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Ae1 19 20_1_beam with cantilvers
- 1. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example LET’S ANALYZE THIS BEAM WITH ONE SPAN AND TWO CANTILEVERS As it is not symmetrical, the vertical reaction forces at A and B are not going to be equal B C
- 2. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example B C B C TO SIMPLIFY THE PROBLEM WE CAN TRANSFORM THE CANTILEVERS IN END MOMENTS AT THE SUPPORTS: TO CALCULATE THE REACTION FORCES WE CAN EQUILIBRATE EACH FORCE AND MOMENT AND ADD THE RESULTS:
- 3. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example ONCE WE KNOW THE VERTICAL REACTION FORCES VALUES, WE CAN DRAW THE SHEAR FORCE DIAGRAM AND THE BENDING MOMENT DIAGRAM
- 4. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example TO CALCULATE THE DEFLECTION IN THE MIDDLE OF THE SPAN WE CAN APPLY THE FORMULAS FOR SIMPLE BEAMS B C
- 5. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE DISTRIBUTED LOAD
- 6. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam exampleEFFECT OF THE PUNCTUAL LOAD
- 7. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE MOMENT AT THE LEFT END
- 8. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE MOMENT AT THE RIGHT END
- 9. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example 𝑣 𝑞 = − 5 ⋅ 𝑞 ⋅ 𝐿4 384 ⋅ 𝐸𝐼 𝑣2𝑃 = − 2𝑃 ⋅ 𝐿3 48 ⋅ 𝐸𝐼 𝑣 𝑀𝑙𝑒𝑓𝑡 = 𝑀𝑙𝑒𝑓𝑡 ⋅ 𝐿2 16 ⋅ 𝐸𝐼 𝑣 𝑀𝑟𝑖𝑔ℎ𝑡 = 𝑀𝑟𝑖𝑔ℎ𝑡 ⋅ 𝐿2 16 ⋅ 𝐸𝐼 DEFLECTION IN THE MIDDLE OF THE SPAN V total = vq + vP + vMleft + vMright = -10,68 mm Beam bending stiffness: EI = 10000 kNm2
- 10. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example TO CALCULATE THE ROTATION AT B WE CAN APPLY AGAIN THE SIMPLE BEAM FORMULAS B C
- 11. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE DISTRIBUTED LOAD
- 12. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE PUNCTUAL LOAD
- 13. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE MOMENT AT THE LEFT END
- 14. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example EFFECT OF THE MOMENT AT THE RIGHT END
- 15. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example 𝑔𝐵𝑞 = − 𝑞 ⋅ 𝐿3 24 ⋅ 𝐸𝐼 𝑔𝐵2𝑃 = − 2𝑃 ⋅ 𝐿2 16 ⋅ 𝐸𝐼 𝑔𝐵 𝑀𝑙𝑒𝑓𝑡 = 𝑀𝑙𝑒𝑓𝑡 ⋅ 𝐿 3 ⋅ 𝐸𝐼 𝑔𝐵 𝑀𝑟𝑖𝑔ℎ𝑡 = 𝑀𝑟𝑖𝑔ℎ𝑡 ⋅ 𝐿 6 ⋅ 𝐸𝐼 B C gB total = gBq + gBP + gB Mleft + gB Mright = -4,275 mrad Beam bending stiffness: EI = 10000 kNm2
- 16. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example 𝑣𝐿𝑒𝑓𝑡 𝑞 = − 𝑞 ⋅ 𝐿4 8 ⋅ 𝐸𝐼 𝑣𝐿𝑒𝑓𝑡 𝑃 = − 𝑃 ⋅ 𝐿3 3 ⋅ 𝐸𝐼 𝑣𝐿𝑒𝑓𝑡 𝑔𝐵 = 𝑔𝐵 ⋅ 𝐿𝑙𝑒𝑓𝑡 TO CALCULATE THE VERTICAL DISPLACEMENT AT THE END OF THE CANTILEVER WE MUST CONSIDER THE EFFECT OF THE CANTILEVER PLUS THE EFFECT OF THE ROTATION AT B (already known) V total = vq + vP + gB x Lcantilever = -0,5 mm
- 17. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam exampleEFFECT OF THE MOMENT LOADS ALONG THE CANTILEVER
- 18. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example 𝑔𝐷 𝑞 = − 𝑞 ⋅ 𝐿3 24 ⋅ 𝐸𝐼 𝑔𝐷2𝑃 = − 2𝑃 ⋅ 𝐿2 16 ⋅ 𝐸𝐼 𝑔𝐷 𝑀𝑙𝑒𝑓𝑡 = 𝑀𝑙𝑒𝑓𝑡 ⋅ 𝐿 6 ⋅ 𝐸𝐼 𝑔𝐷 𝑀𝑟𝑖𝑔ℎ𝑡 = 𝑀𝑟𝑖𝑔ℎ𝑡 ⋅ 𝐿 3 ⋅ 𝐸𝐼 ROTATION AT D (ROLLER) WE REPEAT THE PROCCESS FOR THE RIGHT CANTILEVER
- 19. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example 𝑣𝑅𝑖𝑔ℎ𝑡 𝑞 = − 𝑞 ⋅ 𝐿4 8 ⋅ 𝐸𝐼 𝑣𝑅𝑖𝑔ℎ𝑡 𝑔𝐷 = 𝑔𝐵 ⋅ 𝐿𝑙𝑒𝑓𝑡 VERTICAL DISPLACEMENT AT THE END OF THE CANTILEVER WE REPEAT THE PROCCESS FOR THE RIGHT CANTILEVER V total = vq + vP + gD x Lcantilever = +9,5 mm
- 20. cperez.eps@ceu.es molina.eps@ceu.es STRUCTURAL ANALYSIS I DEGREE IN ARCHITECTURE Year 3 Term 1 simple beam example