Assumptions:The constraints put on the geometry would form theassumptions:1. Beam is initiallystraight, and has aconstant cross-section.2. Beam is made ofhomogeneous materialand the beam has alongitudinal plane of symmetry.3. Resultant of the applied loads lies in the plane of symmetry.4. The geometry of the overall member is such that bending not buckling is the primary cause of failure.5. Elastic limit is nowhere exceeded andE'is same in tension and compression.6. Plane cross - sections remains plane before and after bending.

Thermal EffectsChanges in temperature produce expansion or contraction of structural materials, resulting in thermal strains and thermal stresses. A simple illustration of thermal expansion is shown in Fig. 2-19, where the block of material is unrestrained and therefore free to expand. When the block is heated, every element of the material undergoes thermal strains in all directions, and consequently the dimensions of the block increase. If we take corner A as a fixed reference point and let side AB maintain its original alignment, the block will have the shape shown by the dashed lines.For most structural materials, thermal strain et is proportional to thetemperature change T; that is(2-15)in which a is a property of the material called the coefficient ofthermal expansion. Since strain is a dimensionless quantity, the coefficient of thermal expansion has units equal to the reciprocal oftemperature change. In SI units the dimensions of a can be expressed as either 1/K (the reciprocal of kelvins) or 1/C (the reciprocal of degrees Celsius). The value of a is the same in both cases because a change in temperature is numerically the same in both kelvins and degrees Celsius. In USCS units, the dimensions of a are 1/F (thereciprocal of degrees Fahrenheit).* Typical values of a are listed inTable I-4 of Appendix I (available online).

Bending stress due to thermal loading is due to temperature gradient This is given from the Fournier law of conduction If 1-D heat conduction is assumed Shape of moment is depends on the sign of temperature gradient

Sign conventionUnlike temperature, heat transfer has direction as well as magnitude, and thus it is a vector quantity (Fig. 21). Therefore, we must specify both direction and magnitude in order to describe heat transfer completely at a point.

we can work with a coordinate system and indicatedirection with plus or minus signs. The generally accepted convention is thatheat transfer in the positive direction of a coordinate axis is positive and in the opposite direction it is negative. Therefore, a positive quantity indicates heat transfer in the positive direction and a negative quantity indicates heat transfer in the negative direction.The driving force for any form of heat transfer is the temperature difference, and the larger the temperature difference, the larger the rate of heat transfer.