A beam with a constant height and width is said to be prismatic. When a beam’s width or height (more common) varies, the member is said to be non-prismatic. Horizontal applications of beams are typically at resists the rotation.
TYPES OF LOADS AND BEAMS
Beams can be catalogued into types based on how they are loaded and how they are supported. Loads that are applied to a small section of the beam are simplified by considering the load to be single force placed at a specific point on the beam. These loads are referred to as concentrated loads. Distributed loads (w, usually in units of force per lineal length of the beam) occur over a measurable distance of a beam. For the sake of determining reactions, a distributed load can be simplified in to an equivalent concentrated load by applying the area of the distributed load at the centroid of the distributed load. The weight of the beam can be described as uniform load. A moment is a couple as a result of two equal and opposite forces applied at certain section of the beam. A moment induced on any point can be mathematically described as a force multiplied by at one end and simply supported at the other (see figure 2d).
A continuous beam has more than two simple supports, and a built-in beam (see figure 2f) is fixed at both ends.
The remainder of this report deals only with simple and over-hanging beams loaded with concentrated and uniformly distributed loads.
STATICS-RIGID BODY MECHANICS
were accelerating in some direction the sum of the forces would equal the mass multiplied by the acceleration. Beams are described as either statically determinate or statically indeterminate. A beam is considered to be statically determinate when the support reactions can be solved for with only statics equations. The condition that the deflections due to loads are small enough that the geometry of the initially unloaded beam remains essentially unchanged is implied by the expression “statically indeterminate”. Three equilibrium equations exist for determining the support statically determinate, only two reaction components can exist. The two remaining equilibrium equations become ∑FY = 0∑MZA = 0
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Simply supported, overhanging, and cantilever beams are statically determinate. The other types of beams described above are statically indeterminate. Statically indeterminate beams also require load deformation properties to determine support reactions. When a structure is statically indeterminate at least one member or support is said to be redundant, because after removing all redundancies the structure will become statically determinate. Forces and moments are the internal forces transferred by a transverse cross section (section a, figure 3c) necessary to resist the external forces and remain in equilibrium. Stresses, strains, slopes, and deflections are a result of and a function of the internal forces. The simply supported single span beam in figure 3a is introduced to a uniform load (w) and two concentrated loads (P1) and (P2).
Using the equilibrium equations and a free body diagram the support reactions for the beam in figure 3a will be determined. This example will also show how internal forces (shear and moment) can be found at any point along the beam. This same method is applicable to any statically determinate beam.
Finding the support reactions requires a free body diagram that notes all external forces that act on the beam and all possible reactions that can occur
The steel beam was hung on the hooks at the bottom end of the spring balances. A load hanger was placed at the mid-point of the beam of given span, and the spring balances read. The 2kg weight was placed on the hanger and the deflection in the spring balances read. The load was increased in steps of 2kg up to 16kg and the balances read in each case (i.e. at each incremental loading. All weights were then removed.
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Next, the load hanger weight was placed directly under the spring balance A and the two spring balances read. A 8kg weight was put on the load – hanger and the spring balances read. It was then (i.e. the load) was then put (transferred) to the next 100mm.
The experiment was carried out using steel beam of span 1000mm with a midpoint if 500mm. The load used for the first part was 2kg, 4kg and at intervals of 2kg up to 16kg. For the second part of the experiment, a constant load of 8kg was used.
DISCUSSION AND CONCLUSION
From the results, for the first part of it the experiment, a relation between RA and RB is observed. As both reactions are at equidistance from the load applied, they both share the weight of the load. Thus the magnitude of the reactions are half that of the loading and are equal to each other i.e.
RA = RB = ½ (Weight of Load)
For the second part of the experiment, the position of the load on the beam varies therefore the two reactions vary as well, as the load is borne as a function of the distance of it, from that reaction.
When the load is at A, RA = weight of the load while the reaction RB = 0. As the load is at this point, the reaction RA is maximum (equal to load).
As the load is shifted away from RA, the reaction RA reduces while RB increases, until the load is at point B, in which case RB has the maximum reaction equal to the load, and RA is null or zero. Here an inversely proportional relation is observed.
Comparing the experimental values and those of the theoretical for this part of the experiment, a deviation is seen to occur in values. Nevertheless, this can be as a result (for the experimental part) of zero error on the metre rule of the spring balance as some approximations were made.
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Zero error of the metre rule in measuring the length of the beam was avoided.
It was made sure that the beam was perfectly horizontal
It is thus proven that for every action, there is an equal and opposite reaction
1) Strength of Materials by G.H. Ryder
2) Strength of Material by Beer & Johnson