BPSC AE Civil Paper 5 (Civil) Mock Test - 1 Free Online Test 2026

• The main advantage of using the double integration method for analyzing slopes and deflections in structures is its simplicity, and generality, and gives more accurate results.
• The double integration method is a classical approach that provides a straightforward and systematic way to determine the slope and deflection along the length of a structural member subjected to external loads.
• It may not be as efficient for solving certain complex or irregular structural problems compared to more advanced numerical methods.
• However, for introductory purposes and relatively straightforward structural analyses, the double integration method remains a valuable and widely taught technique.

Additional Information

There are other methods of finding slope deflections other than double integration method.

1. Moment–area method.
2. Mecaulay's method.
3. Conjugate beam method.

Concept:

Area ratio:

It can be defined as the ratio of the maximum cross-sectional area of the cutting edge to the area of the soil sample.

The area ratio can be expressed as

where

D₁ = inner diameter of cutting edge

D₂ = outer diameter of cutting edge

Note:

For stiff formation (A_r)_max = 20%

Soft sensitive clay (Ar)max = 10%

Maximum thickness of cutting edge =

Calculation:

Given, D₂ = 70 mm

Soil is stiff clay, So (Ar)max = 20%

D₁ = 63.9 mm

Maximum thickness of cutting edge =

= = 3.05 mm

Three hinge arch:

• In the case of a three-hinged arch, we have three hinges, two at the support and one at the crown and there are four reaction components in the three-hinged arch.
• One more equation is required in addition to three equations of static equilibrium for evaluating the four reaction components.
• Taking moment about the hinge of all the forces acting on either side of the hinge can set up the required equation, making 3 hinge arch determinate.

Two hinge arch:

• In the two hinge arch, we have 2 hinges at two support of the arch and have four reaction components, as in a 3 hinge arch.
• But here we don't have an additional equation to solve the structure and hence our 2 hinge arch is indeterminate by degree 1.

​Hinge less/ Fixed arch:

• In the fixed arch, both the support are fixed. So, in a fixed arch six reaction forces ( One moment extra on each support ).
• But, we have only 3 equilibrium equations. So, the degree of indeterminacy of fixed arch becomes 6 - 3 = 3.

​​

Concept:

Vertical stress due to concentrated load is given as per Boussinesq’s equation.

where,

σ_z = Stress magnitude, Q = Vertical point load, r = Radial distance, and Z = Vertical distance.

So.

⇒

Here, Z₁ = Z and Z₂ = 2Z

⇒

Free water-cement ratio is either be selected or specified based on the slump value and size of aggregates as per IS methods of concrete mix proportioning. Once the free water-cement ratio is determined, cement content can be calculated using the following formula:

Cement Content =

Note:

1. The formula is used to find out the absolute volume of aggregates, where C and W are cement and water content respectively and S_c is specific gravity of cement.

2. The formula is used to determine the percentage of pulverized ash content in cementing material.

The above two points are based on British DOE method of mix design.

Concept:

Requirements of good earth brick are

1. It should not be mixed with salty water.

2. It must be free from lumps of lime.

3. It should not contain pebbles and organic matter.

4. It should be homogenous.

Following are the constitutions of good brick earth:

(1) Alumina: It is the chief constituent of every kind of clay.

(2) Silica: It exists in clay either as free or combined form.

(3) Lime: A small quantity of lime not exceeding 5 percent is desirable in good brick earth.

(4) Oxides of Iron: A small quantity of oxide of iron to the extent of about 5 to 6 percent is desirable in good brick earth.

(5) Magnesia: A small quantity of magnesia in brick earth imparts yellow tint to the bricks and decreases shrinkage.

Vacuum concrete:

It is the type of concrete in which the entrained air and excess water is removed for improving concrete strength. The water is removed by use of vacuum mats connected to a vacuum pump.

Lightweight concrete:

It can be defined as a type of concrete which includes an expanding agent in it, so that it increases the volume of the mixture while giving additional qualities such as lessened the dead weight.

Pre-stressed concrete:

It is a form of concrete where initial compression is given in the concrete before applying the external load so that stress from external loads are counteracted in the desired way during the service period.

Air entrained concrete:

It has effects on compressive strength of concrete and its workability.

Air entrained concrete increases the workability of concrete without much increase in water-cement ratio.

To maintain the desired compressive strength and workability of concrete together, generally in the case of higher strength concrete, admixtures are used. Air entraining agent is one such concrete admixture to increase the workability without affecting much reduction in compressive strength.

Terzaghi’s bearing capacity factors - Nc, Nq, Nγ depend on the angle of internal friction only and are dimensionless.

Here

Nc = (Nq - 1) cot ϕ

Here

Concept:

Nominal shear stress (τvu)

=

Where V_ueq = Equivalent shear force, b = Width of the beam, and d = effective depth of the beam.

Vueq = Vu + , (For LSM)

Calculation:

Given

Shear force (V_u) = 20 kN

Torque (T_u) = 9 kN-m

Width of the beam (b) = 300 mm

Total depth (D) = 425 mm

Effective cover (d') = 25 mm

Effective depth (d) = D - d' = 400 mm

Equivalent shear force (V_ueq)

= V_u +

Equivalent shear force (Vueq)

= 20 + = 68 kN

Equivalent shear force (Vueq) = 68 kN

Nominal shear stress (τ_vu)

=

Equivalent Nominal shear stress (τvu) = 0.566 N/mm²

Equivalent Nominal shear stress (τvu) = 0.566 MPa

Case 1: Web buckling due to shear will not happen (No need of stiffners)

< 85

Case 2: Transverse stiffners are provided to prevent buckling of web due to diagonal compression which is developed due to shear force

> 85

Case 3: Horizontal stiffeners are provided above NA as they prevent buckling web due to bending compressive stress

> 200

Case 4: Additional horizontal stiffners are provided at NA

> 250

Case 5: Section must be redesigned

> 400

Where tw = Thickness of web, d = Depth of the web

Additional Information

The main purpose of Stiffener in a plate girder is to prevent the buckling of web. In which different type of stiffeners are used.

Stiffeners: Stiffeners are used to make plate girder stiff or rigid.

Different type of stiffeners according to IS 800 : 2007

a) Intermediate transverse web stiffener — to improve the buckling strength of a slender web due to shear.

b) Load carrying stiffener — To prevent local buckling of the web due to concentrated loading.

c) Bearing Stiffener — To prevent local crushing of the web due to concentrated loading.

d) Torsion stiffener — To provide torsional restraint to beams and girders at supports.

e) Diagonal stiffener — To provide local reinforcement to a web under shear and bearing.

f) Tension stiffener — To transmit tensile forces applied to a web through a flange.

Resilience

• Resilience is the total strain energy stored in a given volume of material within the elastic limit.
• On removal of load, this energy is released. In other words, it is the area under the load-deflection curve within the elastic limit.

Toughness

• It is defined as the ability of the material to absorb energy before fracture takes place.

Proof resilience

• The maximum energy which can be stored in a body up to the elastic limit is called proof resilience.​

Modulus of resilience

• It is defined as proof resilience per unit volume.
• It is the area under the stress-strain curve up to the elastic limit.
• 
• 

Fatigue:

(i) When a material is subjected to repeated stresses, it fails at stresses below the yield point stresses. Such type of failure of a material is known as fatigue.

(ii) In materials science, fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading.

Creep:

(i) Creep is the plastic permanent deformation of a structure under constant load for a very long period of time.

(ii) It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increases as they near their melting point.

Resilience

(i) It is the property of materials to absorb energy and to resist shock and impact loads.

(ii) It is measured by the amount of energy absorbed per unit volume within the elastic limit this property is essential for spring materials.

(iii) The resilience of material should be considered when it is subjected to shock loading.

(iv) Resilience is the total strain energy stored in a given volume of material within the elastic limit. On removal of load, this energy is released, Hence it is Recoverable strain energy. In other words, it is the area under the load-deflection curve within the elastic limit.

Plasticity:

(i) This is the ability of material to deform without any rupture by the non-returnable way. After removing the load there are staying permanent deformations.

Endurance limit (σ_e):

(i) It is defined as the maximum value of completely reversed bending stress which a polished standard specimen can withstand without failure, for an infinite number of cycles (usually 106 cycles).

(ii) Hence for steel in fatigue loading the endurance limit is the maximum reversed bending stress it can withstand without failure for an infinite number of cycles.

Concepts:

Unconsolidated undrained(UU) test:

• In the UU test, excess pore water pressure develops in both stages so as there is no drainage, effective stress on the soil will not increase in both stages. i.e. during cell pressure application as well as in the 2nd stage of loading when additional axial stress is applied.
• It is quick to test and maybe complete in 5-10 minutes.
• Such tests are suitable for low permeable soil such as clays with fast loading.
• U test is carried out for evaluation of short-term stability of the structure.

Consolidated Undrained Test (CU test):

• During the first stage of confining pressure, drainage is allowed from the soil sample.
• Hence consolidation will take place. But during vertical shear loading, drainage is not permitted.
• Example: stability analysis of earthen dam during the sudden drawdown.

Consolidation Drain test (CD test):

• Drainage is permitted during both the self-pressure (confining pressure) stage and the shear stage.
• This test is the most time taking and for some soils may take several weeks. So, it is also called ‘slow test;
• CD test is carried out for evaluation of long-term stability of the structure

For linear elastic, isotropic, and homogeneous material, We have relations

E = 2G(1 + μ)

E = 3K (1 – 2μ)

If we know any of two variables from E, G, K, and μ then we can find other variables using these relations.

Hence, If We need to express the stress-strain relationships completely for this material, at least any two of the four must be known.

Additional Information

Elastic Modulus (E)

When the body is loaded within its elastic limit, the ratio of stress and strain is constant. This constant is known as Elastic modulus or Young's Modulus.

Rigidity modulus (G)

When a body is loaded within its elastic limit, the ratio of shear stress and shear strain is constant, this constant is known as the shear modulus.

Bulk modulus (K)

When a body is subjected to three mutually perpendicular like stresses of the same intensity then the ratio of direct stress and the volumetric strain of the body is known as bulk modulus

The relationship between E, K, and, G is:

The effective slenderness ratio of a battened column is increased by 10 % and the effective slenderness ratio of a laced column is increased by 5 %.

Note:

The increase in the effective slenderness ratio of battened columns is two times the increase in the effective slenderness ratio of laced columns.

• As per IS 800: 2007, the effective slenderness ratio of battened columns, shall be taken as 1.1 times the (KL/r), the maximum actual slenderness ratio of the column, to account for shear deformation effects.
• As per CI. 7.6.1.5 of IS 800:2007 states that the effective slenderness ratio of laced columns should be 5 % more than the actual maximum slenderness ratio so that shear deformation effects are accounted for.

Stiffness (k) is defined as the force per unit deflection or moment required per unit rotation.

Stiffness factor for a beam simply supported at both ends is 3EI/L.

Important Points:

For the far end being Hinged Supported:

If a unit rotation is to be caused at an end A for the far end being hinged support. Moment of 3EI / l is to be applied at end and hence stiffness for the member is said to be 3EI / l.

For the far end being Fixed Supported:

If a unit rotation is to be caused at an end A for the far end being fixed supported. Moment of 4EI / l is to be applied at end and hence stiffness for the member is said to be 4EI / l.

Concepts:

The method of joints and method of section both are used for analyzing the truss structures.

Method of Sections

• In this method, we will cut the truss into two sections by passing a cutting plane through the members whose internal forces are to be determined.
• This method permits us to solve directly any member by analyzing the left or the right section of the cutting plane.
• Each section may constitute of non-concurrent force system from which three equilibrium equations can be written.

Σ Fx = 0 and Σ Fy = 0 and Σ M = 0

• Because we can only solve up to three unknowns, it is important not to cut more than three members of the truss. Depending on the type of truss and which members to solve, one may have to repeat the Method of Sections more than once to determine all the desired forces.

Additional InformationMethod of Joints

• The free-body diagram of any joint is a concurrent force system in which the summation of the moment will be of no help.
• Recall that only two equilibrium equations can be written

Σ Fx = 0 and Σ Fy = 0

• This means that to solve completely for the forces acting on a joint, we must select a joint with no more than two unknown forces involved. This can be started by selecting a joint acted on by only two members.
• The method of sections is commonly used when the forces in only a few particular members of a truss are to be determined. The method of sections is always used together with the method of joints to analyze trusses.

Permeability is the property of soil to transmit water through it.

It is usually expressed either as a permeability rate in centimeters per hour (cm/h), millimeters per hour (mm/h), or centimeters per day (cm/d) or as a coefficient of permeability k in meters per second (m/s) or in centimeters per second (cm/s).

For the different soil types as per grain size, the orders of magnitude of permeability are as follows:

Important Points

Permeability (k) =

The factors affecting the permeability of soil can be summarised in the below-tabulated form:

As per IS 13920, the code for earthquake-resistant design, the following are some important provisions:

1. The factored axial stress on the member under earthquake loading should not exceed 0.1f_ck; f_ck is the characteristic compressive strength.
2. The width to depth ratio should be greater than 0.3.
3. The width of a member shall not be less than 200 mm.
4. The overall depth of a beam should not be greater than one-fourth of the clear span.
5. The top, as well as bottom longitudinal reinforcement, shall consist of at least two bars throughout the member length.
6. The reinforcement resisting positive moments at a joint face must be at least half the negative moment reinforcement.
7. The percentage tensile reinforcement should not exceed 2.5 %.

Retaining wall:

• The retaining wall is used to retain earth and resist the lateral pressure of soil at a place with a sudden change in elevation.
• Cantilever type retaining wall is used for heights up to 6m.
• Cantilever type retaining wall has the following components:

1. Stem
2. Toe slab
3. Heel slab

• The stability of a cantilever-type retaining wall against overturning can be obtained by taking moment about the toe of the retaining wall.
• Due to backfill pressure overturning moment is generated which causes the overturning of the retaining wall about the toe of the wall.

Resisting moment against overturning is generated due to the:

1. Self-weight of the wall
2. Backfill over the heel slab

• Let Mr = Resisting moment against overturning and Mo = Overturning moment
• If Mr > MO⇒ Retaining wall is safe against overturning
• If the Heel slab length increases, the Resisting moment also increases as more backfill over the heel slab tends to increase the resistance of the wall against overturning thus wall becomes safe against overturning.

Hence, In the heel slab of retaining wall, reinforcement is provided at the top of the slab.

Concept-

The colours of bricks, as obtained in its natural course of manufacture, depend on the following factors:

• Degree of dryness achieved before burning.
• Natural colour of clay and its chemical composition.
• Nature of sand used in moulding operation.
• Quality of fuel used in burning operation.
• Quantity of air admitted to the kiln during burning.
• Temperature at which bricks are burnt.

Important Points

As per elastic theory of design, the ratio of yield stress to the working stress is called factor of safety, i.e.,

Circumferential or hoop stress:

Longitudinal or axial stress:

where d is the internal diameter and t is the wall thickness of the cylinder.

where d is the internal diameter and t is the wall thickness of the cylinder.

Calculation:

Given:

d = 2 m = 2000 m, t = 10 mm, N = 8, Ultimate strength = 480 MPa

Working Strength (σ_h) = 60 MPa

Maximum or Circumferential or hoop stress,

P = 0.6 MPa

Concept:

For a beam moment carrying capacity depends on the section modulus.

Sectional Modulus (Z):

​Section modulus is defined as the ratio of moment of inertia of a beam about its C.G to the maximum distance of extreme fibre of the beam (ymax) from neutral axis. The strength of a beam is measured by sectional modulus.

Mathematically, Z =

Moment of inertia and the maximum distance of neutral axis from top fiber for a rectangular beam is given by,

and ymax =

Hence,

∴ Z ∝ d2

Calculation:

Given:

d_A = 2d_B

Z_A = 4Z_B

Concept:

Using this equation,

Where,

σ₁ = Major principal stress = σ₃ + σ_d

σ₃ = Cell pressure and σ_d = Axial vertical stress

C = Cohesion and ϕ = Effective angle of inetrnal friction.

Calculation:

Given,

σ₃ = 0 (Soil is laterally unconfined)

ϕ = 0 (For clay soil)

So, σ₁ = σ_d = 20 t/m²

As per the equation,

⇒ 20 = 2C × tan45°

C = 10 t/m²

Concept:

Following are the specifications for steel member exposed to weather:

• When the steelwork is directly exposed to weather and is fully accessible for cleaning and repainting the thickness shall not be less than 6 mm.
• When the steelwork directly exposed to weather and is not accessible for cleaning and repainting the thickness shall not be less than 8 mm.
• When the steelwork is not directly exposed to weather, the thickness of steel in main members shall not be less than 6 mm.
• When the steelwork is not directly exposed to weather, the thickness of steel in secondary members shall not be less than 6 mm.

Power Transmitted by a shaft P

From torsion equation

τ is the allowable shear stress in the shaft, J = Polar moment of inertia, T is the torque acting on the shaft, r is the radius of the shaft

Now from torsion equation

Shear lag occurs when some elements of the member cross-section are not connected.

• Shear lag is a function of the distribution of steel in the section and the length of the load transfer L. The effect of shear lag is less in the large length of the connection.
• Consider an angle section tension member connected with one leg only. Consequently, due to this partial connection, the connected leg will be overloaded and the unconnected leg will not be fully stressed.
• Further, since at the joint/connection more of the load is carried by the connected leg, and it takes the transition distance, for the stress to spread uniformly across the whole angle, stress distribution in the two legs of the section would be different.
• In the transition region, the stress in the connected part of the member may even exceed f_y and go into strain–hardening range; the member may fracture prematurely. Away from joint/connection, the stress distribution is more uniform. In the transition region, the shear transfer lags.
• Since shear lag reduces the effectiveness of the component plates of the tension member that are not connected directly to a gusset plate, the outstanding legs are kept shorter in length. For this reason, unequal angles with long legs connected are preferred.
• It is independent of the type of load and applies to both bolted and welded connections. Of course, bolted connections will be affected more than welded connections because of the reduction of the effective area due to bolt holes. The shear lag can be accounted for by using reduced net area, and is known as effective net area.

Concept:

Design formulas of singly reinforced Beam for limit state design,

1) Limiting depth of neutral axis

2) Actual depth of neutral axis

3) Type of section

xu,max = xu for balanced section

xu,max > xu for under reinforced section

xu,max < xu for over reinforced section

4) Moment of resistance

Mu = 0.36 fck B xu (d - 0.42xu) for compression side

Mu = 0.87 fy Ast (d - 0.42xu) for tensile side

Stress-Strain curve for some materials:

The main constituents of Portland cement are argillaceous and calcareous materials.

Argillaceous materials are the clay, shale, etc.

Calcareous materials are limestone, Chalk, etc.

Characteristics of an Ordinary Portland cement (OPC):

1. It is available in 3 grades – OPC 33, OPC-43, and OPC-53 and the numbers 33, 43, and 53 correspond to 28 days of characteristic compressive strength of cement as obtained from the standard test on cement.
2. Initial setting time shall not be less than 30 minutes and final setting time should not be more than 10 hours.

3. Soundness lies from 5 to 10 mm when calculated from Le-Chatelier’s method.

4. Residue by weight shall not be more than 10 % when sieved on a 90-micron sieve and sieved for 15 minutes continuously.

5. Its tensile strength of 10-15 % of the compressive strength.

6. Its tensile strength should not be less than 2 MPa and 2.5 MPa after 3 days and 7 days respectively.

7. Its compressive strength for 33 grade at 3 days, 7 days, and 28 days is 16 MPa (160 kg/cm2), 22 MPa (220 kg/cm2) and 33 MPa (330 kg/cm2) respectively.

8. Total loss on ignition shall not be more than 5 %.