Behavior of Compression Members
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The elaciticity of an obvject determines the extent to which the object can withold the tensile stress subjected in the form of an external force. Elaciticity depends with the material upon which the mamber is fabricated, where some elements have lower values of transition from the plastic to the elastic nature while others have hagher values. The ranges of these values exhibited by members once subjected to an external force forms the definition of the eccentricity. A bar placed in an axial position could be used to determine its eccentricity since the position allows for uniform subjection of exaternal forces, where a slender bar could bend as a result of the added force. This form of bending is known as buckling, while the nature of the bar reveals an impafection in the member, which is termed as the elasticity. For instance, the figure gives the curviture of a bar in axial positiin subjected to an external force, where the variations in the force applied gives the ideal buckling (Wiliams, 2000).
The bar bends depending on the amount of force applied, while verying the accentricities could be viable for the culculation of the Euler buckling load. This could be done experimentally, where the experimental results for the zero accentricity experiemnt and the maximum accentricity experiment are valuable in revealing the Euler Buckling load. By consideration, the Euler Buckling load is proportional to the product of the square of pie with both the modulus of elaciticity and the 2nd moment of area about the buckling axis, while it is also inveersely proportional to the square of the effective length of the strut. Additionally, the effective length of the strut depends on the flexibility of the rotation. The experiemental investigation of the accentricity and the culculation of the Euler Buckling load forms the essence of this paper (Wiliams, 2000).
Behavior of the Compression Members
The experimental analysis for determination of the behavior of compression members involves the structuring of a strut with known modulus of elaciticity. The strut is places in an axial position against pivots of eqal force of attraction, where the lower pivot and the higher pivots holding the strut in the axial position are adjusted in a way that there is little influence of their individaul weights on the strut. This elaves the bending motions to the force exerted by the wights on the spring, where in this case, weights of 10n were used in an increamenting trend of 5n. this gave the resultant buckling strengnths, which were proportional to the modulus of the strut used and also varying with the amoungt of effert loaded. For instance, the ability of ataining elasticity to gain the transition of the strut from the plastic stage to the elastic state was influenced by an increase in the amount of applied force resulting from an increase in the weights of the loads. The consideration of Zero exentricity involved the application of the knive edge brackets that were housed in the groove. On the convese, the maximum accentricity was obtained through removing the housing in the grooves of the edge brackets for the knives (Megson, 2005).
Different compression members habour different mudulus of elaciticities. This implies that the Euler bucking load for these members differs at varied eccentricities. For instance, varying the weights as exteranl forces increases the extent of deflection, while a plot of the graph of the deflection verses the quotient of the deflection and the Euler bucling load gives a straight line. This curve is also essential in culculatng the initial curviture, which is in this case termed as the accentricity. The initial shape of the strut is defined by the product of the maximum cenrtala mplitude and the sine of the quotient of the product of both pie and the distance of dispklacement along the x axis with the length of the strut. Varaitions of the strut due to increase in the external forces of influence acrued from an increase in the load of the weights gives the initial final position of the strut defined by the central amplitude. This central amplitude is vital in the culculation of the euler buckling load by use of the gradient of the graph of the central amplitude verses the quotient of the curviture and the displacement along the x axis (Megson, 2005).
On the other hand, the accentricity of the strut is culculated from the additional central deflection obtained by increasing the load of the sources of force or stress to the strut. The theoritical value of the Euler Buckling strength is culculated by the fomular: F = (pi^2)*(E)*(I)/ (L^2), where F is the Euler’s buckling load, I is the cross-sectional moment of inertia while E is the Young’s modulus of elasticity for the strut. By consideration, the experiment involved the use of a strut of initial curvature position represented by Y=yo/ (1-p/pE) where Y is the deflecting shape of the strut, yo is the distance of curvature in the initial position, while p is the vertical displacement accrued from the boundary conditions. This formula is a viable tool for the acquaintance of the initial eccentricity or the zero eccentricity in this case. Consequently, the maximum curvature is represented by the equation Y*= ao/(1-p/pE), where Y* is the maximum deflection reached by the highest weights accrued from increased loads. This formula is vital in synthesizing the maximum eccentricity.
The essence of increasing the weights in terms of loads results in the realization of the effect of buckling, which is the critical amount of force that the strut can withhold to acquire a transition from the plastic to the elastic mode of formation. A plot for the graph of the eigenvalues verses the weights gives a straight line, where the gradient of the line is suitable in the acquisition of the Euler’s buckling load. This is also important in the determination of the initial and final position of curvature since the y intercept for the graphs gives both the initial and final positions of the curvature (Ghali, 2003).
By consideration, from the formula: F = (pi^2)*(E)*(I)/ (L^2) using the initial curvature and replacing the values gives the Euler’s buckling load of 136.903. This value is obtainable through using 600mm as the length of the struts with a cross sectional area of 25.4 by 1.6mm. Having the values for E as 205kN/mm2, shows that the modulus of the elasticity for the strut is constant. This implies that in the theoretical value calculation of the Euler’s buckling load for the strut, the modulus of elasticity is constant. This implies that Po= (3.142*205kN/mm2*25.4mm*1.6mm)/600mm), which is equal to 136.9038kips. The value shows a slight deviation from the conventional graphical analysis, which gives 135.089kips as the gradient of the slope correlating to the graphical representation of the Euler’s buckling load. The differences emanate from the fact that the measurements for graphical analysis vary with the net weights of he load, which might not be linear (Ghali, 2003).
Moreover, the differences might have been as a result of lack of uniformity in the application of the forces emanating from the bending curvature. This results in unequal subjection of forces that are geared towards producing the buckling effects. On the other hand, the plot gives the initial curvature as being 1.067, which is acquired from the Y intercept of the graph. This implies that the members of he same group portray similar characteristics in such a way that there is a prior acquaintance to the tensile stress amid constrains of the curvature effects from the load. This implies that the members of he same elements always contain some form of curvature even without the application of an external force of curvature. This is also the value for the eccentricity of the strut at the zero eccentricity, where the Y intercept represents the initial curvature (Milliams, 2005).
The above graph shows that there is a form of force that the structural members made up of different elements exhibit some tensional force that results in the plastic characteristics. These characteristics transit upon exertion of an external force to give rise to the transition of the elements into the elastic nature. This elastic nature is what defines the curvature, which is obtainable through calculation of the Euler’s Buckling strength. This also shows that materials of he same cross-sectional area and same modulus of elasticity harbor the same values of the Euler’s buckling load at zero eccentricity. Moreover, the values for the external force need to be uniform, where it is necessary to use the axial position for the struts to depict the linearity in terms of the external force applied (Ghali, 2003).
This type of eccentricity involves the use of the final position of curvature of the strut, which is elucidated by the bending motion due to the external force. Once the load is placed on to the spring, there is exertion of the external force that culminates in the change in nature of the strut from the plastic nature to the elastic nature. This gives rise to the bending motion, where the plot of Y* verses Y*/pE gives a straight line. The slope of the line gives the Euler’s buckling load, while the Y intercept of the graph gives the final curvature position. This is the amount of force required for the strut to gain the transition from the plastic nature to the elastic nature, which varies with the material making up the strut. For instance, the use of the strut with a modulus of elasticity of 205kN/mm2 for a rod of cross-sectional area of 25.4 by 1.6mm gives a lead into the calculation of the Euler’s buckling load.
This is from the fact that from the formula F = (pi^2)*(E)*(I)/ (L^2), the resultant Euler’s buckling load is equal to 135.88 kips at the maximum eccentricity. By consideration, the value is obtainable from the synthesis that leads to the calculation: (3.1422*250kN/mm2*25.4mm*1.6mm)/600mm. this gives 136.903 kips as the theoretical value for the Euler buckling load. This is the value Euler’s buckling load for the strut at the maximum eccentricity, which corresponds with the value obtained from the calculation from zero eccentricity. The differences in the values obtainable lies in the graphical analysis, where the value obtained in this case as synthesized from the slope of the straight line obtained from the plot of the final curvature with the quotient of the final curvature and the eccentric force (Wiliams, 2000).
From the graphical analysis, it is evident that the Y intercept, which represents the final curviture lies at 0.60. this value is higher than the initial value obtained from the experiemntation with zero eccentricity. It is evident that the values differ in terms of the accentricity, which also provides a aplatform for varied Euler’s buckling load. The nature of he graphs differ in terms of the gradient, where the graphical representation for the zero accentricity is more pronounced than that for the maximum accentricity. This is due to the increase in the load of force, where the result is an increase in the force of applkciation. As the force of stretch increases, it leads to the increase in the extent of stretch. Morever, this results into the reduction in the distance of curviture in the resulting PE values (Megson, 2005).
It is also evident that there is a slight diviation in the values of the Euler buckling constant obtained from the synthesis of the graphical analysis for both the graphical analysis of the Zero eccentricity and the maximum eccentricity with that of the tehoritical value. The value for the tehoritical analysis is slightly grater than the graphical analysis in both cases. This is due to the clarity in the theoritical value, which is determined directly from the values for the buckling constant. The devaitions are as a result of the measurements for the weights, which does not give a linear observation in relation to the extent of stretch caused by the force. In addition, the element of the force of gravity, which in this case is not taken into consideration results in the final influence of the resultant datum (Megson, 2005).
The datum collected from the analysis is a vaible tool for the reflection of both the accentricity of the strut and the Euler’s buckling load, where the varaitions in the laods is in a sound correlation to the influence of stretch. For instance, the initiatl curviture for the strut is due to the normal situational mandate of structural elements belonging to the same group. The natural behaviour of these members shows that they exhibit a form of stretch at an intitial position. This force could be accelerated through increasing the external force. In this case, the exteranl force was increased through an increase in the weights of the laods. This resulted into the increse in the amount of stretch culminating in an increase in the extent of curviture. This also confirms the linearity in the resultant datum with the theoritical basis for the experiement (Milliams, 2005).
The essence of use of standardised loads of known weights takes care of he influence of the gravitational pull, where the mass is defined in terms of the force of gravity acting on the load. On the other hand, stadnardisation of the results was achieved trhough ensuring that the supporting beams do not exert an external force on the spring balance. This was through ensuring that the parts of the supporting beams are not part of the experiemntal analysis (Megson, 2005).
The obtained value for accentricity, which is the Y intercept of the graph of curviture vuses the quotient of the curviture with the displacement as a result of the stretch is 0.17 for the zero eccentricity experiemnt and 0.6 for the maximum eccentricity. On the other hand, the value for the Euler’s accentricity forgraphical analysis is obtainable as 135.88 kips, which is slightly lower than the theoritical value for the Euler’s ccentricity that is 136.9038. these values differ because of the influence of measure of the weights in relation to the gravitational pull. The results obtained from experimental analysis are linear with the fact that the extent of curviture is proportional with the force of stretch, while the force increses with the increse in the mass of he load. This leads to a straight line graph that depicts the initial curviture vurses the quotient of the initial curviture and the extent of stretch, whose gradient gives the Euler’s buckling load while the Y intercept forms the eccentricity.
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