The structural analysis of structures is the determination of the internal forces resulting from the effect of externally applied loads while structural design is the use of these internal stresses to determine the reinforcement and sections of concrete and steel required to effectively bear the imposed loads without failure. Structural analysis requires the knowledge of structural mechanics which include the mechanics of forces, the mechanics of deformation, and the theory of structures (the science of the response of structural systems to external loads).
Structural Analysis Methods
There are two broad methods of analysis of structures namely:
Elastic Analysis
This method is based on the elastic theory and is based on the assumption of Hooke’s law that the relationship between force (F) and displacement (d) is linear. F is directly proportional to d. Besides, displacement is assumed here to be extremely small compared to the geometry of the structure. This method is utilized in the working stress or permissible stress method of design.
The methods under elastic analysis are classified into four:
i. Classical methods: This includes the consistent deformation method, slope deflection method, and method of strain energy.
ii. Relation/iterative method: This includes the moment distribution method and Kani’s method.
iii. Computer methods: This includes the matrix method, finite difference method, and finite element method.
iv. Approximate method: This includes the substitute frame method, portal frame method, cantilever method, and coefficient method (this is where some coefficients mentioned in certain codes are used to obtain bending moments).
Limit State Analysis
This method is based on the plastic or ultimate load theory and utilized in the limit state method of design. This type of analysis deals with the study and behaviour of structures and members at collapses such as the yielding of steel and the crushing of concrete. The aim is to ensure that the ultimate state of yielding of steel and the crushing of concrete is not allowed by the utilization of suitable safety factors. The determination of the suitable safety factors on the other hand requires adequate knowledge of the strength and behaviour of structures on collapse. This philosophy helps to maximize the effective use of these structural elements. The merit of the limit state method lies in the higher load-carrying capacity for an indeterminate structure due to the redistribution of the moment.
Merits of Redistribution of Moments
a. In the case of indeterminate structures, it helps to reduce bending moments in the peak regions such as beam-column junctions or supports of continuous beams, thereby the congestion of reinforcement is reduced making detailing and concreting easier.
b. Reduction in support moment not only helps in reducing steel at supports but utilizes higher moment resisting capacity of a flanged section in the span region.
c. It ensures reinforced failure since the depth of the neutral axis decreases with the increase in the percentage redistribution of moments.
d. It gives distribution of moments along the length of the member makes detailing easier and gives economic design.
e. It is not only reducing the moment at support but, many times, it also does not increase the design moment at mid-span. This can be seen in the case of a continuous beam designed of maximum moments decided by bending moment envelop i.e. diagram for a maximum moment got by all likely loading arrangements consideration.
Structural Design Methods/Philosophies
Reinforced concrete structures can be designed by using one of the philosophies:
1. Working stress method (WSM)
2. Ultimate load method (ULM)
3. Limit state method (LSM).
Working Stress Method (WSM)
This is also known as the Modular Ratio Method: this is the traditional method of design utilised in both reinforced concrete as well for steel structures. Close to about a hundred years old, the method is based on linear elastic theory or classical elastic theory. This method of design was introduced around 1900 and was the first theoretical method accepted by the National Codes of Practice for the design of reinforced concrete sections. In the method, loads are assessed as the working (actual) loads but limit the permissible stresses in the concrete and reinforcement to a fraction of their actual stresses in order to provide an adequate factor of safety. The assumption of linear elastic behavior is also taken into consideration and justifiable since the specified permissible or allowable stresses are kept well below the ultimate strength of the material. The ratio of the yield stress of the steel reinforcement or the cube strength of the concrete, the corresponding permissible or working stress is usually called the factor of safety. The WSM uses a factor of safety of about 3 for the cube strength of concrete and a factor of safety of about 1.8 with respect to the yield strength of steel. Reinforced concrete is a composite material. The WSM assumes strain compatibility, in which the strain in reinforcing steel is assumed to be equal to that in the adjoining concrete to which it is bounded. Consequently, the stress in steel is linearly related to the stress in adjacent concrete by an invariable factor known as modular ratio which is defined as the ratio of modulus of elasticity of steel to that of concrete. The WSM is therefore also known ratio method.
Merits of WSM
WSM has the following merits:
i. Its simplicity both in concept as usual and in design generally results in relatively large sections of structural members in comparison to the ULM. Due to this, structures designed by WSM give better serviceability performance examples (i.e. less deflection, less crack width, etc,) under working loads.
ii. WSM is the only method available when one has to investigate the reinforced concrete selection for service stresses and for the serviceability states of deflection and cracking. It is essential to have knowledge of WSM since it forms a part of limit state design (LSD) for a serviceability condition.
Demerits of WSM
i. The WSM does not show real strength nor give the true factor of safety.
ii. The modular ratio design results in a larger percentage of compressive steel than that given by the limit state design thus leading to an uneconomic design.
iii. Because of the creep and nonlinear stress-strain relationship, concrete does not have a definite modulus of elasticity.
iv. Fails to discriminate between different types of loads so as to act concurrently but contain different uncertainties.
Ultimate Load Method (ULM)
The ultimate load method (ULM) evolved in 1950 as the WSM. In the method, the section is analysed at failure, the actual strength of a section being related to the actual load causing failure, with the latter being determined by applying a factor to the design load. Hence the method is also referred to as the factor method or the ultimate strength method. This makes it possible to use different load factors under combined loading conditions. It is to be carefully noted that satisfactory strength presentation at ultimate loads will not promise satisfactory serviceability in the plastic region (inelastic region) because the ultimate strength of the materials is used in the calculation hence, no variations in materials strength are taken into account; for this reason, it cannot be used for the serviceability states of deflection and cracking. Also, the method does not consider the effects of creep and shrinkage. To summarize, the ultimate load method ensures safety at ultimate loads but disregards the serviceability at service loads.
Merits of ULM
i. While the WSM uses only the nearly linear part of the stress-strain curve, ULM uses fully the actual stress-strain curve. In other words, the stress block parameters are defined by the actual stress-strain curve.
ii. The load factor gives the exact margin of safety against collapse.
iii. The method allows to use of different load factors for different types of combinations thereof.
iv. The failure load computed by ULM matches the experimental results.
v. The method is based on the ultimate strain failure criteria. The method utilises the reserve of strength in the plastic region.
Demerits of ULM
i. The method does not take the serviceability criteria of deflection and cracking.
ii. The utilisation of high strength reinforcing steel and concrete results in an increase in crack width.
iii. The method does not take into consideration the effects of creep and shrinkage.
Limit State Method (LSM)
We have seen that while the WSM gives satisfactory performance of the structure at working loads, it is unrealistic at the ultimate state of collapse. Similarly, while the ULM provides a realistic assessment of safety, it does not guarantee satisfactory requirements at service loads. An ideal method is one that takes into account not only the ultimate strength of the structure but also serviceability and durability requirements. This is where the limit state method of design comes in. A limit can be defined as a state of impending failure, beyond which a structure stops performing its planned or intended function satisfactorily, in terms of either safety or serviceability, i.e. it either collapses or becomes unserviceable. In the limit state method, the structure is considered safe against collapse (that is for ultimate strength) as well as checked for serviceability consequently rendering the structure very fit for its planned use. Thus, the structure is considered at both the working and the ultimate load levels with a view to satisfying the requirements of safety and serviceability.
The aim of structural design is to attain an acceptable likelihood that the structure does not become unfit for which it is intended, i.e., it will not reach a limit state. To ensure the above objectives, the design should be based on characteristics values for material strengths which take into account the loads to be supported and variations of material strengths. The characteristic values have to be based on the availability of statistical data; if such data is not available, then the experience basis is to be considered. Design using the LSM is done by multiplying the working loads by the partial factors of safety and dividing the ultimate material strengths by further partial factors of safety.
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