Structural Engineering Calculations: Beams, Columns, and Shafts

Part 1: Beams

i) A simply supported 6m long beam has two 24kN point loads positioned symmetrically at 2m and 4m from the left hand end of the beam. The beam depth is 300mm, the width is 150mm. The beam is an ‘I’ section beam with a 5mm wall thickness.

Determine:

a) The reaction forces if the beam is supported at each end,

b) the shear force in the centre of the beam by the use of a shear force diagram,

c) the maximum bending moments by producing a bending moment diagram,

d) the maximum induced bending stress within the beam, given that

ii) The metal beam below, used in Science assignment 2, shows a simply supported ‘I’ section beam. The beam carries a point load and a Uniform Distributed load. The beam depth is 400mm, width is 200mm and section thickness of 8mm. Draw a shear force diagram and bending moment diagram and determine the values of,

a) the maximum and minimum shear force

b) the maximum bending moments and the distance of those maximum bending moments from the support on the left of the beam.

c) determine the maximum bending stress induced by the beam.

iii) a) The lintel shown in the diagram will form an encastrè beam which is fixed at each end, it spans a gap of 1.68m. There are three layers of bricks on top forming a uniform distributed load which is one brick wide, each brick is 240mm long and each one applies a force of 350N (including an allowance for mortar).

Calculate the maximum shear force in the supporting wall and bending moments at the centre of the lintel.

b) The wall featured in a) above is to have a 12” hanging basket hung from wall in the centre of the lintel. The basket when full of flowers and properly watered will weigh up to 4.5kg. Taking into account the uniform distributed load of the bricks and the point load of the hanging basket, calculate the new shear force at the supporting walls and the maximum bending moments in the centre of the lintel.

c) The diving board shown in the image extends 2m from the supporting wall. If a person of 80kg stands 200mm from the end of the board ready to dive, calculate the shear force acting at the supporting wall and the maximum bending moments acting in the system. This may be calculated or determined by the use of a shear force and bending moment diagram. Ignore the self-weight of the diving board.

Part 2: Columns

i) If a steel ‘I’ beam (such as RSJ, Universal Beam, Universal Colum and Bull Head), has the following design parameters in the table below. Using the tables provided, justify your selection of a beam that will satisfy the parameters required below.

Second Moment of Area x-x Radius of Gyration Depth of Section 14,136 14.8 355

ii) A 5-metres tall hinged column is manufactured from using a Universal ‘I- Section’ has Slenderness Ratio about the x-axis of 37.3 and a Slenderness Ratio about the y-axis of 65. A load of 2MN is applied on the x-axis 200mm from the centroid. Showing work to:

a) Calculate the radius of gyration for the x-axis and the y-axis.

b) Justify your selection of a suitably sized column that satisfies your calculations using the table for Universal ‘I-Section’ Columns. 

i) A solid shaft, below, is 2.5m long and has a radius of 25mm. The shaft has a tangential force of 225N applied to it. If the Modulus of rigidity (G) is 80GPa determine,

a) The angle of twist, in radians.

b) The maximum shear stress in the shaft (?).

c) The distribution of shear stress in the shaft by hand sketching the diameter of the bar, showing the stress in the end that is twisted. 

ii) Your supervisor would now like you to compare the maximum shear stress in a) above with a hollow shaft with similar dimensions. The hollow shaft is 2.5m long and has an outer radius of 25mm and an inner radius of 20mm. The hollow shaft has the same torque applied as the solid shaft above. If the Modulus of rigidity (G) for the material used is 80GPa determine,

a) The angle of twist of the hollow shaft, in radians.

b) The maximum shear stress in the tube (?).

c) The distribution of shear stress in the hollow shaft by hand sketching the end of the tube that is twisted and showing the stress distribution for the hollow shaft material.

iii) During a torsion experiment the equipment below has been used to gather data about a solid circular shaft produced from an unknown material so that the Shear Modulus (G), or Modulus of Rigidity, can be determined. A solid rod of Ø12mm is fixed at A and a load of 3.25kg is applied to the wheel which has a radius of 150mm. The angular displacement is measured by protractors positioned at B and C, which are 300mm a part. The displacement is determined as the difference between angle B and C, which in this experiment was 1.1° 

Determine:

a) The angle of twist in radians

b) The applied torque (T)

c) The polar second moment of area for the rod (J)

d) The Modulus of rigidity for the material (G)

e) The material that was used in this experiment, using the table below and explain how this relates to the answer you have calculated for the Modulus of Rigidity (G).