Transverse opening in a beam has a reducing effect of the beam stiffness which will cause a significant increase in beam deflection in the region on the opening. In this paper, a new stiffness matrix for a beam element with transverse opening including the effect of shear deformation has been derived. The strain energy principle is used in the derivation process of the stiffness matrix and the fixed-end force vector for the case of a concentrated or a uniformly distributed load is also derived. The accuracy of the obtained results based on the derived stiffness matrix is examined through comparison with that of the finite element method using Abaqus package and a previous study which show a good agreement with high accuracy.
In general, transverse openings are made in beams to allow the passage of the service lines through the structure. Depending on the size and location, opening may reduce flexural stiffness of the beam significantly which will cause an increase in beam deflection as well as a decrease in strength of the beam. When a large opening is located near the support of a beam, the shear stiffness is reduced significantly and the deflection is highly increas- ed in the opening region which will make the beam serviceability unacceptable if it is not taken into account.
There are a limited number of studies that deal with the formulation of stiffness matrix of a beam with a transverse opening or deflection calculation of such beams. Benitez et al. [
Donghua et al. [
The main objective of this paper is to derive an accurate stiffness matrix and a fixed-end force vector for a beam with transverse opening that are useful and simple for matrix analysis and software applications.
Consider a two dimensional (2D) beam element with a transverse opening as shown in
To simplify the derivation of stiffness matrix, it is assumed that the bending moment in the opening region consists of two parts: primary and secondary moments. The primary moment is caused by the normal tension-compression forces couple acting on the section which has a little effect on the total deflection while the secondary moment is caused by the shear force acting on the upper and lower beams. The secondary moment cause a significant increase in deflection in the opening region as shown in
The stiffness coefficients have been found by using Castigliano’s second theorem Boresi and Schmidt [
in which
A = cross-sectional area of the imperforated beam, h = total depth of the beam, b = with of the beam, h1 = depth of the opening, G = shear modulus, E = elastic modulus. After simplifications, carrying out the necessary integrals, and substituting the above expressions into Equation (4), the following expression for the total strain energy can be given:
The partial derivative of the total strain energy with respect to Pi, Qi, and Mi is equal to the corresponding nodal displacements which can be given as follows:
where
The axial stiffness coefficients can be found by setting the axial nodal displacement equal to (1) with all other displacements equal to zero (i.e.
The axial force at node (j), (Pj), can be found from equilibrium condition. Accordingly,
The translational stiffness coefficients corresponding to a unit translational displacement at node (i), (vi), can be found by equating the translational nodal displacement equal to (1) with all other displacements equal to zero (i.e.
Solving the above equation for Pi, Qi, and Mi, the following stiffness coefficients can be obtained:
Similarly, from equilibrium, the following stiffness coefficients can be given:
In the same previous procedure, the rotational stiffness corresponding to a unit rotation at node
The rotational stiffness coefficients can be given as follows:
From equilibrium,
Making use of the symmetry of the stiffness matrix and from equilibrium requirements, the other stiffness matrix coefficients can be obtained as follows:
The (6 × 6) stiffness matrix can be written as follows:
Consider a fixed-end beam with a transverse opening is acting upon by a concentrated load (P) as shown in
Making use of superposition and the condition that the sum of nodal displacements in each direction must equal to zero at the fixed end and neglecting the effect of axial deformation (i.e.
where
in which
for
for
for
Solving Equation (47) for QFi, MFi yields the following:
Consider a fixed-end beam with a transverse opening is acting upon by a concentrated load (W) as shown in
Following the same previous procedure, the following expressions for
Substituting Equations (48), (57) and (58) into Equation (56) the values of QFi and MFi for a uniformly distributed load (W) can be obtain.
To examine the correctness of the derived stiffness matrix and the fixed-end force vectors, the simply supported steel I-beam shown in
Another verification for the obtained results is made by the finite element analysis of the same beam using Abaqus package [
In this article, a new stiffness matrix and fixed-end force vectors for a 2D-beam element have been derived including the effect of shear deformation. It has been found that the existence of the opening enlarges the maximum
deflection of the beam significantly for the studied cases. The ratio of the maximum deflection (at the opening region) to the maximum deflection of the solid beam (no opening) is 3.32 for the case of a concentrated load at midspan and 4.39 for a uniformly distributed load. This is due to the effect of secondary moments acting on the upper and lower beams at the opening region. The maximum ratio for the increase in deflection due to the effect of shear deformation is found in the range of 4.48% - 6.96% for the cases under consideration. Through the verification of results, a good agreement has been observed between the obtained results and that obtained from the finite element analysis in addition to that available in the literature. The derived stiffness matrix and the fixed-end force vectors are useful and simple to use in the matrix structural analysis packages.