## Shear Force and Bending Moment

EXPERIMENT 2 Title : Shear Force and Bending Moment Objective : To determine the shear force and bending moment when concentrated load, symmetrical load and non symmetrical load are applied Introduction The shear force (F) in a beam at any section, X, is the force transverse to the beam tending cause it to shear across the section. The shear force at any section is taken as positive if the right-hand side tends to slide downwards relative to the left hand portion. The negative force tends to cause the right hand portion to slide upwards relative to the left. X W F

Shear force F = Load W but in opposite directions W The bending effect at any section X of a concentrated load W is measured by the applied moment Wx, where x is the perpendicular distance of the line of action of W from section X. This moment is called the bending moment M. x X M = Wx The bending moment is balanced by an equal and opposite moment exerted by the material of the beam at X, called the moment of resistance. The bending moment is positive if its effect makes the beam to sag at the section considered. If the moment tends to make the beam bend upward or hog at the section, it is negative.

For any value of x, the relationship between load W and shearing force F is :- W = dF / dx And the relationship between shearing force and bending moment M is :- F = dM / dx Apparatus and Materials 1. Shear forces apparatus :- 1 set of 80mm x 50mm x 38mm aluminium section with 2 adjustable span support. 2. 1 unit of shear force dynamometer. 3. 2 sets of weight hangers. 4. 1 set of weights. 5. 2m measuring tape. Procedure Shear force and bending moment experiment for concentrated load. 1. The 2 edge supports is set up on the base of the structural test frame at a distance of 800mm from edge to edge. . The shear force apparatus is placed on the supports. 3. The beam is horizontally aligned by adjusting the screws. 4. The weight hanger is placed in the centre of the beam (400mm from the support). 5. The screws is adjusted to repeat horizontal beam alignment. 6. The shear force and bending moment dynamometers is zeroed. 7. The weights is placed as given in table and the shear force and bending is noted. 8. When placing each weight the horizontal beam alignment was carried out and the dynamometers were zeroed. 9. The percentage error for each set of reading is calculated.

Shear force and bending moment experiment for symmetrical load 10. The 2 edge supports is set up on the base of the structural test frame at a distance of 800mm from the edge to edge. 11. The shear force apparatus is placed on the support. 12. The beam is horizontally aligned by adjusting the screws. 13. The weight hangers is placed at a distance of 100mm from the supports. 14. The screws is adjusted to repeat horizontal beam alignment. 15. The shear force and bending moment dynamometers is zeroed. 16. The weights is placed as given in the table and the shear force and bending moment values is noted. 7. When placing each weight the horizontal beam alignment was carried out and the dynamometers were zeroed. 18. The percentage error for each set of reading is calculated. Results : Case 1- Load at Midspan| | | | | | | | | | | Span L, =| 800. 0| mm| | | | | | | | | | | | 0. 800| m | | | | | | | | | | | Lever arm =| 0. 180| m| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | M| | | | | | | | | | | | | | | | | | | | | | | | | V| | | | | | | | | | | Reaction, R | | | | | | | | | | | | | | x| 240| mm| | | | | | | | | | | | | | | | | Experimental Value| Error (%)| | | Experimental Value| | Theoretical Value| V| M| V| M| Load (g)| Load (N)| R (N)| Shear Force, V (g)| Bending Moment (g)| Shear Force, V (N)| Bending Moment (Nm)| (N)| (Nm)| | | 50| 0. 491| 0. 245| 25. 0| 34. 0| 0. 245| 0. 059| 0. 245| 0. 060| 0. 00| 2. 00| 100| 0. 981| 0. 491| 46. 5| 63. 0| 0. 491| 0. 118| 0. 456| 0. 111| 7. 00| 5. 50| 150| 1. 472| 0. 736| 66. 0| 94. 0| 0. 736| 0. 177| 0. 647| 0. 166| 12. 00| 6. 00| 200| 1. 962| 0. 981| 87. 0| 125. 0| 0. 981| 0. 235| 0. 853| 0. 221| 13. 00| 6. 25| 250| 2. 453| 1. 226| 110. 0| 151. 0| 1. 226| 0. 294| 1. 079| 0. 267| 12. 00| 9. 40| 300| 2. 943| 1. 472| 130. | 184. 0| 1. 472| 0. 353| 1. 275| 0. 325| 13. 33| 8. 00| 350| 3. 434| 1. 717| 164. 0| 219. 0| 1. 717| 0. 412| 1. 609| 0. 387| 6. 29| 6. 14| 400| 3. 924| 1. 962| 168. 5| 248. 0| 1. 962| 0. 471| 1. 653| 0. 438| 15. 75| 7. 00| Case 2 Symmetrical Load| | | | | | | P| | | | | | | | Distance ,a=| 100. 0| mm| | | | | | M| | | | | | | | | | V| | | | | | | | | | | | | R| | | | | | | | | | | Experimental Value| Error (%)| | | | Experimental Value| Theoretical Value| V| M| V| M| Load (g)| Load (N)| R (N)| Shear Force, V (g)| Bending Moment (g)| Shear Force, V (N)| Bending Moment (Nm)| (N)| (Nm)| | | 50| 0. 491| 0. 491| 3. | 30. 0| 0. 000| 0. 049| 0. 034| 0. 053| #DIV/0! | 8. 00| 100| 0. 981| 0. 981| -11. 5| 56. 0| 0. 000| 0. 098| -0. 113| 0. 099| #DIV/0! | 0. 80| 150| 1. 472| 1. 472| -14. 0| 86. 0| 0. 000| 0. 147| -0. 137| 0. 152| #DIV/0! | 3. 20| 200| 1. 962| 1. 962| 5. 0| 111. 0| 0. 000| 0. 196| 0. 049| 0. 196| #DIV/0! | 0. 10| 250| 2. 453| 2. 453| -1. 0| 134. 0| 0. 000| 0. 245| -0. 010| 0. 237| #DIV/0! | 3. 52| 300| 2. 943| 2. 943| -4. 0| 157. 0| 0. 000| 0. 294| -0. 039| 0. 277| #DIV/0! | 5. 80| 350| 3. 434| 3. 434| -3. 5| 188. 0| 0. 000| 0. 343| -0. 034| 0. 332| #DIV/0! | 3. 31| 400| 3. 924| 3. 924| 5. 5| 190. 0| 0. 000| 0. 92| 0. 054| 0. 336| #DIV/0! | 14. 50| Case 3 Non Symmetrical Load| (If a > x)| | | | | | | | | | | M| | | V| | | | | | | | | | R| | | | | | | | | | | | | | A distance from right support| 100| mm| | | | | | | | Experimental Value| Error (%)| | | | Experimental Value| Theoretical Value| V| M| V| M| Load (g)| Load (N)| R (N)| Shear Force, V (g)| Bending Moment (g)| Shear Force, V (N)| Bending Moment (Nm)| (N)| (N)| | | 50| 0. 491| 0. 061| 5. 0| 7. 0| 0. 061| 0. 015| 0. 049| 0. 012| 20. 00| 16. 00| 100| 0. 981| 0. 123| 7. 0| 14. 0| 0. 123| 0. 029| 0. 069| 0. 25| 44. 00| 16. 00| 150| 1. 472| 0. 184| 16. 0| 24. 0| 0. 184| 0. 044| 0. 157| 0. 042| 14. 67| 4. 00| 200| 1. 962| 0. 245| 22. 0| 32. 0| 0. 245| 0. 059| 0. 216| 0. 057| 12. 00| 4. 00| 250| 2. 453| 0. 307| 25. 0| 39. 0| 0. 307| 0. 074| 0. 245| 0. 069| 20. 00| 6. 40| 300| 2. 943| 0. 368| 36. 5| 50. 0| 0. 368| 0. 088| 0. 358| 0. 088| 2. 67| 0. 00| 350| 3. 434| 0. 429| 36. 5| 55. 0| 0. 429| 0. 103| 0. 358| 0. 097| 16. 57| 5. 71| 400| 3. 924| 0. 491| 42. 0| 64. 0| 0. 491| 0. 118| 0. 412| 0. 113| 16. 00| 4. 00| Calculation : CASE 1 : Load at Midspan Load (g) = 400 g = 0. 4kg Span, L = 800mm = 0. 8m Lever arm = 0. 18m = 9. 81m/s2 Load (N) = mg = 0. 4*9. 81 = 3. 924N Experimental value of shear force, V (N) = shear force, V (g) * g = (168. 5*9. 81)/1000 =1. 653N Experimental value of bending moment (Nm) = [bending moment, M(g)*g*level arm]/1000 = (248*9. 81*0. 18)/1000 = 0. 438Nm Theoretical value of shear force, V (N) = [Load (g) * g] / (2*1000) = (400*9. 81)/(2000) = 1. 962N Theoretical value of bending moment (Nm) = [Theoretical shear force (N)*x(mm)] / 1000 = (1. 962*240)/1000 = 0. 471Nm X 100% Percentage error = |Experimental value – theoretical value| Theoretical value

X 100% Percentage error of shear force, V = |1. 653 – 1. 962| 1. 962 = 15. 75% X 100% Percentage error of bending moment, M = |0. 438 – 0. 471| 0. 471 = 7% CASE 2 : Symmetrical Load Load = 400g = 0. 4kg Span = 800mm = 0. 8m Lever Arm = 0. 18m g = 9. 81m/s2 a = 100mm = 0. 1m R(N) = (400*9. 81)/1000 = 3. 924N Experimental value of shear force, V (N) = [Shear force, V (g) * g] / 1000 = (5. 5*9. 81)/1000 = 0. 054N Experimental value of bending moment, M (Nm) = [bending moment (g) *g*lever arm] / 1000 = (190*9. 81*0. 18)/1000 = 0. 336Nm Theoretical value of shear force, V (N) R (N) – Load (N) = 3. 924-3. 924 = 0N Theoretical value of bending moment, M (Nm) = R(N) * a = 3. 924*0. 1 = 0. 392Nm X 100% Percentage error of shear force, V (N) = |0. 054 – 0| 0 = undefined X 100% Percentage error of bending moment , M (Nm) = |0. 336 – 0. 392| 0. 392 = 14. 29% CASE 3: Non Symmetrical Load (If a>x) Experimental value of shear force, V(N) = [shear force, V(g) * g] / 1000 = (42*9. 81)/1000 = 0. 412 N Experimental value of bending moment, M (Nm) = [bending moment, M (Nm) * g * lever arm] / 1000 = (64*9. 1*0. 18)/1000 = 0. 113Nm Theoretical value of shear force, V(N) = R(N) = (3. 924*0. 1)/(0. 8) = 0. 491 N Theoretical value of bending moment, M (Nm) = [R(N)*x(mm)]/1000 = (0. 491*240)/1000 = 0. 118 Nm X 100% Percentage error of shear force, V (N) = |0. 412 – 0. 491| 0. 491 = 16. 08% X 100% Percentage error of bending moment , M (Nm) = |0. 113 – 0. 118| 0. 118 = 4. 24% Discussion A beam is a horizontal structural member that is designed to support the applied load, which is perpendicular to the beam. A beam will resists the pplied loading by a combination of internal transverse shear force and bending moment. Shear force is the internal force within the beam. It is caused by any external force acting perpendicular to the beam or force with component tangent to the beam. The shearing force is developed when any part of an beam tends to slide laterally relative to another part of the beam. Bending moment is the tendency of the beam to bend when a load is applied to it or the tendency of the beam to resists the bending. Bending moment is developed whenever there is a force acting perpendicular to the beam. Load 400 mm 400 mm

Throughout this experiment, we will use simply support system to support the beam. The distance between the supports is called the span. For case 1 in the experiment, we place the load at the centre of the beam. In this case, the load is said to be concentrated at a point of the beam, which is at the centre. Therefore, the bending is at the centre of the beam. Based on the table, we can see that for 400g of load, shear force and bending moment we found by the experiment were 1. 653 N and 0. 438 Nm respectively. While the theoretical value that is calculated by using formula were 1. 962 N for shear force and 0. 71 Nm for bending moment. Therefore, we have a percentage error of 15. 75% for shear force and 7% for bending moment. For case 2, two loads were hang on the beam, each with 100mm from one of the support ends. This makes the load were uniformly distributed along the beam. The experimental value for shear force and bending moment were 0. 054 N and 0. 336 Nm respectively. The percentage error calculated based on the theoretical value for bending moment of 0. 392Nm is 14. 29%. The percentage error of shear force for shear force is undefined because of the theoretical value of 0 N. Load 100mm

For Case 3, a load is randomly hang at a point of the beam, which make the load varying distributed. The results we get from the experiment for shear force and bending moment were 0. 412 N and 0. 113 Nm respectively. Therefore, by using the theoretical value of 0. 491 N for shear force, we found that the percentage error was 16. 08%. While the theoretical value of bending moment (0. 118 Nm) gives error of 4. 24%. We can see that there is some error that occurs in the results we taken. There are a few of circumstances that cause the results to be inaccurate, one of it was the flow of air in the laboratory, in other word, wind.

The wind will causes the load hanging on the beam to be unstable, this will cause extra force or removal of parts of the force acting on the beam from the load. Therefore, we should try to avoid wind while the experiment is carrying on. On the other hand, we must always ensure that the load is hung on the right place it should be when the experiment is carrying on. For example, in case 1, we must ensure that the load is exactly at the centre of the beam. We must also always make sure that the beam is horizontally aligned as mentioned in the procedure, and do not forget to set the dynamometer to zero when every load is hung.

Besides that, the vibration of the table and instrument can also cause the results to be inaccurate. Therefore after we applied the load, we must wait for a few moment to let the vibration stop before taking the reading. Other than that, we must also not try to knock or hit the table that the shear force apparatus were located to prevent vibration. Conclusion The experimental value of 400g load for case 1 is 1. 653 N for shear force and 0. 438 Nm for bending moment. The percentage error calculated based on the theoretical value were 15. 75% and 7% respectively The experimental value for case 2 of 400g loads is 0. 054 N for shear force and 0. 36 Nm for bending moment. The percentage error for bending moment is 14. 29% calculated based on the theoretical value. The experimental value for case 3 of 400g load is 0. 412 N for shear force and 0. 113 Nm for bending moment. The percentage error calculated based on the theoretical value are 16. 08% and 4. 24% respectively. Reference http://wiki. answers. com/Q/What_is_shear_force#ixzz1TOhdAuLM http://www. civilcraftstructures. com/civil-subjects/shear-force-and-bending-moment-as-structural-basics/ http://www. wikipedia. com http://zephyr. cit. act. edu. au/toolboxes/buildright/content/bcgbc4010a/04_struct_members/01_beams/page_006. htm