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Non-linear finite element formulations of reinforced concrete haunched beam (RCHB)

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Abstract

Generally, a reinforced concrete haunched beam (RCHB) is used in the structure for increasing shear capacity at support and reduce self-weight of the structure to minimize cost. Large depth at support expresses stronger characteristics of beam. Therefore, classical strong column and weak beam theory may have violated which shows harmful effect of structure during lateral loading. However, in the present study, linear and non-linear formulae are developed for the 5-noded line elements of RCHB and parametric studies are performed under gravity loading. In parametric studies, the depth of the beam is reduced for the formation of haunch with the variations of haunch angles and lengths. Newly developed non-linear formulations are compared with the ETABS (18.1.1) which represents a good-fit of results up to a haunch angle of 7° and results vary after crossing this angle because of some consideration of present formulations responsible for these variations. Also, these formulations are compared with the Godínez-Domínguez et al. (Eng Struct 105: 99–122, 2015) experimental and numerical results which show similar variations of results like ETABS because of the same reason involved herewith. Most of the cases, forces, and displacements of RCHB are deviated from the haunch angle of 3° based on parametric studies. In addition, acceptance criteria meet when displacements of life safety and collapse preventions are 1.6 and 2.3 times of code prescribed maximum allowable displacement. The non-linear model and solution technique of present research can exhibit acceptance level of RCHB because of residual stress generation of the iterative process although results vary from ETABS and previous studies which indicates that further scope will have for the enhancement of present formulations.

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Abbreviations

L :

Length of haunch beam

h y :

Depth of haunch

L i :

Length of haunch

N j :

Nodes

e#i :

Elements

K ei :

Elements stiffness matrixes

I i :

Moment of inertia of elements

E :

Elastic modulus of haunch beam

b :

Width of haunch beam

K G :

Global stiffness matrix in global coordinate

i :

Number of elements (1, 2, 3 and 4)

j :

Number of nodes (1, 2, 3, 4 and 5)

M ij :

Fixed end moments

F ij :

Fixed end forces

δ j :

Vertical translational nodal displacements

θ j :

Rotational nodal displacements

\(F_{ij}^{N}\) :

Nodal forces

\(M_{ij}^{N}\) :

Nodal moments

\(H_{i1}^{N}\) :

Hardening parameter of first stage of iteration

\({\text{d}}M_{i1}^{N}\) :

Change of moment

\({\text{d}}\varepsilon_{pi1}^{N}\) :

Change of plastic curvature

\(M_{pi1}^{N}\) :

Maximum plastic moment

\(M_{ei}\) :

Maximum elastic moment

\({\text{d}}\varepsilon_{i1}^{N}\) :

Change of total curvature

\({\text{d}}\varepsilon_{ei}\) :

Change of elastic curvature

\(M_{0}\) :

Maximum moment at yield of confinement (steel and concrete)

\(EI_{ik}^{N}\).:

Flexural rigidity at kth stage

\(H_{ik}^{N}\) :

Hardening parameter

\(K_{eik}^{N}\) :

Element stiffness matrix

\(K_{Gk}^{N}\) :

Global stiffness matrix

\(\psi_{{\left( {ij} \right)k}}^{N}\) :

Residual forces

\(\varphi_{jk}^{N}\) :

Nodal displacements

\(f_{{\left( {ij} \right)k}}^{N}\) :

Elements fixed end moment and forces

\(\Delta \varphi_{jk}^{N}\) :

Incremental nodal displacements

\(\varphi_{{j\left( {k + 1} \right)}}^{N}\) :

Nodal displacements at (k + 1)th stage

\(\theta_{{j\left( {k + 1} \right)}}^{N}\) :

Nodal rotational displacements

\({\text{d}}M_{ik}^{N}\) :

Changes of moments at (k + 1)th stage

\(\left( {EI} \right)_{{T\left( {ik} \right)}}^{N}\) :

Tangential flexural rigidity

\(H_{{i\left( {k + 1} \right)}}^{N}\) :

Hardening parameter at (k + 1)th stage

\(f_{{\left( {ij} \right)k}}^{NN}\) :

Nodal forces

\(f_{{\left( {ij} \right)k}}^{NF}\) :

Fixed end moments and forces at individual nodes

k :

Successive number loading stage

r :

Maximum number loading stage

\(w_{{\text{c}}}\) :

Unit weight of concrete

\(f_{c}^{^{\prime}}\) :

Cylindrical strength of concrete

N :

Notation of non-linearity

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  1. Department of Civil Engineering, Bangladesh University of Engineering and Technology, BNUS Research Room, Dhaka, 1000, Bangladesh

    Md. Foisal Haque

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  1. Md. Foisal Haque
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Haque, M.F. Non-linear finite element formulations of reinforced concrete haunched beam (RCHB). Asian J Civ Eng 23, 1029–1064 (2022). https://doi.org/10.1007/s42107-022-00462-8

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