EES024 Aula23 MetodoDaRigidezDireta Grelha Profs Rodrigo e Leandro Planilha Excel

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Preenchida
	( i ) Montagem da matriz de rigidez global [K]
	_ Matriz de rigidez global de uma barra de grelha: [k](E) = [R]T·[k'](E)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	kN/m2	b = 	m	Nó inicial ( i )	Nó final ( j )
	ν = 	h = 	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	0.000E+00	kN/m2	I = 	0.0000E+00	m4
	Jt = 	ERROR:#DIV/0!	m4
	Parâmetros de rigidez	l =	0.000	m
	EI = 	0.0000E+00	kNm2	C = 	ERROR:#DIV/0!
	GJt = 	ERROR:#DIV/0!	kNm2	S = 	ERROR:#DIV/0!
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	0	0	1	0	0	0	0	0	1
	øiX	øiY	diZ	øjX	øjY	djZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiY
	[k](E) = 	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	diZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjY
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	djZ
	1	2	3	4	5	6	7	8	9	10	11	12
	1
	2
	3
	4
	5
	[k](E) = 	6
	7
	8
	9
	10
	11
	12
	* Barra (1): [k](1) = [R]T·[k'](1)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	2.00E+08	kN/m2	b = 	0.1	m	Nó inicial ( i )	Nó final ( j )
	ν = 	0.3	h = 	0.2	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	7.692E+07	kN/m2	I = 	6.6667E-05	m4	1	0.0	2.0	3	4.0	0.0
	Jt = 	4.578E-05	m4
	Parâmetros de rigidez	l =	4.472	m
	EI = 	1.3333E+04	kNm2	C = 	0.894
	GJt = 	3.521E+03	kNm2	S = 	-0.447
	0.894	-0.447	0	0	0	0	0.894	0.447	0	0	0	0
	0.447	0.894	0	0	0	0	-0.447	0.894	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	0.894	-0.447	0	0	0	0	0.894	0.447	0
	0	0	0	0.447	0.894	0	0	0	0	-0.447	0.894	0
	0	0	0	0	0	1	0	0	0	0	0	1
	4	5	6	1	2	3
	3.015E+03	4.455E+03	-1.789E+03	5.627E+02	2.700E+03	1.789E+03	4
	4.455E+03	9.698E+03	-3.578E+03	2.700E+03	4.613E+03	3.578E+03	5
	[k](1) = 	-1.789E+03	-3.578E+03	1.789E+03	-1.789E+03	-3.578E+03	-1.789E+03	6
	5.627E+02	2.700E+03	-1.789E+03	3.015E+03	4.455E+03	1.789E+03	1
	2.700E+03	4.613E+03	-3.578E+03	4.455E+03	9.698E+03	3.578E+03	2
	1.789E+03	3.578E+03	-1.789E+03	1.789E+03	3.578E+03	1.789E+03	3
	1	2	3	4	5	6	7	8	9	10	11	12
	3.015E+03	4.455E+03	1.789E+03	5.627E+02	2.700E+03	-1.789E+03	1
	4.455E+03	9.698E+03	3.578E+03	2.700E+03	4.613E+03	-3.578E+03	2
	1.789E+03	3.578E+03	1.789E+03	1.789E+03	3.578E+03	-1.789E+03	3
	5.627E+02	2.700E+03	1.789E+03	3.015E+03	4.455E+03	-1.789E+03	4
	2.700E+03	4.613E+03	3.578E+03	4.455E+03	9.698E+03	-3.578E+03	5
	[k](1) = 	-1.789E+03	-3.578E+03	-1.789E+03	-1.789E+03	-3.578E+03	1.789E+03	6
	7
	8
	9
	10
	11
	12
	* Barra (2): [k](2) = [R]T·[k'](2)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	2.00E+08	kN/m2	b = 	0.1	m	Nó inicial ( i )	Nó final ( j )
	ν = 	0.3	h = 	0.2	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	7.692E+07	kN/m2	I = 	6.6667E-05	m4	3	4.0	0.0	2	4.0	2.0
	Jt = 	4.578E-05	m4
	Parâmetros de rigidez	l =	2.000	m
	EI = 	1.3333E+04	kNm2	C = 	0.000
	GJt = 	3.521E+03	kNm2	S = 	1.000
	0.000	1.000	0	0	0	0	0.000	-1.000	0	0	0	0
	-1.000	0.000	0	0	0	0	1.000	0.000	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	0.000	1.000	0	0	0	0	0.000	-1.000	0
	0	0	0	-1.000	0.000	0	0	0	0	1.000	0.000	0
	0	0	0	0	0	1	0	0	0	0	0	1
	1	2	3	7	8	9
	2.667E+04	0.000E+00	2.000E+04	1.333E+04	0.000E+00	-2.000E+04	1
	0.000E+00	1.761E+03	0.000E+00	0.000E+00	-1.761E+03	0.000E+00	2
	[k](2) = 	2.000E+04	0.000E+00	2.000E+04	2.000E+04	0.000E+00	-2.000E+04	3
	1.333E+04	0.000E+00	2.000E+04	2.667E+04	0.000E+00	-2.000E+04	7
	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	1.761E+03	0.000E+00	8
	-2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	0.000E+00	2.000E+04	9
	1	2	3	4	5	6	7	8	9	10	11	12
	2.667E+04	0.000E+00	2.000E+04	1.333E+04	0.000E+00	-2.000E+04	1
	0.000E+00	1.761E+03	0.000E+00	0.000E+00	-1.761E+03	0.000E+00	2
	2.000E+04	0.000E+00	2.000E+04	2.000E+04	0.000E+00	-2.000E+04	3
	4
	5
	[k](2) = 	6
	1.333E+04	0.000E+00	2.000E+04	2.667E+04	0.000E+00	-2.000E+04	7
	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	1.761E+03	0.000E+00	8
	-2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	0.000E+00	2.000E+04	9
	10
	11
	12
	* Barra (3): [k](3) = [R]T·[k'](3)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	2.00E+08	kN/m2	b = 	0.1	m	Nó inicial ( i )	Nó final ( j )
	ν = 	0.3	h = 	0.2	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	7.692E+07	kN/m2	I = 	6.6667E-05	m4	3	4.0	0.0	4	6.0	0.0
	Jt = 	4.578E-05	m4
	Parâmetros de rigidez	l =	2.000	m
	EI = 	1.3333E+04	kNm2	C = 	1.000
	GJt = 	3.521E+03	kNm2	S = 	0.000
	1.000	0.000	0	0	0	0	1.000	0.000	0	0	0	0
	0.000	1.000	0	0	0	0	0.000	1.000	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	1.000	0.000	0	0	0	0	1.000	0.000	0
	0	0	0	0.000	1.000	0	0	0	0	0.000	1.000	0
	0	0	0	0	0	1	0	0	0	0	0	1
	1	2	3	10	11	12
	1.761E+03	0.000E+00	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	1
	0.000E+00	2.667E+04	-2.000E+04	0.000E+00	1.333E+04	2.000E+04	2
	[k](3) = 	0.000E+00	-2.000E+04	2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	3
	-1.761E+03	0.000E+00	0.000E+00	1.761E+03	0.000E+00	0.000E+00	10
	0.000E+00	1.333E+04	-2.000E+04	0.000E+00	2.667E+04	2.000E+04	11
	0.000E+00	2.000E+04	-2.000E+04	0.000E+00	2.000E+04	2.000E+04	12
	1	2	3	4	5	6	7	8	9	10	11	12
	1.761E+03	0.000E+00	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	1
	0.000E+00	2.667E+04	-2.000E+04	0.000E+00	1.333E+04	2.000E+04	2
	0.000E+00	-2.000E+04	2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	3
	4
	5
	[k](3) = 	6
	7
	8
	9
	-1.761E+03	0.000E+00	0.000E+00	1.761E+03	0.000E+00	0.000E+00	10
	0.000E+00	1.333E+04	-2.000E+04	0.000E+00	2.667E+04	2.000E+04	11
	0.000E+00	2.000E+04	-2.000E+04	0.000E+00	2.000E+04	2.000E+04	12
	1	2	3	4	5	6	7	8	9	10	11	12
	3.144E+04	4.455E+03	2.179E+04	5.627E+02	2.700E+03	-1.789E+03	1.333E+04	0.000E+00	-2.000E+04	-1.761E+03	0.000E+00	0.000E+00	1
	4.455E+03	3.813E+04	-1.642E+04	2.700E+03	4.613E+03	-3.578E+03	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	1.333E+04	2.000E+04	2
	2.179E+04	-1.642E+04	4.179E+04	1.789E+03	3.578E+03	-1.789E+03	2.000E+04	0.000E+00	-2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	3
	5.627E+02	2.700E+03	1.789E+03	3.015E+03	4.455E+03	-1.789E+03	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	4
	2.700E+03	4.613E+03	3.578E+03	4.455E+03	9.698E+03	-3.578E+03	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	5
	[K] = 	-1.789E+03	-3.578E+03	-1.789E+03	-1.789E+03	-3.578E+03	1.789E+03	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	6
	1.333E+04	0.000E+00	2.000E+04	0.000E+00	0.000E+00	0.000E+00	2.667E+04	0.000E+00	-2.000E+04	0.000E+00	0.000E+00	0.000E+00	7
	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	1.761E+03	0.000E+00	0.000E+00	0.000E+00	0.000E+00	8
	-2.000E+04	0.000E+00	-2.000E+04	0.000E+00	0.000E+00	0.000E+00	-2.000E+04	0.000E+00	2.000E+04	0.000E+00	0.000E+00	0.000E+00	9
	-1.761E+03	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	1.761E+03	0.000E+00	0.000E+00	10
	0.000E+00	1.333E+04	-2.000E+04	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	2.667E+04	2.000E+04	11
	0.000E+00	2.000E+04	-2.000E+04	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	2.000E+04	2.000E+04	12
	( ii ) Montagem do vetor de forças globais
	[R]T = 
	q = 	kN/m	1
	l =	m	2
	3
	0.0	d'1	ERROR:#VALUE!	øiX	ERROR:#VALUE!	øiX	4
	0.0	d'2	ERROR:#VALUE!	øiY	ERROR:#VALUE!	øiY	5
	0.0	d'3	ERROR:#VALUE!	diZ	ERROR:#VALUE!	diZ	6
	0.0	d'4	ERROR:#VALUE!	øjX	ERROR:#VALUE!	øjX	7
	0.0	d'5	ERROR:#VALUE!	øjY	ERROR:#VALUE!	øjY	8
	0.0	d'6	ERROR:#VALUE!	djZ	ERROR:#VALUE!	djZ	9
	10
	11
	12
	* Barra
(1):
	0.894	0.447	0	0	0	0
	-0.447	0.894	0	0	0	0
	[R]T = 	0	0	1	0	0	0
	0	0	0	0.894	0.447	0
	0	0	0	-0.447	0.894	0
	0	0	0	0	0	1
	q = 	24.0	kN/m	-17.9	1
	l =	4.472	m	-35.8	2
	-53.7	3
	0.0	d'1	-17.9	4	17.9	4	17.9	4
	-40.0	d'2	-35.8	5	35.8	5	35.8	5
	53.7	d'3	53.7	6	-53.7	6	-53.7	6
	0.0	d'4	17.9	1	-17.9	1	7
	40.0	d'5	35.8	2	-35.8	2	8
	53.7	d'6	53.7	3	-53.7	3	9
	10
	11
	12
	* Barra (2):
	0.000	-1.000	0	0	0	0
	1.000	0.000	0	0	0	0
	[R]T = 	0	0	1	0	0	0
	0	0	0	0.000	-1.000	0
	0	0	0	1.000	0.000	0
	0	0	0	0	0	1
	q = 	0.0	kN/m	0.0	1
	l =	2.000	m	0.0	2
	0.0	3
	0.0	d'1	0.0	1	0.0	1	4
	0.0	d'2	0.0	2	0.0	2	5
	0.0	d'3	0.0	3	0.0	3	6
	0.0	d'4	0.0	7	0.0	7	0.0	7
	0.0	d'5	0.0	8	0.0	8	0.0	8
	0.0	d'6	0.0	9	0.0	9	0.0	9
	10
	11
	12
	* Barra (3):
	1.000	0.000	0	0	0	0
	0.000	1.000	0	0	0	0
	[R]T = 	0	0	1	0	0	0
	0	0	0	1.000	0.000	0
	0	0	0	0.000	1.000	0
	0	0	0	0	0	1
	q = 	24.0	kN/m	0.0	1
	l =	2.000	m	8.0	2
	-24.0	3
	0.0	d'1	0.0	1	0.0	1	4
	-8.0	d'2	-8.0	2	8.0	2	5
	24.0	d'3	24.0	3	-24.0	3	6
	0.0	d'4	0.0	10	0.0	10	7
	8.0	d'5	8.0	11	-8.0	11	8
	24.0	d'6	24.0	12	-24.0	12	9
	0.0	10
	-8.0	11
	-24.0	12
	-17.9	1	1.5	1	-	1
	-27.8	2	0.0	2	-	2
	-77.7	3	-1.0	3	-	3
	17.9	4	0.0	4	R4	4
	35.8	5	0.0	5	R5	5
	-53.7	6	+	0.0	6	+	R6	6
	0.0	7	0.0	7	R7	7
	0.0	8	0.0	8	R8	8
	0.0	9	0.0	9	R9	9
	0.0	10	0.0	10	R10	10
	-8.0	11	0.0	11	R11	11
	-24.0	12	0.0	12	R12	12
	( iii ) Condições de equilíbrio [K]·{D} = {F}
	( iii.1 ) Deslocamentos desconhecidos {Dl}
	[kll]-1 	·	{Fl}	=	{Dl}
	6.789E-05	-2.790E-05	-4.636E-05	-16.4	3.310E-03	rad
	-2.790E-05	4.304E-05	3.146E-05	·	-27.8	=	-3.213E-03	rad
	-4.636E-05	3.146E-05	6.047E-05	-78.7	-4.871E-03	m
	( iii.2 ) Reações de apoio
	{Ff}	-	-	{Pf}	=	{Rf}
	-15.5	17.9	0.0	R4 = 	-33.4	kNm
	-23.3	35.8	0.0	R5 = 	-59.1	kNm
	14.3	-53.7	0.0	R6 = 	68.0	kN
	-53.3	0.0	0.0	R7 = 	-53.3	kNm
	5.7	-	0.0	-	0.0	=	R8 = 	5.7	kNm
	31.2	0.0	0.0	R9 = 	31.2	kN
	-5.8	0.0	0.0	R10 = 	-5.8	kNm
	54.6	-8.0	0.0	R11 = 	62.6	kNm
	33.1	-24.0	0.0	R12 = 	57.1	kN
	( iii.3 ) Esforços internos de extremidade
	[k](E)	·	{d}(E)	=	{f}(E)
	ERROR:#VALUE!
	ERROR:#VALUE!
	·	=	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	[R]	·	{f}(E)	=	{f '}(E)
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	·	ERROR:#VALUE!	=	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	{f '}(E)	+	=	{E'}(E)
	ERROR:#VALUE!	Ti	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mi	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	+	=	Qi	ERROR:#VALUE!	kN
	ERROR:#VALUE!	Tj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Qj	ERROR:#VALUE!	kN
	* Barra (1):
	[k](1)	·	{d}(1)	=	{f}(1)
	3.015E+03	4.455E+03	-1.789E+03	5.627E+02	2.700E+03	1.789E+03	0.000E+00	-15.5
	4.455E+03	9.698E+03	-3.578E+03	2.700E+03	4.613E+03	3.578E+03	0.000E+00	-23.3
	-1.789E+03	-3.578E+03	1.789E+03	-1.789E+03	-3.578E+03	-1.789E+03	·	0.000E+00	=	14.3
	5.627E+02	2.700E+03	-1.789E+03	3.015E+03	4.455E+03	1.789E+03	3.310E-03	-13.1
	2.700E+03	4.613E+03	-3.578E+03	4.455E+03	9.698E+03	3.578E+03	-3.213E-03	-33.8
	1.789E+03	3.578E+03	-1.789E+03	1.789E+03	3.578E+03	1.789E+03	-4.871E-03	-14.3
	[R]	·	{f}(1)	=	{f '}(1)
	0.894	-0.447	0	0	0	0	-15.5	-3.5
	0.447	0.894	0	0	0	0	-23.3	-27.8
	0	0	1	0	0	0	·	14.3	=	14.3
	0	0	0	0.894	-0.447	0	-13.1	3.5
	0	0	0	0.447	0.894	0	-33.8	-36.1
	0	0	0	0	0	1	-14.3	-14.3
	{f '}(1)	+	=	{E'}(1)
	-3.5	0.0	Ti	-3.5	kNm
	-27.8	-40.0	Mi	-67.8	kNm
	14.3	+	53.7	=	Qi	68.0	kN
	3.5	0.0	Tj	3.5	kNm
	-36.1	40.0	Mj	3.9	kNm
	-14.3	53.7	Qj	39.4	kN
	* Barra (2):
	[k](2)	·	{d}(2)	=	{f}(2)
	2.667E+04	0.000E+00	2.000E+04	1.333E+04	0.000E+00	-2.000E+04	3.310E-03	-9.2
	0.000E+00	1.761E+03	0.000E+00	0.000E+00	-1.761E+03	0.000E+00	-3.213E-03	-5.7
	2.000E+04	0.000E+00	2.000E+04	2.000E+04	0.000E+00	-2.000E+04	·	-4.871E-03	=	-31.2
	1.333E+04	0.000E+00	2.000E+04	2.667E+04	0.000E+00	-2.000E+04	0.000E+00	-53.3
	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	1.761E+03	0.000E+00	0.000E+00	5.7
	-2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	0.000E+00	2.000E+04	0.000E+00	31.2
	[R]	·	{f}(2)	=	{f '}(2)
	0.000	1.000	0	0	0	0	-9.2	-5.7
	-1.000	0.000	0	0	0	0	-5.7	9.2
	0	0	1	0	0	0	·	-31.2	=	-31.2
	0	0	0	0.000	1.000	0	-53.3	5.7
	0	0	0	-1.000	0.000	0	5.7	53.3
	0	0	0	0	0	1	31.2	31.2
	{f '}(2)	+	=	{E'}(2)
	-5.7	0.0	Ti	-5.7	kNm
	9.2	0.0	Mi	9.2	kNm
	-31.2	+	0.0	=	Qi	-31.2	kN
	5.7	0.0	Tj	5.7	kNm
	53.3	0.0	Mj	53.3	kNm
	31.2	0.0	Qj	31.2	kN
	* Barra (3):
	[k](3)	·	{d}(3)	=	{f}(3)
	1.761E+03	0.000E+00	0.000E+00	-1.761E+03	0.000E+00	0.000E+00	3.310E-03	5.8
	0.000E+00	2.667E+04	-2.000E+04	0.000E+00	1.333E+04	2.000E+04	-3.213E-03	11.7
	0.000E+00	-2.000E+04	2.000E+04	0.000E+00	-2.000E+04	-2.000E+04	·	-4.871E-03	=	-33.1
	-1.761E+03	0.000E+00	0.000E+00	1.761E+03	0.000E+00	0.000E+00	0.000E+00	-5.8
	0.000E+00	1.333E+04	-2.000E+04	0.000E+00	2.667E+04	2.000E+04	0.000E+00	54.6
	0.000E+00	2.000E+04	-2.000E+04	0.000E+00	2.000E+04	2.000E+04	0.000E+00	33.1
	[R]	·	{f}(3)	=	{f '}(3)
	1.000	0.000	0	0	0	0	5.8	5.8
	0.000	1.000	0	0	0	0	11.7	11.7
	0	0	1	0	0	0	·	-33.1	=	-33.1
	0	0	0	1.000	0.000	0	-5.8	-5.8
	0	0	0	0.000	1.000	0	54.6	54.6
	0	0	0	0	0	1	33.1	33.1
	{f '}(3)	+	=	{E'}(3)
	5.8	0.0	Ti	5.8	kNm
	11.7	-8.0	Mi	3.7	kNm
	-33.1	+	24.0	=	Qi	-9.1	kN
	-5.8	0.0	Tj	-5.8	kNm
	54.6	8.0	Mj	62.6	kNm
	33.1	24.0	Qj	57.1	kN
À preencher
	( i ) Montagem da matriz de rigidez global [K]
	_ Matriz de rigidez global de uma barra de grelha: [k](E) = [R]T·[k'](E)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	kN/m2	b = 	m	Nó inicial ( i )	Nó final ( j )
	ν = 	h = 	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	0.000E+00	kN/m2	I = 	0.0000E+00	m4
	Jt = 	ERROR:#DIV/0!	m4
	Parâmetros de rigidez	l =	0.000	m
	EI = 	0.0000E+00	kNm2	C = 	ERROR:#DIV/0!
	GJt = 	ERROR:#DIV/0!	kNm2	S = 	ERROR:#DIV/0!
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	0	0	1	0	0	0	0	0	1
	øiX	øiY	diZ	øjX	øjY	djZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiY
	[k](E) = 	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	diZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjY
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	djZ
	1	2	3	4	5	6	7	8	9	10	11	12
	1
	2
	3
	4
	5
	[k](E) = 	6
	7
	8
	9
	10
	11
	12
	* Barra (1): [k](1) = [R]T·[k'](1)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	kN/m2	b = 	m	Nó inicial ( i )	Nó final ( j )
	ν = 	h = 	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	0.000E+00	kN/m2	I = 	0.0000E+00	m4
	Jt = 	ERROR:#DIV/0!	m4
	Parâmetros de rigidez	l =	0.000	m
	EI = 	0.0000E+00	kNm2	C = 	ERROR:#DIV/0!
	GJt = 	ERROR:#DIV/0!	kNm2	S = 	ERROR:#DIV/0!
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	0	0	1	0	0	0	0	0	1
	øiX	øiY	diZ	øjX	øjY	djZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiY
	[k](1) = 	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	diZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!
ERROR:#DIV/0!	ERROR:#DIV/0!	øjY
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	djZ
	1	2	3	4	5	6	7	8	9	10	11	12
	1
	2
	3
	4
	5
	[k](1) = 	6
	7
	8
	9
	10
	11
	12
	* Barra (2): [k](2) = [R]T·[k'](2)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	kN/m2	b = 	m	Nó inicial ( i )	Nó final ( j )
	ν = 	h = 	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	0.000E+00	kN/m2	I = 	0.0000E+00	m4
	Jt = 	ERROR:#DIV/0!	m4
	Parâmetros de rigidez	l =	0.000	m
	EI = 	0.0000E+00	kNm2	C = 	ERROR:#DIV/0!
	GJt = 	ERROR:#DIV/0!	kNm2	S = 	ERROR:#DIV/0!
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	0	0	1	0	0	0	0	0	1
	øiX	øiY	diZ	øjX	øjY	djZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiY
	[k](2) = 	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	diZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjY
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	djZ
	1	2	3	4	5	6	7	8	9	10	11	12
	1
	2
	3
	4
	5
	[k](2) = 	6
	7
	8
	9
	10
	11
	12
	* Barra (3): [k](3) = [R]T·[k'](3)·[R]
	Parâmetros materiais	Parâmetros geométricos	Incidência
	E = 	kN/m2	b = 	m	Nó inicial ( i )	Nó final ( j )
	ν = 	h = 	m	Nº.	X (m)	Y (m)	Nº.	X (m)	Y (m)
	G = 	0.000E+00	kN/m2	I = 	0.0000E+00	m4
	Jt = 	ERROR:#DIV/0!	m4
	Parâmetros de rigidez	l =	0.000	m
	EI = 	0.0000E+00	kNm2	C = 	ERROR:#DIV/0!
	GJt = 	ERROR:#DIV/0!	kNm2	S = 	ERROR:#DIV/0!
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0
	[R] = 	0	0	1	0	0	0	[R]T = 	0	0	1	0	0	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0	0	0	0	ERROR:#DIV/0!	ERROR:#DIV/0!	0
	0	0	0	0	0	1	0	0	0	0	0	1
	øiX	øiY	diZ	øjX	øjY	djZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øiY
	[k](3) = 	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	diZ
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjX
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	øjY
	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	ERROR:#DIV/0!	djZ
	1	2	3	4	5	6	7	8	9	10	11	12
	1
	2
	3
	4
	5
	[k](3) = 	6
	7
	8
	9
	10
	11
	12
	1	2	3	4	5	6	7	8	9	10	11	12
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	1
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	2
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	3
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	4
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	5
	[K] = 	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	6
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	7
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	8
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	9
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	10
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	11
	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	0.000E+00	12
	( ii ) Montagem do vetor de forças globais
	[R]T = 
	q = 	kN/m	1
	l =	m	2
	3
	0.0	d'1	ERROR:#VALUE!	øiX	ERROR:#VALUE!	øiX	4
	0.0	d'2	ERROR:#VALUE!	øiY	ERROR:#VALUE!	øiY	5
	0.0	d'3	ERROR:#VALUE!	diZ	ERROR:#VALUE!	diZ	6
	0.0	d'4	ERROR:#VALUE!	øjX	ERROR:#VALUE!	øjX	7
	0.0	d'5	ERROR:#VALUE!	øjY	ERROR:#VALUE!	øjY	8
	0.0	d'6	ERROR:#VALUE!	djZ	ERROR:#VALUE!	djZ	9
	10
	11
	12
	* Barra (1):
	[R]T = 
	q = 	kN/m	1
	l =	m	2
	3
	0.0	d'1	ERROR:#VALUE!	øiX	ERROR:#VALUE!	øiX	4
	0.0	d'2	ERROR:#VALUE!	øiY	ERROR:#VALUE!	øiY	5
	0.0	d'3	ERROR:#VALUE!	diZ	ERROR:#VALUE!	diZ	6
	0.0	d'4	ERROR:#VALUE!	øjX	ERROR:#VALUE!	øjX	7
	0.0	d'5	ERROR:#VALUE!	øjY	ERROR:#VALUE!	øjY	8
	0.0	d'6	ERROR:#VALUE!	djZ	ERROR:#VALUE!	djZ	9
	10
	11
	12
	* Barra (2):
	[R]T = 
	q = 	kN/m	1
	l =	m	2
	3
	0.0	d'1	ERROR:#VALUE!	øiX	ERROR:#VALUE!	øiX	4
	0.0	d'2	ERROR:#VALUE!	øiY	ERROR:#VALUE!	øiY	5
	0.0	d'3	ERROR:#VALUE!	diZ	ERROR:#VALUE!	diZ	6
	0.0	d'4	ERROR:#VALUE!	øjX	ERROR:#VALUE!	øjX	7
	0.0	d'5	ERROR:#VALUE!	øjY	ERROR:#VALUE!	øjY	8
	0.0	d'6	ERROR:#VALUE!	djZ	ERROR:#VALUE!	djZ	9
	10
	11
	12
	* Barra (3):
	[R]T = 
	q = 	kN/m	1
	l =	m	2
	3
	0.0	d'1	ERROR:#VALUE!	øiX	ERROR:#VALUE!	øiX	4
	0.0	d'2	ERROR:#VALUE!	øiY	ERROR:#VALUE!	øiY	5
	0.0	d'3	ERROR:#VALUE!	diZ	ERROR:#VALUE!	diZ	6
	0.0	d'4	ERROR:#VALUE!	øjX	ERROR:#VALUE!	øjX	7
	0.0	d'5	ERROR:#VALUE!	øjY	ERROR:#VALUE!	øjY	8
	0.0	d'6	ERROR:#VALUE!	djZ	ERROR:#VALUE!	djZ	9
	10
	11
	12
	0.0	1	1	1
	0.0	2	2	2
	0.0	3	3	3
	0.0	4	4	4
	0.0	5	5	5
	0.0	6	+	6	+	6
	0.0	7	7	7
	0.0	8	8	8
	0.0	9	9	9
	0.0	10	10	10
	0.0	11	11	11
	0.0	12	12	12
	( iii ) Condições de equilíbrio [K]·{D} = {F}
	( iii.1 ) Deslocamentos desconhecidos {Dl}
	[kll]-1 	·	{Fl}	=	{Dl}
	ERROR:#NUM!	ERROR:#NUM!	ERROR:#NUM!	0.0	ERROR:#NUM!	rad
	ERROR:#NUM!	ERROR:#NUM!	ERROR:#NUM!	·	0.0	=	ERROR:#NUM!	rad
	ERROR:#NUM!	ERROR:#NUM!	ERROR:#NUM!	0.0	ERROR:#NUM!	m
	( iii.2 ) Reações de apoio
	{Ff}	-	-	{Pf}	=	{Rf}
	ERROR:#NUM!	0.0	0.0	R4 = 	ERROR:#NUM!	kNm
	ERROR:#NUM!	0.0	0.0	R5 = 	ERROR:#NUM!	kNm
	ERROR:#NUM!	0.0	0.0	R6 = 	ERROR:#NUM!	kN
	ERROR:#NUM!	0.0	0.0	R7 = 	ERROR:#NUM!	kNm
	ERROR:#NUM!	-	0.0	-	0.0	=	R8 = 	ERROR:#NUM!	kNm
	ERROR:#NUM!	0.0	0.0	R9 = 	ERROR:#NUM!	kN
	ERROR:#NUM!	0.0	0.0	R10 = 	ERROR:#NUM!	kNm
	ERROR:#NUM!	0.0	0.0	R11 = 	ERROR:#NUM!	kNm
	ERROR:#NUM!	0.0	0.0	R12 = 	ERROR:#NUM!	kN
	( iii.3 ) Esforços internos de extremidade
	[k](E)	·	{d}(E)	=	{f}(E)
	ERROR:#VALUE!
	ERROR:#VALUE!
	·	=	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	[R]	·	{f}(E)	=	{f '}(E)
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	·	ERROR:#VALUE!	=	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	{f '}(E)	+	=	{E'}(E)
	ERROR:#VALUE!	Ti	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mi	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	+	=	Qi	ERROR:#VALUE!	kN
	ERROR:#VALUE!	Tj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Qj	ERROR:#VALUE!	kN
	* Barra (1):
	[k](1)	·	{d}(1)	=	{f}(1)
	ERROR:#VALUE!
	ERROR:#VALUE!
	·	=	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	[R]	·	{f}(1)	=	{f '}(1)
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	·	ERROR:#VALUE!	=	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	{f '}(1)	+	=	{E'}(1)
	ERROR:#VALUE!	Ti	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mi	ERROR:#VALUE!
kNm
	ERROR:#VALUE!	+	=	Qi	ERROR:#VALUE!	kN
	ERROR:#VALUE!	Tj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Qj	ERROR:#VALUE!	kN
	* Barra (2):
	[k](2)	·	{d}(2)	=	{f}(2)
	ERROR:#VALUE!
	ERROR:#VALUE!
	·	=	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	[R]	·	{f}(2)	=	{f '}(2)
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	·	ERROR:#VALUE!	=	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	{f '}(2)	+	=	{E'}(2)
	ERROR:#VALUE!	Ti	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mi	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	+	=	Qi	ERROR:#VALUE!	kN
	ERROR:#VALUE!	Tj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Qj	ERROR:#VALUE!	kN
	* Barra (3):
	[k](3)	·	{d}(3)	=	{f}(3)
	ERROR:#VALUE!
	ERROR:#VALUE!
	·	=	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	ERROR:#VALUE!
	[R]	·	{f}(3)	=	{f '}(3)
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	·	ERROR:#VALUE!	=	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	ERROR:#VALUE!	ERROR:#VALUE!
	{f '}(3)	+	=	{E'}(3)
	ERROR:#VALUE!	Ti	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mi	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	+	=	Qi	ERROR:#VALUE!	kN
	ERROR:#VALUE!	Tj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Mj	ERROR:#VALUE!	kNm
	ERROR:#VALUE!	Qj	ERROR:#VALUE!	kN