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A new formulation to calculate the shakedown limit load of Kirchhoff plates under stochastic conditions of strength is developed. Direct structural reliability design by chance con-strained programming is based on the prescribed failure probabilities, which is an effective approach of stochastic programming if it can be formulated as an equivalent deterministic optimization problem. We restrict uncertainty to strength, the loading is still deterministic. A new formulation is derived in case of random strength with lognormal distribution. Upper bound and lower bound shakedown load factors are calculated simultaneously by a dual algorithm.
Stretch-shortening type actions are characterized by lengthening of the pre-activated muscle-tendon unit (MTU) in the eccentric phase immediately followed by muscle shortening. Under 1 g, pre-activity before and muscle activity after ground contact, scale muscle stiffness, which is crucial for the recoil properties of the MTU in the subsequent push-off. This study aimed to examine the neuro-mechanical coupling of the stretch-shortening cycle in response to gravity levels ranging from 0.1 to 2 g. During parabolic flights, 17 subjects performed drop jumps while electromyography (EMG) of the lower limb muscles was combined with ultrasound images of the gastrocnemius medialis, 2D kinematics and kinetics to depict changes in energy management and performance. Neuro-mechanical coupling in 1 g was characterized by high magnitudes of pre-activity and eccentric muscle activity allowing an isometric muscle behavior during ground contact. EMG during pre-activity and the concentric phase systematically increased from 0.1 to 1 g. Below 1 g the EMG in the eccentric phase was diminished, leading to muscle lengthening and reduced MTU stretches. Kinetic energy at take-off and performance were decreased compared to 1 g. Above 1 g, reduced EMG in the eccentric phase was accompanied by large MTU and muscle stretch, increased joint flexion amplitudes, energy loss and reduced performance. The energy outcome function established by linear mixed model reveals that the central nervous system regulates the extensor muscles phase- and load-specifically. In conclusion, neuro-mechanical coupling appears to be optimized in 1 g. Below 1 g, the energy outcome is compromised by reduced muscle stiffness. Above 1 g, loading progressively induces muscle lengthening, thus facilitating energy dissipation.
The compliant nature of distal limb muscle-tendon units is traditionally considered suboptimal in explosive movements when positive joint work is required. However, during accelerative running, ankle joint net mechanical work is positive. Therefore, this study aims to investigate how plantar flexor muscle-tendon behavior is modulated during fast accelerations. Eleven female sprinters performed maximum sprint accelerations from starting blocks, while gastrocnemius muscle fascicle lengths were estimated using ultrasonography. We combined motion analysis and ground reaction force measurements to assess lower limb joint kinematics and kinetics, and to estimate gastrocnemius muscle-tendon unit length during the first two acceleration steps. Outcome variables were resampled to the stance phase and averaged across three to five trials. Relevant scalars were extracted and analyzed using one-sample and two-sample t-tests, and vector trajectories were compared using statistical parametric mapping. We found that an uncoupling of muscle fascicle behavior from muscle-tendon unit behavior is effectively used to produce net positive mechanical work at the joint during maximum sprint acceleration. Muscle fascicles shortened throughout the first and second steps, while shortening occurred earlier during the first step, where negative joint work was lower compared with the second step. Elastic strain energy may be stored during dorsiflexion after touchdown since fascicles did not lengthen at the same time to dissipate energy. Thus, net positive work generation is accommodated by the reuse of elastic strain energy along with positive gastrocnemius fascicle work. Our results show a mechanism of how muscles with high in-series compliance can contribute to net positive joint work.