75 mm i

3.6 (B). A cantilever is to be constructed from a 40 mm x 60 mm T-section beam with a uniform thickness of 5 mm. The cantilever is to carry a u.d.l. over its complete length of 1 m. Determine the maximum u.d.l. that the cantilever can carry if yielding is permitted over the lower part of the web to a depth of 10 mm. ay = 225 MN/m".

3.7 (B). A 305 mm x 127 mm symmetrical I-section has flanges 13 mm thick and a web 5.4 mm thick. Treating the web and flanges as rectangles, calculate the bending moment of resistance of the cross-section (a) at initial yield, (b) for full plasticity of the flanges only, and (c) for full plasticity of the complete cross-section. Yield stress in simple tension and compression = 310 MN/m2. What is the shape factor of the cross-section?

3.8 (B). A steel bar of rectangular section 80 mm by 40 mm is used as a simply supported beam on a span of 1.4 m and point-loaded at mid-span. If the yield stress of the steel is 300 MN/m2 in simple tension and compression and the long edges of the section are vertical, find the load when yielding first occurs.

Assuming that a further increase in load causes yielding to spread in towards the neutral axis with the stress in the yielded part remaining constant at 300 MN/m2, determine the load required to cause yielding for a depth of 10 mm at the top and bottom of the section at mid-span and find the length of beam over which yielding at the top and bottom faces will have occurred. [U.L.] [36.57, 44.6 kN; 0.232 m.[

3.9 (B). A straight bar of steel of rectangular section, 76 mm wide by 25 mm deep, is simply supported at two points 0.61 m apart. It is subjected to a uniform bending moment of 3 kNm over the whole span. Determine the depth of beam over which yielding will occur and make a diagram showing the distribution of bending stress over the full depth of the beam. Yield stress of steel in tension and compression = 280 MN/m2.

Estimate the deflection at mid-span assuming e = 200 GN/m2 for elastic conditions. [5.73, 44.4 mm.]

3.10 (B). A symmetrical I-section beam of length 6 m is simply supported at points 1.2 m from each end and is to carry a u.d.l. ir kN/m run over its entire length. The second moment of area of the cross-section about the neutral axis parallel to the flanges is 6570 cm4 and the beam cross-section dimensions are: flange width and thickness, 154 mm and 13 mm respectively, web thickness 10 mm, overall depth 254 mm.

(a) Determine the value of w to just cause initial yield, stating the position of the transverse section in the beam length at which it occurs.

(b) By how much must w be increased to ensure full plastic penetration of the flanges only, the web remaining elastic?

Take the yield stress of the beam material in simple tension and compression as 340 MN/m2.

3.11 (B). A steel beam of rectangular cross-section. 100 mm wide by 50 mm deep, is bent to the arc of a circle until the material just yields at the outer fibres, top and bottom. Bending takes place about the neutral axis parallel to the 100 mm side. If the yield stress for the steel is 330 MN/m2 in simple tension and compression, determine the applied bending moment and the radius of curvature of the neutral layer. E = 207 GN/m2.

Find how much the bending moment has to be increased so that the stress distribution is as shown in Fig. 3.47.

3.12 (B). A horizontal steel cantilever beam, 2.8 m long and of uniform I-section throughout, has the following cross-sectional dimensions: flanges 150 mm x 25 mm, web 13 mm thick, overall depth 305 mm. It is fixed at one end and free at the other.

(a) Determine the intensity of the u.d.l. which the beam has to carry across its entire length in order to produce fully developed plasticity of the cross-section.

(b) What is the value of the shape factor of the cross-section?

(c) Determine the length of the beam along the top and bottom faces, measured from the fixed end, over which yielding will occur due to the load found in (a).

Yield stress of steel = 330 MN/m2. [106.2 kN/m; 1.16; 0.2 m.]

3.13 (B). A rectangular steel beam, 60 mm deep by 30 mm wide, is supported on knife-edges 2m apart and loaded with two equal point loads at one-third of the span from each end. Find the load at which yielding just begins, the yield stress of the material in simple tension and compression being 300 MN/m2.

If the loads are increased to 25% above this value, estimate how far the yielding penetrates towards the neutral axis, assuming that the yield stress remains constant. [U.L.] [8.1 kN; 8.79 mm.]

3.14 (B). A steel bar of rectangular section, 72 mm deep by 30 mm wide, is used as a beam simply supported at each end over a span of 1.2 m and loaded at mid-span with a point load. The yield stress of the material is 280 MN/m2. Determine the value of the load when yielding first occurs.

Find the load to cause an inward plastic penetration of 12 mm at the top and bottom of the section at mid-span. Also find the length, measured along the top and bottom faces, over which yielding has occurred, and the residual stresses present after unloading. [U.L.] [24.2 kN; 31 kN; 0.26 m, ^79, ±40.7 MN/m2.]

3.15 (B). A symmetrical I-section beam, 300 mm deep, has flanges 125 mm wide by 13 mm thick and a web 8.5 mm thick. Determine:

(a) the applied bending moment to cause initial yield;

(b) the applied bending moment to cause full plasticity of the cross-section;

(c) the shape factor of the cross-section.

Take the yield stress = 250 MN/m2 and assume / = 85 x I06 mm4.

3.16 (B). A rectangular steel beam ab, 20 mm wide by 10 mm deep, is placed symmetrically on two knife-edges C and d. 0.5 m apart, and loaded by applying equal loads at the ends a and b. The steel follows a linear stress/strain law (e = 200 GN/m2) up to a yield stress of 300 MN/m ; at this constant stress considerable plastic deformation occurs. It may be assumed that the properties of the steel are the same in tension and compression.

Calculate the bending moment on the central part of the beam CD when yielding commences and the deflection at the centre relative to the supports.

If the loads are increased until yielding penetrates half-way to the neutral axis, calculate the new value of the bending moment and the corresponding deflection. [U.L.] [100 Nm, 9.375 mm; 137.5 Nm, 103 mm.]

3.17 (B). A steel bar of rectangular material, 75 mm x 25 mm, is used as a simply supported beam on a span of 2 m and is loaded at mid-span. The 75 mm dimension is placed vertically and the yield stress for the material is 240 MN/m2. Find the load when yielding first occurs.

The load is further increased until the bending moment is 20% greater than that which would cause initial yield. Assuming that the increased load causes yielding to spread inwards towards the neutral axis, with the stress in the yielded part remaining at 240 MN/m2, find the depth at the top and bottom of the section at mid-span to which the yielding will extend. Over what length of the beam has yielding occurred?

3.18 (B). The cross-section of a beam is a channel, symmetrical about a vertical centre line. The overall width of the section is 150 mm and the overall depth 100 mm. The thickness of both the horizontal web and each of the vertical flanges is 12 mm. By comparing the behaviour in both the elastic and plastic range determine the shape factor of the section. Work from first principles in both cases. [1.806.]

3.19 (B). The T-section beam shown in Fig. 3.48 is subjected to increased load so that yielding spreads to within 50 mm of the lower edge of the flange. Determine the bending moment required to produce this condition. (7V =240 MN m2. [44 kN m.]

15 mm

50 mm

15 mm

50 mm

3.20 (B). A steel beam of I-section with overall depth 300 mm, flange width 125 mm and length 5 m, is simply supported at each end and carries a uniformly distributed load of 114 kN/m over the full span. Steel reinforcing plates 12 mm thick are welded symmetrically to the outside of the flanges producing a section of overall depth 324 mm. If the plate material is assumed to behave in an elastic-ideally plastic manner, determine the plate width necessary such that yielding has just spread through each reinforcing plate at mid-span under the given load.

Determine also the positions along the reinforcing plates at which the outer surfaces have just reached the yield point. At these sections what is the horizontal shearing stress at the interfaces of the reinforcing plates and the flanges?

Take the yield stress ay = 300 MN/m2 and the second moment of area of the basic I-section to be 80 x 10~6 m4.

3.21 (B). A horizontal cantilever is propped at the free end to the same level as the fixed end. It is required to carry a vertical concentrated load W at any position between the supports. Using the normal assumption of plastic limit design, determine the least favourable position of the load. (Note that the calculation of bending moments under elastic conditions is not required.)

Hence calculate the maximum permissible value of W which may be carried by a rectangular-section cantilever with depth d equal to twice the width over a span L. Assume a load factor of n and a yield stress for the beam material <rv. [0.586 L from built-in end; d3oy / 1.371 Ln.]

3.22 (B). (a) Sketch the idealised stress-strain diagram which is used to establish a quantitative relationship between stress and strain in the plastic range of a ductile material. Include the effect of strain-hardening.

(b) Neglecting strain-hardening, sketch the idealised stress-strain diagram and state, in words, the significance of any alteration you make in the diagram shown for part (a) when calculations are made, say, for pure bending beyond the yield point.

(c) A steel beam of rectangular cross-section, 200 mm wide x 100 mm deep, is bent to the arc of a circle, bending taking place about the neutral axis parallel to the 200 mm side.

Determine the bending moment to be applied such that the stress distribution is as shown in (i) Fig. 3.49(a) and (ii) Fig. 3.49(b).

Take the yield stress of steel in tension and compression as 250 MN/m2. [BP.] [98.3, 125 kN m.]

3.23 (B). (a) A rectangular section beam is 80 mm wide, 120 mm deep and is simply supported at each end over a span of 4 m. Determine the maximum uniformly distributed load that the beam can carry:

(i) if yielding of the beam material is permitted to a depth of 40 mm;

(ii) before complete collapse occurs.

(b) What residual stresses would be present in the beam after unloading from condition (a) (i)?

(c) What external moment must be applied to the beam to hold the deformed bar in a straight position after unloading from condition (a) (i)?

The yield stress of the material of the beam = 280 MN/m2.

[B.P.] [38.8, 40.3 kN/m; ± 123, ± 146 MN/m2; 84.3 kN mj

3.24 (C). A rectangular beam 80 mm wide and 20 mm deep is constructed from a material with a yield stress in tension of 270 MN/m2 and a yield stress in compression of 300 MN/m2. If the beam is now subjected to a pure bending moment find the value required to produce:

(a) initial yield;

(b) initial yield on the compression edge;

(c) a fully plastic section. [1.44; 1.59, 2.27 kN m.]

3.25 (C). Determine the load factor of a propped cantilever carrying a concentrated load W at the centre. Allowable working stress = 150 MN/m2, yield stress = 270 MN/m2. The cantilever is of I-section with dimensions 300 mm x 80 mm x 8 mm. [2.48.]

3.26 (C). A 300 mm x 100 mm beam is carried over a span of 7 m the ends being rigidly built in. Find the maximum point load which can be carried at 3 m from one end and the maximum working stress set up.

/ = 85 x 10"6 m4 and the shape factor = 1.135. [100 kN; 172 MN/m2.]

3.27 (C). A 300 mm x 125 mm I-beam is carried over a span of 20 m the ends being rigidly built in. Find the maximum point load which can be carried at 8 m from one end and the maximum working stress set up. Take a load factor of 1.8 and <jy = 250 MN/m2; z = 56.6 x 10"5 m3 and shape factor A. = 1.11.

3.28 (C). Determine the maximum intensity of loading that can be sustained by a simply supported beam, 75 mm wide x 100 mm deep, assuming perfect elastic-plastic behaviour with a yield stress in tension and compression of 135 MN/m2. The beam span is 2 m.

What will be the distribution of residual stresses in the beam after unloading?

3.29 (C). A short column of 0.05 m square cross-section is subjected to a compressive load of 0.5 MN parallel to but eccentric from the central axis. The column is made from elastic - perfectly plastic material which has a yield stress in tension or compression of 300 MN/m2. Determine the value of the eccentricity which will result in the section becoming just fully plastic. Also calculate the residual stress at the outer surfaces after elastic unloading from the fully plastic state. [10.4 mm; 250, 150 MN/m2.]

330 (C). A rectangular beam 75 mm wide and 200 mm deep is constructed from a material with a yield stress in tension of 270 MN/m2 and a yield stress in compression of 300 MN/m2. If the beam is now subjected to a pure bending moment, determine the value of the moment required to produce (a) initial yield, (b) initial yield on the compression edge, (c) a fully plastic section. [135, 149.2, 213.2 kN m.]

331 (C). Figure 3.50 shows the cross-section of a welded steel structure which forms the shell of a gimbal frame used to support the ship-to-shore transport platform of a dock installation. The section is symmetrical about the vertical centre-line with a uniform thickness of 25 mm throughout.

As a preliminary design study what would you assess as the maximum bending moment which the section can withstand in order to prevent:

(a) initial yielding at any point in the structure if the yield stress for the material is 240 MN/m2,

(b) complete collapse of the structure?

What would be the effect of adverse weather conditions which introduce instantaneous loads approaching, but not exceeding that predicted in (b). Quantify your answers where possible.

State briefly the factors which you would consider important in the selection of a suitable material for such a structure. [309.3 kN m; 423 kN m; local yielding, residual stress max = 279 MN/m2.]

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