TIMOSHENKO BEAM THEORY EXACT SOLUTION FOR BENDING, SECOND-ORDER ANALYSIS, AND STABILITY

Abstract: This paper presents an exact solution to the Timoshenko beam theory (TBT) for bending, secondorder analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory neglects shear deformations. A material law (a moment−shear force−curvature equation) combining bending and shear is presented, together with closed-form solutions based on this material law. A bending analysis of a Timoshenko beam was conducted, and buckling loads were determined on the basis of the bending shear factor. First-order element stiffness matrices were calculated. Finally second-order element stiffness matrices were deduced on the basis of the same principle.


Introduction
First-order analysis of the Timoshenko beam is routine in practice: the principle of virtual work yields accurate results and is easy to apply. Unfortunately, second-order analysis of the Timoshenko beam cannot be modeled with the principle of virtual work. Pirrotta et al. [1] presented an analytical solution for a Timoshenko beam subjected to a uniform load distribution with various boundary conditions. Sang-Ho et al. [2] presented a nonlinear finite element analysis formulation for shear in reinforced concrete Beams: that formulation utilizes an equilibrium-driven shear stress function. Abbas et al. [3] suggested a two-node finite element for analyzing the stability and free vibration of Timoshenko beam: interpolation functions for displacement field and beam rotation were exactly calculated by employing total beam energy and its stationing to shear strain. Hayrullah et al. [4] performed a buckling analysis of a nano sized beam by using Timoshenko beam theory and Eringen's nonlocal elasticity theory: the vertical displacement function and the rotation function are chosen as Fourier series. Onyia et al. [5] presented a finite element formulation for the determination of the critical buckling load of unified beam element that is free from shear locking using the energy method: the proposed technique provides a unified approach for the stability analysis of beams with any end conditions. Jian-Hua Yin [6] proposed a closed-form solution for reinforced Timoshenko beam on elastic foundation subjected to any pressure loading: the effects of geosynthetic shear stiffness and tension modulus and the location of the pressure loading were investigated. In stability analysis Timoshenko and Gere [7] proposed formulas to account for shear stiffness by means of calculation of buckling loads of the associated Euler-Bernoulli beams. Hu et al. [8] used matrix structural analysis to derive a closed-form solution of the second-order element stiffness matrix: the buckling loads of single-span beams were also determined. In this paper a material law combining bending and shear is presented: this material law describes the relationship between the curvature, the bending moment, the bending stiffness, the shear force, and the shear stiffness. Based on this material law closed-form expressions of efforts and deformations are derived, as well as first-order and second-order element stiffness matrices. Stability analysis is conducted, the buckling lengths of single-span systems being determined on the basis of shear factor.

Governing equations 2.1.1 Statics
The sign conventions adopted for the loads, bending moments, shear forces, and displacements are illustrated in Figure 1  Sign convention for loads, bending moments, shear forces, and displacements Specifically, M(x) is the bending moment in the section, V(x) is the shear force, w(x) is the deflection, and q (x) is the distributed load in the positive downward direction.
In first-order analysis the equations of static equilibrium of an infinitesimal element are as follows: (1) According to the Timoshenko beam theory, the bending moment and the shear force are related to the deflection and the rotation (positive in clockwise) of cross section (x) as follows: In these equations E is the elastic modulus, I is the second moment of area,  is the shear correction factor, G is the shear modulus, and A is the cross section area. Equations (1), (3), and (4) also apply in second-order analysis. (4) can be formulated as follows

Material and geometric equations Equation
Differentiating both sides of Equation (5) with respect to x results in the following equation: Substitution of Equation (3) into Equation (6) yields the following: Equation (7) yields the following material law combining bending and shear for the Timoshenko beam: For For a continuously varying cross section along segments of the beam, substituting Equation (1) into Equation (8) yiel yields, In the case of non-uniform heating, the material law (Equation (9)) becomes: (11) Rotation of the cross section (Equations (1) and (5)) yields the following: The governing equations are the equation of static equilibrium (Equation (2)) and the material law (Equation (9) or Equation (10)). The shear force -bending moment relationship (Equation (1)), and the rotation of the cross section (Equation (12)) are used to satisfy the boundary conditions and the continuity equations. Equations (5) to (12) apply as well in first-order analysis and in second-order analysis.

Non uniform heating
The bending shear factor is defined as follows: (13)

First-order analysis of the Timoshenko beam 2.2.1 Beam without elastic Winkler foundation
The application of Equation (2) yields the following formulation of the bending moment: Substitution of Equations (2) and (14) into the material law (Equation (9) for a beam with constant cross section) yields: The integration constants Ci (i = 1, 2, 3, and 4) are determined using the boundary conditions and continuity equations and combining the deflections, the cross section rotations (Equation (12)), the bending moments, and the shear forces (Equation (1)).

Element stiffness matrix
The sign conventions for bending moments, shear forces, displacements, and rotations adopted for use in determining the element stiffness matrix in local coordinates is illustrated in Figure 2.
Considering the sign conventions adopted for bending moments and shear forces in general (see Figure 1) and for bending moments and shear forces in the element stiffness matrix (see Figure 2), we can set following static compatibility boundary conditions in combination with Equations (1) Considering the sign conventions adopted for the displacements and rotations in general (see Figure 1) and for displacements and rotations in the member stiffness matrix (see Figure 2), we can set following geometric compatibility boundary conditions in combination with Equations (20) and (20a): The combination of Equations (16) to (18) and Equations (21) to (28) yields the first-order element stiffness matrix of the Timoshenko beam: The development of Equation (28a) yields the following widely known formulation of the first-order element stiffness matrix of the Timoshenko beam: Assuming the presence of a hinge at the right end, the sign convention for bending moments, shear forces, displacements, and rotations is illustrated in Figure 3.
The element stiffness matrix can be expressed as follows: ; ;

Beam resting on an elastic Winkler foundation
The application of Equation (1) to M(x) yields the shear force, and the combination of M(x) with Equations (12) and (38) yields the rotations of the cross section. The combination of M(x), Equations (12) and (41) yields the rotation of the cross section as follows:

Beam resting on an elastic Winkler foundation with an axial load
In the equation of static equilibrium, the axial force N (the value of which is positive in tension) is assumed to be constant in segments of the beam. The stiffness of the Winkler foundation is denoted by Kw.
The combination of Equation (71) with the material law (Equation (9)) yields the following: Differentiating twice both sides of Equation (72) with respect to x and combining the result with the material law (Equation (9)) yields the following:

Stability of the Timoshenko beam
The equation of static equilibrium (Equation (39)), the material law (Equation (9) or Equation (10)), the bending moment M(x), the deflection w(x), the rotation of the cross section (x), and the transverse force T(x) are used to determine the buckling load, at which the transverse loading q(x) is equal zero. For k  -1, Equations (13) and (47) define the parameters  and k, and Equations (49) to (55) are considered to satisfy the boundary conditions and continuity conditions. For k  -1, Equations (60) to (66) are considered to satisfy the boundary conditions and continuity conditions. The resulting eigenvalue problem is solved to determine the buckling loads.

First-order analysis of Timoshenko beams 3.1.1 Beams subjected to uniformly distributed load Example 1:
Let us calculate the responses of a beam with a constant cross section, simply supported at its ends, and subjected to a uniformly distributed load as shown in Figure 5.
Details of the results are presented in the supplementary material "Spreadsheet S1." Closed-form expressions of single-span systems for various support conditions are presented in Appendix B. Table 2 lists the results obtained with the principle of virtual work and those obtained in the present study.

Beams subjected to concentrated load Example 2:
Let us calculate the responses of a beam with a constant cross section fixed at its left end, simply supported at its right end, and subjected to a concentrated load, as shown in Figure 6.  Table 3 lists the moments at the fixed end (MFEM) and under the load (MuL) for different values of the bending shear factor, calculated as described in this paper and according to the principle of virtual work (exact values).

Second-order analysis of Timoshenko beams 3.2.1 Stability of beams
We determine the buckling loads of single-span beams with various support conditions for different values of the bending shear factor. Details of the analysis and results are listed in Appendix D and in the supplementary materials "Spreadsheet S3," "Spreadsheet S4," "Spreadsheet S5,"and "Spreadsheet S6." The buckling load Ncr is defined as follows: Values of the buckling factor  are listed in Table 4. Closed-form expressions of the matrices expressing the boundary conditions are presented in Appendix D. To determine the buckling loads the determinants of those matrices are set to zero. Closed-form expressions of the buckling factors for a pinned-pinned beam and a fixedfree beam are also presented in Appendix D. We recall that the exact values of the buckling factors  for = 0.0 (corresponding to the Euler-Bernoulli beam) for the conditions SS-SS, F-SS, F-FR and F-F are 1.00, 0.700, 2.00, and 0.500, respectively.

Beams subjected to uniformly distributed load Example 3:
Let us calculate the responses of a beam with a constant cross section, simply supported at its ends and subjected to a uniformly distributed load and an axial force, as shown in Figure 7.  (47)) and  (Equation (13)). Closed-form expressions of the moments are presented in Appendix E. The formulations for the moments at position L/2 in the case of a compressive force with k  -1 (Equation (76)) and in the case of a tensile force or compressive force with k  -1 (Equation (77)) are presented below. The limits of k corresponding to buckling are listed in the supplementary material "Spreadsheet S3". Details of the results are presented in the supplementary material "Spreadsheet S7". Table 5 lists the values of the moments.

3.2.2a Beams subjected to concentrated load Example 4:
Let us calculate the responses of a beam with a constant cross section, fixed at its left end, simply supported at its right end, and subjected to a concentrated load and an axial load, as shown in Figure 8.  Table 6 lists the moments at the fixed end (MFEM) and under the load (MuL) for different values of the bending shear factor and axial force.

Element stiffness matrix
and Details of the calculations are presented in the supplementary material "Spreadsheet S9".
The results are identical. In fact, both formulas are identical since following equivalences exist between the parameters considered by Hu et al. [8] (, ), and those considered in the present study (k,, 1, 2 ): The equations for the determination of the buckling loads (see Appendix D) are also identical to those of Hu et al. [8] (Table 1 of [8]). However, the formula presented by Hu et al. [8] only applies for compressive forces with k  -1.
For the same beam subjected to a tensile force k = 1.5 (Equation (47)), the stiffness matrix is calculated with Equation (67): (81)

Discussion
The material law developed in this study enables the derivation of closed-form solutions for first-order analysis, second-order analysis, and stability of Timoshenko beams. The results show that the calculations conducted as described in this paper yield accurate results for first-order and second-order analysis of Timoshenko beams. Closed-form expressions of second-order element stiffness matrices (the axial force being tensile or compressive) in local coordinates were determined. The determination of element stiffness matrices (ESM) enables the analysis of systems with the direct stiffness method. We showed that ESM can also be determined by the presence of hinges (Equations (33), (59), and (69a) to (70)).

Calculation of bending moments and shear forces:
Influence of tensile force: With increasing tensile force, bending moments decrease (in absolute values), and with increasing bending shear factor, bending moments decrease (in absolute values). Influence of compressive force: With increasing compressive force, bending moments increase (in absolute values), and with increasing bending shear factor, bending moments also increase (in absolute values). Stability of the beam: With increasing bending shear factor, the buckling load decreases (increase of the buckling length). The following aspects not addressed in this study could be examined in future research: ✓ Analysis of linear structures, such as frames, through the transformation of element stiffness matrices from local coordinates in global coordinates. ✓ Second-order analysis of frames free to sidesway with consideration of P- effect. ✓ Use of the direct stiffness method, since element stiffness matrices are presented. ✓ Closed-form expressions of bending moments and deflections for second-order analysis. ✓ Analysis of positions of discontinuity (interior supports, springs, hinges, abrupt change of section), since closed-form expressions of bending moments, shear or transverse forces, rotation of cross sections, and deflection are known.  (1-cos1) 2 -sin1(2-sin1) = 0 → 2-2cos1 + k/1 sin1 = 0 (D7) In the case of a tensile force or a compressive force with k  -1, the bending moment (Equations (40) and (60)) is expressed as follows: (E3) The boundary conditions are described by Equations (E3) and (63). The resulting bending moment is expressed as follows: For k = -1, with Equations (40) and (47), the bending moment is M(x) = pl²/k = -pl². For k = 0 (first-order analysis), the bending moments do not depend on . The value at the position L/2 is pl²/8.