CHAPTER 2 COMPOUND BARS Summary When a compound bar is constructed from members of different materials,lengths and areas and is subjected to an external tensile or compressive load W the load carried by any single member is given by E1A1 W F,二EA Σ where sufixIreferstothesinge member and EA is the sum of all such quantities for all the members. Where the bars have a common length the compound bar can be reduced to a single equivalent bar with an equivalent Young's modulus,termed a combined E. ΣEA Combined E= ZA The free expansion of a bar under a temperature change from Ti to T is a(T2-T)L where a is the coefficient of linear expansion and L is the length of the bar. If this expansion is prevented a stress will be induced in the bar given by a(T-TE To determine the stresses in a compound bar composed of two members of different free lengths two principles are used: (1)The tensile force applied to the short member by the long member is equal in magnitude to the compressive force applied to the long member by the short member. (2)The extension of the short member plus the contraction of the long member equals the difference in free lengths. This difference in free lengths may result from the tightening of a nut or from a temperature change in two members of different material (i.e.different coefficients of expansion)but of equal length initially. If such a bar is then subjected to an additional external load the resultant stresses may be obtained by using the principle of superposition.With this method the stresses in the members 27
CHAPTER 2 COMPOUND BARS Summary When a compound bar is constructed from members of different materials, lengths and areas and is subjected to an external tensile or compressive load W the load carried by any single member is given by ElAl F1 =- L1 w EA c- L EA L where suffix 1 refers to the single member and X - is the sum of all such quantities for all the members. Where the bars have a common length the compound bar can be reduced to a single equivalent bar with an equivalent Young’s modulus, termed a combined E. C EA Combined E = - CA The free expansion of a bar under a temperature change from Tl to T2 is a(T2 -T1)L where a is the coefficient of linear expansion and L is the length of the bar. If this expansion is prevented a stress will be induced in the bar given by a(T2 -T# To determine the stresses in a compound bar composed of two members of different free lengths two principles are used: (1) The tensile force applied to the short member by the long member is equal in magnitude (2) The extension of the short member plus the contraction of the long member equals the This difference in free lengths may result from the tightening of a nut or from a temperature change in two members of different material (Le. different coefficients of expansion) but of equal length initially. If such a bar is then subjected to an additional external load the resultant stresses may be obtained by using the principle ofsuperposition. With this method the stresses in the members 27 to the compressive force applied to the long member by the short member. difference in free lengths
28 Mechanics of Materials $2.1 arising from the separate effects are obtained and the results added,taking account of sign,to give the resultant stresses. N.B.:Discussion in this chapter is concerned with compound bars which are symmetri- cally proportioned such that no bending results. 2.1.Compound bars subjected to external load In certain applications it is necessary to use a combination of elements or bars made from different materials,each material performing a different function.In overhead electric cables, for example,it is often convenient to carry the current in a set of copper wires surrounding steel wires,the latter being designed to support the weight of the cable over large spans.Such combinations of materials are generally termed compound bars.Discussion in this chapter is concerned with compound bars which are symmetrically proportioned such that no bending results. When an external load is applied to such a compound bar it is shared between the individual component materials in proportions depending on their respective lengths,areas and Young's moduli. Consider,therefore,a compound bar consisting of n members,each having a different length and cross-sectional area and each being of a different material;this is shown diagrammatically in Fig.2.1.Let all members have a common extension x,i.e.the load is positioned to produce the same extension in each member. nmember First member Length Ln Length Li Area An Areo A Modulus En Modulus E Load Fn Load F ommon extension x H Fig.2.1.Diagrammatic representation of a compound bar formed of different materials with different lengths,cross-sectional areas and Young's moduli. For the nth member, stress strain =En= FnLn Anxn EnAnx F= (2.10 Ln where F is the force in the nth member and A,and L are its cross-sectional area and length
28 Mechanics of Materials $2.1 arising from the separate effects are obtained and the results added, taking account of sign, to give the resultant stresses. N.B.: Discussion in this chapter is concerned with compound bars which are symmetrically proportioned such that no bending results. 2.1. Compound bars subjected to external load In certain applications it is necessary to use a combination of elements or bars made from different materials, each material performing a different function. In overhead electric cables, for example, it is often convenient to carry the current in a set of copper wires surrounding steel wires, the latter being designed to support the weight of the cable over large spans. Such combinations of materials are generally termed compound burs. Discussion in this chapter is concerned with compound bars which are symmetrically proportioned such that no bending results. When an external load is applied to such a compound bar it is shared between the individual component materials in proportions depending on their respective lengths, areas and Young’s moduli. Consider, therefore, a compound bar consisting of n members, each having a different length and cross-sectional area and each being of a different material; this is shown diagrammatically in Fig. 2.1. Let all members have a common extension x, i.e. the load is positioned to produce the same extension in each member. First member Modulus E, Load 7 n’’mmernber Length L, Area An Modulus E, T’ Load F, Frnmon extension x W Fig. 2.1. Diagrammatic representation of a compound bar formed of different materials with different lengths, cross-sectional areas and Young’s moduli. For the nth member, f‘n Ln E, = - strain Anxn - stress -- where F, is the force in the nth member and A, and Ln are its cross-sectional area and length
§2.2 Compound Bars 29 The total load carried will be the sum of all such loads for all the members i.e. w=-2 (2.2) Now from egn.(2.1)the force in member 1 is given by E1A1x LI But,from eqn.(2.2), W X=- EmAn Ln E1A F1= (2.3) EA i.e.each member carries a portion of the total load W proportional to its EA/L value. If the wires are all of equal length the above equation reduces to EAW F1=ZEA (2.4) The stress in member 1 is then given by F 01= (2.5) A 2.2.Compound bars-“equivalent'or“combined'”modulus In order to determine the common extension of a compound bar it is convenient to consider it as a single bar of an imaginary material with an equivalent or combined modulus E..Here it is necessary to assume that both the extension and the original lengths of the individual members of the compound bar are the same;the strains in all members will then be equal. Now total load on compound bar F:+F2+F3+...+F where F1,F2,etc.,are the loads in members 1,2,etc. But force stress x area G(A1+A2+.,+An)=O1A1+02A2+·+0nAn where o is the stress in the equivalent single bar. Dividing through by the common strain 8, (41+A2+…+A)=2A1+2A2+…+A & i.e. E(A1+A2+.··+An)=E1A1+E2A2+,··+EmAn where E.is the equivalent or combined E of the single bar
52.2 Compound Bars 29 The total load carried will be the sum of all such loads for all the members i.e. Now from eqn. (2.1) the force in member 1 is given by But, from eqn. (2.2), .. F1=- L1 w (2.3) EA c- L i.e. each member carries a portion of the total load W proportional to its EAIL value. If the wires are all of equal length the above equation reduces to The stress in member 1 is then given by 2.2. Compound bars - “equivalent” or “combined” modulus In order to determine the common extension of a compound bar it is convenient to consider it as a single bar of an imaginary material with an equivalent or combined modulus E,. Here it is necessary to assume that both the extension and the original lengths of the individual members of the compound bar are the same; the strains in all members will then be equal. Now total load on compound bar = F1 + Fz + F3 + . . . + F, where F1, F2, etc., are the loads in members 1, 2, etc. But force = stress x area .. a(Al+Az+ ... +An)=~lAl+~zAz+ ... +a,A, where 6 is the stress in the equivalent single bar. Dividing through by the common strain E, d 61 on E E E -(Al + Az + . . . +An) = -Al + :AZ + . . . + -A, i.e. Ec(Ai+Az+ ... +An)=EIA1+EzAz+ ... +E,An where E, is the equivalent or combined E of the single bar
30 Mechanics of Materials §2.3 combined E=E4+E2++EA A1十A2+·..+An ΣEA i.e. Ee= ZA (2.6 With an external load W applied, stress in the equivalent bar= W ΣA and W strain in the equivalent bar= E,ΣA=D ..since stress =E strain WL common extension x= E∑A (2.7) extension of single bar 2.3.Compound bars subjected to temperature change When a material is subjected to a change in temperature its length will change by an amount aLt where a is the coefficient of linear expansion for the material,L is the original length and t the temperature change.(An increase in temperature produces an increase in length and a decrease in temperature a decrease in length except in very special cases of materials with zero or negative coefficients of expansion which need not be considered here.) If,however,the free expansion of the material is prevented by some external force,then a stress is set up in the material.This stress is equal in magnitude to that which would be produced in the bar by initially allowing the free change of length and then applying sufficient force to return the bar to its original length. Now change in length =aLt aLt strain=ot Therefore,the stress created in the material by the application of sufficient force to remove this strain =strain x E =Eat Consider now a compound bar constructed from two different materials rigidly joined together as shown in Fig.2.2 and Fig.2.3(a).For simplicity of description consider that the materials in this case are steel and brass
30 Mechanics of Materials 42.3 .. i.e. EiAi+E,AZ+ . . . +E,An Al+A2+ . . . +An combined E = XEA EA E, = - With an external load W applied, W stress in the equivalent bar = ~ XA and wx strain in the equivalent bar = - = - EJA L .’. since stress strain -- -E WL common extension x = - E,XA = extension of single bar 2.3. Compound bars subjected to temperature change When a material is subjected to a change in temperature its length will change by an amount aLt where a is the coefficient of linear expansion for the material, L is the original length and t the temperature change. (An increase in temperature produces an increase in length and a decrease in temperature a decrease in length except in very special cases of materials with zero or negative coefficients of expansion which need not be considered here.) If, however, the free expansion of the material is prevented by some external force, then a stress is set up in the material. This stress is equal in magnitude to that which would be produced in the bar by initially allowing the free change of length and then applying sufficient force to return the bar to its original length. Now change in length = aLt . aLt .. strain = - = at L Therefore, the stress created in the material by the application of sufficient force to remove this strain = strain x E = Eat Consider now a compound bar constructed from two different materials rigidly joined together as shown in Fig. 2.2 and Fig. 2.3(a). For simplicity of description consider that the materials in this case are steel and brass
§2.3 Compound Bars 31 Steel Brass Step. Fig.2.2. In general,the coefficients of expansion of the two materials forming the compound bar will be different so that as the temperature rises each material will attempt to expand by different amounts.Figure 2.3b shows the positions to which the individual materials will extend if they are completely free to expand(i.e.not joined rigidly together as a compound bar).The extension of any length L is given by aLt Steel (a)Original bar 6ra53 Steel Difference in free lengths (b)Expanded position; members free to expand independenty Extension of steel- Compression of br055 (c)Compound bar Steel expanded position Brass SteeL Fig.2.3.Thermal expansion of compound bar. Thus the difference of "free"expansion lengths or so-called free lengths agLt-asLt=(aB-as)Lt since in this case the coefficient of expansion of the brassis greater than that for the steels. The initial lengths L of the two materials are assumed equal. If the two materials are now rigidly joined as a compound bar and subjected to the same temperature rise,each material will attempt to expand to its free length position but each will be affected by the movement of the other.The higher coefficient of expansion material (brass) will therefore seek to pull the steel up to its free length position and conversely the lower
§2.3 Compound Bars 31 Fig. 2.2. In general, the coefficients of expansion of the two materials forming the compound bar will be different so that as the temperature rises each material will attempt to expand by different amounts. Figure 2.3b shows the positions to which the individual materials will extend if they are completely free to expand (i.e. not joined rigidly together as a compound bar). The extension of any length L is given by cxLt Fig. 2.3. Thermal expansion of compound bar. Thus the difference of "free" expansion lengths or so-called free lengths = IXBLt-IXsLt = (IXB-IXs)Lt since in this case the coefficient of expansion of the brass IXBis greater than that for the steellXs' The initial lengths L of the two materials are assumed equal. If the two materials are now rigidly joined as a compound bar and subjected to the same temperature rise, each material will attempt to expand to its free length position but each will be affected by the movement of the other, The higher coefficient of expansion material (brass) will therefore seek to pull the steel up to its free length position and conversely the lower