When shear force at a point is zero, then bending moment at that point will be

The maximum slope of a cantilever carrying a point load at its free end is at the

The total area under the stress-strain curve of a mild steel specimen tested to failure under tension is a measure of its:

If the section modulus of beam is reduced the bending stress will

Tensile test was performed on a round bar, after fracture it was found that the diameter remains approximately same at fracture. The material under test was

The tangential force per unit area is:

A material in which rupture takes place with little or no plastic deformation is said to be

Tangential stress in a cylinder is given by [symbols have their usual meanings].

Torsional sectional modulus is also known as

Long axially loaded columns tends to deflect about

Differentiate between: (i) Statically determinate and statically indeterminate structures (ii) Bending stress and shear stress of beam

Determine the reactions, and draw the shear force and bending moment diagram for the beam loaded as shown in the Fig. 1

An I-section has a depth of 200 mm, flange width of 120 mm, flange thickness of 15 mm and web thickness of 10 mm. Determine the percentage of the BM and SF are carried by the flange and web individually.

A rectangular block of material is subjected to a tensile stress of $ 110 \text{ N/mm}^2 $ on one plane and a tensile stress of $ 47 \text{ N/mm}^2 $ on the plane right angle to the former. Each of the above stresses is accompanied by a shear stress of $ 63 \text{ N/mm}^2 $ and that associated with the former tensile stress tends to rotate the block anticlockwise. Find: (i) the direction and magnitude of the principal stress and (ii) the magnitude of the greatest shear stress.

At a certain point in a strained material, the intensity of stresses on two planes right angle to each other are $ 20 \text{ N/mm}^2 $ and $ 10 \text{ N/mm}^2 $ both tensile. They are accompanied by shear stress of magnitude $ 10 \text{ N/mm}^2 $. Find graphically or otherwise the location of principal planes and evaluate the principal stresses.

State the assumption of theory of pure bending and derive expression for bending stress in case of simple bending.

Determine the member forces in AB, CH, BH and CG for the truss shown in Fig. 2

A rectangular column of wood 3 m long carries a load of 300 kN. Determine whether or not section of size 200 mm X 150 mm will be able to carry this load if the factor of safety 3 is to be used, assume Euler's formula is applicable $ E = 12.5 \text{ GPa} $ and the permissible stress is 12 MPa. If this section will not be able to carry this load, design a square section to do so.

A solid steel shaft 5 m long is stressed at 80 MPa when twisted through $ 4^\circ $. Using $ G = 83 \text{ GPa} $, compute the shaft diameter. What power can be transmitted by the shaft at 20 Hz?

A cylindrical steel pressure vessel 400 mm in diameter with a wall thickness of 20 mm, is subjected to an internal pressure of 4.5 MN/m². (a) Calculate the tangential and longitudinal stresses in the steel. (b) To what value may the internal pressure be increased if the stress in the steel is limited to $ 120 \text{ MN/m}^2 $? (c) If the internal pressure were increased until the vessel burst, sketch the type of fracture that would occur.

Write the assumption made in Euler's Formula for column and its limitation.

Derive Multi-axial stress-strain relationships among shear stresses and strains for linear isotropic elastic materials.

(a) Castigliano's theorem

(b) Maxwell Bettie's reciprocal theorem

(c) Failure theories

(d) Stability of dams

(e) Yield design

(f) Stress and strain tensor

When shear force at a point is zero, then bending moment at that point will be

The maximum slope of a cantilever carrying a point load at its free end is at the

The total area under the stress-strain curve of a mild steel specimen tested to failure under tension is a measure of its:

If the section modulus of beam is reduced the bending stress will

Tensile test was performed on a round bar, after fracture it was found that the diameter remains approximately same at fracture. The material under test was

The tangential force per unit area is:

A material in which rupture takes place with little or no plastic deformation is said to be

Tangential stress in a cylinder is given by [symbols have their usual meanings].

Torsional sectional modulus is also known as

Long axially loaded columns tends to deflect about

Differentiate between: (i) Statically determinate and statically indeterminate structures (ii) Bending stress and shear stress of beam

Determine the reactions, and draw the shear force and bending moment diagram for the beam loaded as shown in the Fig. 1

An I-section has a depth of 200 mm, flange width of 120 mm, flange thickness of 15 mm and web thickness of 10 mm. Determine the percentage of the BM and SF are carried by the flange and web individually.

A rectangular block of material is subjected to a tensile stress of $ 110 \text{ N/mm}^2 $ on one plane and a tensile stress of $ 47 \text{ N/mm}^2 $ on the plane right angle to the former. Each of the above stresses is accompanied by a shear stress of $ 63 \text{ N/mm}^2 $ and that associated with the former tensile stress tends to rotate the block anticlockwise. Find: (i) the direction and magnitude of the principal stress and (ii) the magnitude of the greatest shear stress.

At a certain point in a strained material, the intensity of stresses on two planes right angle to each other are $ 20 \text{ N/mm}^2 $ and $ 10 \text{ N/mm}^2 $ both tensile. They are accompanied by shear stress of magnitude $ 10 \text{ N/mm}^2 $. Find graphically or otherwise the location of principal planes and evaluate the principal stresses.

State the assumption of theory of pure bending and derive expression for bending stress in case of simple bending.

Determine the member forces in AB, CH, BH and CG for the truss shown in Fig. 2

A rectangular column of wood 3 m long carries a load of 300 kN. Determine whether or not section of size 200 mm X 150 mm will be able to carry this load if the factor of safety 3 is to be used, assume Euler's formula is applicable $ E = 12.5 \text{ GPa} $ and the permissible stress is 12 MPa. If this section will not be able to carry this load, design a square section to do so.

A solid steel shaft 5 m long is stressed at 80 MPa when twisted through $ 4^\circ $. Using $ G = 83 \text{ GPa} $, compute the shaft diameter. What power can be transmitted by the shaft at 20 Hz?

A cylindrical steel pressure vessel 400 mm in diameter with a wall thickness of 20 mm, is subjected to an internal pressure of 4.5 MN/m². (a) Calculate the tangential and longitudinal stresses in the steel. (b) To what value may the internal pressure be increased if the stress in the steel is limited to $ 120 \text{ MN/m}^2 $? (c) If the internal pressure were increased until the vessel burst, sketch the type of fracture that would occur.

Write the assumption made in Euler's Formula for column and its limitation.

Derive Multi-axial stress-strain relationships among shear stresses and strains for linear isotropic elastic materials.

(a) Castigliano's theorem

(b) Maxwell Bettie's reciprocal theorem

(c) Failure theories

(d) Stability of dams

(e) Yield design

(f) Stress and strain tensor

The property of a body to return to its original shape after removal of force is known as

Maximum stress theory is applicable to

Strain is a example of

A hollow prismatic beam of circular section is subjected to a torsional moment, the maximum shear stress occurs at

Consider the following statements for a thick walled cylinder, subjected to an internal pressure, which of the following is correct.

A composite member was formed at $ 20^{\circ}C $ and was made of two materials A and B. If the coefficient of thermal expansion of A is more than that of B and composite member is heated to $ 140^{\circ}C $ then

The shear force and bending moment are zero at the free end of a cantilever, if it carries a

Graphical representation of which one of the following theories is an ellipse?

If the Euler load of a steel column is 1000 KN and crushing load is 1500 KN, the Rankine load is equal to

The degree of indeterminacy of the beam given below is

A rolled steel joist RSJ of I section has top and bottom flanges $ 150 \text{ mm} \times 25 \text{ mm} $ and web of the size $ 300 \text{ mm} \times 12 \text{ mm} $. It is used as a simply supported beam over a span of 4 m to carry an uniformly distributed load of $ 80 \text{ kN/m} $ over its entire span. Draw bending and shearing stresses across the section at $ 1/4^{\text{th}} $ span.

A wooden beam $ 150 \text{ mm} \times 250 \text{ mm} $ is simply supported over a span of 5 m. When a concentrated load W is placed at a distance a, from the left support. The maximum bending stress in beam is $ 11.2 \text{ N/mm}^2 $. Determine W and a.

(a) State the assumptions made in theory of simple bending.

(b) Differentiate between maximum principal stress theory and maximum shear stress theory.

(c) Derive an expression of buckling load for column with one end fixed and the other end free.

(d) Draw and explain the stress-strain curve for brittle materials.

(e) What are the virtual work methods? Write their importance.

Assuming $ E = 160 \text{ GPa} $ and $ G = 100 \text{ GPa} $ for a material, a strain tensor is given as, $ \begin{bmatrix} 0.002 & 0.004 & 0.006 \\ 0.004 & 0.003 & 0 \\ 0.006 & 0 & 0 \end{bmatrix} $. Determine the shear stress, $ \tau_{xy} $.

What are the assumptions of Euler's theory of column? Also derive the expression for the buckling load for hinged condition of column.

A hollow shaft of diameter ratio 3/8 (internal dia. to outer dia.) is to transmit 375 kW power at 100 rpm. The maximum torque being 20% greater than the mean torque. The shear stress is not to exceed $ 60 \text{ N/mm}^2 $ and twist in a length of 4 m is not to exceed $ 2^{\circ} $. Calculate its external and internal diameter which would satisfy both the above conditions. Assume modulus of rigidity $ G = 0.85 \times 10^5 \text{ N/mm}^2 $.

What do you mean by plastic deformation of a material? Discuss the behaviour of the material when loaded beyond the elastic limit.

Draw shear flow diagram and locate shear centre for the channel section.

Differentiate between dam and a retaining wall.

A masonry dam, trapezoidal in cross-section, 4 m high, 1 m wide at its top and 3 m wide at its bottom, retains water on its vertical face to a maximum height of 3.5 m from its base. Determine the maximum and minimum stresses at the base (i) When the reservoir is empty, and (ii) When the reservoir is full. Take the unit weight of masonry as $ 19.62 \text{ kN/m}^3 $.

Determine the maximum deflection of the simply supported beam using double integration method. The beam is made of wood having a modulus of elasticity of $ E = 210 \text{ GPa} $ and cross-section $ 300 \text{ mm} \times 400 \text{ mm} $ in dimension.

Find the forces in the members of the frame shown in Fig. all members have the same sectional area and are made of the same material. Illustrate the forces in the diagram.

The property of a body to return to its original shape after removal of force is known as

Maximum stress theory is applicable to

Strain is a example of

A hollow prismatic beam of circular section is subjected to a torsional moment, the maximum shear stress occurs at

Consider the following statements for a thick walled cylinder, subjected to an internal pressure, which of the following is correct.

A composite member was formed at $ 20^{\circ}C $ and was made of two materials A and B. If the coefficient of thermal expansion of A is more than that of B and composite member is heated to $ 140^{\circ}C $ then

The shear force and bending moment are zero at the free end of a cantilever, if it carries a

Graphical representation of which one of the following theories is an ellipse?

If the Euler load of a steel column is 1000 KN and crushing load is 1500 KN, the Rankine load is equal to

The degree of indeterminacy of the beam given below is

A rolled steel joist RSJ of I section has top and bottom flanges $ 150 \text{ mm} \times 25 \text{ mm} $ and web of the size $ 300 \text{ mm} \times 12 \text{ mm} $. It is used as a simply supported beam over a span of 4 m to carry an uniformly distributed load of $ 80 \text{ kN/m} $ over its entire span. Draw bending and shearing stresses across the section at $ 1/4^{\text{th}} $ span.

A wooden beam $ 150 \text{ mm} \times 250 \text{ mm} $ is simply supported over a span of 5 m. When a concentrated load W is placed at a distance a, from the left support. The maximum bending stress in beam is $ 11.2 \text{ N/mm}^2 $. Determine W and a.

(a) State the assumptions made in theory of simple bending.

(b) Differentiate between maximum principal stress theory and maximum shear stress theory.

(c) Derive an expression of buckling load for column with one end fixed and the other end free.

(d) Draw and explain the stress-strain curve for brittle materials.

(e) What are the virtual work methods? Write their importance.

Assuming $ E = 160 \text{ GPa} $ and $ G = 100 \text{ GPa} $ for a material, a strain tensor is given as, $ \begin{bmatrix} 0.002 & 0.004 & 0.006 \\ 0.004 & 0.003 & 0 \\ 0.006 & 0 & 0 \end{bmatrix} $. Determine the shear stress, $ \tau_{xy} $.

What are the assumptions of Euler's theory of column? Also derive the expression for the buckling load for hinged condition of column.

A hollow shaft of diameter ratio 3/8 (internal dia. to outer dia.) is to transmit 375 kW power at 100 rpm. The maximum torque being 20% greater than the mean torque. The shear stress is not to exceed $ 60 \text{ N/mm}^2 $ and twist in a length of 4 m is not to exceed $ 2^{\circ} $. Calculate its external and internal diameter which would satisfy both the above conditions. Assume modulus of rigidity $ G = 0.85 \times 10^5 \text{ N/mm}^2 $.

What do you mean by plastic deformation of a material? Discuss the behaviour of the material when loaded beyond the elastic limit.

Draw shear flow diagram and locate shear centre for the channel section.

Differentiate between dam and a retaining wall.

A masonry dam, trapezoidal in cross-section, 4 m high, 1 m wide at its top and 3 m wide at its bottom, retains water on its vertical face to a maximum height of 3.5 m from its base. Determine the maximum and minimum stresses at the base (i) When the reservoir is empty, and (ii) When the reservoir is full. Take the unit weight of masonry as $ 19.62 \text{ kN/m}^3 $.

Determine the maximum deflection of the simply supported beam using double integration method. The beam is made of wood having a modulus of elasticity of $ E = 210 \text{ GPa} $ and cross-section $ 300 \text{ mm} \times 400 \text{ mm} $ in dimension.

Find the forces in the members of the frame shown in Fig. all members have the same sectional area and are made of the same material. Illustrate the forces in the diagram.

The energy absorbed in a body, when it is strained within the elastic limits, is known as

The slenderness ratio is (L-length of column and k-least radius of gyration of cross-section about its axis)

Graphical representation of which one of the following theories is an ellipse?

A sudden increase or decrease in shear force diagram between any two points indicates that there is

The angle of twist can be written as

The shear stress varies from centre to the surface of the shaft with

The difference between number of unknown reaction components and the number of available equilibrium equations is

Flexural rigidity is defined as

Principal stresses at a point are 120, -40 and 20 MPa. What is the maximum shear stress at the point?

In simply supported beam, deflection is maximum at

Stress state is given as $ \sigma_x=-29.5 \text{ MPa} $, $ \sigma_y=-29.5 \text{ MPa} $ and $ \tau_{xy}=27 \text{ MPa} $. Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements.

A cantilever beam with a rectangular cross-section has a longitudinal hole drilled throughout its length is shown in Fig. 1. The beam supports a load $ P=600 \text{ N} $. The cross-section is 25 mm wide and 50 mm high and the hole has a diameter of 10 mm. Find the bending stresses at the top of the beam, at the top of the hole and at the bottom of the beam.

A gas storage tank is fabricated by bolting together two half-cylindrical thin shells with two hemispherical shells at end. If the tank is designed to withstand a pressure of 3 MPa, determine the required minimum thickness of the cylindrical and hemispherical shells and the minimum required number of longitudinal bolts per meter length at each side of the cylindrical shell. The tank and the 25 mm diameter bolts are made from material having an allowable normal stress of 150 MPa and 250 MPa, respectively. The tank has an inner diameter of 4 m shown in Fig. 2.

A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque T is applied to the shaft, derive the maximum shear stress in the rubber shown in Fig. 3.

Derive the equations of the elastic curve for the beam using the $ x_1 $ and $ x_2 $ coordinates. EI is constant shown in Fig. 4.

Derive Euler's buckling formula for a column with both end clamped and obtain the effective length as well. Draw the free body diagram with buckled configuration.

The vertical force P acts on the bottom of the plate having a negligible weight. Determine the shortest distance d to the edge of the plate at which it can be applied so that it produces no compressive stresses on the plate at section a-a. The plate has a thickness of 10 mm and P acts along the center line of this thickness shown in Fig. 5.

Draw the shear and moment diagrams for the beam and determine the shear and moment throughout the beam as functions of y shown in Fig. 6.

The energy absorbed in a body, when it is strained within the elastic limits, is known as

The slenderness ratio is (L-length of column and k-least radius of gyration of cross-section about its axis)

Graphical representation of which one of the following theories is an ellipse?

A sudden increase or decrease in shear force diagram between any two points indicates that there is

The angle of twist can be written as

The shear stress varies from centre to the surface of the shaft with

The difference between number of unknown reaction components and the number of available equilibrium equations is

Flexural rigidity is defined as

Principal stresses at a point are 120, -40 and 20 MPa. What is the maximum shear stress at the point?

In simply supported beam, deflection is maximum at

Stress state is given as $ \sigma_x=-29.5 \text{ MPa} $, $ \sigma_y=-29.5 \text{ MPa} $ and $ \tau_{xy}=27 \text{ MPa} $. Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements.

A cantilever beam with a rectangular cross-section has a longitudinal hole drilled throughout its length is shown in Fig. 1. The beam supports a load $ P=600 \text{ N} $. The cross-section is 25 mm wide and 50 mm high and the hole has a diameter of 10 mm. Find the bending stresses at the top of the beam, at the top of the hole and at the bottom of the beam.

A gas storage tank is fabricated by bolting together two half-cylindrical thin shells with two hemispherical shells at end. If the tank is designed to withstand a pressure of 3 MPa, determine the required minimum thickness of the cylindrical and hemispherical shells and the minimum required number of longitudinal bolts per meter length at each side of the cylindrical shell. The tank and the 25 mm diameter bolts are made from material having an allowable normal stress of 150 MPa and 250 MPa, respectively. The tank has an inner diameter of 4 m shown in Fig. 2.

A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque T is applied to the shaft, derive the maximum shear stress in the rubber shown in Fig. 3.

Derive the equations of the elastic curve for the beam using the $ x_1 $ and $ x_2 $ coordinates. EI is constant shown in Fig. 4.

Derive Euler's buckling formula for a column with both end clamped and obtain the effective length as well. Draw the free body diagram with buckled configuration.

The vertical force P acts on the bottom of the plate having a negligible weight. Determine the shortest distance d to the edge of the plate at which it can be applied so that it produces no compressive stresses on the plate at section a-a. The plate has a thickness of 10 mm and P acts along the center line of this thickness shown in Fig. 5.

Draw the shear and moment diagrams for the beam and determine the shear and moment throughout the beam as functions of y shown in Fig. 6.

The property of a body to return to its original shape after removal of the force is known as

The materials which have the same elastic properties in all directions are known as

Which point on the stress-strain curve occurs after yield plateau?

Elastic limit is the point

What is the bending moment at end supports of a simply supported beam?

Torsional sectional modulus is also known as

Which property is undesirable for shaft materials?

The bending stress is

Which stress comes when there is an eccentric load applied?

Maximum slope in a cantilever beam of length L with a moment M at the free end will be

State of stress around a point on a thick bar is defined as $ \sigma_{xx}=100 \text{ MPa} $, $ \sigma_{yy}=-86 \text{ MPa} $, $ \sigma_{zz}=55 \text{ MPa} $, $ \tau_{xy}=60 \text{ MPa} $, $ \tau_{yz}=\tau_{zx}=0 $. Calculate principal stresses, principal planes, maximum shear stress and associated planes.

The bar having a diameter of 20 mm is fixed connected at its ends and supports the axial load P. If the material is elastic perfectly plastic as shown by the stress-strain diagram (Fig. 1), determine the smallest load P needed to cause segment CB to yield. If this load is released, determine the permanent displacement of point C.

A 2014-T6 aluminum tube having a cross-sectional area of $ 500 \text{ mm}^2 $ is used as a sleeve for an A-36 steel bolt having a cross-sectional area of $ 300 \text{ mm}^2 $. When the temperature is $ T_1=30^{\circ}\text{C} $, the nut holds the assembly in a snug position such that the axial force in the bolt is negligible. If the temperature increases to $ T_2=100^{\circ}\text{C} $, determine the force in the bolt and sleeve. Take $ \alpha_{bolt}=12 \times 10^{-6}/^{\circ}\text{C} $, $ \alpha_{sleeve}=23 \times 10^{-6}/^{\circ}\text{C} $, $ E_{bolt}=200 \text{ GPa} $, $ E_{sleeve}=73 \text{ GPa} $.

A motor is connected to a speed reducer by the tubular shaft and coupling. If the motor supplies 20 HP and rotates the shaft at a rate of 600 r.p.m., determine the minimum inner and outer diameters $ d_i $ and $ d_o $ of the shaft if $ d_i/d_o=0.75 $. The shaft is made from a material having an allowable shear stress of $ \tau_{allow}=12 \text{ kPa} $.

Derive Euler's buckling formula for a column with one end clamped and other end free and obtain the effective length as well. Draw the free body diagram with buckled configuration.

If the wide-flange beam is subjected to a shear of $ V=20 \text{ kN} $, determine the shear stress on the web at A (Fig. 2). Indicate the shear-stress components on a volume element located at this point.

Determine the maximum deflection of the simply supported beam using double integration method. The beam is made of wood having a modulus of elasticity of $ E=210 \text{ GPa} $ and cross-section $ 3 \text{ mm} \times 4 \text{ mm} $ in dimension (Fig. 3).

Derive an expression for an equivalent bending moment $ M_e $ that, if applied alone to a solid bar with a circular cross-section, would cause the same maximum shear stress as the combination of an applied moment M and torque T. Assume that the principal stresses are of opposite algebraic signs.

The property of a body to return to its original shape after removal of the force is known as

The materials which have the same elastic properties in all directions are known as

Which point on the stress-strain curve occurs after yield plateau?

Elastic limit is the point

What is the bending moment at end supports of a simply supported beam?

Torsional sectional modulus is also known as

Which property is undesirable for shaft materials?

The bending stress is

Which stress comes when there is an eccentric load applied?

Maximum slope in a cantilever beam of length L with a moment M at the free end will be

State of stress around a point on a thick bar is defined as $ \sigma_{xx}=100 \text{ MPa} $, $ \sigma_{yy}=-86 \text{ MPa} $, $ \sigma_{zz}=55 \text{ MPa} $, $ \tau_{xy}=60 \text{ MPa} $, $ \tau_{yz}=\tau_{zx}=0 $. Calculate principal stresses, principal planes, maximum shear stress and associated planes.

The bar having a diameter of 20 mm is fixed connected at its ends and supports the axial load P. If the material is elastic perfectly plastic as shown by the stress-strain diagram (Fig. 1), determine the smallest load P needed to cause segment CB to yield. If this load is released, determine the permanent displacement of point C.

A 2014-T6 aluminum tube having a cross-sectional area of $ 500 \text{ mm}^2 $ is used as a sleeve for an A-36 steel bolt having a cross-sectional area of $ 300 \text{ mm}^2 $. When the temperature is $ T_1=30^{\circ}\text{C} $, the nut holds the assembly in a snug position such that the axial force in the bolt is negligible. If the temperature increases to $ T_2=100^{\circ}\text{C} $, determine the force in the bolt and sleeve. Take $ \alpha_{bolt}=12 \times 10^{-6}/^{\circ}\text{C} $, $ \alpha_{sleeve}=23 \times 10^{-6}/^{\circ}\text{C} $, $ E_{bolt}=200 \text{ GPa} $, $ E_{sleeve}=73 \text{ GPa} $.

A motor is connected to a speed reducer by the tubular shaft and coupling. If the motor supplies 20 HP and rotates the shaft at a rate of 600 r.p.m., determine the minimum inner and outer diameters $ d_i $ and $ d_o $ of the shaft if $ d_i/d_o=0.75 $. The shaft is made from a material having an allowable shear stress of $ \tau_{allow}=12 \text{ kPa} $.

Derive Euler's buckling formula for a column with one end clamped and other end free and obtain the effective length as well. Draw the free body diagram with buckled configuration.

If the wide-flange beam is subjected to a shear of $ V=20 \text{ kN} $, determine the shear stress on the web at A (Fig. 2). Indicate the shear-stress components on a volume element located at this point.

Determine the maximum deflection of the simply supported beam using double integration method. The beam is made of wood having a modulus of elasticity of $ E=210 \text{ GPa} $ and cross-section $ 3 \text{ mm} \times 4 \text{ mm} $ in dimension (Fig. 3).

Derive an expression for an equivalent bending moment $ M_e $ that, if applied alone to a solid bar with a circular cross-section, would cause the same maximum shear stress as the combination of an applied moment M and torque T. Assume that the principal stresses are of opposite algebraic signs.