UNIT 62: STRENGTHS OF MATERIALS Unit code: K/601/1409 QCF level: 5 Credit value: 15 OUTCOME 2 - TUTORIAL 3

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1 UNIT 6: STRNGTHS O MTRIS Unit code: K/601/1409 QC level: 5 Credit vale: 15 OUTCOM - TUTORI 3 INTRMDIT ND SHORT COMPRSSION MMBRS Be able to determine the behavioral characteristics of loaded beams, colmns and strts Simply spported beams: se of Macalay s method to determine the spport reactions, slope and deflection de to bending in cantilevers and simply spported beams with combined concentrated and niformly distribted loads Reinforced concrete beams: theoretical assmptions; distribtion of stress de to bending Colmns: stress de to asymmetrical bending; middle third rle for rectanglar section colmns and walls; middle qarter rle for circlar section colmns Strts: end fixings; effective length; least radis of gyration of section; slenderness ratio; ler and Ranine-Gordon formlae for determination of critical load Yo shold jdge yor progress by completing the self assessment exercises. 1. INTRODUCTION CONTNTS. INTRMDIT COMPRSSION MMBRS.1 Ranine - Gordon Theory 3 COUMNS 3.1 Offset oads 3. Netral 3.3 Maximm Offset It is assmed that stdents doing this ttorial already familiar with the concepts of second moments of area, bending stress, bending moments and deflection of beams. Stdents shold stdy the ttorial on strts before starting this one. D.J.Dnn Material spplied from 1

2 1. INTRODUCTION Yo shold recall that compression members fall into three grops, long (strts) intermediate and short (colmns). irst we need to now that a material may fail de to exceeding the ltimate (maximm) compressive stress. This is often referred to as the crshing stress. If this was the only factor casing failre the load that prodces it wold be given by the formla = = Ultimate compressive load = ltimate compressive stress = cross sectional area. It is bad practise to apply a load at a point on brittle colmns becase high local stress reslts in that region. steel plate shold be sed to spread the load over the section. Ideally the load is applied at the centre of area and it is assmed that the compressive stress spreads ot evenly over the section. If the load is and the cross sectional area is then the direct (compressive) stress is σ D = -/ (compression is negative) igre 1 In reality, there is often bending associated with the failre and this is especially tre with intermediate members.. INTRMDIT COMPRSSION MMBRS It is fond that steel strts with a slenderness ratio of 80 to 10 fail at smaller loads than predicted by ler. These are intermediate compression members in which compression and bending have an effect on failre..1 RNKIN - GORDON THORY GORDON sggested that for sch members, an empirical formla be sed (based on experimental data). RNKIN modified Gordon's formla. The following shows the reasoning for this formla. π I = ler's critical load. and this applies to strts. = Ultimate compressive load = and this applies to colmns. = ltimate compressive stress. = cross sectional area. Ranine sggested that an intermediate compression member fails de to both bcling and compression to more or less degrees. Based on experimental data, it is fond that a reasonable prediction of the load at failre is given by the reciprocal formla This rearranges to R R = Ranine s Critical load. R The formla indicates that for slender members R dominates and for short members dominates. D.J.Dnn Material spplied from

3 D.J.Dnn Material spplied from 3 URTHR DVOPMNT Consider the frther development of ler's formla for strts....(1) π π stress s critical ler' called stress nominal we convert the force into a area by thecross sectional we divide If π π nto this. for I Sbstitte the above formla I π load for strtsis s critical ler' (S.R.) I I (S.R.) the sbject I to mae Rearrange I S.R. Ratio Slendernes s If we plot this stress against slenderness ratio we get the reslt shown on figre. The graph is called ler's Hyperbola. Next we consider the frther development of the Ranine formla. Sbstitte the following into Ranine s formla. = e = I/ e e R n n where...() 1 1 n a n a n R R

4 lthogh there is a theoretical vale for the constant a based on material properties, it is sal to determine it from experiment. The experimental vale varies slightly from the theoretical. Typical vales of a are Material MPa a Mild Steel 35 1/7500 Wroght Iron 47 1/9000 Cast Iron 557 1/1600 Timber 35 1/3000 GRPHIC RPRSNTTION The diagram shows (eqation ) and R (eqation 1) plotted against slenderness ratio. The ltimate compressive stress is mared on the stress axis. The reslt shows how R tends to as the member becomes short and tends to as the member gets longer. The region of interest is arond the S.R. = 80 point where the correct stress lays between the other two vales. igre WORKD XMP No.1 ind the Ranine critical load for a strt with an I section as shown given = 35 MPa, = 05 MPa and = 16 m. The strt is bilt in rigidly at each end. igre 3 D.J.Dnn Material spplied from 4

5 SOUTION The strt will bend abot the axis of minimm resistance and hence minimm I so we mst determine which is the minimm I. HORIZONT N- The section is symmetrical so sbtract I for the two ct ots from the vale for the oter rectangle. Remember that for a rectangle I = BD 3 /1 abot its centre line. or the oter rectangle I = 50 x 30 3 /1 = x 10 6 mm 4 or one ct ot I = 10 x /1 = 70 x 10 6 mm 4 or the section I = x 10 6 x 70 x 10 6 = x 10 6 mm 4 VRTIC N- Treat this as three rectangles and add them together. Two ends I = x 10 x 50 3 /1 = 6.04 x 10 6 mm 4 Middle I = 300 x 10 3 /1 = 5000 mm 4 dd together I = 6.06 x 10 6 mm 4 The minimm I is abot the vertical axis so se I = 6.06 x 10 6 mm 4 = (50 x 10) x + (300 x 10) = 8000 mm. = (I/) = (6x10 6 /8000) = 57 mm mode n = a = x10 x35x10 R a 1 n 66.N S SSSSMNT XRCIS No.1 1. strt is 0. m diameter and 15 m long. It is pinned at both ends. Calclate ler's critical load. (nswer N). strt has a rectanglar section 0. m x 0.1 m. It is 8 m long. The bottom is bilt in and the top is free. Dring a test, it bcled at 164 N. Given the ltimate compressive stress of the material is 345 MPa : Calclate the Ranine constant "a". ( ) second strt made of the same material has a rectanglar section 0.3 m x 0.m. It is 6 m long and pinned at both ends. Using the constant a fond previosly, find the Ranine bcling load. (nswer 8.5 MN) D.J.Dnn Material spplied from 5

6 3. COUMNS colmn is a thic compression member. Strts fail de to bending bt colmns fail in compression. Colmns are sally made of brittle material which is strong in compression sch as cast iron, stone and concrete. These materials are wea in tension so it is important to ensre that bending does not prodce tensile stresses in them. If the compressive stress is too big, they fail by crmbling and cracing 3.1 OST ODS igre 4 Colmns often spport offset loads and these prodce bending stresses that combine with the compressive stress. This is illstrated in figre 5. igre 5 If a load is applied on the centre of the section, the stress in the colmn will be a direct compressive stress given by D = - / Remember that compressive stresses are always negative. When the load is applied a distance 'x' from the centroid, a bending moment is indced in the colmn as shown. The bending moment is M = x where x is the off set distance. D.J.Dnn Material spplied from 6

7 rom the well nown formla for bending stress we have B = My/I y is the distance from the centroid to the edge of the colmn. The stress prodced will be +ve (tensile) on one edge and -ve (compressive) on the other. On the compressive edge this will add to the direct compressive stress maing it larger so that = B + D = -My/I - / On the tensile edge the reslting stress is Sbstitte M = x = B + D = My/I - / = = xy/i - / WORKD XMP No. colmn is 0.5 m diameter and carries a load of 500 N offset from the centroid by 0.1m. Calclate the extremes of stresses. SOUTION = 500 N x = 0.1 m y = D/ = 0.5 m Tensile dge = B + D = xy/i - / = x 0.1 x 0.5 /( x 0.54/64) /( x 0.5/4) = 1.58 MPa (Tensile) Compressive dge = B + D = -xy/i - / = x 0.1 x 0.5 /( x 0.54/64) /( x 0.5/4) = -6.61MPa (compressive) D.J.Dnn Material spplied from 7

8 3. NUTR XIS The netral axis is the axis of zero stress. In the above example, the stress varied from 1.58 MPa on one edge to MPa on the other edge. Somewhere in between there mst a vale of y which maes the stress zero. This does not occr on the centroid bt is by definition the position of the netral axis. Ideally this axis shold not be on the section at all so that no tensile stress occrs in the colmn. The position of the netral axis can easily be fond by drawing a stress distribtion diagram and then either scaling off the position or calclate it from similar triangles. WORKD XMP No.3 Determine the position of the netral axis for the colmn in example. SOUTION Drawing a graph of stress against position (y) along a diameter we get the figre shown (not drawn to scale). If it is drawn to scale the position of the netral axis may be scaled off. igre 6 Using similar triangles we arrive at the soltion as follows. + B = 0.5 = B /1.58 = B/6.61 (0.5 - B)/1.58 = B/ B = 1.58 B B = m D.J.Dnn Material spplied from 8

9 3.3 MXIMUM OST If a colmn mst not go into tension, then the maximm offset may be calclated. Consider a circlar section first. The combined stress de to compression and bending is: xy σ I If the edge mst not go into tension then the maximm stress will be zero so: xy I Z 0 x(max) I y y or a rond section = πd /4 I = πd 4 /64 and y = D/ If we sbstitte we get 4 4D D x(max) 64 D D/ 8 The load mst be no more than D/8 from the centroid. If the colmn is a rectanglar section I = BD 3 /1 = BD and the critical vale of y is D/ 3 I BD D x(max) when the offset is on the short axis. y 1 BD D 6 When the offset is on the long axis x(max) is B/6. This means the offset mst be within the middle 1/3 of the colmn and this is called the middle third rle. The shaded area on the diagram shows the safe region for applying the load. igre 7 or any standard section sch as those in BS4, the maximm offset is easily fond from x = Z/y althogh for steel sections some tension is allowed. WORKD XMP No.4 colmn is made from an niversal I section 305 x 305 x 97. load of MN is applied on the x axis 00 mm from the centroid. Calclate the stress at the oter edges of the x axis. If the colmn is 5 m tall, what is the slenderness ratio? D.J.Dnn Material spplied from 9

10 SOUTION The offset is x = 0. m and the load = MN rom the table I = 49 x 10-8 m -4 = 13 x 10-4 m y = h/ = m 6 6 xy ( x 10 )(0.)(0.154) x 10 σc 439 MPa -4 I I 13 x xy ( x 10 )(0.)(0.154) x 10 σt 114 MPa -4 I I 13 x 10 There are two radii of gyration. x = y = m Slenderness Ratio abot the x axis is = / x = 5/0.134 =37.3 Slenderness Ratio abot the y axis is = / y = 5/ = 65 These are well below the limit of 10 for steel bt the bending might case collapse and wold be worth checing. S SSSSMNT XRCIS No. 1. colmn is 0.4 m diameter. It has a vertical load of 300 N acting 0.05m from the centroid. Calclate the stresses on the extreme edges. (nswers 0 MPa and MPa).. colmn is 0.3 m diameter. Calclate the offset position of the load which jst prevents the one edge from going into tension. (nswer m). 3. colmn is made from a rectanglar bloc of concrete with a section 600 mm x 300 mm. What is the maximm offset of a point load that jst prevents the edge going into tension. (nswer 50 mm). 4. colmn is made from cast iron tbe 0.4 m otside diameter with a wall 40 mm thic. The top is covered with a flat plate and a vertical load of 70 N is applied to it. Calclate the maximm allowable offset position of the load if the material mst always remain in compression. (nswer 0.08 m) 5. hollow cast iron pillar, 38 cm otside diameter and wall thicness 7.5 cm, carries a load of 75 N along a line parallel to, bt displaced 3 cm from, the axis of the pillar. Determine the maximm and minimm stresses in the pillar. (561 Pa and 1.56 MPa both compressive) What is the maximm allowable eccentricity of the load relative to the axis of the pillar if the stresses are to be compressive at all points of the cross section? ( m) 6. colmn is 4 m tall and made from an niversal I section 15 x 15 x 3. load of 60 N is applied on the x axis 110 mm from the centroid. Calclate the stress at the oter edges of the x axis. (19.6 MPa tensile and 60 MPa compressive, S = 61 abot the x axis and 108 abot y axis) D.J.Dnn Material spplied from 10

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12 SMP O TB OR UNIVRS COUMNS WITH I SCTION Designation Mass per m Depth of Section Width of Section D.J.Dnn Material spplied from 1 Thicness of Depth rea Root between of Web lange Radis illets Section Second Moment rea Radis of Gyration M h b s t r d Ix Iy rx ry Zx Zy Sx Sy g/m mm mm mm mm mm mm cm cm 4 cm 4 cm cm cm 3 cm 3 cm 3 cm 3 356x406x x406x x406x x406x x406x x406x x406x x368x x368x x368x x368x x305x x305x x305x x305x x305x x305x x305x lastic Modls Plastic Modls

13 Designation Mass per m Depth of Section Width of Section Thicness of Depth rea Root between of Web lange Radis illets Section Second Moment rea Radis of Gyration 54x54x x54x x54x x54x x54x x03x x03x x03x x03x x03x x15x x15x x15x lastic Modls Plastic Modls D.J.Dnn Material spplied from