Simplified limit load approximations and their bounds

Mahmood, Sakib Lutful (2015) Simplified limit load approximations and their bounds. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

From a design point of view, a robust simplified limit load solution is the one which is consistently lower bound, yet provides a better estimate compared to the classical lower bound limit load. The robustness is determined by its proximity to the exact limit load. There are several limit load multipliers such as multiplier m"α (Seshadri and Indermohan, 2004), multiplier m"α (Simha and Adibi-Asl, 2012), two bar multiplier (Seshadri and Adibi-Asl, 2007), and multiplier (Seshadri and Hossain, 2009) which provides reasonable estimates of limit loads. However their nature of bounds has not been examined. In this thesis limit load bounds for these multipliers have been investigated. Finally, the nature of bounds of all the limit load multipliers in the literature are summarized, where bounds are either already established or will be addressed in this thesis. The lower bound estimate of the multiplier m"μ relies on the exact distribution of plastic flow parameter. It is found that for an approximate distribution of flow parameter, m"μ is either upper bound or its bounds are not obvious. Since the exact distribution of plastic flow parameter is only available from the limit state stress distribution, the multiplier m"μ could not be established as a lower bound based on the linear elastic analysis. Simha and Adibi-Asl (2012) proposed an inequality relation (m"< m"μ) for lower bound m". It is concluded that the inequality (m"< m"μ ) could not guarantee a lower bound m", when m"μ is estimated from an approximate distribution of plastic flow parameter. In order to investigate limit load bounds of the two bar solution, reference two bar multiplier (which gives bounding limit load over the other two bar configurations) is first identified by performing general two bar analysis. Since a mechanical component or structure can be represented by a suitable multi bar model, a general multi bar analysis is then performed. It is found that the reference two bar multiplier bounds the limit load solution of multi bar models. A correction factor has also been introduced to the reference two bar solution in order to eliminate any possibility of overestimation of limit loads using reference two bar multiplier. Hence the proposed estimate of reference two bar solution provides lower bound limit load. However limit load estimation using this multiplier at times could be conservative (although offers much better accuracy than classical lower bound) compared to the exact limit load. The mαᵀmultiplier which offers better accuracy than the two bar multiplier is also established as a lower bound by investigating exact solution trajectory, utilizing the constraint map construction. Also, it is found that the mαᵀ multiplier bounds the limit load solution of multi bar models, which confirms the lower bound nature of the mαᵀ multiplier. A guideline is proposed to obtain sufficiently accurate lower bound limit load based on a single linear elastic analysis. In terms of elastic modulus adjustment procedure (EMAP), classical lower bound limit load multiplier is susceptible to oscillations with iterations, when sharp modulus adjustments are performed thereby raising convergence issues. On the other hand, more gentle element modulus adjustments turn out to be computationally expensive. In this thesis, the mα-tangent multiplier is used in conjunction with the elastic modulus adjustment procedure for limit load determination. The lower boundedness of the mα-tangent multiplier for any iteration is ensured by incorporating reference volume and peak stress corrections. By the virtue of the faster convergence feature, the mαᵀ-multiplier permits gentler modulus adjustments, and at the same time estimates sufficiently accurate lower bound limit load within a relatively small number of elastic iterations. This minimizes the convergence difficulties usually encountered in EMAP. Simplified techniques on the basis of linear elastic finite element analysis (FEA) assumes elasticperfectly- plastic material model. However, due to strain hardening, a component or a structure can store supplementary strain energy and carry additional load. In this thesis, an iterative elastic modulus adjustment scheme is developed for strain hardening material model, utilizing the “strain energy density” theory. The proposed algorithm is then programmed into repeated linear elastic FEA and implemented to a number of practical components. Moreover, the procedure for elastic modulus adjustment to achieve limit state and elastic-plastic state are explained in parallel, to demonstrate their similarity and diversity.

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