Worked Examples 1
Worked examples involving Bending Stess and Moments of Inertia
Please Note: Reference pages on the bending moments of beams is in preparation and will be published shortly. In the mean time the formulae for the following examples are quoted without proof.The General Equation for bending is used throughout. The proof of this is to be found in "Engineering/Materials/Bending Stress". For convenience the equation is written here and is as follows:
- M is the Bending Moment
- I is the Moment of Inertia of the section
- E is Young's Modulus
- R is the Radius of curvature
- The beam of a symmetrical I section is simply supported over a span of 30 ft. If the maximum permissible stress is 5 tons/sq.in., what concentrated load can be carried at a distance of 10 ft from one support? Whilst it is not stated in the question, it is normal practice to load an I-section with XX as the axis of bending. Thus the Bending Moment is in the YY plane. If the load is W tons then the maximum Bending Moment is given by:-
- An "I" section beam has a depth of 14 ins. and is freely supported at its ends. The Moment of Inertia is . Over what span can a uniform load of 1 ton/ft. be carried if the maximum stress is 4 tons/sq.in. What additional central load can be carried if the maximum stress is 4 tons/sq.in. Bending Moment = where w = weight per unit length
- The cross-section of a cast iron beam is shown in the diagram. The loading is in the plane of the web and the upper portion of the section is in compression. If the maximum permissible stresses are 2 tons/sq in in tension and 3 tons/sq.in. in compression find the moment of resistance of the section and the actual maximum stress.
Since the neutral axis XX passes through the centroid it is necessary to first find its position. This and the position of the total moment of inertia about XX can be most conveniently found by tabulating as follows:
- y is the distance of the centroid of each area from the bottom of the section.
- is the moment of inertia of each area about its own centroid axis parallel to XX
- h is the distance between each centroid axis and XX.
- A 12 in. by 5 in. Steel Beam is to be used as a Cantilever 10 ft. long. If the permissible stress is 8 tons/sq.in. , what uniformly distributed load can be carried? If the Cantilever is to be strengthened by steel plates 1/2 inch thick welded to the top and bottom flanges, find the width of the plates required to withstand an increase of 50% in the load. Calculate as well the length over which the plates should extend for the maximum stress to remain the same. From standard tables for beams If W is the total load in Tons and the length of the beam is measured in inches then:
- The I beam shown in the diagram is simply supported at its ends over a 20 ft. span and carries a central load of 5 tons which acts through the Centroid,. The line of action is as shown. Calculate the maximum stress. The section being symmetrical, the Centroid is at the centre of the web and the principle axes are XX' and YY. For the flange:
- A Cantilever has a free length of 8 ft. It is of "T" section with a flange of 4 ins. by 3/4 ins. and a web of 8 ins. by 1/2 ins.. The flange is in tension. What load per foot run can be applied if the maximum tensile stress is 2 tons/sq.in. What is the compressive stress produced by this loading.
The maximum Bending moment for a cantilever is given by:
TODO: TableThe distance of the Centroid from XX' is