Page 1 Instructional Objectives After reading this chapter the student will be able to 1. Calculate degree of statical indeterminacy of a planar truss 2. Analyse the indeterminate planar truss for external loads 3. Analyse the planar truss for temperature loads 4. Analyse the planar truss for camber and lack of fit of a member. 10.1 Introduction The truss is said to be statically indeterminate when the total number of reactions and member axial forces exceed the total number of static equilibrium equations. In the simple planar truss structures, the degree of indeterminacy can be determined from inspection. Whenever, this becomes tedious, one could use the following formula to evaluate the static indeterminacy of static planar truss (see also section 1.3). j r m i 2 ) ( - + = (10.1) where j m, and rare number of members, joints and unknown reaction components respectively. The indeterminacy in the truss may be external, internal or both. A planar truss is said to be externally indeterminate if the number of reactions exceeds the number of static equilibrium equations available (three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to be internally indeterminate if it has exactly three reaction components and more than members. Finally a truss is both internally and externally indeterminate if it has more than three reaction components and also has more than members. ( 3 2 - j ) ) ( 3 2 - j The basic method for the analysis of indeterminate truss by force method is similar to the indeterminate beam analysis discussed in the previous lessons. Determine the degree of static indeterminacy of the structure. Identify the number of redundant reactions equal to the degree of indeterminacy. The redundants must be so selected that when the restraint corresponding to the redundants are removed, the resulting truss is statically determinate and stable. Select redundant as the reaction component in excess of three and the rest from the member forces. However, one could choose redundant actions completely from member forces. Following examples illustrate the analysis procedure. Example 10.1 Determine the forces in the truss shown in Fig.10.1a by force method. All the members have same axial rigidity. Page 2 Instructional Objectives After reading this chapter the student will be able to 1. Calculate degree of statical indeterminacy of a planar truss 2. Analyse the indeterminate planar truss for external loads 3. Analyse the planar truss for temperature loads 4. Analyse the planar truss for camber and lack of fit of a member. 10.1 Introduction The truss is said to be statically indeterminate when the total number of reactions and member axial forces exceed the total number of static equilibrium equations. In the simple planar truss structures, the degree of indeterminacy can be determined from inspection. Whenever, this becomes tedious, one could use the following formula to evaluate the static indeterminacy of static planar truss (see also section 1.3). j r m i 2 ) ( - + = (10.1) where j m, and rare number of members, joints and unknown reaction components respectively. The indeterminacy in the truss may be external, internal or both. A planar truss is said to be externally indeterminate if the number of reactions exceeds the number of static equilibrium equations available (three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to be internally indeterminate if it has exactly three reaction components and more than members. Finally a truss is both internally and externally indeterminate if it has more than three reaction components and also has more than members. ( 3 2 - j ) ) ( 3 2 - j The basic method for the analysis of indeterminate truss by force method is similar to the indeterminate beam analysis discussed in the previous lessons. Determine the degree of static indeterminacy of the structure. Identify the number of redundant reactions equal to the degree of indeterminacy. The redundants must be so selected that when the restraint corresponding to the redundants are removed, the resulting truss is statically determinate and stable. Select redundant as the reaction component in excess of three and the rest from the member forces. However, one could choose redundant actions completely from member forces. Following examples illustrate the analysis procedure. Example 10.1 Determine the forces in the truss shown in Fig.10.1a by force method. All the members have same axial rigidity. The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The truss is externally determinate the reactions can be evaluated from the equations of statics alone. Select the bar force in member . .e i AD F ADas the redundant. Now cut the member AD to obtain the released structure as shown in Fig. 10.1b. The cut redundant member ADremains in the truss as its deformations need to be included in the calculation of displacements in the released structure. The redundant ( ) AD F consists of the pair of forces acting on the released structure. Page 3 Instructional Objectives After reading this chapter the student will be able to 1. Calculate degree of statical indeterminacy of a planar truss 2. Analyse the indeterminate planar truss for external loads 3. Analyse the planar truss for temperature loads 4. Analyse the planar truss for camber and lack of fit of a member. 10.1 Introduction The truss is said to be statically indeterminate when the total number of reactions and member axial forces exceed the total number of static equilibrium equations. In the simple planar truss structures, the degree of indeterminacy can be determined from inspection. Whenever, this becomes tedious, one could use the following formula to evaluate the static indeterminacy of static planar truss (see also section 1.3). j r m i 2 ) ( - + = (10.1) where j m, and rare number of members, joints and unknown reaction components respectively. The indeterminacy in the truss may be external, internal or both. A planar truss is said to be externally indeterminate if the number of reactions exceeds the number of static equilibrium equations available (three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to be internally indeterminate if it has exactly three reaction components and more than members. Finally a truss is both internally and externally indeterminate if it has more than three reaction components and also has more than members. ( 3 2 - j ) ) ( 3 2 - j The basic method for the analysis of indeterminate truss by force method is similar to the indeterminate beam analysis discussed in the previous lessons. Determine the degree of static indeterminacy of the structure. Identify the number of redundant reactions equal to the degree of indeterminacy. The redundants must be so selected that when the restraint corresponding to the redundants are removed, the resulting truss is statically determinate and stable. Select redundant as the reaction component in excess of three and the rest from the member forces. However, one could choose redundant actions completely from member forces. Following examples illustrate the analysis procedure. Example 10.1 Determine the forces in the truss shown in Fig.10.1a by force method. All the members have same axial rigidity. The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The truss is externally determinate the reactions can be evaluated from the equations of statics alone. Select the bar force in member . .e i AD F ADas the redundant. Now cut the member AD to obtain the released structure as shown in Fig. 10.1b. The cut redundant member ADremains in the truss as its deformations need to be included in the calculation of displacements in the released structure. The redundant ( ) AD F consists of the pair of forces acting on the released structure. Evaluate reactions of the truss by static equations of equilibrium. 5kN (downwards) 5kN (downwards) 15 kN(upwards) Cy Cx Dy R R R = - =- = (1) Please note that the member tensile axial force is taken as positive and horizontal reaction is taken as positive to the right and vertical reaction is taken as positive when acting upwards. When the member cut ends are displaced towards one another then it is taken as positive. The first step in the force method is to calculate displacement ( ) L ? corresponding to redundant bar force in the released structure due to applied external loading. This can be readily done by unit-load method. AD F Page 4 Instructional Objectives After reading this chapter the student will be able to 1. Calculate degree of statical indeterminacy of a planar truss 2. Analyse the indeterminate planar truss for external loads 3. Analyse the planar truss for temperature loads 4. Analyse the planar truss for camber and lack of fit of a member. 10.1 Introduction The truss is said to be statically indeterminate when the total number of reactions and member axial forces exceed the total number of static equilibrium equations. In the simple planar truss structures, the degree of indeterminacy can be determined from inspection. Whenever, this becomes tedious, one could use the following formula to evaluate the static indeterminacy of static planar truss (see also section 1.3). j r m i 2 ) ( - + = (10.1) where j m, and rare number of members, joints and unknown reaction components respectively. The indeterminacy in the truss may be external, internal or both. A planar truss is said to be externally indeterminate if the number of reactions exceeds the number of static equilibrium equations available (three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to be internally indeterminate if it has exactly three reaction components and more than members. Finally a truss is both internally and externally indeterminate if it has more than three reaction components and also has more than members. ( 3 2 - j ) ) ( 3 2 - j The basic method for the analysis of indeterminate truss by force method is similar to the indeterminate beam analysis discussed in the previous lessons. Determine the degree of static indeterminacy of the structure. Identify the number of redundant reactions equal to the degree of indeterminacy. The redundants must be so selected that when the restraint corresponding to the redundants are removed, the resulting truss is statically determinate and stable. Select redundant as the reaction component in excess of three and the rest from the member forces. However, one could choose redundant actions completely from member forces. Following examples illustrate the analysis procedure. Example 10.1 Determine the forces in the truss shown in Fig.10.1a by force method. All the members have same axial rigidity. The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The truss is externally determinate the reactions can be evaluated from the equations of statics alone. Select the bar force in member . .e i AD F ADas the redundant. Now cut the member AD to obtain the released structure as shown in Fig. 10.1b. The cut redundant member ADremains in the truss as its deformations need to be included in the calculation of displacements in the released structure. The redundant ( ) AD F consists of the pair of forces acting on the released structure. Evaluate reactions of the truss by static equations of equilibrium. 5kN (downwards) 5kN (downwards) 15 kN(upwards) Cy Cx Dy R R R = - =- = (1) Please note that the member tensile axial force is taken as positive and horizontal reaction is taken as positive to the right and vertical reaction is taken as positive when acting upwards. When the member cut ends are displaced towards one another then it is taken as positive. The first step in the force method is to calculate displacement ( ) L ? corresponding to redundant bar force in the released structure due to applied external loading. This can be readily done by unit-load method. AD F To calculate displacement , apply external load and calculate member forces as shown in Fig. 10.1b and apply unit virtual load along and calculate member forces ( (see Fig. 10.1c). Thus, ( L ?) () i P AD F ) i v P () AE AE L P P i i v i L 03 . 103 = = ? ? (2) In the next step, apply a real unit load along the redundant and calculate displacement by unit load method. Thus, AD F 11 a () AE E A L P a i i i i v 142 . 24 2 11 = = ? (3) The compatibility condition of the problem is that the relative displacement L ? of the cut member AD due to external loading plus the relative displacement of the member AD caused by the redundant axial forces must be equal to zero . .e i Page 5 Instructional Objectives After reading this chapter the student will be able to 1. Calculate degree of statical indeterminacy of a planar truss 2. Analyse the indeterminate planar truss for external loads 3. Analyse the planar truss for temperature loads 4. Analyse the planar truss for camber and lack of fit of a member. 10.1 Introduction The truss is said to be statically indeterminate when the total number of reactions and member axial forces exceed the total number of static equilibrium equations. In the simple planar truss structures, the degree of indeterminacy can be determined from inspection. Whenever, this becomes tedious, one could use the following formula to evaluate the static indeterminacy of static planar truss (see also section 1.3). j r m i 2 ) ( - + = (10.1) where j m, and rare number of members, joints and unknown reaction components respectively. The indeterminacy in the truss may be external, internal or both. A planar truss is said to be externally indeterminate if the number of reactions exceeds the number of static equilibrium equations available (three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to be internally indeterminate if it has exactly three reaction components and more than members. Finally a truss is both internally and externally indeterminate if it has more than three reaction components and also has more than members. ( 3 2 - j ) ) ( 3 2 - j The basic method for the analysis of indeterminate truss by force method is similar to the indeterminate beam analysis discussed in the previous lessons. Determine the degree of static indeterminacy of the structure. Identify the number of redundant reactions equal to the degree of indeterminacy. The redundants must be so selected that when the restraint corresponding to the redundants are removed, the resulting truss is statically determinate and stable. Select redundant as the reaction component in excess of three and the rest from the member forces. However, one could choose redundant actions completely from member forces. Following examples illustrate the analysis procedure. Example 10.1 Determine the forces in the truss shown in Fig.10.1a by force method. All the members have same axial rigidity. The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The truss is externally determinate the reactions can be evaluated from the equations of statics alone. Select the bar force in member . .e i AD F ADas the redundant. Now cut the member AD to obtain the released structure as shown in Fig. 10.1b. The cut redundant member ADremains in the truss as its deformations need to be included in the calculation of displacements in the released structure. The redundant ( ) AD F consists of the pair of forces acting on the released structure. Evaluate reactions of the truss by static equations of equilibrium. 5kN (downwards) 5kN (downwards) 15 kN(upwards) Cy Cx Dy R R R = - =- = (1) Please note that the member tensile axial force is taken as positive and horizontal reaction is taken as positive to the right and vertical reaction is taken as positive when acting upwards. When the member cut ends are displaced towards one another then it is taken as positive. The first step in the force method is to calculate displacement ( ) L ? corresponding to redundant bar force in the released structure due to applied external loading. This can be readily done by unit-load method. AD F To calculate displacement , apply external load and calculate member forces as shown in Fig. 10.1b and apply unit virtual load along and calculate member forces ( (see Fig. 10.1c). Thus, ( L ?) () i P AD F ) i v P () AE AE L P P i i v i L 03 . 103 = = ? ? (2) In the next step, apply a real unit load along the redundant and calculate displacement by unit load method. Thus, AD F 11 a () AE E A L P a i i i i v 142 . 24 2 11 = = ? (3) The compatibility condition of the problem is that the relative displacement L ? of the cut member AD due to external loading plus the relative displacement of the member AD caused by the redundant axial forces must be equal to zero . .e i 0 11 = + ? AD L F a (4) 103.03 24.142 4.268 kN(compressive) AD F - = =- Now the member forces in the members can be calculated by method of superposition. Thus, ( ) i v AD i i P F P F + = (5) The complete calculations can be done conveniently in a tabular form as shown in the following table. Table 10.1 Computation for example 10.1 Member Length i L Forces in the released truss due to applied loading i P Forces in the released truss due to unit load ( ) i v P () AE L P P i i v i () i i i i v E A L P 2 () i v AD i i P F P F + = m kN kN m m/kN kN AB 5 0 2 / 1 - 0 AE 2 / 5 3.017 BD 5 -15 2 / 1 - AE 2 / 75 AE 2 / 5 -11.983 DC 5 0 2 / 1 - 0 AE 2 / 5 3.017 CA 5 0 2 / 1 - 0 AE 2 / 5 3.017 CB 2 5 2 5 1 AE / 50 AE / 2 5 2.803 AD 2 5 0 1 0 AE / 2 5 -4.268 Total AE 03 . 103 AE 142 . 24 Example 10.2 Calculate reactions and member forces of the truss shown in Fig. 10.2a by force method. The cross sectional areas of the members in square centimeters are shown in parenthesis. Assume . 2 5 N/mm 10 0 . 2 × = ERead More

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