# Direct Stress and Strain Tutorial Questions

Pin jointed frames, generally, transfer the applied loads by inducing axial tensile or compressive forces in the individual members. The magnitude and sense of these forces can be determined by using standard methods of analysis.

The following tutorial questions require that the forces in the individual elements be solved by either the "method of sections" or the "method of joints".

The "method of sections" involves the application of the three equations of static equilibrium to a two dimensional plane frame. An imaginary section line cuts the frame in two. Since there are only three equations of equilibrium, the section through the frame must not include any more than three members for which the internal forces are unknown.

The "method of joints" considers the isolation of each individual joint. For each joint, as the forces are coincident the moment equation is of no value leaving only two equations of equilibrium available to resolve the forces in the members. The equilibrium of each joint must be considered in a sequence that ensures that there are no more than two unknown member forces in each joint under consideration. Notes:

• The analysis associated with these tutorial questions will round all your input values as follows:
• Forces, e.g. vertical and horizontal reactions (kN) - 1 decimal place.
• Dimensions (m) - 3 decimal places.
• Angles (°) - 1 decimal place.
• Numeric answers are required in all the fields. If you calculate a force as being zero, enter zero (0) in the appropriate field.
• You are only able to print the current question as a tutorial question if you have not submitted your answers.
• Once a tutorial question has been printed, you will not be able to submit your answers, or see the solution to that particular question.

Notation:

• ∅ - Diameter of an object (mm).
• σ - Compressive or tensile stress within a material (N/mm²): σT - tensile stress, σC compressive stress.
• ε - Strain, used to measure the deformation or extension of a body that is subjected to a force or set of forces.
• δ - Displacement of a point of an object to an alternate position (mm).
• Δ - Change in length, area or thickness of an object (mm).
• E - Young's Modulus: property that measures stiffness; it defines a relationship between stress and strain (N/mm²).
• ν - Poisson's ratio: the proportional decrease in a lateral measurement to the proportional increase in length.
• C/C - The distance between the centres of objects (mm). A structural steel member is subjected to a tensile load that is transmitted via a bolt placed through a hole at each end. For the given case calculate the following:

• Direct stress at location X due to tensile force.
• Direct stress at location Y due to tensile force.
• For given permissible stress calculate the maximum load that the member can withstand.
• The change in length of the member when the maximum load is applied.

(Given the maximum permissible stress (σmax) = 275N/mm² and the modulus of elasticity (E) = 210kN/mm²)

### Solution

#### Direct Stress

Calculate the stress at each location due to applied force.

σX =
N/mm2
σY =
N/mm2

Calculate the maximum load member can resist given the permissible stress.

F =
kN

#### Change in Length

Calculate the change in length of the member if the maximum load is applied.

ΔL =
mm A vertical steel member ABC is pin-supported at A and loaded by a force P1 at C. A horizontal steel member BDE is pin-supported at point D and is loaded by a force P2 at E. This bars are joined by a pin at point B. For the given case calculate the following:

• The reactions at the supports.
• The displacement at point C.

(Given the value of Young's modulus, E = 210kN/mm²)

### Solution

#### Reaction Forces

Calculate the reaction forces at at the supports.

RA =
kN
RD =
kN

#### Vertical Displacement

Calculate the displacement at C due to the change in length of the member(s).

δC =
mm Three uniform horizontal members labelled AB, BC and CD are pinned together at point B and C. The ends A and D are pinned to rigid foundations. A horizontal force is applied in the X direction that results in displacements to occur at the joints. For the given case calculate the following:

• Reaction force at support A and D
• The total horizontal force at B that produces the stated displacement at B
• The displacement at joint C

(Given the modulus of elasticity = 210kN/mm²)

### Solution

#### Reaction Forces

Calculate the reaction forces at the supports

HA =
kN
HD =
kN

#### Force at B

Calculate the total force at support B, that produces the stated displacement.

FB =
kN

#### Displacement at C

Calculate the displacement at joint C due to this applied force

δC =
mm A steel plate is subjected to tensile stresses in the Z and X direction as displayed. These tensile stresses will result in an change in the area and thickness of the plate. For the given case calculate the following:

• The strain in each direction
• Change in cross-sectional area
• Change in plate thickness

(Given the value of Young's modulus, E = 210kN/mm², Poisson's ratio, ν = 0.3)

### Solution

#### Strain

Calculate the strains present in each direction.

εX =
εY =
εZ =

#### Change in Area

Calculate the change in the cross-sectional area of the block.

ΔA =
mm2

#### Change in Thickness

Calculate the change in thickness of the block.

ΔT =
mm A cylindrical reinforced concrete water tank is filled with water to a specific depth. This water exerts a pressure onto the walls of the tank, the walls are required to be a certain thickness to resist this pressure. A limiting circumferential hoop stress has been provided that can be used to determine this required thickness. For the given case calculate the following:

• The maximum pressure on the wall
• The resultant force due to this water pressure
• The required thickness of the wall to resist this force

### Solution

#### Hydrostatic pressure

Calculate the maximum hydrostatic pressure on the wall of the tank.

P =
N/mm2

#### Resultant force

Calculate the resultant force due to this hydrostatic pressure.

F =
kN

#### Wall Thickness

What is the required wall thickness resist the applied hydrostatic force?

t =
mm A rigid gate AD is pinned at its base at position A, it is in place to retain water in a tank. The gate is tied to a foundation by steel bars BE which are spaced at a given distance centre to centre. For the given case calculate the following:

• Average pressure on the gate
• Resultant hydrostatic force on the gate
• The force per bar
• Displacement at position's B and D

(Given value of gravity = 9.81N/kg, density of water = 1000kg/m³, Young's modulus of elasticity = 210kN/mm²)

### Solution

#### Pressure

Calculate the average pressure that is applied on the gate due to the retained water.

PAvg =
kN/m2

#### Forces

Calculate the equivalent force on the gate as a result of the pressure and subsequently the force present per bar.

FHydro =
kN
FBar =
kN

#### Displacements

Calculate the displacements that will occur at points B and D as a result of the change in length of the bar.

δB =
mm
δD =
mm A long cylindrical pressure vessel contains plate ends that are bolted in place to ensure the pressure vessel remains sealed. The vessel is used to contain a pressurised fluid. For the given case calculate the following:

• The total fluid thrust on one end.
• The longitudinal stress in the wall of the vessel.
• The hoop stress in the wall of the vessel
• The bolt diameter required to ensure tensile stress is not exceeded.

### Solution

#### Total fluid thrust

Calculate the total hydrostatic force on one end of the vessel.

F =
kN

#### Longitudinal stress

Calculate the longitudinal stress that is on the wall of the vessel due to the hydrostatic force.

σLong =
N/mm2

#### Hoop stress

Calculate the hoop stress that is present in the walls of the vessel.

σHoop =
N/mm2

#### Diameter of Bolts

Calculate the required bolt diameter to ensure that the tensile stress in the bolts does not exceed their stress limitations.

Bolt =
mm A rigid bar ABD is pinned to a foundation at A and is supported by a steel cable BC which is pinned to a foundation at C. The bar is subjected to a uniformly distributed load along the entirety of its length. For the given case calculate the following:

• The force in the cable
• The extension of the bar
• The displacement at D
• The shear force at B
• Location of point of contraflexure
• Bending moment at point of contraflexure and point B

(Given the value of Young's modulus, E = 210kN/mm²)

### Solution

#### Force in the Cable

Calculate the force present in the cable due to the UDL on the member.

FCab =
kN

#### Extension of the Cable

Calculate the extension of the cable as a result of this force.

ΔLCable =
mm

#### Displacement at D

Calculate the subsequent displacement at D.

δD =
mm

#### Shear Force

Calculate the maximum and minimum shear force at support B.

@ VMax =
kN
@ VMin =
kN

#### Point of Contraflexure

Calculate the position of the point of contraflexure between point A and B.

X (from A) =
m

#### Bending Moment

Calculate the bending moment at the position of the point of contraflexure.

MEd @ X =
kNm

Calculate the bending moment at point B.

MEd @ B =
kNm