Estruturas Isostáticas - Coordenadoria de Edificações - IFES - Campus VitóriaResolução da Lista Complementar de Exercícios - Esforços Internos
1.
Reações de Apoio
Supondo-se inicialmente as reações de apoio, Vertical ( ), Horizontal ( ) e Momento ( ), todas com sentido positivo:VA HA MA
≔α atan⎛⎜⎝―5
20
⎞⎟⎠
=α 14.04 deg
=0ΣFx
≔HA 0 tonf
=0ΣFy
=0−−VA ⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠⋅1 ――
tonf
m―――10 m
cos ⎛⎝α⎞⎠
≔VA +⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠⋅1 ――
tonf
m―――10 m
cos ⎛⎝α⎞⎠=VA 30.92 tonf
=0ΣMA
=0−−MA ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠5 m ⋅⋅1 ――
tonf
m―――10 m
cos ⎛⎝α⎞⎠15 m
≔MA +⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠5 m ⋅⋅1 ――
tonf
m―――10 m
cos ⎛⎝α⎞⎠15 m
=MA 257.69 ⋅tonf m
Esforços Internos
Esforços Internos
S1
≔N1 −⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
2 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N1 6.50 tonf
≔Q1 −⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
2 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q1 26.00 tonf
≔M1 −+−MA ⋅VA 2 m ⋅⋅2 ――tonf
m―――
2 m
cos ⎛⎝α⎞⎠1 m
=M1 −199.97 ⋅tonf m
S2
≔N2 −⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
5 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N2 5.00 tonf
≔Q2 −⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
5 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q2 20.00 tonf
≔M2 −+−MA ⋅VA 5 m ⋅⋅2 ――tonf
m―――
5 m
cos ⎛⎝α⎞⎠2.5 m
=M2 −128.85 ⋅tonf m
S3
≔N3 −⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
7 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N3 4.00 tonf
≔Q3 −⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
7 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q3 16.00 tonf
≔M2 −+−MA ⋅VA 7 m ⋅⋅2 ――tonf
m―――
7 m
cos ⎛⎝α⎞⎠3.5 m
=M2 −91.74 ⋅tonf m
S4
≔N4 −⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
8 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N4 3.50 tonf
≔Q4 −⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――
8 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q4 14.00 tonf
≔M4 −+−MA ⋅VA 8 m ⋅⋅2 ――tonf
m―――
8 m
cos ⎛⎝α⎞⎠4 m
=M4 −76.28 ⋅tonf m
S5
≔N5 −−⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠ ⋅⋅1 ――
tonf
m―――
1 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N5 2.25 tonf
≔Q5 −−⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠ ⋅⋅1 ――
tonf
m―――
1 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q5 9.00 tonf
≔M5 −−+−MA ⋅VA 11 m ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠6 m ⋅⋅1 ――
tonf
m―――
1 m
cos ⎛⎝α⎞⎠0.5 m
=M5 −41.75 ⋅tonf m
S6
≔N6 −−⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠ ⋅⋅1 ――
tonf
m―――
5 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N6 1.25 tonf
≔Q6 −−⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠ ⋅⋅1 ――
tonf
m―――
5 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q6 5.00 tonf
≔M6 −−+−MA ⋅VA 15 m ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠10 m ⋅⋅1 ――
tonf
m―――
5 m
cos ⎛⎝α⎞⎠2.5 m
=M6 −12.88 ⋅tonf m
S7
≔N7 −−⋅VA sin ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠ ⋅⋅1 ――
tonf
m―――
8 m
cos ⎛⎝α⎞⎠sin ⎛⎝α⎞⎠
=N7 0.50 tonf
≔Q7 −−⋅VA cos ⎛⎝α⎞⎠ ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠ ⋅⋅1 ――
tonf
m―――
8 m
cos ⎛⎝α⎞⎠cos ⎛⎝α⎞⎠
=Q7 2.00 tonf
≔M7 −−+−MA ⋅VA 18 m ⋅⋅2 ――tonf
m―――10 m
cos ⎛⎝α⎞⎠13 m ⋅⋅1 ――
tonf
m―――
8 m
cos ⎛⎝α⎞⎠4 m
=M7 −2.06 ⋅tonf m
2.
Reações de Apoio
Supondo-se inicialmente as reações de apoio, Vertical ( e ) e Horizontal ( ), todas com sentido positivo:VA VB HA
≔β atan⎛⎜⎝――8.66
15
⎞⎟⎠
=β 30.00 deg
=0ΣFx
=0+⋅15 tonf cos ⎛⎝30 deg⎞⎠ HA
≔HB ⋅−15 tonf cos ⎛⎝30 deg⎞⎠
*=HB −12.99 tonf
*uma vez que o valor obtido foi negativo, entende-se que o módulo de e o seu sentido é para a esquerda≔HB 12.99 tonf
=0ΣFy
=0+−−−−+7.5 tonf VA 8 tonf ⋅15 tonf sin ⎛⎝30 deg⎞⎠ 5 tonf ⋅1.5 ――tonf
m―――
8 m
cos ⎛⎝β⎞⎠VB
=V +VA VB
≔V ++++−7.5 tonf 8 tonf ⋅15 tonf sin ⎛⎝30 deg⎞⎠ 5 tonf ⋅1.5 ――tonf
m―――
8 m
cos ⎛⎝β⎞⎠=V 26.86 tonf
=0ΣMA
=0++−−−−−⋅−7.5 tonf 3 m ⋅8 tonf 2 m ⋅⋅15 tonf cos ⎛⎝30 deg⎞⎠⎛⎜⎝――――
⋅4 m 8.66 m
15 m
⎞⎟⎠
⋅⋅15 tonf sin ⎛⎝30 deg⎞⎠ 4 m ⋅5 tonf 7 m ⋅⋅1.5 ――tonf
m―――
8 m
cos ⎛⎝β⎞⎠8 m ⋅VB 12 m ⋅HB 6.95 m
≔VB
⎛⎜⎝
−+++++⋅7.5 tonf 3 m ⋅8 tonf 2 m ⋅⋅15 tonf cos ⎛⎝30 deg⎞⎠⎛⎜⎝―――――
⋅4 m 8.66 m
15 m
⎞⎟⎠
⋅15 tonf sin ⎛⎝30 deg⎞⎠ 4 m ⋅5 tonf 7 m ⋅⋅1.5 ――tonf
m―――
8 m
cos ⎛⎝β⎞⎠8 m ⋅HB 6.95 m
⎞⎟⎠
⎛⎜⎝――
1
12 m
⎞⎟⎠
=VB 12.84 tonf
≔VA −V VB
=VA 14.02 tonf
Esforços Internos
S1≔N1 +−⋅−7.5 tonf sin ⎛⎝β⎞⎠ ⋅VA sin ⎛⎝β⎞⎠ ⋅8 tonf sin ⎛⎝β⎞⎠
=N1 −6.76 tonf
≔Q1 −+⋅7.5 tonf cos ⎛⎝β⎞⎠ ⋅VA cos ⎛⎝β⎞⎠ ⋅8 tonf cos ⎛⎝β⎞⎠
=Q1 11.71 tonf
≔M1 −+⋅7.5 tonf 6 m ⋅VA 3 m ⋅8 tonf 1 m
=M1 79.05 ⋅tonf m
S3
≔N3 +−⋅⋅−1.5 ――tonf
m―――
3 m
cos ⎛⎝β⎞⎠sin ⎛⎝β⎞⎠ ⋅HB cos ⎛⎝β⎞⎠ ⋅VB sin ⎛⎝β⎞⎠
=N3 −7.43 tonf
≔Q3 −−⋅⋅1.5 ――tonf
m―――
3 m
cos ⎛⎝β⎞⎠cos ⎛⎝β⎞⎠ ⋅HB sin ⎛⎝β⎞⎠ ⋅VB cos ⎛⎝β⎞⎠
=Q3 −13.11 tonf
S2
≔N2 +−+−⋅−7.5 tonf sin ⎛⎝β⎞⎠ ⋅VA sin ⎛⎝β⎞⎠ ⋅8 tonf sin ⎛⎝β⎞⎠ ⋅15 tonf cos ⎛⎝ +30 deg β⎞⎠ ⋅⋅1.5 ――tonf
m―――1.5 m
cos ⎛⎝β⎞⎠sin ⎛⎝β⎞⎠
=N2 −12.96 tonf
≔Q2 −−−+⋅7.5 tonf cos ⎛⎝β⎞⎠ ⋅VA cos ⎛⎝β⎞⎠ ⋅8 tonf cos ⎛⎝β⎞⎠ ⋅15 tonf sin ⎛⎝ +30 deg β⎞⎠ ⋅⋅1.5 ――tonf
m―――1.5 m
cos ⎛⎝β⎞⎠cos ⎛⎝β⎞⎠
=Q2 −3.53 tonf
≔M2 −−−−+⋅7.5 tonf 8.5 m ⋅VA 5.5 m ⋅8 tonf 3.5 m ⋅⋅⋅15 tonf cos ⎛⎝30 deg⎞⎠ 1.5 m tan ⎛⎝β⎞⎠ ⋅⋅15 tonf sin ⎛⎝30 deg⎞⎠ 1.5 m ⋅⋅1.5 ――tonf
m―――1.5 m
cos ⎛⎝β⎞⎠0.75 m
=M2 88.40 ⋅tonf m
3.
Reações de Apoio
Supondo-se inicialmente as reações de apoio, Vertical ( e ) e VA VB
Horizontal ( ), todas com sentido positivo:HA
≔μ atan⎛⎜⎝――10 m
10 m
⎞⎟⎠
=μ 45.00 deg
=0ΣFx
=0+5 tonf HA
*≔HB −5.00 tonf
*uma vez que o valor obtido foi negativo, entende-se que o módulo dee o seu sentido é para a esquerda≔HB 5.00 tonf
=0ΣFy
=0+−−+−−VA ⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠10 tonf 7.5 tonf ⋅2 ――
tonf
m―――
3 m
cos ⎛⎝μ⎞⎠8 tonf VB
=+VA VB ++−+⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠10 tonf 7.5 tonf ⋅2 ――
tonf
m―――
3 m
cos ⎛⎝μ⎞⎠8 tonf
=V +VA VB
≔V ++−+⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠10 tonf 7.5 tonf ⋅2 ――
tonf
m―――
3 m
cos ⎛⎝μ⎞⎠8 tonf
=V 26.06 tonf
=0ΣMA
=0++−−−+−⋅⋅−1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠2.5 m ⋅10 tonf 3 m ⋅7.5 tonf 5 m ⋅5 tonf 7 m ⋅⋅2 ――
tonf
m―――
3 m
cos ⎛⎝μ⎞⎠8.5 m ⋅8 tonf 9 m ⋅HB 10 m ⋅VB 10 m
≔VB ――――――――――――――――――――――――――――――――――――――
−+++−+⋅⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠2.5 m ⋅10 tonf 3 m ⋅7.5 tonf 5 m ⋅5 tonf 7 m ⋅⋅2 ――
tonf
m―――
3 m
cos ⎛⎝μ⎞⎠8.5 m ⋅8 tonf 9 m ⋅HB 10 m
10 m=VB 13.93 tonf
≔VA −V VB
=VA 12.13 tonf
Esforços Internos
S1
≔N1 +⋅−VA sin ⎛⎝μ⎞⎠ ⋅⋅1 ――tonf
m―――
2 m
cos ⎛⎝μ⎞⎠sin ⎛⎝μ⎞⎠
=N1 −6.57 tonf
≔Q1 −⋅VA cos ⎛⎝μ⎞⎠ ⋅⋅1 ――tonf
m―――
2 m
cos ⎛⎝μ⎞⎠cos ⎛⎝μ⎞⎠
=Q1 6.57 tonf
≔M1 −⋅VA 2 m ⋅⋅1 ――tonf
m―――
2 m
cos ⎛⎝μ⎞⎠1 m
=M1 21.42 ⋅tonf m
S2
≔N2 −++⋅−VA sin ⎛⎝μ⎞⎠ ⋅⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠sin ⎛⎝μ⎞⎠ ⋅10 tonf sin ⎛⎝μ⎞⎠ ⋅7.5 tonf sin ⎛⎝μ⎞⎠
=N2 −1.81 tonf
≔Q2 +−−⋅VA cos ⎛⎝μ⎞⎠ ⋅⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠cos ⎛⎝μ⎞⎠ ⋅10 tonf cos ⎛⎝μ⎞⎠ ⋅7.5 tonf cos ⎛⎝μ⎞⎠
=Q2 1.81 tonf
≔M2 +−−⋅VA 6 m ⋅⋅1 ――tonf
m―――
5 m
cos ⎛⎝μ⎞⎠3.5 m ⋅10 tonf 3 m ⋅7.5 tonf 1 m
=M2 25.51 ⋅tonf m
S3
≔N3 −+−⋅⋅−2 ――tonf
m―――
2 m
cos ⎛⎝μ⎞⎠sin ⎛⎝μ⎞⎠ ⋅8 tonf sin ⎛⎝μ⎞⎠ ⋅VB sin ⎛⎝μ⎞⎠ ⋅HB cos ⎛⎝μ⎞⎠
=N3 −3.34 tonf
≔Q3 −−+⋅⋅2 ――tonf
m―――
2 m
cos ⎛⎝μ⎞⎠cos ⎛⎝μ⎞⎠ ⋅8 tonf cos ⎛⎝μ⎞⎠ ⋅VB cos ⎛⎝μ⎞⎠ ⋅HB sin ⎛⎝μ⎞⎠
=Q3 −3.73 tonf
≔M3 ++−⋅⋅−2 ――tonf
m―――
2 m
cos ⎛⎝μ⎞⎠1 m ⋅8 tonf 1 m ⋅VB 2 m ⋅HB 2 m
=M3 24.20 ⋅tonf m
4.
Reações de Apoio
≔θ atan⎛⎜⎝――11.55
20
⎞⎟⎠
=θ 30.01 deg
Supondo-se inicialmente as reações de apoio, Vertical ( e ) e Horizontal ( ), todas com sentido positivo:VA VB HA
=0ΣFx
=0−HB 5 tonf
≔HB 5 tonf
=0ΣFy
=0+−−−−+⋅−2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠VA 10 tonf 4 tonf 7.5 tonf ⋅1 ――
tonf
m―――10.50 n
cos ⎛⎝θ⎞⎠VB
=+VA VB ++++⋅2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠10 tonf 4 tonf 7.5 tonf ⋅1 ――
tonf
m―――10.50 m
cos ⎛⎝θ⎞⎠=V +VA VB
≔V ++++⋅2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠10 tonf 4 tonf 7.5 tonf ⋅1 ――
tonf
m―――10.50 m
cos ⎛⎝θ⎞⎠=V 49.79 tonf
=0ΣMA
=0−++−−−−⋅⋅−2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠0.50 m ⋅10 tonf 4 m ⋅4 tonf 10 m ⋅7.5 tonf 12 m ⋅⋅1 ――
tonf
m―――10.50 m
cos ⎛⎝θ⎞⎠11.75 m ⋅VB 13 m ⋅HB 7.51 m ⋅5 tonf 9.82 m
≔VB ―――――――――――――――――――――――――――――――――――――――――
+−++++⋅⋅2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠0.50 m ⋅10 tonf 4 m ⋅4 tonf 10 m ⋅7.5 tonf 12 m ⋅⋅1 ――
tonf
m―――10.50 m
cos ⎛⎝θ⎞⎠11.75 m ⋅HB 7.51 m ⋅5 tonf 9.82 m
13 m=VB 25.55 tonf
≔VA −V VB
=VA 24.25 tonf
Esforços Internos
S1
≔N1 ⋅⋅−2 ――tonf
m―――
1 m
cos ⎛⎝θ⎞⎠sin ⎛⎝θ⎞⎠
=N1 −1.16 tonf
≔Q1 ⋅⋅−2 ――tonf
m―――
1 m
cos ⎛⎝θ⎞⎠cos ⎛⎝θ⎞⎠
=Q1 −2.00 tonf
≔M1 ⋅⋅−2 ――tonf
m―――
1 m
cos ⎛⎝θ⎞⎠0.50 m
=M1 −1.15 ⋅tonf m
S2
≔N2 −−+⋅⋅−2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠sin ⎛⎝θ⎞⎠ ⋅VA sin ⎛⎝θ⎞⎠ ⋅10 tonf sin ⎛⎝θ⎞⎠ ⋅⋅1 ――
tonf
m―――2.50 m
cos ⎛⎝θ⎞⎠sin ⎛⎝θ⎞⎠
=N2 −2.40 tonf
≔Q2 −−+⋅⋅−2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠cos ⎛⎝θ⎞⎠ ⋅VA cos ⎛⎝θ⎞⎠ ⋅10 tonf cos ⎛⎝θ⎞⎠ ⋅⋅1 ――
tonf
m―――2.50 m
cos ⎛⎝θ⎞⎠cos ⎛⎝θ⎞⎠
=Q2 −4.16 tonf
≔M2 −−−⋅VA 9 m ⋅⋅2 ――tonf
m―――
7 m
cos ⎛⎝θ⎞⎠8.50 m ⋅10 tonf 5 m ⋅⋅1 ――
tonf
m―――2.50 m
cos ⎛⎝θ⎞⎠1.25 m
=M2 27.18 ⋅tonf m
S3
≔N3 −−++⋅7.5 tonf sin ⎛⎝θ⎞⎠ ⋅⋅1 ――tonf
m―――
6 m
cos ⎛⎝θ⎞⎠sin ⎛⎝θ⎞⎠ ⋅HB cos ⎛⎝θ⎞⎠ ⋅VB sin ⎛⎝θ⎞⎠ ⋅5 tonf cos ⎛⎝θ⎞⎠
=N3 −5.56 tonf
≔Q3 +−−+⋅7.5 tonf cos ⎛⎝θ⎞⎠ ⋅⋅1 ――tonf
m―――
6 m
cos ⎛⎝θ⎞⎠cos ⎛⎝θ⎞⎠ ⋅HB sin ⎛⎝θ⎞⎠ ⋅VB cos ⎛⎝θ⎞⎠ ⋅5 tonf sin ⎛⎝θ⎞⎠
=Q3 −9.63 tonf
≔M3 −++−⋅−7.5 tonf 1 m ⋅⋅1 ――tonf
m―――
6 m
cos ⎛⎝θ⎞⎠3 m ⋅HB
⎛⎝ ⋅2 m tan ⎛⎝θ⎞⎠⎞⎠ ⋅VB 2 m ⋅5 tonf ⎛⎝ ⋅6 m tan ⎛⎝θ⎞⎠⎞⎠
=M3 11.26 ⋅tonf m
S4
≔N4 −⋅⋅1 ――tonf
m―――
2 m
cos ⎛⎝θ⎞⎠sin ⎛⎝θ⎞⎠ ⋅5 tonf cos ⎛⎝θ⎞⎠
=N4 −3.17 tonf
≔Q4 +⋅⋅1 ――tonf
m―――
2 m
cos ⎛⎝θ⎞⎠cos ⎛⎝θ⎞⎠ ⋅5 tonf sin ⎛⎝θ⎞⎠
=Q4 4.50 tonf
≔M4 −⋅⋅−1 ――tonf
m―――
2 m
cos ⎛⎝θ⎞⎠1 m ⋅5 tonf ⎛⎝ ⋅2 m tan ⎛⎝θ⎞⎠⎞⎠
=M4 −8.08 ⋅tonf m
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