Buckling of Struts - Math Problem Example

Buckling of Struts Author: Course Title: Name of your Institution: [Instructor Name] Date:01/10/2015 Abstract The main purpose was to examine the relationship between the buckling load and strut length for pin –pin -end strut and tinned and pinned -fixed end strut. Various experiment where carried out and the result was tabulated. Graphs have been drawn showing a linear relationship between buckling load verses 1/L2. The cross section area of struts was kept constant. The result for Euler buckling load as calculated has been noted to be affected by slenderness ratio and second moment of area. It is concluded that the effective lengths of the struts will be different from the actual length and it is the effective length that established the buckling load. Table of Contents Abstract 2 Introduction 4 Experimental Procedure 4 Results and Presentation 4 The relationship between buckling load Vs 1/L2 5 Theoretical buckling load 8 Discussion 11 Conclusions 12 References 13 Introduction In most instances struts rupture because of the stress exacted on them and can buckle as result of failure of material’s elastic module. Buckling takes place when struts stiffness becomes zero and can no longer carry any load. This can be summarized as a strut becomes unstable due to load increase leading elasticity collapse (Hulse and Coin, 2000). The main purpose of this report to provide details of an experiment that has been carried and aimed at examining the relationship between elastic deformation by buckling of struts with a range of end conditions and eccentricities of loading. The relationship between shear force and point load will be examined (Wright and McGregor, 2011). Experimental Procedure 1. Log on to the computer and start up to equipment structure 2. Make sure the RHS chucks is loose 3. Connect the software to STR2000 (Run>connect) and set it on a variable force; 4. Click on ‘calibrate’ button and follow the instruction on the screen (use the thumbwheel); 5. Bring back to zero angle 6. Make sure both load and angle on the software are zero, and the pointer is at zero as well; 7. Tighten up the chuck; Results and Presentation The experiment has been carried out and the recordings from the experiment was recorded and as follows Table 1: Pinned-End Strut - Experimental Values Strut length(m) Buckling load(N) Section Shape 2nd Moment of area(x10-9 m4) Material GPa 0.320 -92 Flat section 0.013 69.0 0.420 -52 Flat section 0.013 69.0 0.520 -35 Flat section 0.013 69.0 Table 2: Pinned-Fixed Ends - Experimental Values Strut length(m) Buckling load(N) Section Shape 2nd Moment of area(x10-9 m4) Material(GPa) 0.400 -110 Flat section 0.013 69.0 0.300 -198 Flat section 0.013 69.0 0.500 -66 Flat section 0.013 69.0 The relationship between buckling load Vs 1/L2 In this case we begin by calculating values for 1/L2 and a graph is made to determine the relationship between buckling load verses 1/L2. The calculations and the graphs are shown below; Table 3: Pinned-End Strut - Experimental Values with 1/L2 Strut length(m) Buckling load(N) Section Shape 2nd Moment of area(x10-9 m4) Material GPa 1/L2 0.320 -92 Flat section 0.013 69.0 9.77 0.420 -52 Flat section 0.013 69.0 5.67 0.520 -35 Flat section 0.013 69.0 3.70 The graph for buckling load verses 1/L2 is plotted below Figure 1: Buckling load versus 1/L2 for Pinned-End Strut The graph above shows that there is a leaner linear relationship between buckling load and 1/L2. The gradient for the graph above is (92-35)/(9.77-3.7)= 9.4 The gradient is 9.4 this will enable as establish ratio between each end conditions The slenderness ratio will be calculated using the following formula Slenderness ratio = Let us assume the radius is 0.0006 Table 4: Pinned-End Strut – calculated slenderness ratio Strut length(m) L/R 0.320 533.3 0.420 700 0.520 866.7 Table 5: Pinned-Fixed Ends - Experimental Values with 1/L2 Strut length(m) Buckling load(N) Section Shape 2nd Moment of area(x10-9 m4) Material(GPa) 1/L2 0.400 -110 Flat section 0.013 69.0 6.25 0.300 -198 Flat section 0.013 69.0 11.11 0.500 -66 Flat section 0.013 69.0 4.00 The graph for the relationship between buckling load verses 1/L2 is plotted below for Pinned-Fixed Ends Figure 2: Buckling load versus 1/L2 for Pinned-Fixed Ends The graph above shows that there is a leaner linear relationship between buckling load and 1/L2. The gradient for the graph above is (110-66)/(6.25-4.0)= 19.5 The gradient is 19.5 this will enable as establish ratio between each end conditions. The slenderness ratio will be calculated using the following formula Slenderness ratio = Let us assume the radius is 0.0006 Table 6: Pinned-End Strut – L/R Strut length(m) L/R 0.30 500 0.40 666.7 0.50 833.3 From the result of the graphs above, there gradients and slender ratios it can be noted that pinned-Fixed Ends has higher gradient than pinned –end struts. Theoretical buckling load The axial force is suggested to cause deformation when the force exists work piece. Theoretical buckling load is calculated using the right formulae. This is Euler buckling load which is assumed to be theoretical and it is calculated using the following formulae Buckling Load (N) for pinned-pinned end condition Pe= And Buckling Load (N) for Pinned-Fixed Ends Pe= Where; Pe = Euler buckling load (N) E = Young’s Modulus (Nm-2) I = Second moment of area ( m4) L = length of strut ( m ) This formula is based on Euler-Bernoulli beam theory which considers slenderness ratio of a slender strut. This formula therefore works well in cases of elastic deformation of slender materials. This as been calculated and entered in the table below Buckling Load (N) for pinned-pinned end condition Pe= Pe= = 86.5N Pe= = 50.2 Pe= = 32.7 Pinned-End Strut - Experimental Values Strut length(m) Buckling load(N) 2nd Moment of area(x10-9 m4) Material GPa 1/L2 L/R Buckling Load (N) Theory 0.320 -92 0.013 69.0 9.77 533.3 86.5 0.420 -52 0.013 69.0 5.67 700 50.2 0.520 -35 0.013 69.0 3.70 866.7 32.7 The graph for the relationship comparing theoretical and measured buckling load and 1/L2 The gradient for the theoretical buckling load and strut length is calculated as Gradient for the theoretical buckling load and strut length= =8.86 Buckling Load (N) Theory (Pinned-Fixed Ends) Pe= Pe= = 110.7 Pe= = 196.7 Pe= = 70.8 Pinned-Fixed Ends - Experimental Values Strut length(m) Buckling load(N) 2nd Moment of area(x10-9 m4) Material(GPa) 1/L2 L/R Buckling Load (N) Theory 0.400 -110 0.013 69.0 6.25 500 110.7 0.300 -198 0.013 69.0 11.11 666.7 196.7 0.500 -66 0.013 69.0 4.00 833.3 70.8 The graph for the relationship comparing theoretical and measured buckling load and 1/L2 Pinned -pinned Pinned -fixed Gradient 9.45 19.5 Result Ratio 9.44/9.45 =1 19.5/18.5 =1.05 Theoretical ratio 8.86/8.86 =1 17.7/17.73 =1 From the calculated result ratio, theoretical ratio and gradient shown above it is clear that pinned –fixed strut is strong as compared to pinned to pinned strut as there ratio both theoretical and result is greater than 1. Discussion There is a difference between the theoretical value and the experiment value due to the measurement errors which may occur during the measuring. The combination of pinned and fixed is generally used during the having strong output struts therefore characterizing a certain mechanical condition (McCormac and Csernak, 2011). . However; the buckling of struts depends on the great number of parameters. Among those parameters on may include structures, Young’s Modulus (Nm-2), existence of matrix-rich and matrix-poor region, fiber bridging, specimen lay-up, loading rate, environmental, time and temperature to study the effect on materials buckling (Das, 2010). Conclusions From the result of the experiment for buckling of strut, it was noted that pinned –fixed –end had higher theoretical and experimental ratio value. This meant that it is stronger and requires a large force to be used for it to buckle. This means that it is important to have fixed end strut for structures which are likely to carry heavy materials. References Courtney, T.H. (1990). Mechanical Behavior of Materials, McGraw-Hill, New York, 1990. Das, B. (2010). Principles of Foundation Engineering. New York: CL Engineering Hulse, R, & coin, J. (2000). Structural mechanics (2nd red). New York: Palgrave Macmillan. Türkmen, D. (1995). Experimental Investigation of the Phenomenon of Buckling For Steel And Aluminium Struts. Journal of Engineering Sciences 1995 McCormac, J. & Csernak, S. (2011). Structural Steel Design. New York: Prentice Hall Wright, J. & McGregor, J. (2011). Reinforced Concrete: Mechanics and Design. New York: Prentice Hall Read More