16.1 INTRODUCTION

Bernoulli's bending theory is applicable only for straight beams. In case the beams are curved then the simple bending theory cannot be used. Curved structural elements are common (a crane hook, ring of chain, etc.).

16.2 ASSUMPTIONS

In a curved beam, plane sections before bending remains plane even after bending. This means that the strain variation along the depth of the cross-section of the beam is linear. Consider a curved beam shown in Figure 16.2.

16.3 ANALYSIS OF DIFFERENT CROSS-SECTIONS

The stress analysis of curved beams is developed by Winker Bach. The important aspect to be remembered in the analysis of curved beams is that the neutral axis (NA) and centroidal axis of the cross-section do not coincide.

Referring to Figure 16.2 and writing, PQ is the length of the fi bers located y distant from NA of the cross-section before application of moment.

Let R0 be the radius of the fi ber of the cross-section at NA.

Let QQ' be the increase in the length of the fi ber PQ after the application of moment.

in which, E is modulus of elasticity, y is the distance from the NA as before, and R0 the initial radius of the neutral surface.

The total force acting along the axis of the beam is zero.

Total normal force on cross-section = 0 for pure bending, that is,

∫ydA = Moment of the area from reference axis, that is, axis from which ‘y’ is measured. In the present case, it is A × e.

in which, ‘e’ is the location of NA from centroid of the cross-section.

in which, e is the distance between the NA and the principal axis through the centroid (i.e., being positive for NA to be on the same side of the centroid as the center of curvature).

13.1 INTRODUCTION

In the previous chapters, we have learnt the procedure to determine the stress in structural elements when they are subjected to forces such as axial compression, axial tension, bending, shear, and torsion. These forces rarely occur alone. Frequently, structural elements are subjected to combination of forces such as combined bending and axial load; axial load and torsion; torsion and bending; and torsion, bending, and axial load. In this chapter, we determine the resulting stresses due to combined loads. Different types of stresses developed due to combined loads lead to stresses such as normal stress and shear stress. These normal and shear stresses result in principal stresses. This chapter provides the information regarding such combined stresses.

13.2 SHORT COLUMNS SUBJECTED TO AXIAL COMPRESSION AND BENDING

Short column are axially loaded elements which are very often subjected to combined bending and axial compression. Axial force on a member produces normal stress. Bending moment (BM) produces bending stress, which is also normal to the cross-section. As these two stresses are normal to the cross-section their algebraic addition results in the fi nal stress.

Consider a cross-section subjected to axial load P and BM M. Let A be the cross-sectional area of the cross-section and I be the moment of inertia (MI) of the cross-section. Axial stress is given by. The total cross-section is subjected to axial compressive stress of uniform intensity. BM produces bending stress.

in which y is the location of the fi ber from neutral axis (NA) where the stress is to be found. Stress in the extreme fi ber A is given by

Stress in the extreme fi ber B is given by

A column subjected to eccentric longitudinal load falls under the category of combined bending and axial load case. This case is more common in many structural elements.

The stress analysis of an eccentrically loaded short column is presented here as atypical example of combined bending and axial load.

2.1 INTRODUCTION

Two elastic constants of an elastic body are modulus of elasticity and Poisson's ratio. Modulus of elasticity is the ratio of normal stress to normal strain. It is generally denoted by the letter ‘E’. Units are gigapascal (GPa). The other elastic constant is Poisson's ratio.

Poisson's ratio is defi ned as the ratio of lateral strain to the longitudinal strain. It is denoted by the letter μ.

Consider a bar of length ‘L’ and diameter ‘d’ subjected to an axial tensile stress ‘σ’. ΔL is the longitudinal extension due to load P. Then, the longitudinal strain

Let Δd be the decrease in the lateral dimension of the bar.

The negative sign indicates that positive strain (increase) in longitudinal strain decreases the lateral dimension, thereby compressive strain in the lateral direction is induced.

Rigidity Modulus

Rigidity modulus is the ratio of shear stress to the corresponding shear strain. The unit is gigapascal. This is generally denoted by the letter ‘N’. The shear modulus or rigidity modulus appears in calculation where shape change occurs in the body.

Bulk Modulus

Bulk modulus is the ratio of direct stress or volumetric stress to the volumetric strain. It is denoted by the letter ‘K’ and unit is gigapascal. Generally in solids, the change in volume is very small, but this change in volume in case of fl uids will be considerable. Thus, bulk modulus appears in fl uid mechanics especially in compressible fl uid fl ow problems.

In the case of fl uids, stress at a point in all directions is same. Hence,

The bulk modulus and the rigidity modulus can be expressed in terms of modulus of elasticity and Poisson's ratio. Thus, the elastic constants for a homogeneous and isotropic body are two (2).

2.2 RELATIONSHIP BETWEEN E AND N

Consider a square block of unit width subjected to shear stress τ as shown in Figure 2.3. The shear stress τ induces vertical shear stress on planes AC and DB of same intensity ‘τ’. The shear stress on these planes is referred as complimentary shear stress.

7.1 INTRODUCTION

The beams are subjected to transverse loading deform. The deformed profi le of a beam is referred to as the elastic curve of the beam. The defl ection calculations are necessary as high defl ections cause:

• 1. Undesirable psychological effects (in case of fl oor beams of buildings).

• 2. Misalignment of machine parts.

• 3. Distress in brittle material structures.

Defl ection calculations are also necessary in the analysis of statistically indeterminate structures.

The information about the defl ection characteristics of members is needed in the dynamic analysis of structures. The different methods of calculating defl ections are

• 1. Double integration method.

• 2. Macaulay's method.

• 3. Moment area method.

• 4. Conjugate beam method.

• 5. Energy methods.

7.2 MOMENT-CURVATURE RELATIONSHIP

The elastic curve of a beam subjected to constant moment (pure bending) assumes an arc of a circle. If the bending moment (BM) is varying, associated with shear force (SF) also, the elastic curve varies. Thus, the radius of curvature also varies. In chapter 4, we have seen that

in which EI is fl exural rigidity, M is moment, and 1/R is curvature.

EI is always a positive quantity, thus to the tune of moment, curvature also have sense like positive/negative.

Consider a curve AB shown in Figure 7.1. Let us develop an expression for the radius of curvature (R) or curvature (1/R) of the curve AB from the fundamentals of mathematics. Consider an elemental position ‘ds’ of the curve AB. For this elemental position, radius of curvature can be taken as constant, say ‘R’. ‘δθ’ is the deviation between the tangents drawn at P and Q. PC and QC are the normals drawn to the tangents at P and Q. Thus, <PCQ is also ‘δθ’.

We know that, if ‘θ’ is the inclination of the tangent with the positive x axis measured in the counterclockwise direction.

12.1 INTRODUCTION

Columns are the structural elements that are subjected to axial compressive loads. Columns are referred as struts and stanchions also. Long columns (slender columns) when subjected to axial compressive load are failed by defl ecting laterally undergoing bending rather than failed by direct compression; this type of column failure is called buckling. The phenomenon of buckling can be demonstrated by applying axial compressive load on slender bars (such as plastic rule or wooden rule). Buckling occurs in structures, which are subjected to compressive stress. The compression fl ange of a girder also buckles under bending load.

The axial load on a column that is corresponding to the buckling phenomenon is called critical load or buckling load or crippling load. If Pcr is the critical load of an axial loaded column, then the axial load on the column P can be

• (a) P < Pcr—Column is in stable equilibrium condition and even if the column is slightly displaced in lateral direction the column comes backs to its stable confi guration.

• (b) P = Pcr—Column is in neutral equilibrium condition. There is a slight displacement in the lateral direction of the column in the new confi guration. Column does not come back to original confi guration.

• (c) P > Pcr—Column is in unstable equilibrium; column continuously bends and becomes unserviceable.

The above three equilibrium confi gurations are analogous to the equilibrium of a bar placed on smooth surfaces shown in Figure 12.2.

The basic reason for buckling of a column is due to the in perfection inherently present in the column.

12.2 EULER's THEORY OF BUCKLING

Buckling phenomenon of slender columns was fi rst investigated by Euler in 1744. The assumptions made for the analysis of column are:

• (i) The material of the column is homogenous, isotropic, and elastic, and obeys Hook's law.

• (ii) Axial load passes through the centroid of the column.

• (iii) Column is perfectly straight.

11.1 INTRODUCTION

Springs are the structural elements used to absorb energy or impact. Energy absorption in these elements is due to their ability to deform. Members deform more under loads and absorb more energy. For example, a body falling on a sand heap receives less impact as the falling body moves the sand particles, thereby dissipating energy. If the same body falls on a concrete fl oor, the body receives more impact and very less energy is dissipated. Here, concrete fl oor could not absorb the energy as the particles of concrete are bound together, whereas in case of sand, particles are not bound together allowing energy dissipation. Thus, energy absorbing elements should be capable of deforming without getting ruptured. Coiled springs are the best examples for the energy absorption. Coil springs are of two types:

• 1. Close-coiled helical springs

• 2. Open-coiled helical springs.

In case of a close-coiled helical spring, the pitch (distance between two coils) is very less and angle of helix is zero.

In case of open-coiled helical springs, the gap between the coils (pitch) is considerable and the angle of helix is also fi nite value. The components of helical spring are

11.2 EXPRESSION FOR THE EXTENSION OF A CLOSE-COILED HELICAL SPRING

Consider a close-coiled helical spring subjected to axial load.

The load ‘W’ passes through the center of coil.

The axial load W is transformed on to the cross-section of the coil. Shear force acting on the wire is W, twisting moment acting on the wire T = WR. Compared to wire diameter the radius of the coil is generally high (R > d).

Thus, the shear stress due to twisting moment (WR) is considerable.

The maximum shear stress in the wire of the coil.

17.1 INTRODUCTION

An articulated structure or a truss is composed of a number of bars or members connected by frictionless pins, forming geometrical fi gures which are usually triangles. A truss is said to be statically determinate internally if the number of members

in which m is the total number of members, j is the total number of joints, and r is the total number of equilibrium equations.

In the above equation, the value of r is usually 3 but if there is an additional hinge separating the structure in two parts, r = 3 + 1 = 4, and if there is a link somewhere in the structure, r = 3 + 2 = 5. The frame is said to be perfect if the number m is equal to right side of the equation.

When external loads are applied on the truss, at the joints, the members carry internal forces, usually called the stresses, which may be either tensile or compressive. According to Hook's law, if any axial member of length L carries a force P, it will be deformed by an amount PL/AE, in which A is the area of cross section of the members. In the truss, therefore, the members carrying tension will be elongated, whereas those carrying compression will be shortened. Thus, all the joints will move from their initial position and will occupy their fi nal equilibrium position. The axial deformation in the members of the truss may also be due to temperature changes or due to errors in fabrication or lack of fi t of some members, and the joints may move from their original position. This movement of each joint is defi ned as the defl ection of the joint in the direction of movement. The resolved part of the movement in the vertical direction is called vertical defl ection of the joint, and that in the horizontal direction is known as the horizontal defl ection of the joint. In general, each joint has both vertical as well as horizontal defl ection, unless constrained to move in a given direction.

8.1 INTRODUCTION

Cylinders are the most commonly used structural elements used to convey fl uids under pressure/gravity, maintain pressure, and store the fl uids (e.g., water tanks, gas cylinders). Analysis of cylinders includes the determination of stresses and deformations developed due to internal pressure.

The cylinders are classifi ed into two categories.

• 1. Thin cylinders

• 2. Thick cylinders

Let ‘di’ and ‘t’ be the inner diameter and thickness of the cylinder, respectively. These cylinders are those for which. In this, inner diameter to thickness ratio is less than 20, and the corresponding cylinders are referred as thick cylinders. In case of thin cylinders, the variation of radial stress across the thickness is neglected. Thus, the thin cylinder analysis is approximate one.

In thick cylinder analysis, the variation of radial pressure across the thickness of the cylinder is not neglected. Thus, the thick cylinder analysis is more accurate than thin cylinder analysis. If the thickness of the cylinder is very less, thick cylinder analysis coincides with the thin cylinder analysis.

8.2 ANALYSIS OF THIN CYLINDERS

Internal pressure in cylinder increases the length and perimeter of the cylinders. Thus, it can be inferred that there exists tensile stress in the length direction as well as in the circumferential direction. The stress in longitudinal direction of cylinder is called longitudinal stress denoted by σl, and stress along the circumference or circumferential direction is referred as circumferential stress or hoop stress, denoted by σh. These two stresses are shown in Figure 8.2.

The hoop stress (σh) is sometimes called as bursting stress. Knowing these stresses, the change in the diameter, length and internal volume of the cylinder can be computed.

8.2.1 Expression for Hoop Stress and Longitudinal Stress

Consider a close cylinder, subjected to internal fl uid pressure ‘p’. Let ‘d’ and ‘t’ be the inner diameter and wall thickness of the cylinder, respectively. To evaluate the hoop stress (σh), cut the cylinder horizontally by section (1)-(1) as shown in Figure 8.3.