Structural Analysis Book

Author : Russel Charles Hibbeler

This is most effective book for students and professionals who pursue study and practices in Structural Engineering. The theory and Problems links with practical approaches and real engineering judgements.

Use it for study and Practices only.

R.C Hibbeler

Civil Engineering Formulas

All the Formulas and calculation used in technical & design calculation in Civil Engineering and the related fields of Civil Engineering are incorporated in this Book.

Author: Tyler Gregory Hicks

Used this Book only for Study and Practices.

Civil Engineering Formulas

ASCE 7-22

American Society for Civil Engineer (ASCE) has published the latest version (ASCE 7-22) for Minimum Design Load and Associated Criteria for Building and Other Structure describes the means for determining design loads including dead, live, soil, flood, tsunami, snow, rain, atmospheric ice, seismic, and wind loads and their combinations for general structural design.
Structural engineers, architects, and building code officials will find the structural load requirements essential to their practice.

Download the Document for Study and Design purpose only.

ASCE 7-22

ACI 318-19

aci 318-19

American Concrete Institute (ACI) has published the Building Code Requirements for Structural Concrete which is the guide for Reinforced concrete structure while designing the structure. It provides the background of material and rationale for the code provision.

ACI has revised the guide after every 3 years. ACI 318-19 is the latest version. Originally it is made in US but many other countries has followed this document. It has two parts, both are incorporated in this document:

  • Building Code Requirements for Structural Concrete
  • Commentary on Building Code Requirements on Structural Concrete

Download this Document only for Study purpose and practical help.

ACI 318-19

Balanced Section, Under Reinforced Section & Over Reinforced Section

Parameters Used Here are:
  • Ast= Area of Steel in Tension Zone
  • fst or σst= Stress in Steel
  • fst.u or σst.u= Ultimate Stress in Steel
  • fc or σc = Stress in Concrete
  • fy = Yield strength of steel
  • fck = Characteristics strength of concrete
  • ϵc= Strain in concrete
  • ϵst = Strain in Steel
  • Nc or nor Xu = Critical Neutral Axis
  • N or n= Actual Neutral Axis
  • D= Total Depth of Beam
  • d= Effective depth of beam (from centroid of steel in tension zone to topmost fiber of concrete in compression zone).
  • Note: The value of stress and strain shown in the definition below for each sections of concrete when the section is Design by limit state method (LSM), because it beyond the elast region of stress strain curve.

Balanced Section

Definition: “The RCC Section which is reinforced with such amount of steel that when extreme fiber of compression zone of concrete reaches to its permissible allowable stress and strain value c =0.0035 and fc=0.45fck), the steel provided in tension zone also reaches to its permissible yield allowable strain and stress value st= (0.002+(0.87fy/Es)) and fs= 0.87fy) at same time then that section will be a Balanced or Critical Section.”

  • The failure of such section may be due to compression or tension. So balanced section is basically the combination of both brittle and ductile section.
  • The Neutral Axis of such sections lies in the middle of the section (n=nc) is called Critical Neutral Axis, normally denoted by nc or Xubal.
Lever Arm= d-n/3
  • The Moment of resistance (Mr) of such sections can be determined by Multiplying the lever arm of the stressed section either by Compression force or Tension force because the centroidal of both the areas (Compression and Tension zone) are almost at same distance from neutral axis.

(Moment of Resistance) Mr= Compressive Force*(d-0.42d)

or Mr= Tensile Force*(d-0.42d)

Balanced Section

Over-Reinforced Section

Definition: “The RCC Section which is reinforced with such amount of steel that when extreme fiber of concrete in compression zone reaches to its permissible allowable Strain and stress valuec =0.0035 and fc=0.45fck), while the allowable stresses in steel provided in tension zone doesn’t reaches to its permissible allowable yield strain and stressst= (0.002+(0.87fy/Es)) and fs= 0.87fy) then that section will be an Over-Reinforced Section.”

  • It means that the percentage of reinforcement provided more than the requirements.
  • That RCC element or section will fail due to brittleness which is dangerous to any section.
  • The section is uneconomical due to high percentage of reinforcement provided.
  • In such sections the actual Neutral Axis (NA) will move downward below Critical Neutral Axis (Nc) or n>nc.
  • The Moment of resistance(M r) of such sections will be always more than the Balance section.

            (Moment of resistance) Mr=b.n.(fst/2) *(d-n/3)

Over-balanced Section

Under-Reinforced Section

Definition: “The RCC Section which is reinforced with such amount of steel that when extreme fiber of concrete in compression zone doesn’t reach to its permissible allowable strain and stress valuec =0.0035 and fc=0.45fck), while the allowable stresses in steel provided in tension zone doesn’t reaches to its permissible allowable yield strain and stressst= (0.002+(0.87fy/Es)) and fs= 0.87fy) then that section will be an Under-Reinforced Section.”

This is the most desirable section(Under-Reinforced) because:

  • It means that the percentage of reinforcement provided less than the requirements, so saving steel cost.
  • That RCC element or section will fail in Ductile behavior which is the most desirable condition for any structure.
  • Show enough warning before failure because first steel has to reach its yield stress before concrete.
  • The Moment of resistance of such sections will be always less than the Balance section.
  • In such sections the actual Neutral Axis (NA) will move upward above Critical Neutral Axis (Nc) or n<nc.
  • Moment of resistance (Mr) of such sections can be determined by considering the stress of steel.

           (Moment of Resistance) Mr=fst.Ast(d-n/3)

Under-Reinforced Section

Note: Now You can evaluate Neutral Axis by the sections given below:

Different Sections with the Neutral Axis

Watch the video… if still any any Query you can comment…Thank You


Structural Design Philosophies

Purpose of Structural Design is to design Structures which is economical, safer & serviceable for a specified lifetime.

The three(3) major Design Philosophies used while designing of RCC, Steel or any type of Civil Engineering Structures are discussed here:
  1. ASD (WSD or WSM)
  2. ULM (USD)
  3. LRFD (LSD or LSM)

(1) ASD (WSD)

                   Allowable Stress Design (ASD) also called Working Stress Design/method (WSD) is basically Allowable Strength Design. In this method we apply a Factor of Safety (FoS) on ultimate strength of the material as whole and compare the results with the actual load that actually to be applied during the life span of the structure, if actual load < allowable load then the design is OK. This method only satisfy the serviceability of the structure or Serviceability Limit State (SLS).


  • ASD method uses only the elastic region of stress-strain curve of materials due to which most of the strength acts as a reserve.
  • ASD gives an uneconomical dimensions of the structural element.
  • ASD doesn’t show any sign about collapse of structure, it means doesn’t satisfy the Ultimate Limit State.
  • ASD uses only material strength reduction factors in the form of FoS: Which are Given below:
  1. Factor of Safety (FoS) for Concrete= 3
  2. Factor of Safety (FoS) for Steel= 1.78
Stress-Strain Curve for Concrete

(2) ULM (USD 0r Load factor)

Ultimate Load Method (ULM) or Ultimate Strength Design (USD) or Load Factor is the design philosophy of design structures in which load increasing factor is used while calculating different loads combinations and these factors generally (>1) to increase loading so is to adjust the uncertainties occurring in load applications. Some Load Combination According to ASCE and UBC-97 are Listed here:

ASCE Load Combinations
  • 1.4D
  • 1.2D + 1.6L + 0.5(Lr or S or R)
  • 1.2D + 1.6(Lr or S or R) + (L or 0.5W)
  • 1.2D + 1.0W + L+0.5(Lr or S or R)
  • 1.2D + 1.0E + L +0.2S
  • 0.9D + 1.0W
  • 0.9D + 1.0E
UBC-97 Load Combinations
  • 1.4D                                                       
  • 1.2D + 1.6L +0.5 (Lr or S)           
  • 1.2D +1.6 (Lr or S) + (f1L or 0.8W)       
  • 1.2 D + 1.3W + f1L + 0.5 (Lr or S)     
  • 1.2 D + 1.0E + (f1L + f2S)                   
  • 0.9D ± (1.0E or 1.3W) 

Load factor is the ratio of ultimate strength to the service loads

The ULM makes it possible to consider the effects of different loading acting simultaneously thus solving the shortcomings of WSM (WSD). As the ultimate strength of the material is considered we will get much slender/thinner sections for different elements of the structures e.g columns, beams etc. In this method we use the non-linear method of stress-strain curves for concrete and steel.

Stress-Stain Curve for Steel Rebar


  • This method doesn’t satisfy the Serviceability Limit State (SLS).
  • This method gives very thin sections of members which is vulnerable to severe cracks and damaged.
  • This method only consider load factors while ignoring the strength of materials.

(3) LRFD (LSD or LSM)

                     Load and Resistance Factor Design (LRFD) or Limit State Method/Design (LSM or LSD) is the most advanced method while design any civil engineering structures. It considered both factored loads and material strength reduction factors partially.

There are Two Limit States which is satisfied in LRFD (LSM) Method:

  1. Ultimate Limit State: It considers strength, overturning, fatigue, sliding etc.
  2. Serviceability Limit State: It Considers Cracks width, deflection, vibration etc.

  This method use to multiply partial safety factors for required safety at ultimate load and serviceability at working load.

Partial Safety Factor for Materials:

The strength of concrete in actual structure is taken as (0.67*characteristic strength), i.e 0.67fck. The partial safety factor for (Ultimate limit state) for concrete is 1.5 and that for steel is 1.15. The value for concrete higher because it has more variability and uncertainties compared to steel. The partial safety factor ( for serviceability limit state) for concrete and that for steel is taken is 1. This is taken unity as we are interested in estimating the actual deflections and crack widths during service loads.

Partial Safety Factor for Loads:

Various load Combinations are used and it may be different in different design codes.

For Ultimate Limit States:

  • UL= 1.5(DL+LL)
  • UL=1.5(DL+QL) or 0.9DL+1.5QL (QL: Earthquake/Wind Load)
  • UL=1.2(DL+LL+QL) (1.2 because the probability of three loads reaching its peak together are less)

For Serviceability Limit States:

  • SL= 1.0 (DL+LL)
  • SL=1.0 (DL+QL)
  • SL=1.0DL+0.8LL+0.8QL

The load factor is taken as 0.8 in the third case as the probability of wind load or earthquake load acting with the peak of live load is less. For all other cases we consider the safety factor is 1.0 because we consider the serviceability of structure here.

So Why LRFD/LSM Design Philosophy is more reliable?
  •  LRFD/LSM is a more reliable and statistical based method for predicting both loads and material strengths.
  • Whereas the allowable stress safety factors were based on engineering judgement and past experiences.
  • This method gives you an economical, safer & serviceable structural elements/members.
  • The shortcomings in the previous methods were addressed and rectified here on a more rational basis.

Watch the video given below, this article has been discussed briefly:

========Thank You=======

Factor of Safety (FoS) and Margin of Safety (MoS)

In this blog you will learn about the Factor of Safety (FoS) and Margin of Safety (MoS) that is used in the design of Civil Engineering Structures e.g Buildings, Bridges, Over-head Tanks etc.

Factor of Safety (FoS)

It is the load carrying capacity of any structure or system beyond the expected or actual loads.

Factor of Safety (FoS) used in the design of any civil engineering structures e.g Buildings, Bridges, Over-Head tanks and other RCC, Steel, Wood and composite structures.

The Term Ultimate Stress and Allowable Stress is can be expressed as:

Ultimate Stress = Ultimate Load, Failure Load

Allowable Stress = Allowable load, Design Load

Purpose of FoS

The main purpose of FoS to take some extra loads or to resist some accidental loads by the structure that may occur during the life of that structure. So the structure designer intentionally built the structure stronger then the allowable load. The accidental loads may be due to Earthquakes, Wind, Snow or any other dead or live load that is unexpected in the structure.

Mathematical Representation of FoS

Factor of Safety (FoS) = (Ultimate Stress)/(Allowable Stress)

FoS= ðus/ðas Eq (1)


ðus= 600 N/mm2, ðas= 300 N/mm2

Put the values of ðus and ðas in the above Eq (1)

FoS = (600)/(300) = 2

FoS = 2 , mean that the structure can still take load twice of Allowable Load.

You can derive any value from Eq (1) if there is any two of them are known.

Margin of Safety (MoS)

MoS can be simply defined as it is the margin/gap provided from the allowable stress to ultimate stress. Margin of Safety (MoS) totally depends onFactor of Safety (FoS).

Purpose of MoS

The main purpose of MoS to add some extra margin to adjust or resist an accidental loads by the structure that may occur due to any reason in the structure.Through Margin of Safety (MoS) the Structural designer store some reserve capacity in the structure to take extra loads beyond the allowable load. Those Structures whose margin of safety (MoS) is equal or greater than the value one are more resistant to accidental loads. The MoS value adds the margin to structure to take some additional loads.

Mathematical Representation of MoS

Margin of Safety (MoS) = (Failure Load)/( Design Load) – 1

FoS = (Failure Load)/( Design Load)

Mos = FoS – 1

if FoS = 2

MoS = 2-1 = 1

So MoS = 1 , mean that the structure has still reserve the load carrying capacity upto 100% more than the allowable load.


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