Numerical analysis of RC plane structures: a concentrated nonlinear effect approach

Lemes, Í. J. M.; Barros, R. C.; Silveira, R. A. M.; Silva, A. R. D.; Rocha, P. A. S.

doi:10.1590/1679-78254681

Í. J. M. Lemes ^** Corresponding author

Departamento de Engenharia, Universidade Federal de Lavras, Lavras, MG, Brasil. E-mail: igor.lemes@deg.ufla.br

R. C. Barros

Departamento de Engenharia Civil, Universidade Federal de Ouro Preto, Ouro Preto, MG, Brasil. E-mail: rafaelcesario@hotmail.com, ricardo@em.ufop.br, andreadiassilva@yahoo.com.br, pandrocha@gmail.com

R. A. M. Silveira

Departamento de Engenharia Civil, Universidade Federal de Ouro Preto, Ouro Preto, MG, Brasil. E-mail: rafaelcesario@hotmail.com, ricardo@em.ufop.br, andreadiassilva@yahoo.com.br, pandrocha@gmail.com

A. R. D. Silva

Departamento de Engenharia Civil, Universidade Federal de Ouro Preto, Ouro Preto, MG, Brasil. E-mail: rafaelcesario@hotmail.com, ricardo@em.ufop.br, andreadiassilva@yahoo.com.br, pandrocha@gmail.com

P. A. S. Rocha

Departamento de Engenharia Civil, Universidade Federal de Ouro Preto, Ouro Preto, MG, Brasil. E-mail: rafaelcesario@hotmail.com, ricardo@em.ufop.br, andreadiassilva@yahoo.com.br, pandrocha@gmail.com

* Corresponding author

Parameter	k
Parameter	1	2	3	4	5	6
a	8.7116	-0.3754	11.6985	-10.7167	0.6177	22.9397
b	-0.1978	4.3806	2.8322	3.0191	10.1889	-3.7310
c	1.2333	-22.0048	-4.1654	-4.3927	-15.7592	5.4520
d	0.0011	-0.1823	9.4775	9.7598	5.0682	-0.0189
e	-0.0386	6.2396	-6.7756	-7.1914	-3.2443	-2.9660

Algotithm: Nonlinear structural static analysis
1.	Read the input data: geometric, material and loading properties of the structural system
2.	Form the interaction curves (initial cracking, initial yield, bearing capacity)
3.	Obtain the reference nodal load vector, F_r (loading direction)
4.	t = 0
5.	t₁ = t
6.	Initial conditions: ^t U = 0 and ^t λ = 0
7.	for each load increment do → INCREMENTAL LOAD PROCESS
8.	t = t₁ → Previous load step
9.	t₁ = t + 1 → Current load step
10	Form the structural system tangent stiffness matrix: K
11.	Solve for the tangent displacement vector: δU _r = K^-1F _r
12.	ift₁ = 1 then
13.	Δλ⁰ = (Δλ⁰)₁
14.	else
15.	Define: Δλ⁰ (Eq. 20)
16.	end if
17.	Evaluate the initial incremental displacement vector: ΔU ⁰ = Δλ⁰ δU_r
18.	Update variables in t₁: ^t ₁ λ = ^t λ + Δλ⁰ and ^t₁ U = ^t U + Δ U⁰
19.	fork = 1, nmax do → ITERATIVE PROCESS
20.	Evaluate the internal forces vector: ^t₁ Fi ⁽^k ^-1) = ^t F _i + KΔU ⁽^k^-1)
21.	Evaluate the residual force vector: g⁽ ^k^-1) = ^t₁ λ⁽^k^-1) F_r - ^t ₁ F_i⁽^k^-1)
22.	if \|\| g⁽^k ^-1) \|\| / \|\| Δλ⁽^k ^-1)F_r \|\| ≤ tolerance factor then
23.	Exit the iterative process and go to line 33
24.	end if
25.	if standard Newton-Raphson method then
26.	Update the tangent stiffness matrix K
27.	end if
28.	Update the load parameter correction, δλ^k (Eq. 21)
29.	Evaluate the nodal displacement correction vector: δU^k = δU^k_g + δλ^k δ U^k_r
30.	Update the load parameter and the nodal displacement vector:
	Δλ^k = Δλ⁽ ^k^-1) + δλ^k and ΔU^k = ΔU⁽^k ^-1) + δU^k_g + δλ^k δU ^k_r
	^t₁ λ^k = ^t λ + Δλ^k and ^t₁ U^k = ^t U + ΔU^k
31.	end for
32.	Update the structural system variables (nodal coordinates, internal forces vector and S_p)
33.	end for

Concrete
f_c (kN/cm²)	f_cr (kN/cm²)	γ_c	ε_ci	ε_cu
4.0	0.269	1.5	-0.002	-0.0035
Steel
f_y (kN/cm²)	E_s (kN/cm²)	γ_s	ϕ (cm)	ε_u
35.5	20500	1.15	2	0.01

Concrete					Steel
f_c (kN/cm²)	f_cr (kN/cm²)	E_c (kN/cm²)	ε_ci	ε_cu	f_y (kN/cm²)	E_s (kN/cm²)	ε_u
3.83	0.292	2930	-0.0023	-0.0035	46.5	20000	0.02

Concrete					Steel
f_c (kN/cm²)	f_cr (kN/cm²)	E_c (kN/cm²)	ε_ci	ε_cu	f_y (kN/cm²)	E_s (kN/cm²)	ε_u
3.65	0.32	2777.1	-0.0023	-0.0035	29.3	20000	0.015

Node	Analysis
Node	BM	P
1	0	0
2	6.2	6.2
3	12.24	12.24
4	6.17	6.17
5	0.1	0.1
6	6.08	6.08
7	12.27	12.27
8	12.28	12.28
9	12.27	12.27
10	6.08	6.08
11	0.1	0.1
12	6.17	6.17
13	12.24	12.24
14	6.2	6.2
15	0	0

{\begin{matrix} Δ M_{p i} \\ Δ M_{b i} \end{matrix}} = [\begin{matrix} S_{p i} & - S_{p i} \\ - S_{p i} & S_{p i} \end{matrix}] {\begin{matrix} Δ θ_{p i} \\ Δ θ_{b i} \end{matrix}}

{\begin{matrix} Δ M_{p j} \\ Δ M_{b j} \end{matrix}} = [\begin{matrix} S_{p j} & - S_{p j} \\ - S_{p j} & S_{p j} \end{matrix}] {\begin{matrix} Δ θ_{p j} \\ Δ θ_{b j} \end{matrix}}

{\begin{matrix} Δ M_{b i} \\ Δ M_{b j} \end{matrix}} = [\begin{matrix} k_{22} & k_{23} \\ k_{32} & k_{33} \end{matrix}] {\begin{matrix} Δ θ_{b i} \\ Δ θ_{b j} \end{matrix}}

{\begin{matrix} Δ P \\ Δ M_{p i} \\ Δ M_{p j} \end{matrix}} = [\begin{matrix} k_{11} & 0 & 0 \\ 0 & S_{p i} - \frac{S_{p i}^{2} (S_{p j} + k_{33})}{β} & \frac{S_{p i} k_{23} S_{p j}}{β} \\ 0 & \frac{S_{p j} k_{32} S_{p i}}{β} & S_{p j} - \frac{S_{p j}^{2} (S_{p i} + k_{22})}{β} \end{matrix}] {\begin{matrix} Δ δ \\ Δ θ_{p i} \\ Δ θ_{p j} \end{matrix}}

\begin{matrix} I f & M > M_{c r} : & I_{e q} = {(\frac{M_{c r}}{M})}^{3} \end{matrix} I_{c} + [1 - {(\frac{M_{c r}}{M})}^{3}] I_{c r}, I_{e q} \leq I_{c}

Δ λ^{0} = \pm Δ λ_{1}^{0} \sqrt{| \frac{({}^{1}δ U_{r}^{T}) ({}^{1}δ U_{r})}{({}^{t}δ U_{r}^{T}) ({}^{t}δ U_{r})} |} = \pm Δ λ_{1}^{0} \sqrt{| G S P |}

\begin{array}{l} σ = {\begin{matrix} \frac{f_{c r}}{1 + \sqrt{500 ε}}, & ε_{c r} < ε \\ E_{t r} ε, ​ & 0 < ε \leq ε_{c r} \\ - f_{c} [\frac{2 ε}{ε_{c i}} - {(\frac{ε}{ε_{c i}})}^{2}], & ε_{c i} < ε \leq 0 \\ - f_{c}, & ε_{c i} \leq ε \leq ε_{c u} \end{matrix} \end{array}

σ = {\begin{matrix} E_{s} ε, ​ & | ε | < ε_{1} \\ sgn (ε) f_{y} (c + \sqrt{c^{2} + 45 | ε | - 0.49}), & ε_{1} < | ε | < ε_{y} \\ sgn (ε) f_{y}, & ε_{y} \leq | ε | \leq ε_{u} \end{matrix}

F (X) = [\begin{matrix} N \\ M \end{matrix}] - [\begin{matrix} \sum_{i = 1}^{n_{l a y}} σ [ε_{i} (ε_{0}, ϕ)] A_{i} \\ \sum_{i = 1}^{n_{l a y}} σ [ε_{i} (ε_{0}, ϕ)] y_{i} A_{i} \end{matrix}]

F' = (\frac{\partial F}{\partial X}) = [\begin{matrix} f_{11} = \sum_{i = 1}^{n_{l a y}} E_{T, i} A_{i} & f_{12} = \sum_{i = 1}^{n_{l a y}} E_{T, i} y_{i} A_{i} \\ f_{21} = \sum_{i = 1}^{n_{l a y}} E_{T, i} y_{i} A_{i} & f_{22} = \sum_{i = 1}^{n_{l a y}} E_{T, i} y_{i}^{2} A_{i} \end{matrix}]

[1] * Corresponding author

Brasil

Brasil

Numerical analysis of RC plane structures: a concentrated nonlinear effect approach

Abstract