Exact Stiffness Matrix for Nonprismatic Beams with Parabolic Varying Depth

In this paper, an exact stiffness matrix and fixed-end load vector for nonprismatic beams having parabolic varying depth are derived. The principle of strain energy is used in the derivation of the stiffness matrix. The effect of both shear deformation and the coupling between axial force and the bending moment are considered in the derivation of stiffness matrix. The fixed-end load vector for elements under uniformly distributed or concentrated loads is also derived. The correctness of the derived matrices is verified by numerical examples. It is found that the coupling effect between axial force and bending moment is significant for elements having axial end restraint. It was found that the decrease in bending moment was in the range of 31.72%-42.29% in case of including the effect of axial force for the studied case. For midspan deflection, the decrease was 46.07% due to the effect of axial force generated at supports as a result of axial restraint


INTRODUCTION
Members with variable depth are used in many engineering structures such as highway bridges, buildings, as well as in many mechanical components and aerospace engineering structures. In civil engineering construction, nonprismatic members are frequently used to optimize material distribution and stresses, increase the overall stability and stiffness, reduce the dead load positive moment and deflection, and sometimes to satisfy architectural requirements. Accordingly, the analysis of structures having nonprismatic elements is of interest in structural, mechanical, and aerospace engineering. The analysis of nonprismatic members is covered in several publications (e.g., Timoshenko and Young 1965;AL-Gahtani 1996;Al-Gahtani and Khan 1998;Luo et al 2007). Most of the available publications deal with the analysis of tapered members only. Some particular cases (e.g., Timoshenko and Young 1965;AL-Gahtani 1996) deal with the analysis of nonprismatic beams having parabolic varying depth. However, these cases are limited to the analysis only (no stiffness matrix derivation) of such type of members involving lengthy and tedious calculations which are not applicable for use in the analysis packages in which the analysis is based on matrix operations. In addition, the available analytical solutions do not consider the effect of shear deformation and the axial forcebending moment interaction.
The other alternative publications deal with the numerical methods of analysis such as the finite element method (e.g., Bathe 1996) in which the member is discretized to a number of elements and the stiffness matrices of the elements are assembled to obtain the stiffness matrix for the whole member. The main disadvantage resulting from member discretization is the large number of input data required even for simple structures. The purpose of this paper is to present an exact stiffness matrix for nonprismatic beam elements with parabolic varying depth including the effect of shear deformation and the axial force-bending moment interaction. The correctness of the derived stiffness matrix and the fixed-end load vector is examined through numerical examples.

PROBLEM STATEMENT
Consider a nonprismatic Euler-Bernoulli beam element of length L as shown in Fig. 1(a). The element is rectangular in cross-section and has a parabolic varying depth and constant width. Three degrees of freedom are assumed at each node. Only the deformation in the plane of the element and the bending moment about the centroidal main axis are considered. The positive direction of displacements and forces are as shown in Fig.  1 The stiffness components corresponding to the degrees of freedom shown in Fig. 1(b) can be obtained by using Castigliano ' s second theorem (Boresi, A.P. and Schmidt, R.J. 2003), which states that the deflection caused by an external force is equal to the partial derivative of the strain energy (U) with respect to that force. The total strain energy (U) for the element shown in Fig.  1(a) including the strain energy caused by bending moment, shear and axial forces can be given by where M x , Q x , P x , I x , h x , are the bending moment, shear force, axial force, moment of inertia and the depth of the element at the distance x respectively; b, E, G, are the width of the element, Young ' s and shear modulus of elasticity, respectively. The bending moment M x , shear force Q x , and the axial force P x can be found from equilibrium as follows The moment of inertia and the depth of the element cross section at a distance x can be given by 3 2 3 2 3 0 0 (1 ) where h 0 =the minimum depth of the element (at the origin), c=(h 1 -h 0 )/(h 0 L 2 ), and h 1 = the maximum depth of the element. Substituting eqs. (2a), (2b), (2c), and (3) into eq. (1) and after integrations, the following exact expression for the strain energy can be given as 2 2 2 0 0 0 1 0 2 0 2 3 0 4 5 (4) (6) The partial derivative of the strain energy (U) with respect to P i , Q i , and M i can be given respectively as follows where u i , v i , and i θ are the displacement components in horizontal ,vertical directions, and rotation angle at node (i) respectively.
The stiffness coefficient (k ij ) of an element can be defined as the force or moment at node (i) required to induce a unit displacement or rotation at node (j) with all other displacements equal to zero. Therefore, eqs. (7), (8), and (9) will be used to derive the stiffness matrix of the element.

AXIAL STIFFNESS
Applying a unit axial displacement at node (i) with all other displacements equal to zero (i.e. put u i =1, v i = 0, and 0 i θ = in the displacements vector δ ), the stiffness coefficients corresponding to that displacement can be found by solving eq. (11) for the column matrix F , hence From equilibrium, the force vector (or stiffness coefficients) at node (j) corresponding to the unit axial displacement at node (i) can be given as

FLEXURAL STIFFNESS
Following the same procedure given for the derivation of axial stiffness, the flexural (translational and rotational) stiffness coefficients can be obtained by applying a unit lateral displacement or a unit rotation (with all other displacements equal to zero) to obtain the translational or rotational stiffness coefficients, respectively.
Hence, by substituting u i =0, v i = 1, and 0 i θ = in the displacements vector δ , the translational stiffness coefficients can be given as From equilibrium, the stiffness coefficients or the force vector at node (j) corresponding to the unit lateral displacement at node (i) (i.e. u i =0, v i = 1, and 0 i θ = ) can be given as Similarly, the rotational stiffness coefficients can be obtained by substituting u i =0, v i = 0, and Taking advantage of the symmetry characteristic in the stiffness matrix, and from equilibrium requirements, the other coefficients of the 6*6 stiffness matrix can be given as follows where λ is given by eq. (15) Similar results can be obtained for the stiffness coefficients given by eqs. (18a)-(20f) by using the same procedure presented before. Therefore, substituting for M x , Q x , and P x in the strain energy expression (eq. (1)) interms of the nodal force vector at node (j) and following the same previous procedure will yield the same expressions given in eqs. (18a) For a beam element having an orientation as shown in Fig. 2, the stiffness coefficients can be . (2 ) 0 0

FIXED-END LOAD VECTOR DUE TO UNIFORM LOAD
Consider a nonprismatic beam element with a parabolic varying depth under a uniform load q as shown in Fig. 3(a). By using the principle of superposition and knowing that the sum of all displacement components in each direction at the fixed end must be zero, the flexibility matrix equation corresponding to node (i) can be written as The right hand side of eq. (28) represents the free-end displacement vector at node (i) due to applied load q.
From equilibrium, the fixed-end reactions at node (j) can be given as

FIXED-END LOAD VECTOR DUE TO CONCENTRATED LOAD
For a beam element loaded by a concentrated load (P) at an arbitrary location defined by a distance (L 1 ) from the left support as shown in Fig. 3(b), the fixed-end load vector can be derived by using the same procedure given before. The flexibility matrix equation for this case can be given as where ( ) 0 1 2 2 2 2 1 1 1 0 2 2 1 1 1 1 2 8 (1 ) (1 ) (1 ) and all other constants are previously defined.
Solving eq. (39) yields the following expressions for the unknowns fixed-end reactions at node (i) interms of the flexibility matrix coefficients And from equilibrium, the fixed-end reactions at node (j) can be given as

NUMERICAL EXAMPLES
To verify the correctness of the derived matrices, the following examples are considered.

Example 1
Consider the beam shown in Fig. 4 which has a single span of length, L=1units and fixed at both ends. The beam is carrying a uniformly distributed load, q=1. The depth of the beam is h 0 =1units at the left end and increase parabolically to h 1 =2 units at the right end and has a unit width, b=1 units. The beam was analyzed by Khan and Al-Gahtani (1995) by using the boundary integral method (BIM). Using the same dimensionless data adopted by Al-Gahtani and Khan, the beam is reanalyzed by using the derived expressions for

Example 2
Consider a three-span continuous bridge girder having a parabolic varying depth as shown in Fig.  5. The depth of the girder varies from h=2.5 units at both ends and midspan to h=7.5 units at interior supports. This problem has been analyzed by Timoshenko and Young 1965 ;Al-Gahtani and Khan (1998).Using the same dimensionless data, the problem is reanalyzed by using the derived stiffness matrices and the fixed-end load vector. The analysis results (at nodes 1,2,3,4,and 5) are presented in Table 2 together with those obtained by Timoshenko and Young (1965) ; Al-Gahtani and Khan (1998). It can be seen that when the girder is restraint against horizontal (axial) displacement at supports (i.e. all supports are hinges), the results diverge significantly from that obtained by other methods as given in the last column of Table 2. This is due to the coupling effect between the axial force generated from axial restraint and the bending moment which reduces the displacement, rotations, and bending moments.

Example 3
Finally, consider the beam shown in Fig. 6 which has a span of unit length and a depth varies parabolically from h=1.0 units at left end to h=2.0 units at the right end. The beam is supported at the left end on a translational spring with a stiffness constant of K=10 , fixed at the right end and carrying a concentrated load P=1.0 at the left end. The beam is analyzed by using the derived stiffness matrix interms of the given dimensionless data. The analysis results are presented in Table 3 in which the third column show the results when the shear deformation is considered. The results show a significant effect for shear deformation. This is due to the large translational stiffness relative to the rotational stiffness for this beam.

SUMMARY AND CONCLUSIONS
In this paper, an exact stiffness matrix and fixedend load vector for beams with parabolic varying depth are derived. An exact integrations were carried out to obtain the strain energy equation including bending, shear, and axial strain energies which is used to obtain the exact expressions for the coefficients of the stiffness matrix. The correctness of the derived expressions is examined through numerical examples. It is found that the derived stiffness matrices and the equivalent load vector are efficient for the analysis of structures having members with parabolic varying depth. Furthermore, the derived matrices can be used in the structural analysis softwars as compared to the available analytical solutions. The obtained results show a significant effect for axial force-bending moment coupling in continuous beams with axial restraint. , φ φ = variables defined by eq. (6) and eq. (41) respectively; and ψ = variable defined in eq. (37).

NOTATION
The following symbols are used in this paper: