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In this paper, for the first time, the nonlinear motion characteristics of a hinged-hinged third-order shear deformable microbeam are examined, based on the modified couple stress theory and the third-order shear deformation theory. The extensibility of the microbeam is modelled by taking into account the longitudinal displacement. The nonlinear equations governing the longitudinal, transverse, and rotational motions are derived by means of Hamilton's principle in conjunction with the modified couple stress theory (to take into account small-scale effects). The three coupled nonlinear partial differential equations are discretized via the Galerkin method and the resulting set of ordinary differential equations is solved by means of the pseudo-arclength continuation technique and via direct time-integration. The effects of the system parameters on the behaviour of the microbeam are studied. Results are presented in the form of frequency-responses and force-responses. Points of interest in the parameter space are also highlighted in the form of time histories, phase-plane portraits, and fast Fourier transforms (FFTs). Moreover, the similarities and differences in the response of the system obtained via the modified couple stress and classical continuum mechanics theories are discussed.

Elastic-plastic, closed-form solutions were developed recently by the author, to capture the nonlinear response of laterally loaded rigid piles. Presented in compact form, the solutions are convenient to use, and sufficiently accurate despite using only two input parameters of the net limiting force per unit length pu along the pile, and a subgrade modulus k. Nevertheless, piles may be subjected to limited cap-restraints or loading below ground surface, which alter the response remarkably.This paper provides explicit expressions for estimating loading capacity of anchored piles and develops new solutions for lateral piles with cap-rotation by stipulating a constant pu or a linear increasing pu (Gibson pu) with depth. Lateral loading capacity Ho (at the tip-yield state and yield at rotation point state) and maximum bending moment Mm (at the tip-yield state) are presented against loading locations, and in form of the lateral capacity Ho-. Mo (applied moment) locus. The capacity is consistent with available solutions for anchored piles, and caissons with either pu profile, allowing a united approach from lateral piles to anchored piles. The new solutions are also presented in charts to highlight the impact of rotational stiffness of pile-cap on nonlinear response, offering a united approach for free-head piles through fixed-head piles.Several advantages of the solutions are identified against the prevalent p-. y curve based approach. To estimate the key parameter pu, values of the resistance factor Np (=ratio of pile-soil limiting resistance over the undrained shear strength su) are deduced using the current expressions against available normalised pile capacity involving the impact of gapping (between pile and soil), pile movement mode, pile slenderness ratio, inclined loading angle (anchored piles) and batter angles (lateral piles). The Np is characterised by: (i) An increase from 5.6-8.6 to 10.14-11.6, as gapping is eliminated around lateral piles and caissons, and

In this study, the rotational resistance of a spiral-type capsule rotating inside a small intestine is investigated by in vitro experiments and analytical modeling, on which a limited literature is available. The results presented exhibit viscoelastic nature of the intestinal tissue. The significance of various spiral structures and rotating speeds is quantitatively evaluated from the propulsion point of view. Also, an analytical torque model is proposed and subsequently validated. The close match between the experimental results and numerical results from the model shows that the model is reasonably accurate to estimate the rotational resistance torque of the small intestine. Both the experimental and modeling works provide a useful guide to determine the torque required for a spiral-type endoscopic capsule operating in a 'really' small intestine. Therefore, the proposed torque model can be used in the design and optimization of in-body robotic systems, which can remotely be articulated using magnetic actuation.

In this study, the rotational resistance of a spiral-type capsule rotating inside a small intestine is investigated by in vitro experiments and analytical modeling, on which a limited literature is available. The results presented exhibit viscoelastic nature of the intestinal tissue. The significance of various spiral structures and rotating speeds is quantitatively evaluated from the propulsion point of view. Also, an analytical torque model is proposed and subsequently validated. The close match between the experimental results and numerical results from the model shows that the model is reasonably accurate to estimate the rotational resistance torque of the small intestine. Both the experimental and modeling works provide a useful guide to determine the torque required for a spiral-type endoscopic capsule operating in a 'really' small intestine. Therefore, the proposed torque model can be used in the design and optimization of in-body robotic systems, which can remotely be articulated using magnetic actuation.

The aim of the present investigation was to examine the critical vortex-induced vibration (VIV) point for cylinders in a rotational system. For translational systems, the existence of a critical mass ratio for cylinders undergoing vortex-induced vibration has been well established. At mass ratios below this critical point, the reduced velocity at VIV lock-out tends to infinity, resulting in the resonance forever condition as described by the authors first reporting this phenomenon. The approach adopted in the present investigation involved measuring the VIV response of a positively buoyant pivoted cylinder being towed at very high reduced velocity. High reduced velocity was attained by establishing a very low system natural frequency. The key finding of this study is the presence of a critical point with a value similar to that of the critical mass ratio in translational systems. This critical point appears to be governed by the force moment ratio rather than the mass moment of inertia ratio. In the present study, a critical force moment ratio of 0.512 ± 0.002 was determined as the onset point for the infinite VIV lock-in regime.

In this study, the rotational resistance of a spiral-type capsule rotating inside a small intestine is investigated by in vitro experiments and analytical modeling, on which a limited literature is available. The results presented exhibit viscoelastic nature of the intestinal tissue. The significance of various spiral structures and rotating speeds is quantitatively evaluated from the propulsion point of view. Also, an analytical torque model is proposed and subsequently validated. The close match between the experimental results and numerical results from the model shows that the model is reasonably accurate to estimate the rotational resistance torque of the small intestine. Both the experimental and modeling works provide a useful guide to determine the torque required for a spiral-type endoscopic capsule operating in a 'really' small intestine. Therefore, the proposed torque model can be used in the design and optimization of in-body robotic systems, which can remotely be articulated using magnetic actuation.

The aim of the present investigation was to examine the critical vortex-induced vibration (VIV) point for cylinders in a rotational system. For translational systems, the existence of a critical mass ratio for cylinders undergoing vortex-induced vibration has been well established. At mass ratios below this critical point, the reduced velocity at VIV lock-out tends to infinity, resulting in the resonance forever condition as described by the authors first reporting this phenomenon. The approach adopted in the present investigation involved measuring the VIV response of a positively buoyant pivoted cylinder being towed at very high reduced velocity. High reduced velocity was attained by establishing a very low system natural frequency. The key finding of this study is the presence of a critical point with a value similar to that of the critical mass ratio in translational systems. This critical point appears to be governed by the force moment ratio rather than the mass moment of inertia ratio. In the present study, a critical force moment ratio of 0.512 ± 0.002 was determined as the onset point for the infinite VIV lock-in regime.

The aim of the present investigation was to examine the critical vortex-induced vibration (VIV) point for cylinders in a rotational system. For translational systems, the existence of a critical mass ratio for cylinders undergoing vortex-induced vibration has been well established. At mass ratios below this critical point, the reduced velocity at VIV lock-out tends to infinity, resulting in the resonance forever condition as described by the authors first reporting this phenomenon. The approach adopted in the present investigation involved measuring the VIV response of a positively buoyant pivoted cylinder being towed at very high reduced velocity. High reduced velocity was attained by establishing a very low system natural frequency. The key finding of this study is the presence of a critical point with a value similar to that of the critical mass ratio in translational systems. This critical point appears to be governed by the force moment ratio rather than the mass moment of inertia ratio. In the present study, a critical force moment ratio of 0.512 ± 0.002 was determined as the onset point for the infinite VIV lock-in regime.

In this study, the rotational resistance of a spiral-type capsule rotating inside a small intestine is investigated by in vitro experiments and analytical modeling, on which a limited literature is available. The results presented exhibit viscoelastic nature of the intestinal tissue. The significance of various spiral structures and rotating speeds is quantitatively evaluated from the propulsion point of view. Also, an analytical torque model is proposed and subsequently validated. The close match between the experimental results and numerical results from the model shows that the model is reasonably accurate to estimate the rotational resistance torque of the small intestine. Both the experimental and modeling works provide a useful guide to determine the torque required for a spiral-type endoscopic capsule operating in a 'really' small intestine. Therefore, the proposed torque model can be used in the design and optimization of in-body robotic systems, which can remotely be articulated using magnetic actuation.

The impact of 3D rotational errors in patient positioning on dose delivered target volumes and critical structures in IMRT was studied. Patient rotational errors ranging from -30 to +30 was introduced to IMRT treatment plans of pelvis, head and neck and brain treatment sites and the impact of rotational error on DVH metrics was assessed. The magnitude of impact of rotational error on the error in dose delivered to the target volume and critical structures depends on the location of the structures from plan isocentre. In studied plans, a maximum percentage difference of up to -9.8(1s=13.4) % in D95 to PTV was observed for head and neck treatments. Similarly, in Brain treatments a maximum difference of up to 24.0(1s=33.0) % in maximum dose of Optic chiasm was observed. The results suggest that failure to correct patient's rotational error results in under-dosage to target volumes and over-dosage to the critical structures in some specific treatment scenarios.

Elastic-plastic, closed-form solutions were developed recently by the author, to capture the nonlinear response of laterally loaded rigid piles. Presented in compact form, the solutions are convenient to use, and sufficiently accurate despite using only two input parameters of the net limiting force per unit length pu along the pile, and a subgrade modulus k. Nevertheless, piles may be subjected to limited cap-restraints or loading below ground surface, which alter the response remarkably.This paper provides explicit expressions for estimating loading capacity of anchored piles and develops new solutions for lateral piles with cap-rotation by stipulating a constant pu or a linear increasing pu (Gibson pu) with depth. Lateral loading capacity Ho (at the tip-yield state and yield at rotation point state) and maximum bending moment Mm (at the tip-yield state) are presented against loading locations, and in form of the lateral capacity Ho-. Mo (applied moment) locus. The capacity is consistent with available solutions for anchored piles, and caissons with either pu profile, allowing a united approach from lateral piles to anchored piles. The new solutions are also presented in charts to highlight the impact of rotational stiffness of pile-cap on nonlinear response, offering a united approach for free-head piles through fixed-head piles.Several advantages of the solutions are identified against the prevalent p-. y curve based approach. To estimate the key parameter pu, values of the resistance factor Np (=ratio of pile-soil limiting resistance over the undrained shear strength su) are deduced using the current expressions against available normalised pile capacity involving the impact of gapping (between pile and soil), pile movement mode, pile slenderness ratio, inclined loading angle (anchored piles) and batter angles (lateral piles). The Np is characterised by: (i) An increase from 5.6-8.6 to 10.14-11.6, as gapping is eliminated around lateral piles and caissons, and

The aim of the present investigation was to examine the critical vortex-induced vibration (VIV) point for cylinders in a rotational system. For translational systems, the existence of a critical mass ratio for cylinders undergoing vortex-induced vibration has been well established. At mass ratios below this critical point, the reduced velocity at VIV lock-out tends to infinity, resulting in the resonance forever condition as described by the authors first reporting this phenomenon. The approach adopted in the present investigation involved measuring the VIV response of a positively buoyant pivoted cylinder being towed at very high reduced velocity. High reduced velocity was attained by establishing a very low system natural frequency. The key finding of this study is the presence of a critical point with a value similar to that of the critical mass ratio in translational systems. This critical point appears to be governed by the force moment ratio rather than the mass moment of inertia ratio. In the present study, a critical force moment ratio of 0.512 ± 0.002 was determined as the onset point for the infinite VIV lock-in regime.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

Prediction of the seismic rotational displacements of retaining wall under passive condition is an important aspect of design in earthquake prone region. In this paper, the pseudo-dynamic method is used to compute the rotational displacements of rigid retaining wall supporting cohesionless backfill under seismic loading for the passive earth pressure condition. The proposed method considers time, phase difference and effect of amplification in shear and primary waves propagating through both the backfill and the retaining wall. The influence of ground motion characteristics on rotational displacement of the wall is evaluated. Also the effects of variation of parameters like wall friction angle, soil friction angle, amplification factor, shear wave velocity, primary wave velocity, period of lateral shaking, horizontal and vertical seismic accelerations on the rotational displacements are studied. The rotational displacement of the wall increases substantially with increase in amplification of both shear and primary waves, time of input motion, period of lateral shaking and decreases with increase in soil friction angle, wall friction angle. The rotational displacements of the wall also increase when the effect of wall inertia is taken into account. Results are provided in graphical form.

A coupled-channel Schrödinger equation (CSE) model of photodissociation, which includes the effects of all interactions between the , , and and the and states, is employed to study the effects of rotation on the lowest- band oscillator strengths and predissociationlinewidths. Significant rotational dependences are found which are in excellent agreement with recent experimental results, where comparisons are possible. New extreme-ultraviolet (EUV)photoabsorption spectra of the key transition of are also presented and analyzed, revealing apredissociationlinewidth peaking near . This behavior can be reproduced only if the triplet structure of the state is included explicitly in the CSE-model calculations, with a spin-orbit constant for the diffuse level which accidentally predissociates . The complex rotational behavior of the and other bands may be an important component in the modeling of EUV transmission through nitrogen-rich planetary atmospheres.

A coupled-channel Schrödinger equation (CSE) model of photodissociation, which includes the effects of all interactions between the , , and and the and states, is employed to study the effects of rotation on the lowest- band oscillator strengths and predissociationlinewidths. Significant rotational dependences are found which are in excellent agreement with recent experimental results, where comparisons are possible. New extreme-ultraviolet (EUV)photoabsorption spectra of the key transition of are also presented and analyzed, revealing apredissociationlinewidth peaking near . This behavior can be reproduced only if the triplet structure of the state is included explicitly in the CSE-model calculations, with a spin-orbit constant for the diffuse level which accidentally predissociates . The complex rotational behavior of the and other bands may be an important component in the modeling of EUV transmission through nitrogen-rich planetary atmospheres.

A coupled-channel Schrödinger equation (CSE) model of photodissociation, which includes the effects of all interactions between the , , and and the and states, is employed to study the effects of rotation on the lowest- band oscillator strengths and predissociationlinewidths. Significant rotational dependences are found which are in excellent agreement with recent experimental results, where comparisons are possible. New extreme-ultraviolet (EUV)photoabsorption spectra of the key transition of are also presented and analyzed, revealing apredissociationlinewidth peaking near . This behavior can be reproduced only if the triplet structure of the state is included explicitly in the CSE-model calculations, with a spin-orbit constant for the diffuse level which accidentally predissociates . The complex rotational behavior of the and other bands may be an important component in the modeling of EUV transmission through nitrogen-rich planetary atmospheres.

Rotational temperatures derived from the OH(8–3) band may vary by ~18K depending on the choice of transition probabilities. This is of concern when absolute temperatures or trends determined in combination with measurements of other hydroxyl bands are important. In this paper, measurements of the OH(8–3) temperature-insensitive Q/P and R/P line intensity ratios are used to select the most appropriate transition probabilities for use with this band. Aurora, airglow and solar and telluric absorption in the OH(8–3) band are also investigated. Water vapour absorption of P1(4), airglow or auroral contamination of P1(2) and solar absorption in the vicinity of P1(5) are concerns to be considered when deriving rotational temperatures from this band. A comparison is made of temperatures derived from OH(6–2) and OH(8–3) spectra collected alternately at Davis (69° S, 78° E) in 1990. An average difference of ~4K is found, with OH(8–3) temperatures being warmer, but a difference of this magnitude is within the two sigma uncertainty limit of the measurements.

Rotational temperatures derived from the OH(8–3) band may vary by ~18K depending on the choice of transition probabilities. This is of concern when absolute temperatures or trends determined in combination with measurements of other hydroxyl bands are important. In this paper, measurements of the OH(8–3) temperature-insensitive Q/P and R/P line intensity ratios are used to select the most appropriate transition probabilities for use with this band. Aurora, airglow and solar and telluric absorption in the OH(8–3) band are also investigated. Water vapour absorption of P1(4), airglow or auroral contamination of P1(2) and solar absorption in the vicinity of P1(5) are concerns to be considered when deriving rotational temperatures from this band. A comparison is made of temperatures derived from OH(6–2) and OH(8–3) spectra collected alternately at Davis (69° S, 78° E) in 1990. An average difference of ~4K is found, with OH(8–3) temperatures being warmer, but a difference of this magnitude is within the two sigma uncertainty limit of the measurements.

Rotational temperatures derived from the OH(8–3) band may vary by ~18K depending on the choice of transition probabilities. This is of concern when absolute temperatures or trends determined in combination with measurements of other hydroxyl bands are important. In this paper, measurements of the OH(8–3) temperature-insensitive Q/P and R/P line intensity ratios are used to select the most appropriate transition probabilities for use with this band. Aurora, airglow and solar and telluric absorption in the OH(8–3) band are also investigated. Water vapour absorption of P1(4), airglow or auroral contamination of P1(2) and solar absorption in the vicinity of P1(5) are concerns to be considered when deriving rotational temperatures from this band. A comparison is made of temperatures derived from OH(6–2) and OH(8–3) spectra collected alternately at Davis (69° S, 78° E) in 1990. An average difference of ~4K is found, with OH(8–3) temperatures being warmer, but a difference of this magnitude is within the two sigma uncertainty limit of the measurements.

It has been pointed out in a previous paper by the authors that conservative internal moments of a spatial beam are of the so-called fourth kind, and that the rotation variables which are energy-conjugate with these moments are vectorial rotations. Vectorial rotations of a spatial Euler–Bernoulli beam have nonlinear relationships with its transverse displacement derivatives. This implies that strictly speaking, the first partial derivative of the strain energy with respect to a transverse displacement derivative is not a bending moment (even if we ignore the axial deformation), and that modifications should be introduced to the conventional Hermitian shape functions employed in the Rayleigh–Ritz method of finite element analysis. On the other hand, the neglect of the rotational behaviour of nodal moments has led to an incorrect stability matrix in the literature, and it is shown through numerical examples that this incorrect stability matrix cannot detect the flexural-torsional buckling load of spatial structures in which the members are not connected collinearly. The validity of the Wagner hypothesis on the coupling of axial and torsional deformations for a spatial beam-column is illustrated through a numerical example. Finally, one issue related to the Updated Lagrangian formulation addressed in this paper is the oft-used assumption of a straight configuration at the last known state.

It has been pointed out in a previous paper by the authors that conservative internal moments of a spatial beam are of the so-called fourth kind, and that the rotation variables which are energy-conjugate with these moments are vectorial rotations. Vectorial rotations of a spatial Euler–Bernoulli beam have nonlinear relationships with its transverse displacement derivatives. This implies that strictly speaking, the first partial derivative of the strain energy with respect to a transverse displacement derivative is not a bending moment (even if we ignore the axial deformation), and that modifications should be introduced to the conventional Hermitian shape functions employed in the Rayleigh–Ritz method of finite element analysis. On the other hand, the neglect of the rotational behaviour of nodal moments has led to an incorrect stability matrix in the literature, and it is shown through numerical examples that this incorrect stability matrix cannot detect the flexural-torsional buckling load of spatial structures in which the members are not connected collinearly. The validity of the Wagner hypothesis on the coupling of axial and torsional deformations for a spatial beam-column is illustrated through a numerical example. Finally, one issue related to the Updated Lagrangian formulation addressed in this paper is the oft-used assumption of a straight configuration at the last known state.

It has been pointed out in a previous paper by the authors that conservative internal moments of a spatial beam are of the so-called fourth kind, and that the rotation variables which are energy-conjugate with these moments are vectorial rotations. Vectorial rotations of a spatial Euler–Bernoulli beam have nonlinear relationships with its transverse displacement derivatives. This implies that strictly speaking, the first partial derivative of the strain energy with respect to a transverse displacement derivative is not a bending moment (even if we ignore the axial deformation), and that modifications should be introduced to the conventional Hermitian shape functions employed in the Rayleigh–Ritz method of finite element analysis. On the other hand, the neglect of the rotational behaviour of nodal moments has led to an incorrect stability matrix in the literature, and it is shown through numerical examples that this incorrect stability matrix cannot detect the flexural-torsional buckling load of spatial structures in which the members are not connected collinearly. The validity of the Wagner hypothesis on the coupling of axial and torsional deformations for a spatial beam-column is illustrated through a numerical example. Finally, one issue related to the Updated Lagrangian formulation addressed in this paper is the oft-used assumption of a straight configuration at the last known state.

It has been pointed out in a previous paper by the authors that conservative internal moments of a spatial beam are of the so-called fourth kind, and that the rotation variables which are energy-conjugate with these moments are vectorial rotations. Vectorial rotations of a spatial Euler–Bernoulli beam have nonlinear relationships with its transverse displacement derivatives. This implies that strictly speaking, the first partial derivative of the strain energy with respect to a transverse displacement derivative is not a bending moment (even if we ignore the axial deformation), and that modifications should be introduced to the conventional Hermitian shape functions employed in the Rayleigh–Ritz method of finite element analysis. On the other hand, the neglect of the rotational behaviour of nodal moments has led to an incorrect stability matrix in the literature, and it is shown through numerical examples that this incorrect stability matrix cannot detect the flexural-torsional buckling load of spatial structures in which the members are not connected collinearly. The validity of the Wagner hypothesis on the coupling of axial and torsional deformations for a spatial beam-column is illustrated through a numerical example. Finally, one issue related to the Updated Lagrangian formulation addressed in this paper is the oft-used assumption of a straight configuration at the last known state.