Stresses Due to Point Load 3 P P z = z 2 R 2 R - PDF document

Stresses Due to Point Load ⎡ ⎤ 3 P P z σ = ⎢ ⎥ z π 2 R 2 ⎣ R 3 ⎦ ⎡ ⎤ 2 P 3r z ( ) R σ = − − υ ⎢ 1 2 ⎥ π + r 2 3 2 R ⎣ R R z ⎦ z R ⎡ ⎤ P ( ) R z σ = − υ − 1 2 ⎢ ⎥ t π + 2 ⎣ ⎦ 2 R R z R r Wednesday, Sept. 5 1 CIVL 7166 Stresses Due to Circular Load ⎡ ⎤ 3 ⎛ ⎞ z ⎢ ⎥ 2a σ = − ⎜ q 1 ⎟ z ⎢ ⎥ ⎝ + ⎠ a 2 z 2 ⎣ ⎦ q ⎡ ⎤ ( ) 3 + υ ⎛ ⎞ 2 1 z q z σ = ⎢ + υ − + ⎜ ⎥ 1 2 ⎟ r ⎢ + + ⎥ 2 2 2 ⎝ 2 2 ⎠ ⎣ a z a z ⎦ z ⎡ ( ) ⎤ + υ ⎛ ⎞ 3 q 2 1 z z σ = ⎢ + υ − + ⎜ ⎥ ⎟ 1 2 t ⎢ ⎥ 2 2 + 2 ⎝ 2 + 2 ⎠ a z a z ⎣ ⎦ Wednesday, Sept. 5 2 CIVL 7166 Hooke’s Law 1 ( ) ⎡ ⎤ ε = σ − ν σ + σ ⎣ ⎦ z z r t E 1 ( ) ε = ⎡ σ − ν σ + σ ⎤ ⎣ ⎦ r r z t E 1 ( ) ε = ⎡ σ − ν σ + σ ⎤ ⎣ ⎦ t t r z E Wednesday, Sept. 5 3 CIVL 7166 1

Strains Due to Circular Load ( ) ⎡ ⎤ 3 + υ υ ⎛ ⎞ q 1 2 z z ⎢ ⎥ ε = − υ + − ⎜ ⎟ 1 2 z ⎢ ⎥ E 2 + 2 ⎝ 2 + 2 ⎠ a z a z ⎣ ⎦ ( ) ⎡ ( ) ⎤ 3 + υ − υ ⎛ ⎞ q 1 2 1 z z ε = ⎢ − υ − + ⎜ ⎥ ⎟ 1 2 r ⎢ ⎥ 2E + ⎝ + ⎠ 2 2 2 2 ⎣ a z a z ⎦ ⎡ ⎤ ( ) ( ) + υ − υ ⎛ ⎞ 3 q 1 2 1 z z ε = ⎢ − υ − + ⎜ ⎥ 1 2 ⎟ t ⎢ ⎥ + + 2E 2 2 ⎝ 2 2 ⎠ a z a z ⎣ ⎦ Wednesday, Sept. 5 4 CIVL 7166 Deflections Due to Circular Load ( ) ( ) ∞ + υ ⎡ ⎤ − υ qa 1 a 1 2 ∫ = ε = + + − ⎢ 2 2 ⎥ w dz a z z z + E ⎣ 2 2 a ⎦ a z z ( ) − υ 2 2qa 1 = w o E Wednesday, Sept. 5 5 CIVL 7166 Rigid Loading 2a q qa σ = z − 2 2 2 a r Wednesday, Sept. 5 6 CIVL 7166 2

Rigid vs. Flexible Loading Flexible Plate Flexible Pl ate Rigid Plate Rigid Plate ( ) ( ) − υ π − υ 2 2 2qa 1 qa 1 = = w w o o E 2E ( ) − υ 2 qa 1 2 π flexible rigid w 4 w E = = = ⇒ = = o o 1.27 0.79 ( ) π rigid flexible w − υ w 4 2 π qa 1 o o 2 E Wednesday, Sept. 5 7 CIVL 7166 Multiple Wheel Loads L R z r L r R A ( ) ( ) σ = σ + σ L R r ,z r ,z A A L A R Wednesday, Sept. 5 8 CIVL 7166 Ahlvin and Ulery (1962) [ ] σ = + p A B z ( ) ⎡ ⎤ σ = υ + + − υ p 2 ⎣ 1 2 ⎦ = function A C F E r ( ) σ = ⎡ υ − + − υ ⎤ p 2 1 2 E = modulus ⎣ A D E ⎦ t ( ) + υ ⎡ ⎤ pa 1 z ( ) = + − υ w 1 ⎢ A H ⎥ o ⎣ ⎦ E a Wednesday, Sept. 5 9 CIVL 7166 3