the problem, we get: Mmax = 5 kN/m x (6 m)^2 / 8 = 112.5 kNm.
Therefore, the maximum bending moment in the beam is 112.5 kNm.
Question 2: A steel bar with a diameter of 20 mm is subjected to a tensile load of 100 kN.
Determine the stress and strain in the bar.
Solution:
The stress in the bar can be calculated using the formula: Stress = Force / Area where Force is the applied load, and Area is the cross-sectional area of the bar.
The cross-sectional area of the bar can be calculated using the formula: Area = πd^2/4 where d is the diameter of the bar.
Substituting the values given in the problem, we get: Area = π x (20 mm)^2 / 4 = 314.16 mm^2
Stress = 100 kN / 314.16 mm^2 = 318.3 N/mm^2
The strain in the bar can be calculated using the formula: Strain = Change in length / Original length
The change in length can be calculated using the formula: Change in length = Force x Length / (Area x Modulus of elasticity) where Length is the original length of the bar, and Modulus of elasticity is a material property.
Substituting the values given in the problem, we get: Change in length = 100 kN x 1000 mm / (314.16 mm^2 x 200 GPa) = 0.159 mm.
The original length of the bar is assumed to be 1000 mm, so the strain is: Strain = 0.159 mm / 1000 mm = 0.000159.
Therefore, the stress in the bar is 318.3 N/mm^2, and the strain is 0.000159.
Question 3:
A reinforced concrete beam has a span of 5 meters and carries a concentrated load of 30 kN at a distance of 2 meters from one end. Calculate the maximum shear force and bending moment in the beam.
Solution:
The maximum shear force occurs at the point where the concentrated load is applied and can be calculated using the formula: Vmax = F where F is the concentrated load.
Substituting the value given in the problem, we get: Vmax = 30 kN.
The maximum bending moment occurs at the center of the span and can be calculated using the formula: Mmax = F x (L - a) x a / L where L is the span, and a is the distance from one end to the point where the concentrated load is applied.
Substituting the values given in the problem, we get: Mmax = 30 kN x (5 m - 2 m) x 2 m / 5 m = 36 kNm.
Therefore, the maximum shear force in the beam is 30 kN, and the maximum bending moment is 36 kNm.
Question 4: A rectangular concrete column with a cross-sectional area of 300 mm x 600 mm is subjected to a compressive load of 800 kN. Determine the stress and strain in the column.
Solution:
The stress in the column can be calculated using the formula:
Stress = Force / Area, where Force is the applied load, and Area is the crosssectional area of the column.
Substituting the values given in the problem, we get: Area = 300 mm x 600 mm = 180000 mm^2
Stress = 800 kN / 180000 mm^2 = 4.44 N/mm^2.
The strain in the column can be calculated using the formula: Strain = Change in length / Original length
The change in length can be calculated using the formula:
Change in length = Stress x Original length / Modulus of elasticity, where Modulus of elasticity is a material property.
Assuming the original length of the column is 3000 mm and the modulus of elasticity of concrete is 30 GPa, we get: Change in length = 4.44 N/mm^2 x 3000 mm / 30 GPa = 0.000444.
Therefore, the stress in the column is 4.44 N/mm^2, and the strain is 0.000444.
Question 5:
A steel truss bridge has a span of 50 meters and is designed to carry a maximum load of 500 kN. The dead load of the bridge is 100 kN, and the live load is assumed to be uniformly distributed over the span. Determine the maximum allowable live load per meter of the span.
Solution:
The maximum allowable live load can be determined using the formula:
Maximum allowable live load = (Maximum load - Dead load) / Live load factor
The live load factor is a safety factor that accounts for the uncertainty in the actual live load. A typical value for the live load factor for a bridge is 1.5.
Substituting the values given in the problem, we get:
Maximum allowable live load = (500 kN - 100 kN) / 1.5 = 267 kN.
The maximum allowable live load per meter of the span can be calculated by dividing the maximum allowable live load by the span:
Maximum allowable live load per meter = 267 kN / 50 m = 5.34 kN/m.
Therefore, the maximum allowable live load per meter of the span is 5.34 kN/m.