Lecture mechanics of materials chapter four axial members
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Lecture mechanics of materials chapter four axial members
M. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members ns for the design and analysis of axial members.2.Develop the discipline to draw frec-body diagrams and approximate defonned shapes 111 the design and analysis of stale* tinesTile tensile forces supporting the weight of the Mackinaw bridge (Figure 4 la) act along the longimdinal axis of each cable. Lecture mechanics of materials chapter four axial members The compressive forces raising the weight of the dump on a truck act along the axis of the hydraulic cylinders. The cables and hydraulic cylinders areLecture mechanics of materials chapter four axial members
axial members, long straight bodies on which the forces are applied along the longitudinal axis. Connecting rods in an engine, struts in aircraft engM. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members gure 4.1 Axial members: (a) Cables of Mackinaw bridge, (b) Hydraulic cy linders in a dump truck.Tins chapter develops the simplest theory' for axial members, following the logic shown 111 Figure 3.15 but subject to the limitations described in Section 3.13. We can then apply the formulas to statical Lecture mechanics of materials chapter four axial members ly determinate and indeterminate strucnires. The two most important tools in our analysis will be frcc-body diagrams and approximate defonned shapes.4Lecture mechanics of materials chapter four axial members
.1 PRELUDE TO THEORYAs a prelude to theory, we consider two numerical examples solved using the logic discussed in Section 3.2 Their solution will higM. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members ot Materials: Axial MembersF = 1000 N +350 N = 1350 N h == 14.81 mm(1000 N +350 N)ANS. F=135ON A = 14.81 mmCOMMENTS1.Both bars, ữrespective of the material, were subjected to the same axial strain. This is the fundamental kinematic assumption in the development of the theory for axial members, discu Lecture mechanics of materials chapter four axial members ssed in Section 4.2.2.Tire sum on the right in Equation (E8) can be written y G, A.-Í.. where Ợ, is foe normal stress in the .'th bar. AX,is the crossLecture mechanics of materials chapter four axial members
-sectional area of the rth bar. and >1=2 reflects that we have two bars in this problem. If we had n bars attached to the ngid plate, then the total aM. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members initesimal aiea d.4) as we try to fit an infinite numbei of bars on the same plate, resulting in a continuous body. The sum then becomes an integral, as discussed in Section 4.1.1.3.If the external force were located at any point other than that given by the value of A, then the plate would rotate T Lecture mechanics of materials chapter four axial members hus, for pure axial problems with no bending, a point on the cross section must be found such that the internal moment from the axial stress distributLecture mechanics of materials chapter four axial members
ion is zero. To emphasize this, consider the left side of Equation (E9). which can be written as y‘‘ ,V!GI Ad,, where V, is the coordinate of the rth M. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members he problem IS of pure axial deformation, as discussed tn Section 4.1.1.4.Even though the strains in both bars were the same in both cases, the stresses were different when E changed. Case (a) corresponds to a homogeneous cross section, whereas case (b) is analogous to a laminated ba: in which the no Lecture mechanics of materials chapter four axial members n-homogeneity affects die stress distribution.4.1.1 Internal Axial ForceIn this section we formalize the key observation made in Example 4.1: the normLecture mechanics of materials chapter four axial members
al stress can be replaced by an equivalent internal axial force using an integral over the cross-sectional area Figure 4.4 shows the statically equivaM. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members on IS to be replaced by only an axial force at the origin, then the internal moments V and M. must be zero at die origin, and from Figure 4.4 we obtainI'nriolfrvfi: lil^.VuirArictilHCikếruv./íaKiístMdlííiỊ yơxx d-i = 0(4.2a)J :ơxx(ỈA=Ồ(4.2b).4Equations (4.1). (4.2a). and (4.2b) are independent of th Lecture mechanics of materials chapter four axial members e material models because the}' represent static equivalency between the normal stress on the cross section and internal axial force. If we were to coLecture mechanics of materials chapter four axial members
nsider a lammatcd cross section or nonlinear material, dien it would affect die value and distribution of ơ„ acioss the cross section, but Equation (4M. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its application Lecture mechanics of materials chapter four axial members e used to determine the location at winch dK internal and external forces have to act for pure axial problem without bending, as discussed in Section 4.2.6.AUỊUII 2Ú12M. (ableMedrauicsof -Materials: Axial MembersI Lecture mechanics of materials chapter four axial members M. VableMechanics of Materials: Axial MembersCHAPTER FOURAXIAL MEMBERSLearning objectives1.Understand the theory. Its limitations, and its applicationGọi ngay
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