【正文】 ion in longitudinal andlateral stability and controllability, as well as to increased stresseson the container structure (Bauer,1975). The handling and stabilitylimits of tank trucks are thus dependent upon factors other thannormal trucking practices. These factors include tank geometry。is toLiquidtheusuati. Thegoodpaperreddisturbancerigid containersfew decades.andonater oscillationpoolsto both highway safety and the environment (Botkin,1970).Tank trucks employed in general purpose chemical transportation encounter partial fill conditions due to the varyingweight density of the products and the laws governing axle loads,while those employed in fuel transportation encounter partial fillApart from heavy vehicle design factors, the dynamic stabilitylimit of tank trucks are directly related to dynamic load shift. Thedynamic load transfer encountered during a braking or turningmanoeuvre is a plex function of fluid slosh, fill level, tankgeometry, vehicle weight and dimension, suspension and tireproperties. The study of the sloshing behaviour of liquids withina moving container involves highly plex dynamic modelingand analyses. Slibar and Troger (1977) have characterized thesloshing liquid cargo in the roll plane as two lumped masses* Corresponding author.Contents lists availableEuropean Journal ofEuropean Journal of Mechanics A/Solids 28 (2020) 1026–1034Email address: (M. Bouazara).limits of partially filled liquid cargo vehicles are known to besignificantly lower than those of conventional rigid cargo vehiclesduetotheuniquedynamicinteractionsbetweenthevehicleandthesloshing liquid cargo. The forces and moments arising from a directional manoeuvre yield considerable dynamic load shifts in the rolland pitch planes due to the sloshing of the liquid cargo within thepartially filled tank. The dynamic load shift affects the directionalstability of the partially filled tank trucks in an adverse manner.When dangerous goods are hauled it canpose an unreasonable riskturning, braking and lane change。height of the centre of gravity (cg)。Fr。 ih240。 (4)Starting from Eq. (4), the total derivative of the pressure will havethe following form:dP 188。C0raxdx C0raydy C0razdz(5)where (x, y, z) and (ax, ay, az) are respectively, the coordinates andaccelerations of the liquid centre of mass.Thedynamicloadtransfercausedbythemovementof theliquidcargowithinapartiallyfilledtankisevaluatedfromtheverticalandlateral displacements of its centre of mass and the variation of theinertia matrix. The procedure used in this research is to calculateMechanics A/Solids 28 (2020) 1026–1034 1027coupled through a linear spring and a viscous damper. Abramson(1966) has investigated the fluid slosh in spacecraft fuel tankscaused by lateral motion using a similar mechanical model basedon a liberalized Euler equation. Mallikarjunarao (1982) has investigated the directional and roll dynamic response characteristics ofa double tanker configuration used in gasoline transportationthrough the development and analysis of a threedimensionalmodel of the tank vehicle. The study, however, did not consider theeffects of a moving liquid cargo under partial fill conditions, as thetank was assumed to be pletely full or empty.Popov (1991) has modeled liquid movement using the finitedifferences method. He also studied the optimization of the shapeof the tank. The study concluded that for some liquids with lowviscosity the vibrations of the liquid can be considered not atten(1993),Nichkawdeetal.(2020)havesimulatedthe movement of the liquid with a regular pendulum and a fixedmass. The mass of the pendulum simulated the effects of the firstmode of the fluid in movement. The fixed mass simulated theinertia and the weight of the remaining liquid. For the pendulumparameters, he adapted the equations given by Budiansky (1960)for describing the dynamic effects of liquids and the dynamiceffects of a pendulum with a simple mass.In this study analytical and numerical models are formulatedbased on the Navier–Stokes equations with some assumptions forthe analytical model. In the second part, a full, simplified vehiclemodel is developed. It is coupled with the liquid model as a multibody system.2. Liquid analytical modelThe aim of this work consists in developing a simple andplete model that represents the movement of liquids in a tankunder various conditions. There is a large variety of conditions forwhich the movement of the liquid can be configured. Typicalexamples are longitudinal and lateral movements along a straightline or cornering (Grundelius, 2020). The present model exploitsequations that govern the movement of liquids known as the Navier–Stokes equations. However, these equations cannot be solvedanalytically. The ideal models are obtained assuming a small oscillation based on the work of Frandsen (2020). The following sectiondescribestheapplicationoftheNavier–Stokesequationstotheflowof a liquid. It is a system of nonlinear threedimensional partialequations. The first equation describes the conservation of mass.vrvt254。0 (1)For viscous and Newtonian liquids the momentum equation isobtained by:rC18vVvt254。C0VP 254。V240。 F (2)where r is the density of liquid。 l represents the factor ofvolume pression and F is the external body force.To develop the analytical model, one needs to apply someassumptions. In this study we assume that the liquid is inpressible (l188。vVvt254。C01rVP 254。s222。g as shown in Fig. 1.The shift in the centre of the mass and the variation of theinertia matrix is calculated according to the volume integral, by thefollowing equation as:xi188。x2kC17dV。1。 a2r2254。 m 188。m1(8)The equation for the volume fraction is not solved for the primaryphase (gas). The volume of gas fraction is calculated starting fromthe constrai