Structural Analysis II Structural Analysis Trigonometry Concepts Vectors

Structural Analysis II • • Structural Analysis Trigonometry Concepts Vectors Equilibrium Reactions Static Determinancy and Stability Free Body Diagrams Calculating Bridge Member Forces

Learning Objectives • Generate a free body diagram • Calculate internal member forces using the Method of Joints

Free Body Diagram • Key to structural analysis 1) Draw a simple sketch of the isolated structure, dimensions, angles and x-y coordinate system 2) Draw and label all loads on the structure 3) Draw and label reactions at each support

Structural Analysis Problem • Calculate the internal member forces on this nutcracker truss if the finger is pushing down with a force of eight newtons.

Structural Analysis Solution Draw the Free Body Diagram Step 1: Draw simple sketch with dimensions, angles, and x-y coordinate system c y x 70 o a Nutcracker truss formed by tied ends m 12 c 40 o 70 o b Corresponding sketch

Structural Analysis Solution Draw the Free Body Diagram Step 2: Draw and label all loads on the structure 8 N y c x a Nutcracker truss with 8 N load m 12 c 40 o 70 o b Added to free body diagram

Structural Analysis Solution Draw the Free Body Diagram Step 3: Draw and label all reactions at each support • The truss is in equilibrium so there must reactions at the two supports. They are named Ra and Rb. 8 N y c x m 12 c 40 o 70 o a Ra 70 o b Rb

Structural Analysis Solution Method of Joints • Use the Method of Joints to calculate the internal member forces of the truss 1. Isolate one joint from the truss 2. Draw a free body diagram of this joint 3. Separate every force and reaction into x and y components 4. Solve the equilibrium equations 5. Repeat for all joints

Structural Analysis Solution Method of Joints y 8 N x c 40 o a Step 1: Isolate one joint Step 2: Draw the free body diagram 70 o Ra = 4 N 1 70 o y 2 cm b Rb = 4 N Fac a c 70 o Fab b Ra = 4 N x

Structural Analysis Solution Method of Joints Step 3: Separate every force and reaction into x and y components y First analyse Ra • x-component = 0 N • y-component = 4 N a x Ra = 4 N

Structural Analysis Solution Method of Joints Step 3: Separate every force and reaction into x and y components y Next analyse Fab • x-component = Fab • y-component = 0 N a Fab b x

Structural Analysis Solution Method of Joints Step 3: Separate every force and reaction into x and y components y Lastly, analyse Fac • x-component = Fac*cos 70˚ N • y-component = Fac*sin 70˚ N Fac a c 70 o x

Summary of Force Components, Node ‘a’ Force Name Free Body Diagram x- component y-component Ra y a Fab Fac y x Ra = 4 N 0 N 4 N x Fab 0 N Fac a 70 o x Fac * cos 70˚ N Fac * sin 70˚ N

Structural Analysis Solution Method of Joints Step 4: Solve y-axis equilibrium equations • The bridge is not moving, so ΣFy = 0 • From the table, ΣFy = 4 N + Fac * cos 70˚ = 0 • Fac = ( -4 N / cos 70˚ ) = -4. 26 N • Internal Fac has magnitude 4. 26 N in compression

Structural Analysis Solution Method of Joints Step 4: Solve x-axis equilibrium equations • The bridge is not moving, so ΣFx = 0 • From the table, ΣFx = Fab + Fac * sin 70˚ = 0 Fab = - ( -4. 26 N / sin 70˚ ) = 1. 45 N • Internal Fab has magnitude 1. 45 N in tension

Structural Analysis Solution Method of Joints Tabulated Force Solutions Member Force Magnitude AB 4. 26 N, compression BC 1. 45 N, tension AC (not yet calculated)

Structural Analysis Solution Method of Joints Step 5: Repeat for other joints Step 1: Isolate one joint Step 2: Draw the free body diagram y 8 N x y 8 N c m 12 c 40 o 70 o a 70 o c Fac = -4. 26 N 40 o b a Ra Fbc x Rb b

Structural Analysis Solution Method of Joints Step 3: Separate every force and reaction into x and y components y First analyse Rc • y-component is -8 N • x-component is 0 N 8 N c x

Structural Analysis Solution Method of Joints Step 3: Separate every force and reaction into x and y components y Next analyse Fac • x-component is –(Fac * sin 20˚) = - (-4. 26 N * 0. 34) = 1. 46 N c Fac 20 o • y-component is –(Fac * cos 20˚) = - (-4. 26 N * 0. 94) = 4. 00 N a x

Structural Analysis Solution Method of Joints Step 3: Separate every force and reaction into x and y components y Lastly analyse Fbc • y-component = –(Fbc * cos 20˚) c x Fbc • x-component = (Fbc * sin 20˚) 20 o b

Summary of Force Components, Node ‘c’ Force Name Free Body Diagram Rc 8 N c Fac Fbc c c Fac 20 o a x- component y-component 0. 00 N -8. 00 N 1. 46 N 4. 00 N 20 o Fbc b Fbc * sin 20˚ N -Fbc * cos 20˚ N

Structural Analysis Solution Method of Joints Step 4: Solve y-axis equilibrium equations • The bridge is not moving, so ΣFy = 0 • From the table, ΣFy = -8. 00 N + 4. 00 N - Fbc * cos 20˚ = 0 Fbc = -4. 26 N • Internal Fbc has magnitude 4. 26 N in compression

Structural Analysis Solution Method of Joints Step 4: Solve x-axis equilibrium equations • The bridge is not moving, so ΣFx = 0 • From the table, ΣFx = 1. 46 N + Fbc * sin 20˚ = 0 Fbc = -4. 26 N • This verifies the ΣFy = 0 equilibrium equation and also the symmetry property

Structural Analysis Solution Method of Joints Tabulated Force Solutions Member Force Magnitude AB 4. 26 N, compression BC 1. 45 N, tension AC 4. 26 N, compression

Acknowledgements • This presentation is based on Learning Activity #3, Analyze and Evaluate a Truss from the book by Colonel Stephen J. Ressler, P. E. , Ph. D. , Designing and Building File-Folder Bridges